Subgroups — #72234201

Order 2048.

At least 10968 conjugacy classes of subgroups; only the 786 normal subgroups are shown.

The subgroup lattice has 436 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234197	C₂	1	1	yes	maximal, characteristic, complemented
2	#72234193	C₂	1	1	yes	maximal, characteristic, complemented
3	#72233917	C₂	1	1	yes	maximal, characteristic, complemented
4	#72233911	C₂	1	1	yes	maximal, characteristic, complemented
5	#72233902	C₂	1	1	yes	maximal, characteristic, complemented
6	#72233262	C₂	1	2	yes	maximal, complemented
7	#72232863	C₂	1	1	yes	maximal, characteristic, complemented
8	#72232739	C₂	1	2	yes	maximal, complemented
9	#72232033	C₂	1	1	yes	maximal, characteristic
10	#72225779	C₂	1	2	yes	maximal, complemented
11	#72219306	C₂	1	2	yes	maximal, complemented
12	C₂×#63011	C₂²	1	1	yes	characteristic
13	C₂³×D₄²	C₄	1	1	yes	characteristic, complemented
14	#7737413	C₄	1	1	yes	characteristic, complemented
15	#7737411	C₂²	1	1	yes	characteristic
16	#7733150	C₄	1	2	yes	complemented
17	#7733149	C₂²	1	2	yes	—
18	#7711201	C₄	1	1	yes	characteristic, complemented
19	#7708349	C₄	1	2	yes	complemented
20	#7671946	C₂²	1	1	yes	characteristic
21	#7670203	C₂²	1	2	yes	—
22	#7627211	C₄	1	1	yes	characteristic, complemented
23	C₂×#33348	C₂²	1	1	yes	characteristic, complemented
24	#494742	C₂²	1	1	yes	characteristic, complemented
25	#494722	C₂²	1	1	yes	characteristic, complemented
26	#494700	C₂²	1	2	yes	complemented
27	#494682	C₂²	1	2	yes	complemented
28	#493441	C₂²	1	1	yes	characteristic, complemented
29	#493430	C₂²	1	1	yes	characteristic, complemented
30	#493425	C₂²	1	2	yes	complemented
31	#493425	C₂²	1	1	yes	characteristic, complemented
32	#493424	C₂²	1	2	yes	complemented
33	#493424	C₂²	1	2	yes	complemented
34	#359605	C₂²	1	1	yes	characteristic, complemented
35	#359601	C₂²	1	1	yes	characteristic, complemented
36	#359593	C₂²	1	2	yes	complemented
37	#359591	C₂²	1	2	yes	complemented
38	#214616	C₂²	1	2	yes	complemented
39	#213906	C₂²	1	2	yes	complemented
40	#213854	C₂²	1	2	yes	complemented
41	C₂×#3498	C₂³	1	1	yes	characteristic
42	C₂×#3484	D₄	1	1	yes	characteristic
43	C₂×#3484	C₂³	1	1	yes	characteristic
44	C₂×#3484	C₂³	1	2	yes	—
45	C₂×#3549	D₄	1	1	yes	characteristic
46	C₂×#3543	C₂×C₄	1	1	yes	characteristic
47	C₂×#3543	D₄	1	1	yes	characteristic
48	C₂×#3537	D₄	1	2	yes	—
49	C₂×#3532	C₂×C₄	1	2	yes	—
50	C₂×#3532	D₄	1	2	yes	—
51	C₂×#3481	D₄	1	1	yes	characteristic
52	C₂²×D₄²	C₂×C₄	1	1	yes	characteristic
53	C₂²×D₄²	C₂×C₄	1	1	yes	characteristic
54	C₂²×D₄²	C₂×C₄	1	2	yes	—
55	C₂²×D₄²	D₄	1	1	yes	characteristic
56	C₂³×#127	C₂×C₄	1	1	yes	characteristic
57	C₂³×#120	C₂×C₄	1	2	yes	—
58	C₂³×#120	D₄	1	2	yes	—
59	C₂×#2888	C₂³	1	1	yes	characteristic, complemented
60	C₂×#2883	C₂³	1	1	yes	characteristic, complemented
61	#59126	D₄	1	2	yes	—
62	#59089	D₄	1	4	yes	—
63	#59041	D₄	1	2	yes	—
64	#58990	D₄	1	4	yes	—
65	#52004	D₄	1	2	yes	—
66	#52004	D₄	1	2	yes	—
67	#49545	D₄	1	4	yes	—
68	#47827	D₄	1	4	yes	—
69	#37232	C₂×C₄	1	2	yes	—
70	#36714	C₂×C₄	1	2	yes	—
71	#36702	C₂×C₄	1	2	yes	—
72	#36624	D₄	1	1	yes	characteristic
73	C₂×#2846	D₄	1	2	yes	—
74	#35528	C₂×C₄	1	1	yes	characteristic
75	#35528	D₄	1	1	yes	characteristic
76	#35527	D₄	1	1	yes	characteristic
77	#35527	C₂³	1	1	yes	characteristic
78	#35390	D₄	1	2	yes	—
79	#35348	C₂×C₄	1	2	yes	—
80	#35348	D₄	1	2	yes	—
81	#35346	C₂³	1	2	yes	—
82	#35274	D₄	1	1	yes	characteristic
83	#35272	D₄	1	1	yes	characteristic
84	#35222	C₂×C₄	1	1	yes	characteristic
85	#35222	D₄	1	1	yes	characteristic
86	#34981	D₄	1	2	yes	—
87	#34981	D₄	1	2	yes	—
88	#34980	C₂×C₄	1	2	yes	—
89	#34980	D₄	1	2	yes	—
90	#34364	C₂×C₄	1	1	yes	characteristic
91	#34356	C₂×C₄	1	2	yes	—
92	#34210	C₂×C₄	1	1	yes	characteristic
93	#34198	C₂×C₄	1	2	yes	—
94	#34198	D₄	1	2	yes	—
95	C₄×#525	D₄	1	1	yes	characteristic
96	#20592	C₂³	1	2	yes	complemented
97	#20592	C₂³	1	1	yes	characteristic, complemented
98	#20586	C₂³	1	1	yes	characteristic, complemented
99	#20585	C₂³	1	2	yes	complemented
100	C₂⁷	C₂²⋊C₄	1	2	yes	complemented
101	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
102	C₂²×H(5,𝔽₂)	C₂⁴	1	1	yes	characteristic
103	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
104	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
105	C₂⁴×D₄	C₂×D₄	1	2	yes	—
106	C₂×H(3,𝔽₄)	C₂²⋊C₄	1	2	yes	—
107	C₂×#560	C₂²⋊C₄	1	2	yes	—
108	C₂×#560	C₂²⋊C₄	1	2	yes	—
109	C₂×#559	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×#551	C₂²⋊C₄	1	2	yes	—
111	C₂×D₄²	C₂²⋊C₄	1	2	yes	—
112	C₂×D₄²	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×D₄²	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×D₄²	C₂²⋊C₄	1	2	yes	—
115	C₂×D₄²	C₂²⋊C₄	1	1	yes	characteristic
116	C₂×D₄²	C₂²⋊C₄	1	2	yes	complemented
117	C₂×D₄²	C₂²⋊C₄	1	2	yes	complemented
118	C₂×D₄²	C₂²⋊C₄	1	4	yes	complemented
119	C₂×D₄²	C₂²⋊C₄	1	4	yes	complemented
120	C₂×#535	C₂²⋊C₄	1	1	yes	characteristic, complemented
121	C₂×#535	C₂²⋊C₄	1	1	yes	characteristic, complemented
122	C₂×#535	C₂²×C₄	1	1	yes	characteristic
123	C₂×#535	C₂×D₄	1	1	yes	characteristic
124	C₂×#535	C₂×D₄	1	1	yes	characteristic
125	C₂×#535	C₂×D₄	1	1	yes	characteristic
126	C₂×#535	C₂×D₄	1	1	yes	characteristic
127	C₂×#535	C₂×D₄	1	1	yes	characteristic
128	C₂×#535	C₂×D₄	1	1	yes	characteristic
129	C₂×#535	C₂×D₄	1	1	yes	characteristic
130	C₂×#535	C₂×D₄	1	1	yes	characteristic
131	C₂×#535	C₂×D₄	1	1	yes	characteristic
132	C₂×#535	C₂×D₄	1	2	yes	—
133	C₂×#535	C₂×D₄	1	2	yes	—
134	C₂×#534	C₂²⋊C₄	1	2	yes	complemented
135	C₂×#534	C₂²×C₄	1	2	yes	—
136	C₂×#534	C₂×D₄	1	2	yes	—
137	C₂×#534	C₂×D₄	1	2	yes	—
138	C₂×#534	C₂×D₄	1	2	yes	—
139	C₂×#534	C₂×D₄	1	2	yes	—
140	C₂×#534	C₂×D₄	1	2	yes	—
141	C₂×#532	C₂×D₄	1	1	yes	characteristic
142	C₂×#532	C₂×D₄	1	1	yes	characteristic
143	C₂×#532	C₂×D₄	1	1	yes	characteristic
144	C₂²×#127	C₂²⋊C₄	1	1	yes	characteristic, complemented
145	C₂²×#127	C₂²⋊C₄	1	1	yes	characteristic, complemented
146	C₂²×#127	C₂²×C₄	1	1	yes	characteristic
147	C₂²×#127	C₂²×C₄	1	1	yes	characteristic
148	C₂²×#127	C₂×D₄	1	1	yes	characteristic
149	C₂²×#127	C₂×D₄	1	1	yes	characteristic
150	C₂×#525	C₂²⋊C₄	1	1	yes	characteristic, complemented
151	C₂×#525	C₂×D₄	1	1	yes	characteristic
152	C₂×#525	C₂×D₄	1	1	yes	characteristic
153	C₂×#525	C₂×D₄	1	2	yes	—
154	C₂×#525	C₂×D₄	1	2	yes	—
155	C₂²×#120	C₂²⋊C₄	1	2	yes	complemented
156	C₂²×#120	C₂²⋊C₄	1	2	yes	complemented
157	C₂²×#120	C₂²⋊C₄	1	2	yes	complemented
158	C₂²×#120	C₂²⋊C₄	1	2	yes	complemented
159	C₂²×#120	C₂²×C₄	1	2	yes	—
160	C₂²×#120	C₂²×C₄	1	2	yes	—
161	C₂²×#120	C₂×D₄	1	2	yes	—
162	C₂²×#120	C₂×D₄	1	2	yes	—
163	C₂²×#120	C₂×D₄	1	2	yes	—
164	C₂²×#120	C₂×D₄	1	2	yes	—
165	C₂²×#120	C₂×D₄	1	2	yes	—
166	C₂²×#120	C₂×D₄	1	2	yes	—
167	C₂²×#120	C₂×D₄	1	2	yes	—
168	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic, complemented
169	C₂×#518	C₂²×C₄	1	1	yes	characteristic
170	C₂×#518	C₂×D₄	1	2	yes	—
171	C₂×C₄×#41	C₂×D₄	1	1	yes	characteristic
172	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
173	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	complemented
174	C₂³×C₄²	C₂²⋊C₄	1	1	yes	characteristic, complemented
175	#2709	C₂²⋊C₄	1	4	yes	complemented
176	#2697	C₂²⋊C₄	1	4	yes	complemented
177	#2684	C₂²⋊C₄	1	4	yes	complemented
178	#2679	C₂²⋊C₄	1	4	yes	complemented
179	#2669	C₂²⋊C₄	1	4	yes	complemented
180	#2669	C₂²⋊C₄	1	4	yes	complemented
181	#2433	C₂×D₄	1	4	yes	—
182	#2431	C₂×D₄	1	4	yes	—
183	#2404	C₂×D₄	1	4	yes	—
184	#2404	C₂×D₄	1	4	yes	—
185	#2403	C₂×D₄	1	4	yes	—
186	#2403	C₂×D₄	1	4	yes	—
187	#2380	C₂²⋊C₄	1	2	yes	complemented
188	#2375	C₂²⋊C₄	1	4	yes	complemented
189	#2366	C₂²⋊C₄	1	2	yes	complemented
190	#2358	C₂²⋊C₄	1	4	yes	complemented
191	#2315	C₂²⋊C₄	1	2	yes	complemented
192	#2315	C₂²⋊C₄	1	2	yes	complemented
193	#2309	C₂²⋊C₄	1	1	yes	characteristic, complemented
194	#2309	C₂²⋊C₄	1	1	yes	characteristic, complemented
195	#2309	C₂²⋊C₄	1	1	yes	characteristic, complemented
196	#2309	C₂²×C₄	1	1	yes	characteristic
197	#2308	C₂²⋊C₄	1	2	yes	complemented
198	#2308	C₂²⋊C₄	1	2	yes	complemented
199	#2308	C₂²×C₄	1	2	yes	—
200	#2308	C₂×D₄	1	2	yes	—
201	#2308	C₂×D₄	1	2	yes	—
202	#2297	C₂²⋊C₄	1	1	yes	characteristic, complemented
203	#2297	C₂×D₄	1	1	yes	characteristic
204	#2297	C₂×D₄	1	1	yes	characteristic
205	#2296	C₂²⋊C₄	1	1	yes	characteristic, complemented
206	#2285	C₂²⋊C₄	1	1	yes	characteristic, complemented
207	#2280	C₂²⋊C₄	1	1	yes	characteristic, complemented
208	C₂×#379	C₂²⋊C₄	1	2	yes	complemented
209	C₂×#379	C₂²⋊C₄	1	2	yes	complemented
210	C₂×#379	C₂²⋊C₄	1	2	yes	complemented
211	#2273	C₂²⋊C₄	1	2	yes	complemented
212	C₄×#117	C₂²⋊C₄	1	1	yes	characteristic, complemented
213	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
214	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
215	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
216	C₂⁶	#120	1	2	yes	—
217	C₂²×#41	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂²×#41	#120	1	1	yes	characteristic
219	C₂²×#41	#120	1	1	yes	characteristic
220	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
221	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
222	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
223	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
224	C₂²×#41	C₂²×D₄	1	2	yes	—
225	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
229	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
230	C₂³×D₄	#120	1	2	yes	—
231	C₂³×D₄	#120	1	1	yes	characteristic
232	C₂³×D₄	#120	1	1	yes	characteristic
233	C₂³×D₄	#120	1	2	yes	—
234	C₂³×D₄	#120	1	4	yes	—
235	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
236	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
237	C₂³×D₄	C₂²×D₄	1	2	yes	—
238	C₂³×D₄	C₂²×D₄	1	2	yes	—
239	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
240	C₂³×D₄	C₂²×D₄	1	2	yes	—
241	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂⁴×C₄	#120	1	2	yes	—
243	D₄²	#99	1	4	yes	complemented
244	D₄²	#99	1	8	yes	complemented
245	D₄²	#99	1	4	yes	complemented
246	C₂×C₄⋊Q₈	#120	1	1	yes	characteristic
247	C₂×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×#127	C₂×C₂²⋊C₄	1	4	yes	—
254	C₂×#127	#120	1	1	yes	characteristic
255	C₂×#127	#120	1	1	yes	characteristic
256	C₂×#127	#120	1	1	yes	characteristic
257	C₂×#127	#120	1	2	yes	—
258	C₂×#124	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×#124	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×#124	#120	1	1	yes	characteristic
261	C₂×#124	#120	1	1	yes	characteristic
262	C₂×#124	#120	1	1	yes	characteristic
263	C₂×#124	#120	1	1	yes	characteristic
264	C₂×#124	#120	1	2	yes	—
265	C₂×#123	C₂×C₂²⋊C₄	1	2	yes	—
266	C₂×#123	C₂×C₂²⋊C₄	1	2	yes	—
267	C₂×#123	#120	1	1	yes	characteristic
268	C₂×#123	#120	1	1	yes	characteristic
269	C₂×#123	#120	1	2	yes	—
270	C₂×#123	#120	1	2	yes	—
271	C₂×#123	#120	1	2	yes	—
272	C₂×#121	#120	1	4	yes	—
273	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
274	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
275	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
276	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
277	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
278	C₂×#120	C₂×C₂²⋊C₄	1	2	yes	—
279	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
280	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
281	C₂×#120	#120	1	2	yes	—
282	C₂×#120	#120	1	2	yes	—
283	C₂×#120	#120	1	2	yes	—
284	C₂×#120	#120	1	1	yes	characteristic
285	C₂×#120	#120	1	2	yes	—
286	C₂×#120	#120	1	2	yes	—
287	C₂×#120	#120	1	2	yes	—
288	C₂×#120	#120	1	2	yes	—
289	C₂×#120	#120	1	2	yes	—
290	C₂×#120	#120	1	2	yes	—
291	C₂×#120	#120	1	1	yes	characteristic
292	C₂×#120	#120	1	2	yes	—
293	C₂×#120	#120	1	2	yes	—
294	C₂×#120	#120	1	4	yes	—
295	C₂×#120	#120	1	4	yes	—
296	C₂×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
297	C₂×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
298	C₂×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
299	C₂×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
300	C₂×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
301	C₂×#117	#120	1	2	yes	—
302	C₂×#117	#120	1	2	yes	—
303	C₂×#117	#120	1	1	yes	characteristic
304	C₂×#117	#120	1	1	yes	characteristic
305	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
306	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
307	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
308	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
309	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
310	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
311	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
312	C₂²×C₂²⋊C₄	#120	1	2	yes	—
313	C₂²×C₂²⋊C₄	#120	1	2	yes	—
314	C₂²×C₂²⋊C₄	#120	1	4	yes	—
315	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
316	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
317	C₂²×C₄²	#120	1	1	yes	characteristic
318	C₂²×C₄²	#120	1	1	yes	characteristic
319	#391	#120	1	2	yes	—
320	#390	C₂×C₂²⋊C₄	1	4	yes	—
321	#390	#120	1	2	yes	—
322	#386	C₂×C₂²⋊C₄	1	4	yes	—
323	#386	C₂×C₂²⋊C₄	1	4	yes	—
324	#386	#120	1	4	yes	—
325	#386	#120	1	4	yes	—
326	#381	#120	1	4	yes	—
327	#379	#120	1	4	yes	—
328	C₄×C₂²⋊C₄	#120	1	4	yes	—
329	C₂⁵	#379	1	2	yes	—
330	C₂⁵	#379	1	2	yes	—
331	C₂⁵	#379	1	2	yes	—
332	C₂⁵	#379	1	1	yes	characteristic
333	C₂⁵	#379	1	2	yes	—
334	C₂⁵	#379	1	4	yes	—
335	C₂⁵	C₂²×C₂²⋊C₄	1	2	yes	—
336	C₂⁵	C₂×#120	1	2	yes	—
337	C₂⁵	C₂×#120	1	2	yes	—
338	C₂²×Q₈	#379	1	1	yes	characteristic
339	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
340	C₂²×Q₈	C₂×#120	1	1	yes	characteristic
341	C₂²×D₄	#379	1	1	yes	characteristic
342	C₂²×D₄	#379	1	1	yes	characteristic
343	C₂²×D₄	#379	1	1	yes	characteristic
344	C₂²×D₄	#379	1	2	yes	—
345	C₂²×D₄	C₂²×C₂²⋊C₄	1	2	yes	—
346	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
347	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
348	C₂²×D₄	C₂×#120	1	2	yes	—
349	C₂²×D₄	C₂×#120	1	1	yes	characteristic
350	C₂²×D₄	C₂×#120	1	2	yes	—
351	C₂²×D₄	C₂×#120	1	2	yes	—
352	C₂²×D₄	C₂×#120	1	2	yes	—
353	C₂²×D₄	C₂×#120	1	2	yes	—
354	C₂²×D₄	C₂×#120	1	1	yes	characteristic
355	C₂²×D₄	C₂×#120	1	1	yes	characteristic
356	C₂²×D₄	C₂×#120	1	1	yes	characteristic
357	C₂³×C₄	#379	1	1	yes	characteristic
358	C₂³×C₄	#379	1	2	yes	—
359	C₂³×C₄	#379	1	4	yes	—
360	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
361	C₂³×C₄	C₂×#120	1	2	yes	—
362	C₂³×C₄	C₂×#120	1	1	yes	characteristic
363	C₂³×C₄	C₂×#120	1	1	yes	characteristic
364	C₂³×C₄	C₂×#120	1	2	yes	—
365	C₂³×C₄	C₂×#120	1	2	yes	—
366	#127	C₂×#99	1	2	yes	—
367	#127	C₂×#99	1	2	yes	—
368	#127	C₂×#99	1	4	yes	—
369	#120	C₂×#99	1	4	yes	—
370	#120	C₂×#99	1	4	yes	—
371	#120	C₂×#99	1	4	yes	—
372	#120	C₂×#99	1	4	yes	—
373	#120	B(4,𝔽₂)	1	2	yes	—
374	#120	#458	1	2	yes	—
375	C₂×C₄⋊C₄	#379	1	2	yes	—
376	C₂×C₂²⋊C₄	#379	1	2	yes	—
377	C₂×C₂²⋊C₄	#379	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	#379	1	2	yes	—
379	C₂×C₂²⋊C₄	#379	1	2	yes	—
380	C₂×C₂²⋊C₄	#379	1	2	yes	—
381	C₂×C₂²⋊C₄	#379	1	2	yes	—
382	C₂×C₂²⋊C₄	#379	1	1	yes	characteristic
383	C₂×C₂²⋊C₄	#379	1	4	yes	—
384	C₂×C₂²⋊C₄	#379	1	4	yes	—
385	C₂×C₄²	#379	1	1	yes	characteristic
386	C₂×C₄²	#379	1	1	yes	characteristic
387	C₂×C₄²	#379	1	1	yes	characteristic
388	C₂×C₄²	#379	1	1	yes	characteristic
389	C₂×C₄²	#379	1	2	yes	—
390	C₂×C₄²	#379	1	2	yes	—
391	C₂⁴	#1781	1	4	yes	—
392	C₂⁴	#1781	1	4	yes	—
393	C₂⁴	#1790	1	2	yes	—
394	C₂⁴	C₂×#379	1	2	yes	—
395	C₂⁴	C₂×#379	1	2	yes	—
396	C₂⁴	C₂×#379	1	1	yes	characteristic
397	C₂⁴	#2846	1	1	yes	characteristic
398	C₂⁴	#2846	1	2	yes	—
399	C₂⁴	#2846	1	1	yes	characteristic
400	C₂×D₄	#1781	1	2	yes	—
401	C₂×D₄	#1781	1	2	yes	—
402	C₂×D₄	C₂²×#99	1	1	yes	characteristic
403	C₂×D₄	C₂²×#99	1	2	yes	—
404	C₂×D₄	C₂²×#99	1	1	yes	characteristic
405	C₂×D₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
406	C₂×D₄	C₂×#458	1	1	yes	characteristic
407	C₂×D₄	#3026	1	1	yes	characteristic
408	C₂×D₄	#3026	1	1	yes	characteristic
409	C₂²×C₄	C₂×#379	1	1	yes	characteristic
410	C₂²×C₄	C₂×#379	1	1	yes	characteristic
411	C₂²×C₄	#2846	1	1	yes	characteristic
412	C₂²×C₄	#2846	1	2	yes	—
413	C₂²×C₄	#2846	1	1	yes	characteristic
414	C₂²⋊C₄	#1781	1	4	yes	—
415	C₂²⋊C₄	#1781	1	4	yes	—
416	C₂²⋊C₄	#1790	1	2	yes	—
417	C₄²	#1781	1	2	yes	—
418	C₄²	#1781	1	2	yes	—
419	C₂³	#14679	1	1	yes	characteristic
420	C₂³	C₂×#1781	1	2	yes	—
421	C₂³	C₂×#1781	1	2	yes	—
422	C₂³	C₂×#1790	1	1	yes	characteristic
423	C₂³	#20609	1	1	yes	characteristic
424	C₂³	#32898	1	1	yes	characteristic
425	C₂³	#32900	1	1	yes	characteristic
426	C₂×C₄	C₂×#1781	1	1	yes	characteristic
427	C₂×C₄	C₂×#1781	1	1	yes	characteristic
428	C₂×C₄	#32898	1	1	yes	characteristic
429	C₂×C₄	#32900	1	1	yes	characteristic
430	C₂²	#392109	1	1	yes	characteristic
431	C₂²	#392125	1	1	yes	characteristic
432	C₂²	#392135	1	1	yes	characteristic
433	C₂	#72225822	1	2	yes	—
434	C₂	#72234164	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#392135
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂⁴
prime radical	#72232033	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72234197	C₂	semidirect
2. #72234193	C₂	semidirect
3. #72233917	C₂	semidirect
4. #72233911	C₂	semidirect
5. #72233902	C₂	semidirect
6. #72233262	C₂	semidirect
7. #72232863	C₂	semidirect
8. #72232739	C₂	semidirect
10. #72225779	C₂	semidirect
11. #72219306	C₂	semidirect
13. C₂³×D₄²	C₄	semidirect
14. #7737413	C₄	semidirect
16. #7733150	C₄	semidirect
18. #7711201	C₄	semidirect
19. #7708349	C₄	semidirect
22. #7627211	C₄	semidirect
23. C₂×#33348	C₂²	semidirect
24. #494742	C₂²	semidirect
25. #494722	C₂²	semidirect
26. #494700	C₂²	semidirect
27. #494682	C₂²	semidirect
28. #493441	C₂²	semidirect
29. #493430	C₂²	semidirect
30. #493425	C₂²	semidirect
31. #493425	C₂²	semidirect
32. #493424	C₂²	semidirect
33. #493424	C₂²	semidirect
34. #359605	C₂²	semidirect
35. #359601	C₂²	semidirect
36. #359593	C₂²	semidirect
37. #359591	C₂²	semidirect
38. #214616	C₂²	semidirect
39. #213906	C₂²	semidirect
40. #213854	C₂²	semidirect
59. C₂×#2888	C₂³	semidirect
60. C₂×#2883	C₂³	semidirect
96. #20592	C₂³	semidirect
97. #20592	C₂³	semidirect
98. #20586	C₂³	semidirect
99. #20585	C₂³	semidirect
100. C₂⁷	C₂²⋊C₄	semidirect
103. C₂⁴×D₄	C₂²⋊C₄	semidirect
116. C₂×D₄²	C₂²⋊C₄	semidirect
117. C₂×D₄²	C₂²⋊C₄	semidirect
118. C₂×D₄²	C₂²⋊C₄	semidirect
119. C₂×D₄²	C₂²⋊C₄	semidirect
120. C₂×#535	C₂²⋊C₄	semidirect
121. C₂×#535	C₂²⋊C₄	semidirect
134. C₂×#534	C₂²⋊C₄	semidirect
144. C₂²×#127	C₂²⋊C₄	semidirect
145. C₂²×#127	C₂²⋊C₄	semidirect
150. C₂×#525	C₂²⋊C₄	semidirect
155. C₂²×#120	C₂²⋊C₄	semidirect
156. C₂²×#120	C₂²⋊C₄	semidirect
157. C₂²×#120	C₂²⋊C₄	semidirect
158. C₂²×#120	C₂²⋊C₄	semidirect
168. C₂×#518	C₂²⋊C₄	semidirect
172. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
173. C₂³×C₂²⋊C₄	C₂²⋊C₄	semidirect
174. C₂³×C₄²	C₂²⋊C₄	semidirect
175. #2709	C₂²⋊C₄	semidirect
176. #2697	C₂²⋊C₄	semidirect
177. #2684	C₂²⋊C₄	semidirect
178. #2679	C₂²⋊C₄	semidirect
179. #2669	C₂²⋊C₄	semidirect
180. #2669	C₂²⋊C₄	semidirect
187. #2380	C₂²⋊C₄	semidirect
188. #2375	C₂²⋊C₄	semidirect
189. #2366	C₂²⋊C₄	semidirect
190. #2358	C₂²⋊C₄	semidirect
191. #2315	C₂²⋊C₄	semidirect
192. #2315	C₂²⋊C₄	semidirect
193. #2309	C₂²⋊C₄	semidirect
194. #2309	C₂²⋊C₄	semidirect
195. #2309	C₂²⋊C₄	semidirect
197. #2308	C₂²⋊C₄	semidirect
198. #2308	C₂²⋊C₄	semidirect
202. #2297	C₂²⋊C₄	semidirect
205. #2296	C₂²⋊C₄	semidirect
206. #2285	C₂²⋊C₄	semidirect
207. #2280	C₂²⋊C₄	semidirect
208. C₂×#379	C₂²⋊C₄	semidirect
209. C₂×#379	C₂²⋊C₄	semidirect
210. C₂×#379	C₂²⋊C₄	semidirect
211. #2273	C₂²⋊C₄	semidirect
212. C₄×#117	C₂²⋊C₄	semidirect
243. D₄²	#99	semidirect
244. D₄²	#99	semidirect
245. D₄²	#99	semidirect

Maximal subgroups

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

subgroup	index
#72234197	2
#72234193	2
#72233917	2
#72233911	2
#72233902	2
#72233262	2
#72232863	2
#72232739	2
#72232033	2
#72225779	2
#72219306	2