Subgroups — #72282395

Order 2048.

At least 9662 conjugacy classes of subgroups; only the 902 normal subgroups are shown.

The subgroup lattice has 516 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72282383	C₂	1	1	yes	maximal, characteristic, complemented
2	#72282374	C₂	1	1	yes	maximal, characteristic, complemented
3	#72282365	C₂	1	1	yes	maximal, characteristic, complemented
4	#72282355	C₂	1	4	yes	maximal, complemented
5	#72282344	C₂	1	1	yes	maximal, characteristic, complemented
6	#72282336	C₂	1	1	yes	maximal, characteristic, complemented
7	#72282329	C₂	1	1	yes	maximal, characteristic, complemented
8	#72282315	C₂	1	1	yes	maximal, characteristic, complemented
9	#72282307	C₂	1	1	yes	maximal, characteristic, complemented
10	#72282289	C₂	1	1	yes	maximal, characteristic, complemented
11	#72282264	C₂	1	1	yes	maximal, characteristic, complemented
12	#72282259	C₂	1	4	yes	maximal, complemented
13	#72282253	C₂	1	1	yes	maximal, characteristic, complemented
14	#72282048	C₂	1	4	yes	maximal, complemented
15	#72281537	C₂	1	4	yes	maximal, complemented
16	C₂×#130740	C₂	1	1	yes	maximal, characteristic, complemented
17	#72278529	C₂	1	1	yes	maximal, characteristic, complemented
18	#72278314	C₂	1	1	yes	maximal, characteristic, complemented
19	C₂×D₄×B(4,𝔽₂)	C₂	1	1	yes	maximal, characteristic
20	C₂×D₄×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
21	#7708375	C₂²	1	1	yes	characteristic, complemented
22	C₂×D₄×#120	C₂²	1	1	yes	characteristic
23	#7523321	C₂²	1	2	yes	complemented
24	#7513981	C₂²	1	2	yes	complemented
25	#7200271	C₂²	1	2	yes	complemented
26	#7199984	C₂²	1	2	yes	complemented
27	#6634505	C₂²	1	1	yes	characteristic, complemented
28	#6632009	C₂²	1	1	yes	characteristic, complemented
29	#6629120	C₂²	1	1	yes	characteristic, complemented
30	#6629108	C₂²	1	1	yes	characteristic, complemented
31	C₂×#32926	C₂²	1	1	yes	characteristic
32	D₄×B(4,𝔽₂)	C₂²	1	4	yes	—
33	D₄×B(4,𝔽₂)	C₂²	1	4	yes	—
34	#6487535	C₂²	1	1	yes	characteristic
35	#6470078	C₂²	1	1	yes	characteristic
36	C₂×#25771	C₂²	1	1	yes	characteristic
37	C₂×#24260	C₂²	1	1	yes	characteristic
38	#6408082	C₂²	1	1	yes	characteristic, complemented
39	C₂×D₄×#99	C₂²	1	1	yes	characteristic
40	C₂×C₄×B(4,𝔽₂)	C₂²	1	1	yes	characteristic
41	#6347464	C₂²	1	1	yes	characteristic, complemented
42	#6347433	C₂²	1	1	yes	characteristic, complemented
43	#500953	C₂²	1	2	yes	complemented
44	#500952	C₂²	1	2	yes	complemented
45	#500943	C₂²	1	2	yes	complemented
46	#500942	C₂²	1	2	yes	complemented
47	#495417	C₂²	1	1	yes	characteristic, complemented
48	#495407	C₂²	1	1	yes	characteristic, complemented
49	C₂×#12793	C₂²	1	1	yes	characteristic, complemented
50	#486402	C₂²	1	2	yes	complemented
51	#486401	C₂²	1	2	yes	complemented
52	#486396	C₂²	1	2	yes	complemented
53	#486395	C₂²	1	2	yes	complemented
54	#482118	C₂²	1	1	yes	characteristic, complemented
55	#482117	C₂²	1	1	yes	characteristic, complemented
56	C₂×#11880	C₂²	1	1	yes	characteristic, complemented
57	C₂×#11873	C₂²	1	1	yes	characteristic, complemented
58	C₂×#11471	C₂²	1	1	yes	characteristic, complemented
59	#347298	C₂²	1	4	yes	complemented
60	#347250	C₂²	1	4	yes	complemented
61	#346542	C₂²	1	4	yes	complemented
62	#346526	C₂²	1	4	yes	complemented
63	C₂×#8383	C₂²	1	1	yes	characteristic
64	C₂×#8375	C₂²	1	1	yes	characteristic
65	#185384	C₂²	1	1	yes	characteristic, complemented
66	#185331	C₂²	1	1	yes	characteristic, complemented
67	#184930	C₂²	1	1	yes	characteristic, complemented
68	#184905	C₂²	1	1	yes	characteristic, complemented
69	#144274	C₂²	1	4	yes	complemented
70	#144272	C₂²	1	4	yes	complemented
71	#144266	C₂²	1	4	yes	complemented
72	#144264	C₂²	1	4	yes	complemented
73	#142738	C₂²	1	4	yes	complemented
74	#142736	C₂²	1	4	yes	complemented
75	#142730	C₂²	1	4	yes	complemented
76	#142728	C₂²	1	4	yes	complemented
77	#139546	C₂²	1	4	yes	complemented
78	#139544	C₂²	1	4	yes	complemented
79	#138894	C₂²	1	4	yes	complemented
80	#138892	C₂²	1	4	yes	complemented
81	#138886	C₂²	1	4	yes	complemented
82	#138884	C₂²	1	4	yes	complemented
83	#136819	C₂²	1	4	yes	complemented
84	#136817	C₂²	1	4	yes	complemented
85	#130740	C₂²	1	4	yes	complemented
86	#130740	C₂²	1	4	yes	complemented
87	#127353	C₂²	1	1	yes	characteristic, complemented
88	#127335	C₂²	1	1	yes	characteristic, complemented
89	#127075	C₂²	1	1	yes	characteristic, complemented
90	#127071	C₂²	1	1	yes	characteristic, complemented
91	C₂⁵×D₄	D₄	1	1	yes	characteristic
92	D₄×H(5,𝔽₂)	C₂³	1	4	yes	—
93	C₂×#3496	C₂³	1	1	yes	characteristic
94	C₂²×D₄²	C₂³	1	1	yes	characteristic
95	C₂×C₄×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
96	C₂×#3208	D₄	1	1	yes	characteristic, complemented
97	C₂×#3187	D₄	1	1	yes	characteristic, complemented
98	C₂×#3101	D₄	1	1	yes	characteristic, complemented
99	C₂²×B(4,𝔽₂)	D₄	1	1	yes	characteristic
100	C₂²×#450	D₄	1	1	yes	characteristic, complemented
101	#59199	C₂³	1	1	yes	characteristic, complemented
102	#58998	C₂³	1	1	yes	characteristic, complemented
103	#45494	C₂³	1	1	yes	characteristic, complemented
104	#45493	C₂³	1	1	yes	characteristic, complemented
105	C₂×#2847	D₄	1	1	yes	characteristic
106	D₄×#120	D₄	1	2	yes	—
107	D₄×#120	D₄	1	2	yes	—
108	D₄×#120	C₂³	1	2	yes	—
109	D₄×#120	C₂³	1	2	yes	—
110	#35172	C₂³	1	1	yes	characteristic
111	#35010	C₂³	1	1	yes	characteristic
112	C₂×#2433	D₄	1	1	yes	characteristic
113	C₂×#2408	C₂³	1	1	yes	characteristic
114	C₂×D₄×C₂²⋊C₄	D₄	1	1	yes	characteristic
115	C₂×C₄×#120	C₂³	1	1	yes	characteristic
116	#32976	C₂³	1	2	yes	complemented
117	#32975	C₂³	1	2	yes	complemented
118	#32971	C₂³	1	2	yes	complemented
119	#32969	C₂³	1	2	yes	complemented
120	#32926	C₂³	1	4	yes	—
121	#31667	C₂³	1	2	yes	complemented
122	#31261	C₂³	1	2	yes	complemented
123	#31058	C₂³	1	2	yes	complemented
124	#30500	C₂³	1	2	yes	complemented
125	#25881	D₄	1	2	yes	complemented
126	#25881	C₂³	1	2	yes	—
127	#25771	C₂³	1	4	yes	—
128	#25753	D₄	1	2	yes	complemented
129	#25753	C₂³	1	2	yes	—
130	#24260	D₄	1	2	yes	—
131	#24260	D₄	1	2	yes	—
132	#24260	C₂³	1	2	yes	—
133	#24260	C₂³	1	2	yes	—
134	#24033	D₄	1	2	yes	complemented
135	#24033	C₂³	1	2	yes	—
136	#23849	D₄	1	2	yes	complemented
137	#23849	C₂³	1	2	yes	—
138	C₂×#2044	D₄	1	2	yes	—
139	C₂×#2021	D₄	1	2	yes	—
140	#23306	C₂³	1	2	yes	complemented
141	#22842	C₂³	1	2	yes	complemented
142	D₄×#99	C₂³	1	4	yes	—
143	#21803	D₄	1	1	yes	characteristic, complemented
144	#21786	C₂³	1	1	yes	characteristic
145	#21758	D₄	1	1	yes	characteristic, complemented
146	#21738	C₂³	1	1	yes	characteristic
147	C₄×B(4,𝔽₂)	D₄	1	2	yes	—
148	C₄×B(4,𝔽₂)	D₄	1	2	yes	—
149	C₄×B(4,𝔽₂)	C₂³	1	2	yes	—
150	C₄×B(4,𝔽₂)	C₂³	1	2	yes	—
151	C₂×#1889	D₄	1	1	yes	characteristic
152	C₂×#1867	D₄	1	1	yes	characteristic
153	#20732	D₄	1	1	yes	characteristic, complemented
154	C₂×#1800	C₂³	1	1	yes	characteristic
155	C₂×#1799	C₂³	1	1	yes	characteristic
156	C₂×#1799	C₂³	1	1	yes	characteristic
157	C₂×#1790	D₄	1	1	yes	characteristic
158	C₂×#1786	D₄	1	1	yes	characteristic, complemented
159	C₂×C₄×#99	C₂³	1	1	yes	characteristic
160	#18395	C₂³	1	2	yes	complemented
161	#18388	C₂³	1	2	yes	complemented
162	#18203	C₂³	1	2	yes	complemented
163	#18195	C₂³	1	2	yes	complemented
164	C₂×#1460	C₂³	1	1	yes	characteristic
165	C₂×#1459	C₂³	1	1	yes	characteristic
166	#13299	C₂³	1	2	yes	complemented
167	#13298	C₂³	1	2	yes	complemented
168	#13296	C₂³	1	2	yes	complemented
169	#13295	C₂³	1	2	yes	complemented
170	#12793	C₂³	1	2	yes	complemented
171	#12793	C₂³	1	2	yes	complemented
172	#11901	C₂³	1	2	yes	complemented
173	#11900	C₂³	1	2	yes	complemented
174	#11899	C₂³	1	2	yes	complemented
175	#11898	C₂³	1	2	yes	complemented
176	#11880	C₂³	1	4	yes	complemented
177	#11873	C₂³	1	4	yes	complemented
178	#11471	C₂³	1	2	yes	complemented
179	#11471	C₂³	1	2	yes	complemented
180	#8383	C₂³	1	4	yes	—
181	#8375	C₂³	1	4	yes	—
182	#8353	C₂³	1	4	yes	complemented
183	#8350	C₂³	1	4	yes	complemented
184	#8318	C₂³	1	4	yes	complemented
185	#8316	C₂³	1	4	yes	complemented
186	#7237	C₂³	1	4	yes	complemented
187	#7236	C₂³	1	4	yes	complemented
188	#7205	C₂³	1	4	yes	complemented
189	#7204	C₂³	1	4	yes	complemented
190	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
191	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
192	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
193	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
194	C₂⁴×D₄	C₂×D₄	1	2	yes	—
195	C₂⁴×D₄	C₂×D₄	1	2	yes	—
196	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
197	C₂³×SD₁₆	C₂×D₄	1	2	yes	complemented
198	#3496	C₂×D₄	1	2	yes	—
199	#3496	C₂⁴	1	2	yes	—
200	C₂×D₄²	C₂×D₄	1	2	yes	—
201	C₂×D₄²	C₂×D₄	1	2	yes	—
202	C₂×D₄²	C₂×D₄	1	2	yes	—
203	C₂×D₄²	C₂⁴	1	2	yes	—
204	C₂×#534	C₂×D₄	1	2	yes	—
205	C₂×#533	C₂×D₄	1	1	yes	characteristic, complemented
206	C₂²×#127	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₂²⋊Q₈	C₂×D₄	1	1	yes	characteristic, complemented
208	C₂²×#121	C₂×D₄	1	1	yes	characteristic
209	C₂²×#121	C₂×D₄	1	1	yes	characteristic
210	C₂²×#120	C₂×D₄	1	1	yes	characteristic
211	C₄×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
212	C₄×H(5,𝔽₂)	C₂⁴	1	2	yes	—
213	C₂×#518	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
215	C₂²×C₄×D₄	C₂⁴	1	1	yes	characteristic
216	C₂×#492	C₂×D₄	1	2	yes	complemented
217	C₂×#486	C₂×D₄	1	2	yes	complemented
218	C₂×#481	C₂×D₄	1	2	yes	complemented
219	C₂×#476	C₂×D₄	1	2	yes	complemented
220	C₂×#461	C₂×D₄	1	2	yes	complemented
221	C₂×B(4,𝔽₂)	SD₁₆	1	8	yes	complemented
222	C₂×B(4,𝔽₂)	C₂×D₄	1	2	yes	—
223	C₂×B(4,𝔽₂)	C₂×D₄	1	2	yes	—
224	C₂×#448	C₂×D₄	1	2	yes	complemented
225	C₂×#431	C₂×D₄	1	1	yes	characteristic, complemented
226	C₂×#431	C₂×D₄	1	1	yes	characteristic, complemented
227	C₂²×#102	C₂×D₄	1	2	yes	complemented
228	C₂²×#99	C₂×D₄	1	1	yes	characteristic
229	C₂²×#99	C₂×D₄	1	1	yes	characteristic
230	C₂²×C₂²⋊C₈	C₂×D₄	1	1	yes	characteristic, complemented
231	C₂²×C₂²⋊C₈	C₂×D₄	1	1	yes	characteristic, complemented
232	#2847	C₂×D₄	1	2	yes	—
233	#2847	C₂×D₄	1	2	yes	—
234	#2563	C₂×D₄	1	1	yes	characteristic, complemented
235	#2563	C₂⁴	1	1	yes	characteristic
236	#2542	C₂×D₄	1	1	yes	characteristic, complemented
237	#2542	C₂⁴	1	1	yes	characteristic
238	#2487	C₂×D₄	1	1	yes	characteristic, complemented
239	#2487	C₂⁴	1	1	yes	characteristic
240	#2433	C₂×D₄	1	2	yes	—
241	#2433	C₂×D₄	1	2	yes	—
242	#2409	C₂×D₄	1	1	yes	characteristic, complemented
243	#2409	C₂⁴	1	1	yes	characteristic
244	#2408	C₂×D₄	1	2	yes	—
245	#2408	C₂⁴	1	2	yes	—
246	D₄×C₂²⋊C₄	C₂×D₄	1	2	yes	—
247	D₄×C₂²⋊C₄	C₂×D₄	1	2	yes	—
248	C₄×#120	C₂×D₄	1	1	yes	characteristic
249	C₄×#120	C₂×D₄	1	1	yes	characteristic
250	C₄×#120	C₂×D₄	1	1	yes	characteristic
251	C₄×#120	C₂×D₄	1	1	yes	characteristic
252	C₄×#120	C₂×D₄	1	1	yes	characteristic
253	C₄×#120	C₂×D₄	1	1	yes	characteristic
254	C₄×#120	C₂⁴	1	1	yes	characteristic
255	C₄×#120	C₂⁴	1	1	yes	characteristic
256	C₂×#386	C₂×D₄	1	1	yes	characteristic
257	C₂×#385	C₂×D₄	1	1	yes	characteristic
258	#2285	C₂×D₄	1	1	yes	characteristic, complemented
259	C₂×#380	C₂×D₄	1	1	yes	characteristic
260	C₂×#380	C₂×D₄	1	1	yes	characteristic, complemented
261	C₂×#379	C₂×D₄	1	1	yes	characteristic
262	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
263	#2044	C₂×D₄	1	4	yes	—
264	#2044	C₂×D₄	1	4	yes	—
265	#2021	C₂×D₄	1	4	yes	—
266	#2021	C₂×D₄	1	4	yes	—
267	#1982	C₂×D₄	1	2	yes	complemented
268	#1982	C₂×D₄	1	2	yes	complemented
269	#1954	C₂×D₄	1	2	yes	complemented
270	#1954	C₂⁴	1	2	yes	—
271	#1893	C₂×D₄	1	2	yes	complemented
272	#1893	C₂×D₄	1	2	yes	complemented
273	#1889	C₂×D₄	1	2	yes	—
274	#1889	C₂×D₄	1	2	yes	—
275	#1869	C₂×D₄	1	2	yes	complemented
276	#1869	C₂⁴	1	2	yes	—
277	#1867	C₂×D₄	1	2	yes	—
278	#1867	C₂×D₄	1	2	yes	—
279	#1800	C₂⁴	1	2	yes	—
280	#1799	C₂×D₄	1	2	yes	—
281	#1799	C₂⁴	1	2	yes	—
282	#1799	C₂⁴	1	2	yes	—
283	#1790	C₂×D₄	1	2	yes	—
284	#1790	C₂×D₄	1	2	yes	—
285	#1786	C₂×D₄	1	2	yes	complemented
286	#1786	C₂×D₄	1	2	yes	complemented
287	C₄×#102	C₂×D₄	1	2	yes	complemented
288	C₄×#102	C₂×D₄	1	2	yes	complemented
289	C₄×#99	C₂×D₄	1	2	yes	—
290	C₄×#99	C₂⁴	1	2	yes	—
291	C₂×#342	C₂×D₄	1	2	yes	—
292	#1460	C₂⁴	1	2	yes	—
293	#1459	C₂⁴	1	2	yes	—
294	C₂⁶	#120	1	1	yes	characteristic
295	C₂×H(5,𝔽₂)	C₂×SD₁₆	1	4	yes	complemented
296	C₂×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
297	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic, complemented
298	C₂³×D₄	#120	1	2	yes	—
299	C₂³×D₄	#120	1	2	yes	—
300	C₂³×D₄	#120	1	1	yes	characteristic
301	C₂³×D₄	#120	1	1	yes	characteristic
302	C₂³×D₄	#120	1	1	yes	characteristic
303	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
304	C₂³×D₄	C₂²×D₄	1	2	yes	—
305	C₂³×D₄	C₂²×D₄	1	2	yes	—
306	C₂³×D₄	C₂²×D₄	1	2	yes	—
307	C₂⁴×C₄	#120	1	1	yes	characteristic
308	C₂⁴×C₄	#120	1	1	yes	characteristic
309	C₂⁴×C₄	#120	1	1	yes	characteristic
310	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
311	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
312	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
313	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
314	C₂³×C₈	C₂²×D₄	1	2	yes	complemented
315	#534	C₂²×D₄	1	4	yes	—
316	C₂×C₄⋊Q₈	C₂²×D₄	1	1	yes	characteristic, complemented
317	C₂×C₄⋊Q₈	C₂²×D₄	1	1	yes	characteristic, complemented
318	C₂×#127	#120	1	2	yes	—
319	C₂×#127	C₂²×D₄	1	2	yes	—
320	C₂×#121	#120	1	2	yes	—
321	C₂×#121	#120	1	2	yes	—
322	C₂×#121	C₂²×D₄	1	2	yes	—
323	C₂×#121	C₂²×D₄	1	2	yes	—
324	C₂×#120	#120	1	1	yes	characteristic
325	C₂×#120	#120	1	1	yes	characteristic
326	C₂×#120	C₂×SD₁₆	1	4	yes	—
327	C₂×#120	AGL(1,ℤ/8)	1	4	yes	—
328	C₂×#120	C₂²×D₄	1	1	yes	characteristic
329	C₂×#120	C₂²×D₄	1	1	yes	characteristic
330	#518	C₂²×D₄	1	2	yes	—
331	C₂×C₄×D₄	#120	1	2	yes	—
332	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
333	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
335	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
336	C₂×C₄×D₄	C₂²×D₄	1	2	yes	—
337	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
338	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
339	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
340	C₂×C₄×D₄	C₂⁵	1	1	yes	characteristic
341	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
342	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic, complemented
343	C₂²×C₄⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
344	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
345	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
346	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
347	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
348	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
349	B(4,𝔽₂)	C₂×SD₁₆	1	8	yes	—
350	B(4,𝔽₂)	C₂×SD₁₆	1	8	yes	complemented
351	C₂×C₄⋊C₈	C₂²×D₄	1	2	yes	complemented
352	C₂×#99	C₂×SD₁₆	1	4	yes	complemented
353	C₂×#99	C₂²×D₄	1	2	yes	—
354	C₂×#99	C₂²×D₄	1	2	yes	—
355	C₂×C₂²⋊C₈	C₂²×D₄	1	2	yes	complemented
356	C₂×C₄×C₈	C₂²×D₄	1	2	yes	complemented
357	#386	#120	1	1	yes	characteristic
358	#386	#120	1	1	yes	characteristic
359	#386	C₂²×D₄	1	1	yes	characteristic
360	#386	C₂²×D₄	1	1	yes	characteristic
361	#385	#120	1	1	yes	characteristic
362	#385	#120	1	1	yes	characteristic
363	#385	C₂²×D₄	1	1	yes	characteristic
364	#385	C₂²×D₄	1	1	yes	characteristic
365	#380	C₂²×D₄	1	2	yes	—
366	#380	C₂²×D₄	1	1	yes	characteristic, complemented
367	#380	C₂²×D₄	1	1	yes	characteristic, complemented
368	#379	#120	1	1	yes	characteristic
369	#379	#120	1	1	yes	characteristic
370	#379	C₂²×D₄	1	1	yes	characteristic
371	#379	C₂²×D₄	1	1	yes	characteristic
372	C₄×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic, complemented
373	C₄×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic, complemented
374	C₄×C₂²⋊C₄	#120	1	1	yes	characteristic
375	C₄×C₂²⋊C₄	#120	1	1	yes	characteristic
376	C₄×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
377	C₄×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
378	#342	C₂²×D₄	1	4	yes	—
379	C₂⁵	#450	1	4	yes	complemented
380	C₂⁵	C₂×#120	1	1	yes	characteristic
381	C₂⁵	C₂×#120	1	1	yes	characteristic
382	C₂⁵	C₂×#120	1	1	yes	characteristic
383	C₂⁵	D₄²	1	1	yes	characteristic
384	H(5,𝔽₂)	C₂²×SD₁₆	1	4	yes	—
385	C₂²×D₄	B(4,𝔽₂)	1	1	yes	characteristic
386	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
387	C₂²×D₄	B(4,𝔽₂)	1	1	yes	characteristic
388	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
389	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
390	C₂²×D₄	C₂×#120	1	1	yes	characteristic
391	C₂²×D₄	C₂×#120	1	2	yes	—
392	C₂²×D₄	C₂×#120	1	2	yes	—
393	C₂²×D₄	C₂×#120	1	2	yes	—
394	C₂²×D₄	D₄²	1	1	yes	characteristic
395	C₂²×D₄	D₄²	1	1	yes	characteristic
396	C₂²×D₄	C₂²×SD₁₆	1	2	yes	—
397	C₂²×D₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
398	C₂²×D₄	#577	1	2	yes	—
399	C₂²×D₄	#577	1	2	yes	—
400	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
401	C₂³×C₄	C₂×#120	1	1	yes	characteristic
402	C₂³×C₄	C₂×#120	1	1	yes	characteristic
403	C₂³×C₄	C₂×#120	1	1	yes	characteristic
404	C₂³×C₄	C₂×#120	1	1	yes	characteristic
405	C₂³×C₄	C₂×#120	1	1	yes	characteristic
406	C₂³×C₄	C₂×#120	1	1	yes	characteristic
407	C₂³×C₄	#534	1	1	yes	characteristic
408	C₂³×C₄	#534	1	1	yes	characteristic
409	C₂³×C₄	D₄²	1	1	yes	characteristic
410	C₂³×C₄	D₄²	1	1	yes	characteristic
411	C₂³×C₄	D₄²	1	1	yes	characteristic
412	C₂³×C₄	D₄²	1	1	yes	characteristic
413	C₂³×C₄	D₄²	1	1	yes	characteristic
414	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
415	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
416	#120	#450	1	4	yes	complemented
417	#120	#450	1	4	yes	complemented
418	#120	C₂²×SD₁₆	1	2	yes	—
419	#120	C₂²×SD₁₆	1	2	yes	—
420	#120	C₂×AGL(1,ℤ/8)	1	2	yes	—
421	#120	C₂×AGL(1,ℤ/8)	1	2	yes	—
422	C₂×C₄⋊C₄	C₂×#120	1	2	yes	—
423	C₂×C₄⋊C₄	#534	1	1	yes	characteristic
424	C₂×C₄⋊C₄	D₄²	1	1	yes	characteristic
425	C₂×C₄⋊C₄	D₄²	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	#450	1	4	yes	complemented
428	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
429	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
434	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
436	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	C₂³×D₄	1	1	yes	characteristic
439	C₂×C₂²⋊C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
440	C₂×C₄²	C₂×#120	1	1	yes	characteristic
441	C₂×C₄²	D₄²	1	1	yes	characteristic
442	C₂×C₄²	D₄²	1	1	yes	characteristic
443	C₂×C₄²	C₂³×D₄	1	1	yes	characteristic
444	#99	C₂²×SD₁₆	1	4	yes	—
445	C₂⁴	#2403	1	1	yes	characteristic
446	C₂⁴	#2403	1	1	yes	characteristic
447	C₂⁴	#2403	1	1	yes	characteristic
448	C₂⁴	C₂×#450	1	2	yes	—
449	C₂⁴	C₂×#450	1	2	yes	—
450	C₂⁴	C₂×#450	1	2	yes	—
451	C₂⁴	#3009	1	2	yes	—
452	C₂⁴	#3265	1	2	yes	—
453	C₂⁴	D₄×SD₁₆	1	2	yes	—
454	C₂⁴	C₂²×#120	1	1	yes	characteristic
455	C₂⁴	C₂×D₄²	1	1	yes	characteristic
456	C₂×D₄	C₂×#450	1	2	yes	—
457	C₂×D₄	#3009	1	2	yes	—
458	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
459	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
460	C₂×D₄	C₂×B(4,𝔽₂)	1	4	yes	—
461	C₂×D₄	#3265	1	2	yes	—
462	C₂×D₄	#3265	1	2	yes	—
463	C₂×D₄	D₄×SD₁₆	1	2	yes	—
464	C₂×D₄	D₄×SD₁₆	1	2	yes	—
465	C₂×D₄	C₂³×SD₁₆	1	1	yes	characteristic
466	C₂×D₄	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
467	C₂×D₄	C₂×#577	1	1	yes	characteristic
468	C₂×D₄	C₂×#577	1	1	yes	characteristic
469	C₂²×C₄	#2403	1	1	yes	characteristic
470	C₂²×C₄	#2403	1	1	yes	characteristic
471	C₂²×C₄	#2403	1	1	yes	characteristic
472	C₂²×C₄	#2403	1	1	yes	characteristic
473	C₂²×C₄	#2403	1	1	yes	characteristic
474	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
475	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
476	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
477	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
478	C₂²×C₄	#3265	1	2	yes	—
479	C₂²×C₄	D₄×SD₁₆	1	2	yes	—
480	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
481	C₂²×C₄	C₂×#534	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄²	1	1	yes	characteristic
485	C₂²⋊C₄	C₂×#450	1	2	yes	—
486	C₂²⋊C₄	C₂×#450	1	2	yes	—
487	C₂³	#23849	1	2	yes	—
488	C₂³	#23849	1	2	yes	—
489	C₂³	#23849	1	2	yes	—
490	C₂³	#23882	1	2	yes	—
491	C₂³	#23900	1	1	yes	characteristic
492	C₂³	#23900	1	1	yes	characteristic
493	C₂³	C₂×#2403	1	1	yes	characteristic
494	C₂³	C₂²×#450	1	1	yes	characteristic
495	C₂³	C₂×#3009	1	1	yes	characteristic
496	C₂³	C₂×#3265	1	1	yes	characteristic
497	C₂³	C₂×D₄×SD₁₆	1	1	yes	characteristic
498	D₄	#33551	1	4	yes	—
499	C₂×C₄	#23849	1	2	yes	—
500	C₂×C₄	#23900	1	1	yes	characteristic
501	C₂×C₄	#23900	1	1	yes	characteristic
502	C₂×C₄	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
503	C₂×C₄	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
504	C₂×C₄	C₂×#3265	1	1	yes	characteristic
505	C₂×C₄	C₂×D₄×SD₁₆	1	1	yes	characteristic
506	C₂²	#248456	1	2	yes	—
507	C₂²	#248456	1	2	yes	—
508	C₂²	C₂×#23849	1	1	yes	characteristic
509	C₂²	C₂×#23882	1	1	yes	characteristic
510	C₂²	C₂×#23900	1	1	yes	characteristic
511	C₄	C₂×#33551	1	2	yes	—
512	C₂	#72218077	1	1	yes	characteristic
513	C₂	#72231746	1	1	yes	characteristic
514	C₂	C₂×#248456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#23900
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂⁵
prime radical	C₂×D₄×B(4,𝔽₂)	C₂
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72282383	C₂	semidirect
2. #72282374	C₂	semidirect
3. #72282365	C₂	semidirect
4. #72282355	C₂	semidirect
5. #72282344	C₂	semidirect
6. #72282336	C₂	semidirect
7. #72282329	C₂	semidirect
8. #72282315	C₂	semidirect
9. #72282307	C₂	semidirect
10. #72282289	C₂	semidirect
11. #72282264	C₂	semidirect
12. #72282259	C₂	semidirect
13. #72282253	C₂	semidirect
14. #72282048	C₂	semidirect
15. #72281537	C₂	semidirect
16. C₂×#130740	C₂	semidirect
17. #72278529	C₂	semidirect
18. #72278314	C₂	semidirect
21. #7708375	C₂²	semidirect
23. #7523321	C₂²	semidirect
24. #7513981	C₂²	semidirect
25. #7200271	C₂²	semidirect
26. #7199984	C₂²	semidirect
27. #6634505	C₂²	semidirect
28. #6632009	C₂²	semidirect
29. #6629120	C₂²	semidirect
30. #6629108	C₂²	semidirect
38. #6408082	C₂²	semidirect
41. #6347464	C₂²	semidirect
42. #6347433	C₂²	semidirect
43. #500953	C₂²	semidirect
44. #500952	C₂²	semidirect
45. #500943	C₂²	semidirect
46. #500942	C₂²	semidirect
47. #495417	C₂²	semidirect
48. #495407	C₂²	semidirect
49. C₂×#12793	C₂²	semidirect
50. #486402	C₂²	semidirect
51. #486401	C₂²	semidirect
52. #486396	C₂²	semidirect
53. #486395	C₂²	semidirect
54. #482118	C₂²	semidirect
55. #482117	C₂²	semidirect
56. C₂×#11880	C₂²	semidirect
57. C₂×#11873	C₂²	semidirect
58. C₂×#11471	C₂²	semidirect
59. #347298	C₂²	semidirect
60. #347250	C₂²	semidirect
61. #346542	C₂²	semidirect
62. #346526	C₂²	semidirect
65. #185384	C₂²	semidirect
66. #185331	C₂²	semidirect
67. #184930	C₂²	semidirect
68. #184905	C₂²	semidirect
69. #144274	C₂²	semidirect
70. #144272	C₂²	semidirect
71. #144266	C₂²	semidirect
72. #144264	C₂²	semidirect
73. #142738	C₂²	semidirect
74. #142736	C₂²	semidirect
75. #142730	C₂²	semidirect
76. #142728	C₂²	semidirect
77. #139546	C₂²	semidirect
78. #139544	C₂²	semidirect
79. #138894	C₂²	semidirect
80. #138892	C₂²	semidirect
81. #138886	C₂²	semidirect
82. #138884	C₂²	semidirect
83. #136819	C₂²	semidirect
84. #136817	C₂²	semidirect
85. #130740	C₂²	semidirect
86. #130740	C₂²	semidirect
87. #127353	C₂²	semidirect
88. #127335	C₂²	semidirect
89. #127075	C₂²	semidirect
90. #127071	C₂²	semidirect
96. C₂×#3208	D₄	semidirect
97. C₂×#3187	D₄	semidirect
98. C₂×#3101	D₄	semidirect
100. C₂²×#450	D₄	semidirect
101. #59199	C₂³	semidirect
102. #58998	C₂³	semidirect
103. #45494	C₂³	semidirect
104. #45493	C₂³	semidirect
116. #32976	C₂³	semidirect
117. #32975	C₂³	semidirect
118. #32971	C₂³	semidirect
119. #32969	C₂³	semidirect
121. #31667	C₂³	semidirect
122. #31261	C₂³	semidirect
123. #31058	C₂³	semidirect
124. #30500	C₂³	semidirect
125. #25881	D₄	semidirect
128. #25753	D₄	semidirect
134. #24033	D₄	semidirect
136. #23849	D₄	semidirect
140. #23306	C₂³	semidirect
141. #22842	C₂³	semidirect
143. #21803	D₄	semidirect
145. #21758	D₄	semidirect
153. #20732	D₄	semidirect
158. C₂×#1786	D₄	semidirect
160. #18395	C₂³	semidirect
161. #18388	C₂³	semidirect
162. #18203	C₂³	semidirect
163. #18195	C₂³	semidirect
166. #13299	C₂³	semidirect
167. #13298	C₂³	semidirect
168. #13296	C₂³	semidirect
169. #13295	C₂³	semidirect
170. #12793	C₂³	semidirect
171. #12793	C₂³	semidirect
172. #11901	C₂³	semidirect
173. #11900	C₂³	semidirect
174. #11899	C₂³	semidirect
175. #11898	C₂³	semidirect
176. #11880	C₂³	semidirect
177. #11873	C₂³	semidirect
178. #11471	C₂³	semidirect
179. #11471	C₂³	semidirect
182. #8353	C₂³	semidirect
183. #8350	C₂³	semidirect
184. #8318	C₂³	semidirect
185. #8316	C₂³	semidirect
186. #7237	C₂³	semidirect
187. #7236	C₂³	semidirect
188. #7205	C₂³	semidirect
189. #7204	C₂³	semidirect
197. C₂³×SD₁₆	C₂×D₄	semidirect
205. C₂×#533	C₂×D₄	semidirect
207. C₂²×C₂²⋊Q₈	C₂×D₄	semidirect
216. C₂×#492	C₂×D₄	semidirect
217. C₂×#486	C₂×D₄	semidirect
218. C₂×#481	C₂×D₄	semidirect
219. C₂×#476	C₂×D₄	semidirect
220. C₂×#461	C₂×D₄	semidirect
221. C₂×B(4,𝔽₂)	SD₁₆	semidirect
224. C₂×#448	C₂×D₄	semidirect
225. C₂×#431	C₂×D₄	semidirect
226. C₂×#431	C₂×D₄	semidirect
227. C₂²×#102	C₂×D₄	semidirect
230. C₂²×C₂²⋊C₈	C₂×D₄	semidirect
231. C₂²×C₂²⋊C₈	C₂×D₄	semidirect
234. #2563	C₂×D₄	semidirect
236. #2542	C₂×D₄	semidirect
238. #2487	C₂×D₄	semidirect
242. #2409	C₂×D₄	semidirect
258. #2285	C₂×D₄	semidirect
260. C₂×#380	C₂×D₄	semidirect
267. #1982	C₂×D₄	semidirect
268. #1982	C₂×D₄	semidirect
269. #1954	C₂×D₄	semidirect
271. #1893	C₂×D₄	semidirect
272. #1893	C₂×D₄	semidirect
275. #1869	C₂×D₄	semidirect
285. #1786	C₂×D₄	semidirect
286. #1786	C₂×D₄	semidirect
287. C₄×#102	C₂×D₄	semidirect
288. C₄×#102	C₂×D₄	semidirect
295. C₂×H(5,𝔽₂)	C₂×SD₁₆	semidirect
297. C₂³×Q₈	C₂²×D₄	semidirect
314. C₂³×C₈	C₂²×D₄	semidirect
316. C₂×C₄⋊Q₈	C₂²×D₄	semidirect
317. C₂×C₄⋊Q₈	C₂²×D₄	semidirect
342. C₂²×C₄⋊C₄	C₂²×D₄	semidirect
350. B(4,𝔽₂)	C₂×SD₁₆	semidirect
351. C₂×C₄⋊C₈	C₂²×D₄	semidirect
352. C₂×#99	C₂×SD₁₆	semidirect
355. C₂×C₂²⋊C₈	C₂²×D₄	semidirect
356. C₂×C₄×C₈	C₂²×D₄	semidirect
366. #380	C₂²×D₄	semidirect
367. #380	C₂²×D₄	semidirect
372. C₄×C₄⋊C₄	C₂²×D₄	semidirect
373. C₄×C₄⋊C₄	C₂²×D₄	semidirect
379. C₂⁵	#450	semidirect
416. #120	#450	semidirect
417. #120	#450	semidirect
427. C₂×C₂²⋊C₄	#450	semidirect

Maximal subgroups

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

subgroup	index
#72282383	2
#72282374	2
#72282365	2
#72282355	2
#72282344	2
#72282336	2
#72282329	2
#72282315	2
#72282307	2
#72282289	2
#72282264	2
#72282259	2
#72282253	2
#72282048	2
#72281537	2
C₂×#130740	2
#72278529	2
#72278314	2
C₂×D₄×B(4,𝔽₂)	2