Subgroups — #72314708

Order 2048.

At least 6864 conjugacy classes of subgroups; only the 662 normal subgroups are shown.

The subgroup lattice has 308 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314665	C₂	1	1	yes	maximal, characteristic, complemented
2	#72258235	C₂	1	8	yes	maximal, complemented
3	#72258019	C₂	1	1	yes	maximal, characteristic, complemented
4	#72256048	C₂	1	1	yes	maximal, characteristic, complemented
5	#72256043	C₂	1	1	yes	maximal, characteristic, complemented
6	#72255841	C₂	1	1	yes	maximal, characteristic, complemented
7	#72255841	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251527	C₂	1	1	yes	maximal, characteristic, complemented
9	#72240132	C₂	1	1	yes	maximal, characteristic, complemented
10	#72238342	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237402	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#7266798	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218215	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#139750	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#144378	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#139731	C₂	1	1	yes	maximal, characteristic, complemented
17	#72216336	C₂	1	8	yes	maximal, complemented
18	#10586422	C₂²	1	1	yes	characteristic
19	#7266798	C₂²	1	8	yes	complemented
20	C₂×#32873	C₂²	1	1	yes	characteristic, complemented
21	C₂×#23895	C₂²	1	1	yes	characteristic
22	C₂×#23895	C₂²	1	1	yes	characteristic
23	C₂×#23900	C₂²	1	1	yes	characteristic
24	C₂×#23900	C₂²	1	1	yes	characteristic
25	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
26	#6437445	C₂²	1	1	yes	characteristic
27	C₂×#21776	C₂²	1	1	yes	characteristic
28	C₂×#21776	C₂²	1	1	yes	characteristic
29	C₂×#21776	C₂²	1	1	yes	characteristic, complemented
30	C₂×#21746	C₂²	1	1	yes	characteristic, complemented
31	#616961	C₂²	1	1	yes	characteristic
32	#511812	C₂²	1	4	yes	complemented
33	#511811	C₂²	1	4	yes	complemented
34	#493481	C₂²	1	4	yes	complemented
35	#493480	C₂²	1	4	yes	complemented
36	C₂×#12707	C₂²	1	1	yes	characteristic
37	C₂×#12707	C₂²	1	1	yes	characteristic
38	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
39	C₂×#12705	C₂²	1	1	yes	characteristic
40	C₂×#12705	C₂²	1	1	yes	characteristic
41	#493370	C₂²	1	1	yes	characteristic
42	#493370	C₂²	1	1	yes	characteristic
43	#493370	C₂²	1	1	yes	characteristic
44	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
45	#347047	C₂²	1	4	yes	complemented
46	#347045	C₂²	1	4	yes	complemented
47	#188050	C₂²	1	1	yes	characteristic
48	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
49	#187944	C₂²	1	1	yes	characteristic
50	#187944	C₂²	1	1	yes	characteristic
51	#187944	C₂²	1	1	yes	characteristic, complemented
52	#144639	C₂²	1	8	yes	complemented
53	#144634	C₂²	1	4	yes	complemented
54	#144633	C₂²	1	4	yes	complemented
55	#144388	C₂²	1	4	yes	complemented
56	#144386	C₂²	1	4	yes	complemented
57	#144378	C₂²	1	8	yes	complemented
58	#139760	C₂²	1	4	yes	complemented
59	#139759	C₂²	1	4	yes	complemented
60	#139750	C₂²	1	8	yes	complemented
61	#139743	C₂²	1	8	yes	complemented
62	#139743	C₂²	1	8	yes	complemented
63	#139731	C₂²	1	8	yes	complemented
64	#139720	C₂²	1	4	yes	complemented
65	#139718	C₂²	1	4	yes	complemented
66	#135516	C₂²	1	8	yes	complemented
67	#128033	C₂²	1	1	yes	characteristic, complemented
68	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
69	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
70	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
71	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
72	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
73	#128009	C₂²	1	1	yes	characteristic, complemented
74	#128009	C₂²	1	1	yes	characteristic, complemented
75	#62832	C₂³	1	2	yes	—
76	#62824	C₂³	1	2	yes	—
77	C₂×#3484	C₂³	1	1	yes	characteristic
78	C₂×#3484	C₂³	1	1	yes	characteristic
79	C₂×#3484	C₂³	1	1	yes	characteristic
80	C₂²×#535	C₂³	1	1	yes	characteristic
81	C₂³×#120	C₂³	1	1	yes	characteristic
82	C₂×#3026	D₄	1	2	yes	complemented
83	C₂×#2883	D₄	1	2	yes	complemented
84	C₂×#2846	D₄	1	2	yes	complemented
85	#32873	C₂³	1	4	yes	—
86	#23907	C₂³	1	2	yes	—
87	#23906	C₂³	1	2	yes	—
88	#23900	D₄	1	4	yes	complemented
89	#23900	C₂³	1	4	yes	—
90	#23900	C₂³	1	4	yes	—
91	#23900	C₂³	1	4	yes	—
92	#23895	D₄	1	4	yes	complemented
93	#23895	C₂³	1	4	yes	—
94	#23895	C₂³	1	4	yes	—
95	#21776	D₄	1	4	yes	complemented
96	#21776	C₂³	1	4	yes	—
97	#21776	C₂³	1	4	yes	—
98	#21776	C₂³	1	4	yes	—
99	#21746	C₂³	1	4	yes	—
100	C₂×#1913	D₄	1	2	yes	complemented
101	C₂×#1898	D₄	1	2	yes	complemented
102	C₂×#1828	D₄	1	2	yes	complemented
103	C₂×#1783	C₂³	1	1	yes	characteristic
104	C₂×#1781	C₂³	1	1	yes	characteristic
105	C₂×#1781	C₂³	1	1	yes	characteristic
106	C₂×#1781	C₂³	1	1	yes	characteristic
107	C₂×#1781	C₂³	1	1	yes	characteristic
108	C₂×#1781	C₂³	1	1	yes	characteristic
109	C₂×#1781	C₂³	1	1	yes	characteristic
110	C₂×#1781	C₂³	1	1	yes	characteristic
111	C₂×#1781	C₂³	1	1	yes	characteristic
112	#15259	C₂³	1	2	yes	—
113	#15258	C₂³	1	2	yes	—
114	#14335	C₂³	1	1	yes	characteristic
115	#12707	C₂³	1	4	yes	—
116	#12707	C₂³	1	4	yes	—
117	#12707	C₂³	1	4	yes	—
118	#12705	C₂³	1	4	yes	—
119	#12705	C₂³	1	4	yes	—
120	#12694	C₂³	1	2	yes	—
121	#12694	C₂³	1	2	yes	—
122	#12694	C₂³	1	2	yes	—
123	#12693	C₂³	1	2	yes	—
124	#12693	C₂³	1	2	yes	—
125	#12693	C₂³	1	2	yes	—
126	#12687	C₂³	1	4	yes	—
127	#8546	C₂³	1	2	yes	—
128	#8544	C₂³	1	2	yes	—
129	#8541	C₂³	1	4	yes	—
130	#8534	C₂³	1	4	yes	—
131	#8534	C₂³	1	4	yes	—
132	#8534	C₂³	1	4	yes	—
133	#7316	C₂³	1	4	yes	—
134	#7314	C₂³	1	4	yes	—
135	#7310	C₂³	1	4	yes	—
136	#7310	C₂³	1	4	yes	—
137	#7310	C₂³	1	4	yes	—
138	#7307	C₂³	1	4	yes	—
139	#7307	C₂³	1	4	yes	—
140	#7304	C₂³	1	4	yes	—
141	C₂⁷	C₂×D₄	1	1	yes	characteristic
142	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
143	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
144	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic, complemented
145	C₂³×#41	#41	1	1	yes	characteristic, complemented
146	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
147	#3484	C₂×D₄	1	2	yes	—
148	#3484	C₂×D₄	1	2	yes	—
149	#3484	C₂×D₄	1	2	yes	—
150	#3484	C₂×D₄	1	2	yes	—
151	#3484	#41	1	2	yes	complemented
152	#3484	C₂⁴	1	2	yes	—
153	#3484	C₂⁴	1	2	yes	—
154	#3484	C₂⁴	1	2	yes	—
155	C₂×#534	C₂×D₄	1	2	yes	—
156	C₂²×#127	#41	1	1	yes	characteristic, complemented
157	C₂²×#127	C₂⁴	1	1	yes	characteristic
158	C₂²×#124	#41	1	1	yes	characteristic, complemented
159	C₂²×#124	C₂⁴	1	1	yes	characteristic
160	C₂²×#120	C₂×D₄	1	2	yes	—
161	C₂²×#120	C₂×D₄	1	2	yes	—
162	C₂²×#120	C₂⁴	1	2	yes	—
163	C₂×#518	C₂×D₄	1	1	yes	characteristic
164	C₂×#518	C₂×D₄	1	1	yes	characteristic
165	C₂×#518	#41	1	1	yes	characteristic, complemented
166	C₂²×#117	#41	1	1	yes	characteristic, complemented
167	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
168	#3026	C₂×D₄	1	8	yes	complemented
169	#2883	C₂×D₄	1	8	yes	complemented
170	C₂²×#99	C₂×D₄	1	2	yes	—
171	C₂²×#99	C₂×D₄	1	2	yes	—
172	C₂²×#99	C₂×D₄	1	2	yes	—
173	#2846	C₂×D₄	1	8	yes	complemented
174	C₂×#379	C₂×D₄	1	2	yes	—
175	#1913	C₂×D₄	1	8	yes	complemented
176	#1898	C₂×D₄	1	8	yes	complemented
177	#1828	C₂×D₄	1	8	yes	complemented
178	#1783	C₂⁴	1	2	yes	—
179	#1781	C₂×D₄	1	2	yes	—
180	#1781	C₂×D₄	1	2	yes	—
181	#1781	C₂×D₄	1	2	yes	—
182	#1781	C₂×D₄	1	2	yes	—
183	#1781	C₂⁴	1	2	yes	—
184	#1781	C₂⁴	1	2	yes	—
185	#1781	C₂⁴	1	2	yes	—
186	#1781	C₂⁴	1	2	yes	—
187	#1781	C₂⁴	1	2	yes	—
188	#1781	C₂⁴	1	2	yes	—
189	#1781	C₂⁴	1	2	yes	—
190	#1781	C₂⁴	1	2	yes	—
191	#1438	C₂⁴	1	2	yes	—
192	C₂⁶	C₂²×D₄	1	1	yes	characteristic
193	C₂⁶	C₂²×D₄	1	2	yes	—
194	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×H(5,𝔽₂)	#120	1	2	yes	—
196	C₂×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
197	C₂×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
198	C₂×H(5,𝔽₂)	C₂×#41	1	2	yes	complemented
199	C₂³×Q₈	C₂×#41	1	1	yes	characteristic, complemented
200	C₂³×D₄	#120	1	2	yes	—
201	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
202	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
203	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
204	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
205	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
206	C₂³×D₄	C₂×#41	1	1	yes	characteristic
207	C₂³×D₄	C₂×#41	1	1	yes	characteristic
208	C₂³×D₄	C₂×#41	1	1	yes	characteristic
209	C₂³×D₄	C₂×#41	1	1	yes	characteristic, complemented
210	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
213	C₂³×D₄	C₂⁵	1	1	yes	characteristic
214	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
215	#534	#120	1	4	yes	—
216	#534	C₂²×D₄	1	4	yes	—
217	C₂×#120	#120	1	4	yes	—
218	C₂×#120	C₂²×D₄	1	4	yes	—
219	#518	#120	1	2	yes	—
220	#518	C₂²×D₄	1	2	yes	—
221	#518	C₂²×D₄	1	2	yes	—
222	#518	C₂×#41	1	2	yes	complemented
223	C₂²×C₂²⋊C₄	#120	1	2	yes	—
224	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
225	C₂²×C₄²	C₂×#41	1	1	yes	characteristic, complemented
226	C₂²×C₄²	C₂×#41	1	1	yes	characteristic, complemented
227	C₂×#99	C₂²×D₄	1	4	yes	—
228	C₂×#99	C₂²×D₄	1	4	yes	—
229	C₂×#99	C₂²×D₄	1	4	yes	—
230	#379	C₂²×D₄	1	4	yes	—
231	C₂⁵	B(4,𝔽₂)	1	2	yes	—
232	C₂⁵	C₂×#120	1	1	yes	characteristic
233	C₂⁵	C₂×#120	1	1	yes	characteristic
234	C₂⁵	#535	1	1	yes	characteristic
235	C₂⁵	#535	1	1	yes	characteristic
236	C₂⁵	#546	1	1	yes	characteristic
237	C₂⁵	#546	1	1	yes	characteristic
238	C₂⁵	#546	1	1	yes	characteristic
239	C₂⁵	#546	1	1	yes	characteristic
240	C₂⁵	#546	1	1	yes	characteristic
241	C₂⁵	#546	1	1	yes	characteristic
242	C₂⁵	#546	1	1	yes	characteristic
243	C₂⁵	C₂³×D₄	1	1	yes	characteristic
244	C₂⁵	C₂×H(5,𝔽₂)	1	1	yes	characteristic
245	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
246	C₂²×D₄	C₂×#120	1	1	yes	characteristic
247	C₂²×D₄	C₂×#120	1	1	yes	characteristic
248	C₂²×D₄	C₂×#120	1	4	yes	—
249	C₂²×D₄	#525	1	1	yes	characteristic
250	C₂²×D₄	#525	1	1	yes	characteristic
251	C₂²×D₄	#535	1	1	yes	characteristic
252	C₂²×D₄	#535	1	1	yes	characteristic
253	C₂²×D₄	#546	1	1	yes	characteristic
254	C₂²×D₄	#546	1	1	yes	characteristic
255	C₂²×D₄	#546	1	1	yes	characteristic
256	C₂²×D₄	#546	1	1	yes	characteristic
257	C₂²×D₄	#546	1	1	yes	characteristic
258	C₂²×D₄	#546	1	1	yes	characteristic
259	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
260	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
261	C₂²×D₄	C₂²×#41	1	1	yes	characteristic
262	C₂²×D₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
263	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
264	C₂³×C₄	#525	1	1	yes	characteristic
265	C₂³×C₄	#525	1	1	yes	characteristic
266	C₂³×C₄	#546	1	1	yes	characteristic
267	C₂³×C₄	#546	1	1	yes	characteristic
268	C₂³×C₄	#546	1	1	yes	characteristic
269	C₂³×C₄	C₂²×#41	1	1	yes	characteristic
270	C₂×C₂²⋊C₄	C₂×#120	1	4	yes	—
271	C₂⁴	#2404	1	1	yes	characteristic
272	C₂⁴	#2404	1	1	yes	characteristic
273	C₂⁴	#2404	1	1	yes	characteristic
274	C₂⁴	#2404	1	1	yes	characteristic
275	C₂⁴	#2404	1	1	yes	characteristic
276	C₂⁴	#2404	1	1	yes	characteristic
277	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
278	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
279	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
280	C₂⁴	C₂²×#120	1	1	yes	characteristic
281	C₂⁴	C₂×#535	1	1	yes	characteristic
282	C₂⁴	C₂×#546	1	1	yes	characteristic
283	C₂⁴	C₂×#546	1	1	yes	characteristic
284	C₂⁴	C₂×#546	1	1	yes	characteristic
285	C₂²×C₄	#2404	1	1	yes	characteristic
286	C₂²×C₄	#2404	1	1	yes	characteristic
287	C₂²×C₄	C₂×B(4,𝔽₂)	1	4	yes	—
288	C₂²×C₄	C₂×#525	1	1	yes	characteristic
289	C₂²×C₄	C₂×#546	1	1	yes	characteristic
290	C₂³	#23984	1	2	yes	—
291	C₂³	#23984	1	1	yes	characteristic
292	C₂³	#23984	1	1	yes	characteristic
293	C₂³	#23996	1	1	yes	characteristic
294	C₂³	#23996	1	1	yes	characteristic
295	C₂³	#23996	1	1	yes	characteristic
296	C₂³	#23996	1	1	yes	characteristic
297	C₂³	C₂×#2404	1	1	yes	characteristic
298	C₂³	C₂²×B(4,𝔽₂)	1	2	yes	—
299	C₂²	#251827	1	1	yes	characteristic
300	C₂²	#251827	1	1	yes	characteristic
301	C₂²	#251827	1	1	yes	characteristic
302	C₂²	#251827	1	1	yes	characteristic
303	C₂²	C₂×#23996	1	1	yes	characteristic
304	C₂²	C₂×#23996	1	1	yes	characteristic
305	C₂²	C₂×#23984	1	1	yes	characteristic
306	C₂	C₂×#251827	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#251827
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72314665	C₂	semidirect
2. #72258235	C₂	semidirect
3. #72258019	C₂	semidirect
4. #72256048	C₂	semidirect
5. #72256043	C₂	semidirect
6. #72255841	C₂	semidirect
7. #72255841	C₂	semidirect
8. #72251527	C₂	semidirect
9. #72240132	C₂	semidirect
10. #72238342	C₂	semidirect
11. #72237402	C₂	semidirect
12. C₂×#7266798	C₂	semidirect
13. #72218215	C₂	semidirect
14. C₂×#139750	C₂	semidirect
15. C₂×#144378	C₂	semidirect
16. C₂×#139731	C₂	semidirect
17. #72216336	C₂	semidirect
19. #7266798	C₂²	semidirect
20. C₂×#32873	C₂²	semidirect
25. C₂×#23900	C₂²	semidirect
29. C₂×#21776	C₂²	semidirect
30. C₂×#21746	C₂²	semidirect
32. #511812	C₂²	semidirect
33. #511811	C₂²	semidirect
34. #493481	C₂²	semidirect
35. #493480	C₂²	semidirect
38. C₂×#12707	C₂²	semidirect
44. C₂×#12687	C₂²	semidirect
45. #347047	C₂²	semidirect
46. #347045	C₂²	semidirect
48. C₂×#8541	C₂²	semidirect
51. #187944	C₂²	semidirect
52. #144639	C₂²	semidirect
53. #144634	C₂²	semidirect
54. #144633	C₂²	semidirect
55. #144388	C₂²	semidirect
56. #144386	C₂²	semidirect
57. #144378	C₂²	semidirect
58. #139760	C₂²	semidirect
59. #139759	C₂²	semidirect
60. #139750	C₂²	semidirect
61. #139743	C₂²	semidirect
62. #139743	C₂²	semidirect
63. #139731	C₂²	semidirect
64. #139720	C₂²	semidirect
65. #139718	C₂²	semidirect
66. #135516	C₂²	semidirect
67. #128033	C₂²	semidirect
68. C₂×#7316	C₂²	semidirect
69. C₂×#7314	C₂²	semidirect
70. C₂×#7310	C₂²	semidirect
71. C₂×#7310	C₂²	semidirect
72. C₂×#7310	C₂²	semidirect
73. #128009	C₂²	semidirect
74. #128009	C₂²	semidirect
82. C₂×#3026	D₄	semidirect
83. C₂×#2883	D₄	semidirect
84. C₂×#2846	D₄	semidirect
88. #23900	D₄	semidirect
92. #23895	D₄	semidirect
95. #21776	D₄	semidirect
100. C₂×#1913	D₄	semidirect
101. C₂×#1898	D₄	semidirect
102. C₂×#1828	D₄	semidirect
144. C₂²×H(5,𝔽₂)	#41	semidirect
145. C₂³×#41	#41	semidirect
151. #3484	#41	semidirect
156. C₂²×#127	#41	semidirect
158. C₂²×#124	#41	semidirect
165. C₂×#518	#41	semidirect
166. C₂²×#117	#41	semidirect
168. #3026	C₂×D₄	semidirect
169. #2883	C₂×D₄	semidirect
173. #2846	C₂×D₄	semidirect
175. #1913	C₂×D₄	semidirect
176. #1898	C₂×D₄	semidirect
177. #1828	C₂×D₄	semidirect
198. C₂×H(5,𝔽₂)	C₂×#41	semidirect
199. C₂³×Q₈	C₂×#41	semidirect
209. C₂³×D₄	C₂×#41	semidirect
222. #518	C₂×#41	semidirect
225. C₂²×C₄²	C₂×#41	semidirect
226. C₂²×C₄²	C₂×#41	semidirect

Maximal subgroups

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

subgroup	index
#72314665	2
#72258235	2
#72258019	2
#72256048	2
#72256043	2
#72255841	2
#72251527	2
#72240132	2
#72238342	2
#72237402	2
C₂×#7266798	2
#72218215	2
C₂×#139750	2
C₂×#144378	2
C₂×#139731	2
#72216336	2