Subgroups — #72234097

Order 2048.

At least 7920 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	#72234074	C₂	1	1	yes	maximal, characteristic, complemented
2	#72234067	C₂	1	1	yes	maximal, characteristic, complemented
3	#72234063	C₂	1	1	yes	maximal, characteristic, complemented
4	#72234051	C₂	1	1	yes	maximal, characteristic, complemented
5	#72233901	C₂	1	1	yes	maximal, characteristic, complemented
6	#72233845	C₂	1	1	yes	maximal, characteristic, complemented
7	#72233828	C₂	1	1	yes	maximal, characteristic, complemented
8	#72233813	C₂	1	1	yes	maximal, characteristic, complemented
9	#72233798	C₂	1	1	yes	maximal, characteristic, complemented
10	#72233591	C₂	1	1	yes	maximal, characteristic, complemented
11	#72233399	C₂	1	1	yes	maximal, characteristic, complemented
12	#72233227	C₂	1	1	yes	maximal, characteristic, complemented
13	#72233047	C₂	1	1	yes	maximal, characteristic, complemented
14	#72231082	C₂	1	1	yes	maximal, characteristic, complemented
15	#72231013	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#56346	C₂²	1	1	yes	characteristic
17	C₂×#46350	C₂²	1	1	yes	characteristic
18	#6491100	C₂²	1	1	yes	characteristic, complemented
19	#6454047	C₂²	1	1	yes	characteristic, complemented
20	#6393515	C₂²	1	1	yes	characteristic, complemented
21	#6347256	C₂²	1	1	yes	characteristic, complemented
22	#6346631	C₂²	1	1	yes	characteristic
23	#6346599	C₂²	1	1	yes	characteristic, complemented
24	#4861302	C₂²	1	1	yes	characteristic
25	#4378714	C₂²	1	1	yes	characteristic
26	#640973	C₂²	1	1	yes	characteristic
27	#619533	C₂²	1	1	yes	characteristic
28	#516285	C₂²	1	1	yes	characteristic
29	#496330	C₂²	1	1	yes	characteristic, complemented
30	#493998	C₂²	1	1	yes	characteristic, complemented
31	#482377	C₂²	1	1	yes	characteristic, complemented
32	#482360	C₂²	1	1	yes	characteristic
33	#482360	C₂²	1	1	yes	characteristic
34	#482357	C₂²	1	1	yes	characteristic, complemented
35	#481985	C₂²	1	1	yes	characteristic, complemented
36	#481961	C₂²	1	1	yes	characteristic, complemented
37	#480708	C₂²	1	1	yes	characteristic, complemented
38	#480382	C₂²	1	1	yes	characteristic, complemented
39	#201743	C₂²	1	1	yes	characteristic, complemented
40	#201549	C₂²	1	1	yes	characteristic
41	#201386	C₂²	1	1	yes	characteristic, complemented
42	#197340	C₂²	1	1	yes	characteristic, complemented
43	#196338	C₂²	1	1	yes	characteristic, complemented
44	#186816	C₂²	1	1	yes	characteristic, complemented
45	#128200	C₂²	1	1	yes	characteristic, complemented
46	#128086	C₂²	1	1	yes	characteristic, complemented
47	#106763	C₂²	1	1	yes	characteristic, complemented
48	#106107	C₂²	1	1	yes	characteristic, complemented
49	#105582	C₂²	1	1	yes	characteristic, complemented
50	#105580	C₂²	1	1	yes	characteristic, complemented
51	#62773	C₂³	1	1	yes	characteristic
52	C₂³×#127	D₄	1	1	yes	characteristic, complemented
53	#60230	C₂³	1	1	yes	characteristic
54	C₂×#2735	D₄	1	1	yes	characteristic, complemented
55	C₂×#2693	D₄	1	1	yes	characteristic, complemented
56	C₂×#2675	D₄	1	1	yes	characteristic, complemented
57	C₂×#2668	D₄	1	1	yes	characteristic, complemented
58	C₂×#2812	D₄	1	1	yes	characteristic, complemented
59	C₂×#2812	D₄	1	1	yes	characteristic, complemented
60	C₂×#2535	C₂³	1	1	yes	characteristic
61	C₂×#2539	D₄	1	1	yes	characteristic, complemented
62	C₂×#2408	C₂³	1	1	yes	characteristic
63	C₂×#2408	C₂³	1	1	yes	characteristic
64	C₂×#2403	D₄	1	1	yes	characteristic, complemented
65	C₂×#2403	D₄	1	1	yes	characteristic, complemented
66	C₂×#2635	C₂³	1	1	yes	characteristic
67	C₂×#2321	C₂³	1	1	yes	characteristic
68	C₂×#2290	C₂³	1	1	yes	characteristic
69	C₂×#2275	D₄	1	1	yes	characteristic, complemented
70	#20726	C₂³	1	1	yes	characteristic
71	#20722	C₂³	1	1	yes	characteristic
72	#20722	C₂³	1	1	yes	characteristic
73	#18212	D₄	1	2	yes	complemented
74	#18210	D₄	1	2	yes	complemented
75	#17364	C₂³	1	1	yes	characteristic
76	#17362	C₂³	1	1	yes	characteristic
77	#17362	C₂³	1	1	yes	characteristic
78	#17141	D₄	1	1	yes	characteristic, complemented
79	#17141	D₄	1	1	yes	characteristic, complemented
80	#17082	D₄	1	1	yes	characteristic, complemented
81	#17082	D₄	1	1	yes	characteristic, complemented
82	#15947	D₄	1	1	yes	characteristic
83	#15947	D₄	1	1	yes	characteristic
84	#15266	D₄	1	1	yes	characteristic
85	#15266	D₄	1	1	yes	characteristic
86	#14682	D₄	1	1	yes	characteristic, complemented
87	#14168	D₄	1	1	yes	characteristic, complemented
88	#14136	D₄	1	1	yes	characteristic, complemented
89	#13959	D₄	1	1	yes	characteristic, complemented
90	#13959	C₂³	1	1	yes	characteristic
91	#13953	D₄	1	1	yes	characteristic, complemented
92	#13902	D₄	1	1	yes	characteristic, complemented
93	#13902	D₄	1	1	yes	characteristic, complemented
94	#13868	D₄	1	1	yes	characteristic, complemented
95	#13781	D₄	1	1	yes	characteristic, complemented
96	#11875	D₄	1	4	yes	complemented
97	#11774	D₄	1	4	yes	complemented
98	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
99	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
100	#3501	C₂×D₄	1	1	yes	characteristic, complemented
101	#3496	C₂×D₄	1	1	yes	characteristic, complemented
102	C₂²×#121	C₂×D₄	1	1	yes	characteristic
103	C₂²×#120	C₂×D₄	1	1	yes	characteristic
104	C₂²×#120	C₂×D₄	1	1	yes	characteristic
105	C₂²×#120	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×D₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×D₄	C₂⁴	1	1	yes	characteristic
108	C₂²×#117	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
111	#2535	C₂×D₄	1	2	yes	—
112	#2535	C₂×D₄	1	2	yes	—
113	#2535	C₂×D₄	1	2	yes	—
114	#2535	#41	1	2	yes	—
115	C₂×#400	C₂×D₄	1	1	yes	characteristic
116	C₂×#399	C₂×D₄	1	1	yes	characteristic
117	C₂×#397	C₂×D₄	1	1	yes	characteristic
118	C₂×#397	C₂×D₄	1	1	yes	characteristic
119	C₂×#394	C₂×D₄	1	1	yes	characteristic
120	C₂×#394	C₂×D₄	1	1	yes	characteristic
121	C₂×#394	C₂×D₄	1	1	yes	characteristic
122	C₂×#392	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×#386	C₂×D₄	1	1	yes	characteristic
128	C₂×#385	C₂×D₄	1	1	yes	characteristic
129	C₂×#383	C₂×D₄	1	1	yes	characteristic
130	C₂×#382	C₂×D₄	1	1	yes	characteristic
131	C₂×#381	C₂×D₄	1	1	yes	characteristic
132	C₂×#380	C₂×D₄	1	1	yes	characteristic
133	C₂×#380	C₂×D₄	1	1	yes	characteristic
134	C₂×#379	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×C₄×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×#376	C₂×D₄	1	1	yes	characteristic
142	C₂×#376	C₂×D₄	1	1	yes	characteristic
143	#1799	C₂×D₄	1	2	yes	complemented
144	C₄×#99	C₂×D₄	1	2	yes	complemented
145	#1465	C₂×D₄	1	2	yes	complemented
146	#1460	C₂×D₄	1	2	yes	complemented
147	#1459	C₂×D₄	1	2	yes	complemented
148	#1445	C₂×D₄	1	1	yes	characteristic
149	#1445	C₂×D₄	1	1	yes	characteristic
150	#1438	C₂×D₄	1	1	yes	characteristic
151	#1438	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	C₂³×D₄	#120	1	1	yes	characteristic
156	C₂³×D₄	#120	1	1	yes	characteristic
157	C₂³×D₄	#120	1	1	yes	characteristic
158	C₂³×D₄	#120	1	1	yes	characteristic
159	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
160	C₂⁴×C₄	#120	1	1	yes	characteristic
161	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
162	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×D₄	#121	1	2	yes	—
164	C₂×C₄×D₄	#127	1	2	yes	—
165	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×D₄	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×D₄	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
175	C₂²×C₄⋊C₄	C₂²×D₄	1	1	yes	characteristic
176	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
177	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
179	C₂²×C₂²⋊C₄	#120	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×C₄²	#120	1	1	yes	characteristic
183	C₂²×C₄²	#120	1	1	yes	characteristic
184	C₂²×C₄²	#120	1	1	yes	characteristic
185	C₂²×C₄²	#120	1	1	yes	characteristic
186	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
187	#397	#120	1	2	yes	—
188	#397	#120	1	2	yes	—
189	#397	#121	1	2	yes	—
190	#397	#121	1	2	yes	—
191	#392	#120	1	2	yes	—
192	#392	#120	1	2	yes	—
193	#392	#121	1	2	yes	—
194	#392	#121	1	2	yes	—
195	#382	#121	1	2	yes	—
196	#382	#127	1	2	yes	—
197	C₂×#95	#120	1	1	yes	characteristic
198	C₂×#95	#120	1	1	yes	characteristic
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	C₂×#120	1	1	yes	characteristic
208	C₂⁵	C₂×#120	1	1	yes	characteristic
209	C₂²×D₄	#392	1	2	yes	—
210	C₂²×D₄	D₄²	1	1	yes	characteristic
211	C₂²×D₄	D₄²	1	1	yes	characteristic
212	C₂²×D₄	D₄²	1	1	yes	characteristic
213	C₂²×D₄	#546	1	1	yes	characteristic
214	C₂³×C₄	C₂×#120	1	1	yes	characteristic
215	C₂³×C₄	C₂×#120	1	1	yes	characteristic
216	C₂³×C₄	C₂×#120	1	1	yes	characteristic
217	C₂³×C₄	C₂×#120	1	1	yes	characteristic
218	C₂³×C₄	C₂×#120	1	1	yes	characteristic
219	C₂³×C₄	D₄²	1	1	yes	characteristic
220	C₂³×C₄	D₄²	1	1	yes	characteristic
221	C₂³×C₄	D₄²	1	1	yes	characteristic
222	C₂³×C₄	D₄²	1	1	yes	characteristic
223	C₂³×C₄	D₄²	1	1	yes	characteristic
224	C₂³×C₄	D₄²	1	1	yes	characteristic
225	C₂³×C₄	#546	1	1	yes	characteristic
226	C₂³×C₄	#546	1	1	yes	characteristic
227	C₂×C₄⋊C₄	#392	1	2	yes	—
228	C₂×C₄⋊C₄	D₄²	1	1	yes	characteristic
229	C₂×C₄⋊C₄	D₄²	1	1	yes	characteristic
230	C₂×C₄⋊C₄	D₄²	1	1	yes	characteristic
231	C₂×C₄⋊C₄	#546	1	1	yes	characteristic
232	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
233	C₂×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
234	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
235	C₂×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
236	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
237	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
238	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	#534	1	1	yes	characteristic
240	C₂×C₄²	B(4,𝔽₂)	1	2	yes	—
241	C₂×C₄²	C₂×#121	1	1	yes	characteristic
242	C₂×C₄²	C₂×#127	1	1	yes	characteristic
243	C₂×C₄²	#535	1	1	yes	characteristic
244	C₂×C₄²	#535	1	1	yes	characteristic
245	#95	#392	1	2	yes	—
246	#95	#392	1	2	yes	—
247	C₂⁴	#2403	1	1	yes	characteristic
248	C₂⁴	#2403	1	1	yes	characteristic
249	C₂⁴	#2410	1	1	yes	characteristic
250	C₂⁴	#2410	1	1	yes	characteristic
251	C₂⁴	#2846	1	1	yes	characteristic
252	C₂²×C₄	C₂×#392	1	1	yes	characteristic
253	C₂²×C₄	#2403	1	1	yes	characteristic
254	C₂²×C₄	#2403	1	1	yes	characteristic
255	C₂²×C₄	#2408	1	1	yes	characteristic
256	C₂²×C₄	#2408	1	1	yes	characteristic
257	C₂²×C₄	#2410	1	1	yes	characteristic
258	C₂²×C₄	#2410	1	1	yes	characteristic
259	C₂²×C₄	#2433	1	1	yes	characteristic
260	C₂²×C₄	#2433	1	1	yes	characteristic
261	C₂²×C₄	#2613	1	1	yes	characteristic
262	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
263	C₂²×C₄	#3025	1	1	yes	characteristic
264	C₂²×C₄	#3026	1	1	yes	characteristic
265	C₂²×C₄	#3027	1	1	yes	characteristic
266	C₂³	#14715	1	1	yes	characteristic
267	C₂³	#23882	1	2	yes	—
268	C₂³	#32898	1	1	yes	characteristic
269	C₂³	#32899	1	1	yes	characteristic
270	C₂×C₄	#24260	1	1	yes	characteristic
271	C₂×C₄	#24262	1	1	yes	characteristic
272	C₂×C₄	#25771	1	1	yes	characteristic
273	C₂×C₄	#25773	1	1	yes	characteristic
274	C₂²	#393934	1	1	yes	characteristic
275	C₂²	#394102	1	1	yes	characteristic
276	C₂²	#394106	1	1	yes	characteristic
277	C₂²	#511790	1	1	yes	characteristic
278	C₂²	#511794	1	1	yes	characteristic
279	C₂²	#511846	1	1	yes	characteristic
280	C₂²	#511846	1	1	yes	characteristic
281	C₂	#72233152	1	1	yes	characteristic
282	C₂	#72234095	1	1	yes	characteristic
283	C₂	#72234096	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#393934
commutator, Frattini, cosocle	C₂²×C₄×D₄	C₂⁴
LCS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2290 ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72234074	C₂	semidirect
2. #72234067	C₂	semidirect
3. #72234063	C₂	semidirect
4. #72234051	C₂	semidirect
5. #72233901	C₂	semidirect
6. #72233845	C₂	semidirect
7. #72233828	C₂	semidirect
8. #72233813	C₂	semidirect
9. #72233798	C₂	semidirect
10. #72233591	C₂	semidirect
11. #72233399	C₂	semidirect
12. #72233227	C₂	semidirect
13. #72233047	C₂	semidirect
14. #72231082	C₂	semidirect
15. #72231013	C₂	semidirect
18. #6491100	C₂²	semidirect
19. #6454047	C₂²	semidirect
20. #6393515	C₂²	semidirect
21. #6347256	C₂²	semidirect
23. #6346599	C₂²	semidirect
29. #496330	C₂²	semidirect
30. #493998	C₂²	semidirect
31. #482377	C₂²	semidirect
34. #482357	C₂²	semidirect
35. #481985	C₂²	semidirect
36. #481961	C₂²	semidirect
37. #480708	C₂²	semidirect
38. #480382	C₂²	semidirect
39. #201743	C₂²	semidirect
41. #201386	C₂²	semidirect
42. #197340	C₂²	semidirect
43. #196338	C₂²	semidirect
44. #186816	C₂²	semidirect
45. #128200	C₂²	semidirect
46. #128086	C₂²	semidirect
47. #106763	C₂²	semidirect
48. #106107	C₂²	semidirect
49. #105582	C₂²	semidirect
50. #105580	C₂²	semidirect
52. C₂³×#127	D₄	semidirect
54. C₂×#2735	D₄	semidirect
55. C₂×#2693	D₄	semidirect
56. C₂×#2675	D₄	semidirect
57. C₂×#2668	D₄	semidirect
58. C₂×#2812	D₄	semidirect
59. C₂×#2812	D₄	semidirect
61. C₂×#2539	D₄	semidirect
64. C₂×#2403	D₄	semidirect
65. C₂×#2403	D₄	semidirect
69. C₂×#2275	D₄	semidirect
73. #18212	D₄	semidirect
74. #18210	D₄	semidirect
78. #17141	D₄	semidirect
79. #17141	D₄	semidirect
80. #17082	D₄	semidirect
81. #17082	D₄	semidirect
86. #14682	D₄	semidirect
87. #14168	D₄	semidirect
88. #14136	D₄	semidirect
89. #13959	D₄	semidirect
91. #13953	D₄	semidirect
92. #13902	D₄	semidirect
93. #13902	D₄	semidirect
94. #13868	D₄	semidirect
95. #13781	D₄	semidirect
96. #11875	D₄	semidirect
97. #11774	D₄	semidirect
100. #3501	C₂×D₄	semidirect
101. #3496	C₂×D₄	semidirect
143. #1799	C₂×D₄	semidirect
144. C₄×#99	C₂×D₄	semidirect
145. #1465	C₂×D₄	semidirect
146. #1460	C₂×D₄	semidirect
147. #1459	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72234074	2
#72234067	2
#72234063	2
#72234051	2
#72233901	2
#72233845	2
#72233828	2
#72233813	2
#72233798	2
#72233591	2
#72233399	2
#72233227	2
#72233047	2
#72231082	2
#72231013	2