Subgroups — #72235929

Order 2048.

At least 8652 conjugacy classes of subgroups; only the 320 normal subgroups are shown.

The subgroup lattice has 285 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72235926	C₂	1	1	yes	maximal, characteristic, complemented
2	#72235925	C₂	1	1	yes	maximal, characteristic, complemented
3	#72235924	C₂	1	1	yes	maximal, characteristic, complemented
4	#72235923	C₂	1	1	yes	maximal, characteristic, complemented
5	#72235922	C₂	1	1	yes	maximal, characteristic, complemented
6	#72235919	C₂	1	1	yes	maximal, characteristic, complemented
7	#72235868	C₂	1	1	yes	maximal, characteristic, complemented
8	#72235864	C₂	1	1	yes	maximal, characteristic, complemented
9	#72235820	C₂	1	1	yes	maximal, characteristic, complemented
10	#72235808	C₂	1	1	yes	maximal, characteristic, complemented
11	#72235729	C₂	1	1	yes	maximal, characteristic, complemented
12	#72235690	C₂	1	1	yes	maximal, characteristic, complemented
13	#72232667	C₂	1	1	yes	maximal, characteristic, complemented
14	#72231851	C₂	1	1	yes	maximal, characteristic, complemented
15	#72231809	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#46996	C₂²	1	1	yes	characteristic
17	#7642350	C₂²	1	1	yes	characteristic, complemented
18	C₂×#35347	C₂²	1	1	yes	characteristic
19	#6451874	C₂²	1	1	yes	characteristic, complemented
20	#6346594	C₂²	1	1	yes	characteristic
21	#6346594	C₂²	1	1	yes	characteristic, complemented
22	#6346594	C₂²	1	1	yes	characteristic, complemented
23	#4866548	C₂²	1	1	yes	characteristic
24	#4335643	C₂²	1	1	yes	characteristic
25	#635987	C₂²	1	1	yes	characteristic
26	#618316	C₂²	1	1	yes	characteristic
27	#587363	C₂²	1	1	yes	characteristic
28	#482549	C₂²	1	1	yes	characteristic
29	#482549	C₂²	1	1	yes	characteristic
30	#482549	C₂²	1	1	yes	characteristic, complemented
31	#482549	C₂²	1	1	yes	characteristic, complemented
32	#482549	C₂²	1	1	yes	characteristic, complemented
33	#482549	C₂²	1	1	yes	characteristic, complemented
34	#480250	C₂²	1	1	yes	characteristic, complemented
35	#480250	C₂²	1	1	yes	characteristic, complemented
36	#359705	C₂²	1	1	yes	characteristic, complemented
37	#201462	C₂²	1	1	yes	characteristic
38	#201462	C₂²	1	1	yes	characteristic, complemented
39	#201462	C₂²	1	1	yes	characteristic, complemented
40	C₂×#8549	C₂²	1	1	yes	characteristic, complemented
41	C₂×#8542	C₂²	1	1	yes	characteristic, complemented
42	#187945	C₂²	1	1	yes	characteristic, complemented
43	#185255	C₂²	1	1	yes	characteristic, complemented
44	#184796	C₂²	1	1	yes	characteristic, complemented
45	#128157	C₂²	1	1	yes	characteristic, complemented
46	#128157	C₂²	1	1	yes	characteristic, complemented
47	#127215	C₂²	1	1	yes	characteristic, complemented
48	#127215	C₂²	1	1	yes	characteristic, complemented
49	#107115	C₂²	1	1	yes	characteristic, complemented
50	#106326	C₂²	1	1	yes	characteristic, complemented
51	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
52	C₂×#3490	C₂³	1	1	yes	characteristic
53	C₂²×#525	D₄	1	1	yes	characteristic, complemented
54	C₂³×#121	C₂³	1	1	yes	characteristic
55	C₂×#2760	D₄	1	1	yes	characteristic, complemented
56	C₂×#2760	D₄	1	1	yes	characteristic, complemented
57	C₂×#2840	D₄	1	1	yes	characteristic, complemented
58	C₂×#2406	D₄	1	1	yes	characteristic, complemented
59	C₂×#2406	D₄	1	1	yes	characteristic, complemented
60	C₂×#2404	D₄	1	1	yes	characteristic, complemented
61	C₂×#2404	C₂³	1	1	yes	characteristic
62	C₂×#2404	C₂³	1	1	yes	characteristic
63	C₂×#2404	C₂³	1	1	yes	characteristic
64	C₂×#2403	D₄	1	1	yes	characteristic
65	C₂×#2403	D₄	1	1	yes	characteristic
66	C₂×#2403	D₄	1	1	yes	characteristic, complemented
67	#34176	C₂³	1	1	yes	characteristic
68	C₂×#2308	D₄	1	1	yes	characteristic, complemented
69	C₂×#2308	C₂³	1	1	yes	characteristic
70	C₂×#1819	C₂³	1	1	yes	characteristic
71	C₂×#1819	C₂³	1	1	yes	characteristic
72	#18133	D₄	1	2	yes	complemented
73	#18132	D₄	1	2	yes	complemented
74	C₂×#1462	C₂³	1	1	yes	characteristic
75	C₂×#1462	C₂³	1	1	yes	characteristic
76	C₂×#1462	C₂³	1	1	yes	characteristic
77	C₂×#1462	C₂³	1	1	yes	characteristic
78	#17198	D₄	1	1	yes	characteristic, complemented
79	#17105	D₄	1	1	yes	characteristic, complemented
80	#17105	D₄	1	1	yes	characteristic, complemented
81	#17102	D₄	1	1	yes	characteristic, complemented
82	#15930	D₄	1	1	yes	characteristic, complemented
83	#15561	D₄	1	1	yes	characteristic, complemented
84	#15467	D₄	1	1	yes	characteristic, complemented
85	#15267	D₄	1	1	yes	characteristic, complemented
86	#14681	D₄	1	1	yes	characteristic, complemented
87	#14649	D₄	1	1	yes	characteristic, complemented
88	#14609	D₄	1	1	yes	characteristic, complemented
89	#14533	D₄	1	1	yes	characteristic, complemented
90	#14341	C₂³	1	1	yes	characteristic
91	#14207	D₄	1	1	yes	characteristic, complemented
92	#13868	D₄	1	1	yes	characteristic, complemented
93	#13868	C₂³	1	1	yes	characteristic
94	#13867	D₄	1	1	yes	characteristic
95	#13867	D₄	1	1	yes	characteristic
96	#11839	D₄	1	4	yes	complemented
97	#9475	D₄	1	4	yes	complemented
98	C₂³×#41	C₂×D₄	1	1	yes	characteristic
99	C₂³×#41	C₂×D₄	1	1	yes	characteristic
100	C₂³×#41	C₂×D₄	1	1	yes	characteristic
101	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
102	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
103	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
104	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
105	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×#543	C₂×D₄	1	1	yes	characteristic, complemented
107	C₂×#534	C₂×D₄	1	1	yes	characteristic, complemented
108	C₂²×#117	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
111	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₂²⋊C₄	C₂⁴	1	1	yes	characteristic
113	#2404	C₂×D₄	1	2	yes	—
114	#2404	C₂×D₄	1	2	yes	—
115	#2404	C₂×D₄	1	2	yes	—
116	#2404	#41	1	2	yes	—
117	C₂×#393	C₂×D₄	1	1	yes	characteristic
118	C₂×#393	C₂×D₄	1	1	yes	characteristic
119	C₂×#393	C₂×D₄	1	1	yes	characteristic
120	C₂×#393	C₂×D₄	1	1	yes	characteristic
121	C₂×#393	C₂×D₄	1	1	yes	characteristic
122	C₂×#393	C₂×D₄	1	1	yes	characteristic
123	C₂×#392	C₂×D₄	1	1	yes	characteristic
124	C₂×#392	C₂×D₄	1	1	yes	characteristic
125	C₂×#392	C₂×D₄	1	1	yes	characteristic
126	C₂×#392	C₂×D₄	1	1	yes	characteristic
127	C₂×#392	C₂×D₄	1	1	yes	characteristic
128	C₂×#392	C₂×D₄	1	1	yes	characteristic
129	C₂×#381	C₂×D₄	1	1	yes	characteristic
130	C₂×#381	C₂×D₄	1	1	yes	characteristic
131	C₂×#381	C₂×D₄	1	1	yes	characteristic
132	C₂×#381	C₂×D₄	1	1	yes	characteristic
133	C₂×#381	C₂×D₄	1	1	yes	characteristic
134	C₂×#381	C₂×D₄	1	1	yes	characteristic
135	C₂×#379	C₂×D₄	1	1	yes	characteristic
136	C₂×#379	C₂×D₄	1	1	yes	characteristic
137	C₂×#379	C₂×D₄	1	1	yes	characteristic
138	C₂×#379	C₂×D₄	1	1	yes	characteristic
139	#2273	C₂×D₄	1	2	yes	complemented
140	#1819	C₂×D₄	1	2	yes	complemented
141	#1462	C₂×D₄	1	2	yes	complemented
142	#1462	C₂×D₄	1	2	yes	complemented
143	C₂×#323	C₂×D₄	1	2	yes	complemented
144	#1440	C₂×D₄	1	1	yes	characteristic
145	#1440	C₂×D₄	1	1	yes	characteristic
146	#1440	C₂×D₄	1	1	yes	characteristic
147	#1440	C₂×D₄	1	1	yes	characteristic
148	#1437	C₂×D₄	1	1	yes	characteristic
149	#1437	C₂×D₄	1	1	yes	characteristic
150	#1437	C₂×D₄	1	1	yes	characteristic
151	#1437	C₂×D₄	1	1	yes	characteristic
152	C₂⁶	#120	1	1	yes	characteristic
153	C₂⁶	#120	1	1	yes	characteristic
154	C₂⁶	#120	1	1	yes	characteristic
155	C₂⁶	C₂²×D₄	1	1	yes	characteristic
156	C₂²×#41	#120	1	2	yes	—
157	C₂²×#41	#121	1	2	yes	—
158	C₂³×Q₈	#120	1	1	yes	characteristic
159	C₂³×Q₈	#120	1	1	yes	characteristic
160	C₂³×Q₈	#120	1	1	yes	characteristic
161	C₂³×D₄	#120	1	1	yes	characteristic
162	C₂³×D₄	#120	1	1	yes	characteristic
163	C₂³×D₄	#120	1	1	yes	characteristic
164	C₂⁴×C₄	#120	1	1	yes	characteristic
165	C₂⁴×C₄	#120	1	1	yes	characteristic
166	C₂⁴×C₄	#120	1	1	yes	characteristic
167	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
168	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
169	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
170	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
171	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
172	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
173	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
174	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
175	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
176	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
177	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
179	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
180	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
181	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
182	C₂²×C₂²⋊C₄	C₂×#41	1	1	yes	characteristic
183	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂²×C₄²	#120	1	1	yes	characteristic
188	C₂²×C₄²	#120	1	1	yes	characteristic
189	#393	#120	1	2	yes	—
190	#393	#121	1	2	yes	—
191	#393	#121	1	2	yes	—
192	#393	#127	1	2	yes	—
193	#392	#120	1	2	yes	—
194	#392	#121	1	2	yes	—
195	#392	#121	1	2	yes	—
196	#392	#127	1	2	yes	—
197	#381	#120	1	2	yes	—
198	#381	#121	1	2	yes	—
199	C₂×#95	#120	1	1	yes	characteristic
200	C₂×#95	#120	1	1	yes	characteristic
201	C₂×#95	#120	1	1	yes	characteristic
202	C₂×#95	#120	1	1	yes	characteristic
203	C₂×#95	#120	1	1	yes	characteristic
204	C₂×#95	#120	1	1	yes	characteristic
205	C₂×#95	#120	1	1	yes	characteristic
206	C₂×#95	#120	1	1	yes	characteristic
207	C₂⁵	B(4,𝔽₂)	1	2	yes	—
208	C₂⁵	B(4,𝔽₂)	1	2	yes	—
209	C₂⁵	C₂×#120	1	1	yes	characteristic
210	C₂⁵	C₂×#120	1	1	yes	characteristic
211	C₂⁵	C₂×#120	1	1	yes	characteristic
212	C₂⁵	D₄²	1	1	yes	characteristic
213	C₂⁵	D₄²	1	1	yes	characteristic
214	C₂⁵	D₄²	1	1	yes	characteristic
215	C₂⁵	#546	1	1	yes	characteristic
216	C₂²×Q₈	#392	1	2	yes	—
217	C₂²×D₄	#392	1	2	yes	—
218	C₂³×C₄	C₂×#120	1	1	yes	characteristic
219	C₂³×C₄	C₂×#120	1	1	yes	characteristic
220	C₂³×C₄	C₂×#120	1	1	yes	characteristic
221	C₂³×C₄	C₂×#120	1	1	yes	characteristic
222	C₂³×C₄	C₂×#120	1	1	yes	characteristic
223	C₂³×C₄	C₂×#121	1	1	yes	characteristic
224	C₂³×C₄	#534	1	1	yes	characteristic
225	C₂³×C₄	#534	1	1	yes	characteristic
226	C₂³×C₄	D₄²	1	1	yes	characteristic
227	C₂³×C₄	D₄²	1	1	yes	characteristic
228	C₂³×C₄	D₄²	1	1	yes	characteristic
229	C₂³×C₄	#546	1	1	yes	characteristic
230	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
232	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
233	C₂×C₂²⋊C₄	C₂×#127	1	1	yes	characteristic
234	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
235	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
236	C₂×C₂²⋊C₄	#535	1	1	yes	characteristic
237	C₂×C₂²⋊C₄	#535	1	1	yes	characteristic
238	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
240	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
241	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
242	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
243	C₂×C₂²⋊C₄	D₄²	1	1	yes	characteristic
244	C₂×C₂²⋊C₄	#546	1	1	yes	characteristic
245	C₂×C₂²⋊C₄	#546	1	1	yes	characteristic
246	#95	#392	1	2	yes	—
247	#95	#392	1	2	yes	—
248	C₂⁴	#2403	1	1	yes	characteristic
249	C₂⁴	#2403	1	1	yes	characteristic
250	C₂⁴	#2408	1	1	yes	characteristic
251	C₂⁴	#2410	1	1	yes	characteristic
252	C₂⁴	#2410	1	1	yes	characteristic
253	C₂⁴	#2433	1	1	yes	characteristic
254	C₂⁴	#2846	1	1	yes	characteristic
255	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
256	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
257	C₂⁴	#3026	1	1	yes	characteristic
258	C₂⁴	#3026	1	1	yes	characteristic
259	C₂²×C₄	C₂×#392	1	1	yes	characteristic
260	C₂²×C₄	#2403	1	1	yes	characteristic
261	C₂²×C₄	#2403	1	1	yes	characteristic
262	C₂²×C₄	#2408	1	1	yes	characteristic
263	C₂²×C₄	#2410	1	1	yes	characteristic
264	C₂²×C₄	#2410	1	1	yes	characteristic
265	C₂²×C₄	#2433	1	1	yes	characteristic
266	C₂²×C₄	#2613	1	1	yes	characteristic
267	C₂³	#14715	1	1	yes	characteristic
268	C₂³	#23900	1	1	yes	characteristic
269	C₂³	#23903	1	1	yes	characteristic
270	C₂³	#24404	1	1	yes	characteristic
271	C₂³	#24404	1	1	yes	characteristic
272	C₂³	#32898	1	1	yes	characteristic
273	C₂³	#32898	1	1	yes	characteristic
274	C₂²	#394066	1	1	yes	characteristic
275	C₂²	#394106	1	1	yes	characteristic
276	C₂²	#394108	1	1	yes	characteristic
277	C₂²	#511790	1	1	yes	characteristic
278	C₂²	#511790	1	1	yes	characteristic
279	C₂²	#511790	1	1	yes	characteristic
280	C₂²	#511793	1	1	yes	characteristic
281	C₂	#72213329	1	1	yes	characteristic
282	C₂	#72218099	1	1	yes	characteristic
283	C₂	#72235200	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#394066
commutator, Frattini, cosocle	C₂³×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#121 ◁ G
LECS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72235926	C₂	semidirect
2. #72235925	C₂	semidirect
3. #72235924	C₂	semidirect
4. #72235923	C₂	semidirect
5. #72235922	C₂	semidirect
6. #72235919	C₂	semidirect
7. #72235868	C₂	semidirect
8. #72235864	C₂	semidirect
9. #72235820	C₂	semidirect
10. #72235808	C₂	semidirect
11. #72235729	C₂	semidirect
12. #72235690	C₂	semidirect
13. #72232667	C₂	semidirect
14. #72231851	C₂	semidirect
15. #72231809	C₂	semidirect
17. #7642350	C₂²	semidirect
19. #6451874	C₂²	semidirect
21. #6346594	C₂²	semidirect
22. #6346594	C₂²	semidirect
30. #482549	C₂²	semidirect
31. #482549	C₂²	semidirect
32. #482549	C₂²	semidirect
33. #482549	C₂²	semidirect
34. #480250	C₂²	semidirect
35. #480250	C₂²	semidirect
36. #359705	C₂²	semidirect
38. #201462	C₂²	semidirect
39. #201462	C₂²	semidirect
40. C₂×#8549	C₂²	semidirect
41. C₂×#8542	C₂²	semidirect
42. #187945	C₂²	semidirect
43. #185255	C₂²	semidirect
44. #184796	C₂²	semidirect
45. #128157	C₂²	semidirect
46. #128157	C₂²	semidirect
47. #127215	C₂²	semidirect
48. #127215	C₂²	semidirect
49. #107115	C₂²	semidirect
50. #106326	C₂²	semidirect
51. C₂³×H(5,𝔽₂)	D₄	semidirect
53. C₂²×#525	D₄	semidirect
55. C₂×#2760	D₄	semidirect
56. C₂×#2760	D₄	semidirect
57. C₂×#2840	D₄	semidirect
58. C₂×#2406	D₄	semidirect
59. C₂×#2406	D₄	semidirect
60. C₂×#2404	D₄	semidirect
66. C₂×#2403	D₄	semidirect
68. C₂×#2308	D₄	semidirect
72. #18133	D₄	semidirect
73. #18132	D₄	semidirect
78. #17198	D₄	semidirect
79. #17105	D₄	semidirect
80. #17105	D₄	semidirect
81. #17102	D₄	semidirect
82. #15930	D₄	semidirect
83. #15561	D₄	semidirect
84. #15467	D₄	semidirect
85. #15267	D₄	semidirect
86. #14681	D₄	semidirect
87. #14649	D₄	semidirect
88. #14609	D₄	semidirect
89. #14533	D₄	semidirect
91. #14207	D₄	semidirect
92. #13868	D₄	semidirect
96. #11839	D₄	semidirect
97. #9475	D₄	semidirect
106. C₂×#543	C₂×D₄	semidirect
107. C₂×#534	C₂×D₄	semidirect
139. #2273	C₂×D₄	semidirect
140. #1819	C₂×D₄	semidirect
141. #1462	C₂×D₄	semidirect
142. #1462	C₂×D₄	semidirect
143. C₂×#323	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72235926	2
#72235925	2
#72235924	2
#72235923	2
#72235922	2
#72235919	2
#72235868	2
#72235864	2
#72235820	2
#72235808	2
#72235729	2
#72235690	2
#72232667	2
#72231851	2
#72231809	2