Subgroups — #72234424

Order 2048.

At least 14412 conjugacy classes of subgroups; only the 1036 normal subgroups are shown.

The subgroup lattice has 565 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234421	C₂	1	1	yes	maximal, characteristic, complemented
2	#72233187	C₂	1	1	yes	maximal, characteristic, complemented
3	#72233169	C₂	1	1	yes	maximal, characteristic, complemented
4	#72233135	C₂	1	2	yes	maximal, complemented
5	#72233117	C₂	1	2	yes	maximal, complemented
6	#72233091	C₂	1	1	yes	maximal, characteristic, complemented
7	#72233051	C₂	1	2	yes	maximal, complemented
8	#72233038	C₂	1	1	yes	maximal, characteristic, complemented
9	#72232947	C₂	1	4	yes	maximal, complemented
10	#72232895	C₂	1	1	yes	maximal, characteristic, complemented
11	#72232823	C₂	1	2	yes	maximal, complemented
12	#72232813	C₂	1	2	yes	maximal, complemented
13	#72232390	C₂	1	1	yes	maximal, characteristic, complemented
14	#72219306	C₂	1	2	yes	maximal, complemented
15	#72219249	C₂	1	2	yes	maximal, complemented
16	#72218099	C₂	1	4	yes	maximal, complemented
17	#72216646	C₂	1	2	yes	maximal, complemented
18	C₂×#60399	C₂²	1	2	yes	complemented
19	C₂×#60399	C₂²	1	1	yes	characteristic, complemented
20	#7829344	C₂²	1	1	yes	characteristic
21	#7829278	C₂²	1	1	yes	characteristic, complemented
22	#7819715	C₂²	1	2	yes	complemented
23	#7819605	C₂²	1	2	yes	—
24	#7711204	C₂²	1	1	yes	characteristic, complemented
25	#7708349	C₂²	1	2	yes	complemented
26	#7671874	C₂²	1	1	yes	characteristic
27	#7670203	C₂²	1	2	yes	—
28	#7659311	C₂²	1	1	yes	characteristic, complemented
29	#7659307	C₂²	1	1	yes	characteristic
30	C₂²×#2679	C₂²	1	1	yes	characteristic, complemented
31	C₂×#33551	C₂²	1	4	yes	complemented
32	C₂×#33348	C₂²	1	2	yes	complemented
33	#6628896	C₂²	1	2	yes	complemented
34	#6628876	C₂²	1	1	yes	characteristic, complemented
35	C₂×#23900	C₂²	1	4	yes	complemented
36	#6370747	C₂²	1	1	yes	characteristic, complemented
37	#498984	C₂²	1	2	yes	complemented
38	#498927	C₂²	1	4	yes	complemented
39	#498890	C₂²	1	2	yes	complemented
40	#498886	C₂²	1	4	yes	complemented
41	C₂×#13343	C₂²	1	4	yes	complemented
42	C₂×#13343	C₂²	1	2	yes	complemented
43	C₂×#13343	C₂²	1	2	yes	complemented
44	#494721	C₂²	1	2	yes	complemented
45	#494700	C₂²	1	2	yes	complemented
46	#494682	C₂²	1	2	yes	complemented
47	#494582	C₂²	1	4	yes	complemented
48	#494575	C₂²	1	4	yes	complemented
49	#493867	C₂²	1	1	yes	characteristic, complemented
50	C₂×#12753	C₂²	1	4	yes	complemented
51	#493424	C₂²	1	2	yes	complemented
52	C₂×#12687	C₂²	1	4	yes	complemented
53	#492900	C₂²	1	1	yes	characteristic, complemented
54	#394472	C₂²	1	2	yes	complemented
55	#394273	C₂²	1	4	yes	complemented
56	#394064	C₂²	1	4	yes	complemented
57	#393975	C₂²	1	2	yes	complemented
58	#374036	C₂²	1	1	yes	characteristic, complemented
59	#374004	C₂²	1	2	yes	complemented
60	#364532	C₂²	1	4	yes	complemented
61	#364526	C₂²	1	2	yes	complemented
62	#359888	C₂²	1	1	yes	characteristic, complemented
63	#359593	C₂²	1	1	yes	characteristic, complemented
64	#359591	C₂²	1	2	yes	complemented
65	#359591	C₂²	1	2	yes	complemented
66	#359563	C₂²	1	2	yes	complemented
67	#149481	C₂²	1	2	yes	complemented
68	#149444	C₂²	1	2	yes	complemented
69	#147815	C₂²	1	4	yes	complemented
70	#147811	C₂²	1	4	yes	complemented
71	#147783	C₂²	1	2	yes	complemented
72	#147779	C₂²	1	2	yes	complemented
73	#147779	C₂²	1	4	yes	complemented
74	#147779	C₂²	1	2	yes	complemented
75	#147732	C₂²	1	4	yes	complemented
76	#147569	C₂²	1	2	yes	complemented
77	#147567	C₂²	1	4	yes	complemented
78	#146692	C₂²	1	4	yes	complemented
79	#146675	C₂²	1	2	yes	complemented
80	#146609	C₂²	1	4	yes	complemented
81	#146593	C₂²	1	4	yes	complemented
82	#146465	C₂²	1	2	yes	complemented
83	C₂×H(7,𝔽₂)	C₂³	1	1	yes	characteristic
84	C₂×#3484	C₂³	1	2	yes	—
85	C₂×#3484	C₂³	1	2	yes	—
86	C₂×#3484	C₂³	1	1	yes	characteristic
87	C₂×#3532	C₂³	1	2	yes	—
88	C₂×#3444	D₄	1	1	yes	characteristic, complemented
89	C₂³×#120	D₄	1	2	yes	complemented
90	C₂³×#120	C₂³	1	2	yes	—
91	C₂×#3026	C₂³	1	4	yes	—
92	C₂²×B(4,𝔽₂)	C₂³	1	4	yes	—
93	C₂²×B(4,𝔽₂)	C₂³	1	4	yes	—
94	C₂×#3019	C₂³	1	2	yes	—
95	C₂×#2888	C₂³	1	1	yes	characteristic
96	C₂×#2883	C₂³	1	1	yes	characteristic
97	C₂×#2883	C₂³	1	2	yes	—
98	C₂×#2883	C₂³	1	2	yes	—
99	#60045	D₄	1	1	yes	characteristic, complemented
100	#59814	D₄	1	1	yes	characteristic, complemented
101	#59179	D₄	1	2	yes	complemented
102	#58990	D₄	1	2	yes	complemented
103	#52163	D₄	1	1	yes	characteristic, complemented
104	C₂×#2679	D₄	1	1	yes	characteristic, complemented
105	C₂×#2679	C₂³	1	1	yes	characteristic
106	C₂×#2679	C₂³	1	1	yes	characteristic
107	C₂×#2679	C₂³	1	2	yes	—
108	#51940	D₄	1	2	yes	complemented
109	#51815	D₄	1	2	yes	complemented
110	#50856	D₄	1	2	yes	complemented
111	#50646	D₄	1	2	yes	complemented
112	#49562	D₄	1	2	yes	complemented
113	#49545	D₄	1	2	yes	complemented
114	#46287	D₄	1	1	yes	characteristic, complemented
115	#46089	D₄	1	1	yes	characteristic, complemented
116	#45452	D₄	1	2	yes	complemented
117	#45439	D₄	1	2	yes	complemented
118	#37297	D₄	1	1	yes	characteristic, complemented
119	#37269	D₄	1	2	yes	complemented
120	#37134	C₂³	1	1	yes	characteristic
121	#37129	D₄	1	1	yes	characteristic, complemented
122	#37129	C₂³	1	1	yes	characteristic
123	#36714	D₄	1	2	yes	complemented
124	#36714	C₂³	1	2	yes	—
125	#36702	C₂³	1	2	yes	—
126	#36626	D₄	1	1	yes	characteristic, complemented
127	C₂×#2846	D₄	1	2	yes	complemented
128	#35556	D₄	1	1	yes	characteristic, complemented
129	#35533	D₄	1	1	yes	characteristic, complemented
130	#35515	C₂³	1	1	yes	characteristic
131	#35513	C₂³	1	1	yes	characteristic
132	#35390	D₄	1	2	yes	complemented
133	#35348	D₄	1	2	yes	complemented
134	#35348	C₂³	1	2	yes	—
135	#35346	C₂³	1	2	yes	—
136	#35256	D₄	1	1	yes	characteristic, complemented
137	#35254	D₄	1	1	yes	characteristic, complemented
138	#35223	D₄	1	1	yes	characteristic, complemented
139	#35222	C₂³	1	1	yes	characteristic
140	#34981	D₄	1	2	yes	complemented
141	#34981	D₄	1	2	yes	complemented
142	#34980	D₄	1	2	yes	complemented
143	#34980	C₂³	1	2	yes	—
144	#34408	C₂³	1	2	yes	—
145	#34356	C₂³	1	2	yes	—
146	#34269	C₂³	1	1	yes	characteristic
147	#34258	C₂³	1	1	yes	characteristic
148	#34200	D₄	1	1	yes	characteristic, complemented
149	#34198	D₄	1	2	yes	complemented
150	#34198	C₂³	1	2	yes	—
151	C₂²×#390	C₂³	1	1	yes	characteristic
152	C₂×#2127	C₂³	1	4	yes	—
153	C₂×#2121	C₂³	1	4	yes	—
154	C₂×#2121	C₂³	1	4	yes	—
155	C₂²×AΣL(1,𝔽₁₆)	C₂³	1	4	yes	—
156	C₂²×AΣL(1,𝔽₁₆)	C₂³	1	4	yes	—
157	C₂²×AΣL(1,𝔽₁₆)	C₂³	1	4	yes	—
158	C₂²×AΣL(1,𝔽₁₆)	C₂³	1	4	yes	—
159	C₂×#1913	C₂³	1	4	yes	—
160	C₂×#1911	C₂³	1	2	yes	—
161	C₂×#1889	C₂³	1	4	yes	—
162	C₂×#1889	C₂³	1	4	yes	—
163	#20616	C₂³	1	2	yes	—
164	C₂×#1792	C₂³	1	2	yes	—
165	C₂×#1792	C₂³	1	2	yes	—
166	#20609	C₂³	1	4	yes	—
167	#20592	C₂³	1	1	yes	characteristic
168	C₂×#1783	C₂³	1	2	yes	—
169	#20585	C₂³	1	2	yes	—
170	C₂×#1781	C₂³	1	4	yes	—
171	#12582	C₂³	1	2	yes	—
172	#12582	C₂³	1	2	yes	—
173	C₂×#1393	C₂³	1	2	yes	—
174	C₂×#1393	C₂³	1	2	yes	—
175	C₂×#1393	C₂³	1	4	yes	—
176	#12574	C₂³	1	4	yes	—
177	#12574	C₂³	1	4	yes	—
178	#10769	C₂³	1	2	yes	—
179	#10768	C₂³	1	4	yes	—
180	#10754	C₂³	1	4	yes	—
181	#10749	C₂³	1	2	yes	—
182	#10749	C₂³	1	2	yes	—
183	#10740	C₂³	1	2	yes	—
184	C₂⁷	C₂×D₄	1	2	yes	complemented
185	C₂²×H(5,𝔽₂)	C₂⁴	1	2	yes	—
186	C₂²×H(5,𝔽₂)	C₂⁴	1	1	yes	characteristic
187	C₂⁴×D₄	C₂×D₄	1	2	yes	—
188	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
189	C₂×H(3,𝔽₄)	C₂×D₄	1	2	yes	complemented
190	C₂×#560	C₂×D₄	1	2	yes	complemented
191	C₂×#551	C₂×D₄	1	2	yes	complemented
192	C₂×D₄²	C₂×D₄	1	2	yes	complemented
193	C₂×#537	C₂×D₄	1	1	yes	characteristic
194	C₂×#535	C₂×D₄	1	2	yes	—
195	C₂×#535	C₂×D₄	1	2	yes	—
196	C₂×#535	C₂×D₄	1	1	yes	characteristic
197	C₂×#535	C₂×D₄	1	1	yes	characteristic
198	C₂×#535	C₂×D₄	1	1	yes	characteristic, complemented
199	C₂×#535	C₂⁴	1	1	yes	characteristic
200	C₂×#534	C₂×D₄	1	2	yes	—
201	C₂×#534	C₂×D₄	1	2	yes	—
202	C₂×#534	C₂×D₄	1	2	yes	—
203	C₂×#534	C₂×D₄	1	2	yes	—
204	C₂×#534	C₂×D₄	1	2	yes	—
205	C₂×#534	C₂×D₄	1	2	yes	complemented
206	C₂×#534	C₂⁴	1	2	yes	—
207	C₂×#533	C₂×D₄	1	1	yes	characteristic, complemented
208	C₂×#532	C₂×D₄	1	1	yes	characteristic
209	C₂×#525	C₂×D₄	1	2	yes	—
210	C₂×#525	C₂×D₄	1	2	yes	—
211	C₂×#525	C₂×D₄	1	1	yes	characteristic
212	C₂×#525	C₂×D₄	1	1	yes	characteristic
213	C₂×#525	C₂×D₄	1	1	yes	characteristic, complemented
214	C₂²×#120	C₂×D₄	1	2	yes	—
215	C₂²×#120	C₂×D₄	1	2	yes	—
216	C₂²×#120	C₂×D₄	1	2	yes	—
217	C₂²×#120	C₂×D₄	1	2	yes	—
218	C₂²×#120	C₂×D₄	1	2	yes	—
219	C₂²×#120	C₂×D₄	1	2	yes	—
220	C₂²×#120	C₂×D₄	1	2	yes	—
221	C₂²×#120	C₂×D₄	1	2	yes	complemented
222	C₂²×#120	C₂×D₄	1	2	yes	complemented
223	C₂²×#120	C₂×D₄	1	2	yes	complemented
224	C₂²×#120	C₂×D₄	1	2	yes	complemented
225	C₂²×#120	C₂⁴	1	2	yes	—
226	C₂²×#120	C₂⁴	1	2	yes	—
227	C₂×#520	C₂×D₄	1	1	yes	characteristic
228	C₂×#518	C₂×D₄	1	2	yes	—
229	C₂×#518	C₂⁴	1	2	yes	—
230	C₂×#518	C₂⁴	1	1	yes	characteristic
231	C₂×C₄×#41	C₂×D₄	1	1	yes	characteristic
232	C₂×C₄×#41	C₂×D₄	1	1	yes	characteristic, complemented
233	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic, complemented
234	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	complemented
235	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	complemented
236	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic, complemented
237	C₂²×#100	C₂⁴	1	2	yes	—
238	C₂²×#100	C₂⁴	1	2	yes	—
239	C₂²×#99	C₂⁴	1	4	yes	—
240	C₂²×#99	C₂⁴	1	4	yes	—
241	#2804	C₂×D₄	1	1	yes	characteristic, complemented
242	#2748	C₂×D₄	1	2	yes	complemented
243	#2724	C₂×D₄	1	1	yes	characteristic
244	#2697	C₂×D₄	1	2	yes	—
245	#2697	C₂×D₄	1	2	yes	—
246	#2697	C₂×D₄	1	2	yes	—
247	#2684	C₂×D₄	1	2	yes	complemented
248	#2684	C₂×D₄	1	2	yes	complemented
249	#2683	C₂×D₄	1	1	yes	characteristic, complemented
250	#2679	C₂×D₄	1	1	yes	characteristic
251	#2679	C₂×D₄	1	1	yes	characteristic
252	#2679	C₂×D₄	1	2	yes	—
253	#2679	C₂×D₄	1	2	yes	—
254	#2679	C₂×D₄	1	2	yes	—
255	#2679	C₂×D₄	1	1	yes	characteristic
256	#2679	C₂×D₄	1	1	yes	characteristic, complemented
257	#2679	C₂×D₄	1	2	yes	complemented
258	#2679	C₂×D₄	1	2	yes	complemented
259	#2679	C₂×D₄	1	2	yes	complemented
260	#2679	C₂×D₄	1	2	yes	complemented
261	#2679	C₂×D₄	1	2	yes	complemented
262	#2679	C₂×D₄	1	1	yes	characteristic, complemented
263	#2669	C₂×D₄	1	2	yes	complemented
264	#2669	C₂×D₄	1	2	yes	complemented
265	#2435	C₂×D₄	1	2	yes	—
266	#2433	C₂×D₄	1	2	yes	—
267	#2432	C₂×D₄	1	2	yes	—
268	#2431	C₂×D₄	1	2	yes	—
269	#2404	C₂×D₄	1	2	yes	—
270	#2404	C₂×D₄	1	2	yes	—
271	#2404	C₂×D₄	1	2	yes	—
272	#2404	C₂×D₄	1	2	yes	—
273	#2403	C₂×D₄	1	2	yes	—
274	#2403	C₂×D₄	1	2	yes	—
275	#2403	C₂×D₄	1	2	yes	—
276	#2403	C₂×D₄	1	2	yes	—
277	#2376	C₂×D₄	1	2	yes	complemented
278	#2375	C₂×D₄	1	2	yes	complemented
279	#2375	C₂×D₄	1	2	yes	complemented
280	#2375	C₂×D₄	1	2	yes	complemented
281	#2374	C₂×D₄	1	2	yes	complemented
282	#2358	C₂×D₄	1	2	yes	complemented
283	#2358	C₂×D₄	1	2	yes	complemented
284	#2358	C₂×D₄	1	2	yes	complemented
285	#2317	C₂×D₄	1	1	yes	characteristic, complemented
286	#2316	C₂×D₄	1	1	yes	characteristic, complemented
287	#2315	C₂×D₄	1	2	yes	complemented
288	#2315	C₂×D₄	1	2	yes	complemented
289	#2311	C₂×D₄	1	1	yes	characteristic
290	#2311	C₂×D₄	1	1	yes	characteristic
291	#2311	C₂×D₄	1	1	yes	characteristic
292	#2311	C₂×D₄	1	1	yes	characteristic
293	#2311	C₂×D₄	1	1	yes	characteristic, complemented
294	#2309	C₂×D₄	1	1	yes	characteristic
295	#2309	C₂×D₄	1	1	yes	characteristic
296	#2309	C₂×D₄	1	1	yes	characteristic, complemented
297	#2309	C₂⁴	1	1	yes	characteristic
298	#2308	C₂×D₄	1	2	yes	—
299	#2308	C₂×D₄	1	2	yes	—
300	#2308	C₂×D₄	1	2	yes	complemented
301	#2308	C₂×D₄	1	2	yes	complemented
302	#2308	C₂⁴	1	2	yes	—
303	#2297	C₂×D₄	1	1	yes	characteristic
304	#2297	C₂×D₄	1	1	yes	characteristic
305	#2297	C₂×D₄	1	1	yes	characteristic
306	#2297	C₂×D₄	1	1	yes	characteristic
307	#2296	C₂×D₄	1	1	yes	characteristic, complemented
308	#2296	C₂×D₄	1	1	yes	characteristic, complemented
309	C₂×#390	C₂×D₄	1	1	yes	characteristic
310	C₂×#390	C₂×D₄	1	1	yes	characteristic
311	C₂×#390	C₂×D₄	1	1	yes	characteristic, complemented
312	C₂×#390	C₂×D₄	1	1	yes	characteristic, complemented
313	C₂×#390	C₂⁴	1	1	yes	characteristic
314	C₂×#390	C₂⁴	1	1	yes	characteristic
315	#2280	C₂×D₄	1	1	yes	characteristic, complemented
316	C₂×#379	C₂×D₄	1	2	yes	complemented
317	C₂×#379	C₂×D₄	1	2	yes	complemented
318	C₂×#379	C₂×D₄	1	2	yes	complemented
319	#2274	C₂×D₄	1	1	yes	characteristic, complemented
320	#2273	C₂×D₄	1	2	yes	complemented
321	C₂⁶	#120	1	2	yes	complemented
322	C₂⁶	C₂²×D₄	1	2	yes	—
323	C₂⁶	C₂²×D₄	1	2	yes	—
324	C₂⁶	C₂²×D₄	1	2	yes	—
325	C₂×H(5,𝔽₂)	H(5,𝔽₂)	1	4	yes	complemented
326	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
327	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
328	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
329	C₂²×#41	C₂²×D₄	1	2	yes	—
330	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
331	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
332	C₂³×D₄	#120	1	2	yes	complemented
333	C₂³×D₄	#120	1	2	yes	complemented
334	C₂³×D₄	#120	1	2	yes	complemented
335	C₂³×D₄	#120	1	2	yes	complemented
336	C₂³×D₄	C₂²×D₄	1	2	yes	—
337	C₂³×D₄	C₂²×D₄	1	2	yes	—
338	C₂³×D₄	C₂²×D₄	1	2	yes	—
339	C₂³×D₄	C₂²×D₄	1	2	yes	—
340	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
341	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
342	C₂³×D₄	C₂²×D₄	1	2	yes	—
343	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
344	C₂³×D₄	C₂⁵	1	1	yes	characteristic
345	C₂⁴×C₄	#120	1	2	yes	complemented
346	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
347	C₂×C₄⋊Q₈	#120	1	1	yes	characteristic, complemented
348	C₂×#127	#120	1	1	yes	characteristic, complemented
349	C₂×#127	C₂²×D₄	1	2	yes	—
350	C₂×#124	#120	1	1	yes	characteristic, complemented
351	C₂×#124	#120	1	1	yes	characteristic, complemented
352	C₂×#124	C₂²×D₄	1	2	yes	—
353	C₂×#123	#120	1	1	yes	characteristic
354	C₂×#123	#120	1	1	yes	characteristic
355	C₂×#123	#120	1	2	yes	—
356	C₂×#123	#120	1	2	yes	complemented
357	C₂×#123	#120	1	2	yes	complemented
358	C₂×#123	C₂²×D₄	1	2	yes	—
359	C₂×#123	C₂²×D₄	1	2	yes	—
360	C₂×C₂²⋊Q₈	#120	1	2	yes	complemented
361	C₂×#121	#120	1	2	yes	complemented
362	C₂×#120	#120	1	2	yes	—
363	C₂×#120	#120	1	2	yes	—
364	C₂×#120	#120	1	1	yes	characteristic
365	C₂×#120	#120	1	2	yes	—
366	C₂×#120	#120	1	1	yes	characteristic
367	C₂×#120	#120	1	2	yes	—
368	C₂×#120	#120	1	2	yes	—
369	C₂×#120	#120	1	2	yes	complemented
370	C₂×#120	#120	1	2	yes	complemented
371	C₂×#120	#120	1	2	yes	complemented
372	C₂×#120	#120	1	2	yes	complemented
373	C₂×#120	#120	1	2	yes	complemented
374	C₂×#120	#120	1	2	yes	complemented
375	C₂×#120	#120	1	2	yes	complemented
376	C₂×#120	#120	1	2	yes	complemented
377	C₂×#120	#120	1	2	yes	complemented
378	C₂×#120	#120	1	2	yes	complemented
379	C₂×#120	C₂²×D₄	1	2	yes	—
380	C₂×#120	C₂²×D₄	1	2	yes	—
381	C₂×#120	C₂²×D₄	1	2	yes	—
382	C₂×#120	C₂²×D₄	1	2	yes	—
383	C₂×#120	C₂²×D₄	1	2	yes	—
384	C₂×#120	C₂²×D₄	1	2	yes	—
385	C₂×#120	C₂²×D₄	1	2	yes	—
386	C₂×#120	C₂²×D₄	1	2	yes	—
387	C₂×#120	C₂²×D₄	1	2	yes	—
388	C₂×#120	C₂²×D₄	1	2	yes	—
389	C₂×#117	#120	1	1	yes	characteristic, complemented
390	C₂×#117	#120	1	1	yes	characteristic, complemented
391	C₂×#117	#120	1	2	yes	complemented
392	C₂×#117	#120	1	2	yes	complemented
393	C₂×#117	#120	1	1	yes	characteristic, complemented
394	C₂×#117	#120	1	1	yes	characteristic, complemented
395	C₂×#117	C₂²×D₄	1	1	yes	characteristic
396	C₂×#117	C₂²×D₄	1	1	yes	characteristic
397	C₂×#117	C₂²×D₄	1	1	yes	characteristic
398	C₂×#117	C₂²×D₄	1	1	yes	characteristic
399	C₂×#117	C₂²×D₄	1	1	yes	characteristic
400	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic, complemented
401	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic, complemented
402	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
403	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
404	C₂²×C₂²⋊C₄	#120	1	2	yes	complemented
405	C₂²×C₂²⋊C₄	#120	1	2	yes	complemented
406	C₂²×C₂²⋊C₄	#120	1	2	yes	complemented
407	C₂²×C₂²⋊C₄	#120	1	2	yes	complemented
408	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
409	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
410	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
411	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
412	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
413	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
414	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
415	C₂²×C₄²	#120	1	1	yes	characteristic, complemented
416	C₂²×C₄²	#120	1	1	yes	characteristic, complemented
417	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
418	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
419	C₂×#99	H(5,𝔽₂)	1	8	yes	complemented
420	#391	#120	1	1	yes	characteristic, complemented
421	#391	#120	1	1	yes	characteristic, complemented
422	#391	#120	1	1	yes	characteristic, complemented
423	#391	#120	1	2	yes	complemented
424	#391	#120	1	1	yes	characteristic, complemented
425	#390	#120	1	1	yes	characteristic, complemented
426	#390	#120	1	1	yes	characteristic, complemented
427	#390	#120	1	1	yes	characteristic, complemented
428	#390	#120	1	2	yes	complemented
429	#390	#120	1	1	yes	characteristic, complemented
430	#390	C₂²×D₄	1	1	yes	characteristic
431	#390	C₂²×D₄	1	1	yes	characteristic
432	#390	C₂²×D₄	1	1	yes	characteristic
433	#390	C₂²×D₄	1	1	yes	characteristic
434	#390	C₂²×D₄	1	1	yes	characteristic
435	#390	C₂²×D₄	1	1	yes	characteristic
436	#390	C₂²×D₄	1	2	yes	—
437	#390	C₂²×D₄	1	2	yes	—
438	#390	C₂²×D₄	1	1	yes	characteristic
439	#390	C₂²×D₄	1	1	yes	characteristic
440	#386	#120	1	2	yes	complemented
441	#386	#120	1	2	yes	complemented
442	#386	#120	1	2	yes	complemented
443	#386	#120	1	2	yes	complemented
444	#386	C₂²×D₄	1	2	yes	—
445	#386	C₂²×D₄	1	2	yes	—
446	#386	C₂²×D₄	1	2	yes	—
447	#386	C₂²×D₄	1	2	yes	—
448	#381	#120	1	2	yes	complemented
449	#381	#120	1	2	yes	complemented
450	#379	#120	1	2	yes	complemented
451	#379	#120	1	2	yes	complemented
452	C₄×C₂²⋊C₄	#120	1	2	yes	complemented
453	C₄×C₂²⋊C₄	#120	1	2	yes	complemented
454	C₂⁵	C₂×#120	1	2	yes	—
455	C₂⁵	C₂×#120	1	1	yes	characteristic
456	C₂⁵	C₂×#120	1	2	yes	—
457	C₂⁵	C₂×#120	1	2	yes	—
458	C₂⁵	C₂×#120	1	2	yes	—
459	C₂⁵	C₂×#120	1	2	yes	—
460	C₂⁵	C₂×#120	1	2	yes	—
461	C₂⁵	C₂×#120	1	2	yes	—
462	C₂⁵	C₂×#120	1	2	yes	—
463	C₂⁵	C₂³×D₄	1	2	yes	—
464	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
465	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
466	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
467	C₂²×D₄	C₂×#120	1	1	yes	characteristic
468	C₂²×D₄	C₂×#120	1	2	yes	—
469	C₂²×D₄	C₂×#120	1	2	yes	—
470	C₂²×D₄	C₂×#120	1	2	yes	—
471	C₂²×D₄	C₂×#120	1	1	yes	characteristic
472	C₂²×D₄	C₂×#120	1	2	yes	—
473	C₂²×D₄	C₂×#120	1	2	yes	—
474	C₂²×D₄	C₂×#120	1	1	yes	characteristic
475	C₂²×D₄	C₂×#120	1	1	yes	characteristic
476	C₂²×D₄	C₂×#120	1	1	yes	characteristic
477	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
478	C₂²×D₄	C₂³×D₄	1	2	yes	—
479	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
480	C₂²×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
481	C₂²×D₄	C₂×H(5,𝔽₂)	1	4	yes	—
482	C₂²×D₄	#585	1	2	yes	—
483	C₂³×C₄	C₂×#120	1	1	yes	characteristic
484	C₂³×C₄	C₂×#120	1	2	yes	—
485	C₂³×C₄	C₂×#120	1	1	yes	characteristic
486	C₂³×C₄	C₂×#120	1	1	yes	characteristic
487	C₂³×C₄	C₂×#120	1	2	yes	—
488	C₂³×C₄	C₂×#120	1	2	yes	—
489	C₂³×C₄	C₂×#120	1	2	yes	—
490	C₂³×C₄	C₂×#120	1	2	yes	—
491	C₂³×C₄	C₂×#120	1	2	yes	—
492	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
493	#120	B(4,𝔽₂)	1	4	yes	—
494	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
495	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
496	C₂×C₄⋊C₄	C₂×#120	1	2	yes	—
497	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
498	C₂×C₄⋊C₄	C₂×#120	1	1	yes	characteristic
499	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
500	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
501	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
502	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
503	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
504	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
505	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
506	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
507	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
508	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
509	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
510	C₂×C₄²	C₂×#120	1	2	yes	—
511	C₂×C₄²	C₂×#120	1	1	yes	characteristic
512	C₂×C₄²	C₂×#120	1	1	yes	characteristic
513	C₂×C₄²	C₂×#120	1	1	yes	characteristic
514	C₂×C₄²	C₂×#120	1	1	yes	characteristic
515	C₂×C₄²	C₂×#120	1	1	yes	characteristic
516	C₂×C₄²	C₂×#120	1	1	yes	characteristic
517	C₂⁴	#2846	1	2	yes	—
518	C₂⁴	#2846	1	1	yes	characteristic
519	C₂⁴	#2846	1	1	yes	characteristic
520	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
521	C₂⁴	C₂²×#120	1	2	yes	—
522	C₂⁴	C₂²×#120	1	2	yes	—
523	C₂⁴	C₂²×#120	1	1	yes	characteristic
524	C₂⁴	#3484	1	4	yes	—
525	C₂⁴	#3496	1	4	yes	—
526	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
527	C₂×D₄	#3026	1	2	yes	—
528	C₂×D₄	#3484	1	4	yes	—
529	C₂×D₄	#3484	1	1	yes	characteristic
530	C₂×D₄	#3484	1	2	yes	—
531	C₂×D₄	#3485	1	1	yes	characteristic
532	C₂×D₄	#3488	1	2	yes	—
533	C₂×D₄	#3496	1	4	yes	—
534	C₂×D₄	#3498	1	2	yes	—
535	C₂²×C₄	#2846	1	2	yes	—
536	C₂²×C₄	#2846	1	1	yes	characteristic
537	C₂²×C₄	#2846	1	1	yes	characteristic
538	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
539	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
540	C₂²×C₄	#3488	1	2	yes	—
541	C₂²×C₄	#3498	1	2	yes	—
542	C₂²⋊C₄	C₂×B(4,𝔽₂)	1	2	yes	—
543	C₂³	#32898	1	2	yes	—
544	C₂³	#35346	1	2	yes	—
545	C₂³	#35347	1	2	yes	—
546	C₂³	#35680	1	4	yes	—
547	C₂³	#35680	1	2	yes	—
548	C₂³	C₂×#2846	1	1	yes	characteristic
549	C₂³	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
550	C₂³	C₂×#3026	1	1	yes	characteristic
551	C₂³	#60399	1	2	yes	—
552	C₂×C₄	#32898	1	2	yes	—
553	C₂×C₄	#35512	1	1	yes	characteristic
554	C₂×C₄	#35527	1	1	yes	characteristic
555	C₂×C₄	#35680	1	2	yes	—
556	C₂²	#4928938	1	1	yes	characteristic
557	C₂²	#6533135	1	2	yes	—
558	C₂²	#6533339	1	2	yes	—
559	C₂²	C₂×#32898	1	1	yes	characteristic
560	C₂²	#6628896	1	1	yes	characteristic
561	C₂	#72216366	1	1	yes	characteristic
562	C₂	#72233203	1	1	yes	characteristic
563	C₂	#72234419	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#6628896
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#3484 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #72234421	C₂	semidirect
2. #72233187	C₂	semidirect
3. #72233169	C₂	semidirect
4. #72233135	C₂	semidirect
5. #72233117	C₂	semidirect
6. #72233091	C₂	semidirect
7. #72233051	C₂	semidirect
8. #72233038	C₂	semidirect
9. #72232947	C₂	semidirect
10. #72232895	C₂	semidirect
11. #72232823	C₂	semidirect
12. #72232813	C₂	semidirect
13. #72232390	C₂	semidirect
14. #72219306	C₂	semidirect
15. #72219249	C₂	semidirect
16. #72218099	C₂	semidirect
17. #72216646	C₂	semidirect
18. C₂×#60399	C₂²	semidirect
19. C₂×#60399	C₂²	semidirect
21. #7829278	C₂²	semidirect
22. #7819715	C₂²	semidirect
24. #7711204	C₂²	semidirect
25. #7708349	C₂²	semidirect
28. #7659311	C₂²	semidirect
30. C₂²×#2679	C₂²	semidirect
31. C₂×#33551	C₂²	semidirect
32. C₂×#33348	C₂²	semidirect
33. #6628896	C₂²	semidirect
34. #6628876	C₂²	semidirect
35. C₂×#23900	C₂²	semidirect
36. #6370747	C₂²	semidirect
37. #498984	C₂²	semidirect
38. #498927	C₂²	semidirect
39. #498890	C₂²	semidirect
40. #498886	C₂²	semidirect
41. C₂×#13343	C₂²	semidirect
42. C₂×#13343	C₂²	semidirect
43. C₂×#13343	C₂²	semidirect
44. #494721	C₂²	semidirect
45. #494700	C₂²	semidirect
46. #494682	C₂²	semidirect
47. #494582	C₂²	semidirect
48. #494575	C₂²	semidirect
49. #493867	C₂²	semidirect
50. C₂×#12753	C₂²	semidirect
51. #493424	C₂²	semidirect
52. C₂×#12687	C₂²	semidirect
53. #492900	C₂²	semidirect
54. #394472	C₂²	semidirect
55. #394273	C₂²	semidirect
56. #394064	C₂²	semidirect
57. #393975	C₂²	semidirect
58. #374036	C₂²	semidirect
59. #374004	C₂²	semidirect
60. #364532	C₂²	semidirect
61. #364526	C₂²	semidirect
62. #359888	C₂²	semidirect
63. #359593	C₂²	semidirect
64. #359591	C₂²	semidirect
65. #359591	C₂²	semidirect
66. #359563	C₂²	semidirect
67. #149481	C₂²	semidirect
68. #149444	C₂²	semidirect
69. #147815	C₂²	semidirect
70. #147811	C₂²	semidirect
71. #147783	C₂²	semidirect
72. #147779	C₂²	semidirect
73. #147779	C₂²	semidirect
74. #147779	C₂²	semidirect
75. #147732	C₂²	semidirect
76. #147569	C₂²	semidirect
77. #147567	C₂²	semidirect
78. #146692	C₂²	semidirect
79. #146675	C₂²	semidirect
80. #146609	C₂²	semidirect
81. #146593	C₂²	semidirect
82. #146465	C₂²	semidirect
88. C₂×#3444	D₄	semidirect
89. C₂³×#120	D₄	semidirect
99. #60045	D₄	semidirect
100. #59814	D₄	semidirect
101. #59179	D₄	semidirect
102. #58990	D₄	semidirect
103. #52163	D₄	semidirect
104. C₂×#2679	D₄	semidirect
108. #51940	D₄	semidirect
109. #51815	D₄	semidirect
110. #50856	D₄	semidirect
111. #50646	D₄	semidirect
112. #49562	D₄	semidirect
113. #49545	D₄	semidirect
114. #46287	D₄	semidirect
115. #46089	D₄	semidirect
116. #45452	D₄	semidirect
117. #45439	D₄	semidirect
118. #37297	D₄	semidirect
119. #37269	D₄	semidirect
121. #37129	D₄	semidirect
123. #36714	D₄	semidirect
126. #36626	D₄	semidirect
127. C₂×#2846	D₄	semidirect
128. #35556	D₄	semidirect
129. #35533	D₄	semidirect
132. #35390	D₄	semidirect
133. #35348	D₄	semidirect
136. #35256	D₄	semidirect
137. #35254	D₄	semidirect
138. #35223	D₄	semidirect
140. #34981	D₄	semidirect
141. #34981	D₄	semidirect
142. #34980	D₄	semidirect
148. #34200	D₄	semidirect
149. #34198	D₄	semidirect
184. C₂⁷	C₂×D₄	semidirect
189. C₂×H(3,𝔽₄)	C₂×D₄	semidirect
190. C₂×#560	C₂×D₄	semidirect
191. C₂×#551	C₂×D₄	semidirect
192. C₂×D₄²	C₂×D₄	semidirect
198. C₂×#535	C₂×D₄	semidirect
205. C₂×#534	C₂×D₄	semidirect
207. C₂×#533	C₂×D₄	semidirect
213. C₂×#525	C₂×D₄	semidirect
221. C₂²×#120	C₂×D₄	semidirect
222. C₂²×#120	C₂×D₄	semidirect
223. C₂²×#120	C₂×D₄	semidirect
224. C₂²×#120	C₂×D₄	semidirect
232. C₂×C₄×#41	C₂×D₄	semidirect
233. C₂³×C₄⋊C₄	C₂×D₄	semidirect
234. C₂³×C₂²⋊C₄	C₂×D₄	semidirect
235. C₂³×C₂²⋊C₄	C₂×D₄	semidirect
236. C₂³×C₄²	C₂×D₄	semidirect
241. #2804	C₂×D₄	semidirect
242. #2748	C₂×D₄	semidirect
247. #2684	C₂×D₄	semidirect
248. #2684	C₂×D₄	semidirect
249. #2683	C₂×D₄	semidirect
256. #2679	C₂×D₄	semidirect
257. #2679	C₂×D₄	semidirect
258. #2679	C₂×D₄	semidirect
259. #2679	C₂×D₄	semidirect
260. #2679	C₂×D₄	semidirect
261. #2679	C₂×D₄	semidirect
262. #2679	C₂×D₄	semidirect
263. #2669	C₂×D₄	semidirect
264. #2669	C₂×D₄	semidirect
277. #2376	C₂×D₄	semidirect
278. #2375	C₂×D₄	semidirect
279. #2375	C₂×D₄	semidirect
280. #2375	C₂×D₄	semidirect
281. #2374	C₂×D₄	semidirect
282. #2358	C₂×D₄	semidirect
283. #2358	C₂×D₄	semidirect
284. #2358	C₂×D₄	semidirect
285. #2317	C₂×D₄	semidirect
286. #2316	C₂×D₄	semidirect
287. #2315	C₂×D₄	semidirect
288. #2315	C₂×D₄	semidirect
293. #2311	C₂×D₄	semidirect
296. #2309	C₂×D₄	semidirect
300. #2308	C₂×D₄	semidirect
301. #2308	C₂×D₄	semidirect
307. #2296	C₂×D₄	semidirect
308. #2296	C₂×D₄	semidirect
311. C₂×#390	C₂×D₄	semidirect
312. C₂×#390	C₂×D₄	semidirect
315. #2280	C₂×D₄	semidirect
316. C₂×#379	C₂×D₄	semidirect
317. C₂×#379	C₂×D₄	semidirect
318. C₂×#379	C₂×D₄	semidirect
319. #2274	C₂×D₄	semidirect
320. #2273	C₂×D₄	semidirect
321. C₂⁶	#120	semidirect
325. C₂×H(5,𝔽₂)	H(5,𝔽₂)	semidirect
332. C₂³×D₄	#120	semidirect
333. C₂³×D₄	#120	semidirect
334. C₂³×D₄	#120	semidirect
335. C₂³×D₄	#120	semidirect
345. C₂⁴×C₄	#120	semidirect
347. C₂×C₄⋊Q₈	#120	semidirect
348. C₂×#127	#120	semidirect
350. C₂×#124	#120	semidirect
351. C₂×#124	#120	semidirect
356. C₂×#123	#120	semidirect
357. C₂×#123	#120	semidirect
360. C₂×C₂²⋊Q₈	#120	semidirect
361. C₂×#121	#120	semidirect
369. C₂×#120	#120	semidirect
370. C₂×#120	#120	semidirect
371. C₂×#120	#120	semidirect
372. C₂×#120	#120	semidirect
373. C₂×#120	#120	semidirect
374. C₂×#120	#120	semidirect
375. C₂×#120	#120	semidirect
376. C₂×#120	#120	semidirect
377. C₂×#120	#120	semidirect
378. C₂×#120	#120	semidirect
389. C₂×#117	#120	semidirect
390. C₂×#117	#120	semidirect
391. C₂×#117	#120	semidirect
392. C₂×#117	#120	semidirect
393. C₂×#117	#120	semidirect
394. C₂×#117	#120	semidirect
400. C₂²×C₄⋊C₄	#120	semidirect
401. C₂²×C₄⋊C₄	#120	semidirect
404. C₂²×C₂²⋊C₄	#120	semidirect
405. C₂²×C₂²⋊C₄	#120	semidirect
406. C₂²×C₂²⋊C₄	#120	semidirect
407. C₂²×C₂²⋊C₄	#120	semidirect
415. C₂²×C₄²	#120	semidirect
416. C₂²×C₄²	#120	semidirect
419. C₂×#99	H(5,𝔽₂)	semidirect
420. #391	#120	semidirect
421. #391	#120	semidirect
422. #391	#120	semidirect
423. #391	#120	semidirect
424. #391	#120	semidirect
425. #390	#120	semidirect
426. #390	#120	semidirect
427. #390	#120	semidirect
428. #390	#120	semidirect
429. #390	#120	semidirect
440. #386	#120	semidirect
441. #386	#120	semidirect
442. #386	#120	semidirect
443. #386	#120	semidirect
448. #381	#120	semidirect
449. #381	#120	semidirect
450. #379	#120	semidirect
451. #379	#120	semidirect
452. C₄×C₂²⋊C₄	#120	semidirect
453. C₄×C₂²⋊C₄	#120	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#72234421	2
#72233187	2
#72233169	2
#72233135	2
#72233117	2
#72233091	2
#72233051	2
#72233038	2
#72232947	2
#72232895	2
#72232823	2
#72232813	2
#72232390	2
#72219306	2
#72219249	2
#72218099	2
#72216646	2