Subgroups — #72233155

Order 2048.

At least 9350 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

The subgroup lattice has 286 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72233145	C₂	1	1	yes	maximal, characteristic, complemented
2	#72233140	C₂	1	1	yes	maximal, characteristic, complemented
3	#72233085	C₂	1	1	yes	maximal, characteristic, complemented
4	#72233080	C₂	1	1	yes	maximal, characteristic, complemented
5	#72233056	C₂	1	1	yes	maximal, characteristic, complemented
6	#72233037	C₂	1	1	yes	maximal, characteristic, complemented
7	#72233004	C₂	1	1	yes	maximal, characteristic, complemented
8	#72232999	C₂	1	1	yes	maximal, characteristic, complemented
9	#72232754	C₂	1	1	yes	maximal, characteristic, complemented
10	#72232637	C₂	1	1	yes	maximal, characteristic, complemented
11	#72231473	C₂	1	1	yes	maximal, characteristic, complemented
12	#72231451	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218840	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218838	C₂	1	1	yes	maximal, characteristic, complemented
15	#72216587	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#47827	C₂²	1	1	yes	characteristic
17	C₂×#58990	C₂²	1	1	yes	characteristic
18	#6470225	C₂²	1	1	yes	characteristic, complemented
19	#6470225	C₂²	1	1	yes	characteristic, complemented
20	#6470019	C₂²	1	1	yes	characteristic
21	#6437621	C₂²	1	1	yes	characteristic, complemented
22	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
23	#6393523	C₂²	1	1	yes	characteristic, complemented
24	#4867137	C₂²	1	1	yes	characteristic
25	#688032	C₂²	1	1	yes	characteristic
26	#647670	C₂²	1	1	yes	characteristic
27	#616960	C₂²	1	1	yes	characteristic
28	#561507	C₂²	1	1	yes	characteristic
29	#496225	C₂²	1	1	yes	characteristic, complemented
30	#496225	C₂²	1	1	yes	characteristic, complemented
31	#496225	C₂²	1	1	yes	characteristic, complemented
32	#496225	C₂²	1	1	yes	characteristic, complemented
33	#496189	C₂²	1	1	yes	characteristic
34	#496189	C₂²	1	1	yes	characteristic
35	#493357	C₂²	1	1	yes	characteristic, complemented
36	#493357	C₂²	1	1	yes	characteristic, complemented
37	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
38	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
39	#359501	C₂²	1	1	yes	characteristic, complemented
40	#359442	C₂²	1	1	yes	characteristic, complemented
41	#201664	C₂²	1	1	yes	characteristic, complemented
42	#201664	C₂²	1	1	yes	characteristic, complemented
43	#201293	C₂²	1	1	yes	characteristic
44	#186835	C₂²	1	1	yes	characteristic, complemented
45	#128009	C₂²	1	1	yes	characteristic, complemented
46	#128009	C₂²	1	1	yes	characteristic, complemented
47	#108238	C₂²	1	1	yes	characteristic, complemented
48	#106660	C₂²	1	1	yes	characteristic, complemented
49	#106654	C₂²	1	1	yes	characteristic, complemented
50	#106654	C₂²	1	1	yes	characteristic, complemented
51	C₂×#3484	C₂³	1	1	yes	characteristic
52	C₂²×D₄²	D₄	1	1	yes	characteristic, complemented
53	C₂²×D₄²	D₄	1	1	yes	characteristic, complemented
54	C₂²×#534	C₂³	1	1	yes	characteristic
55	C₂³×#120	D₄	1	1	yes	characteristic, complemented
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂×#2709	D₄	1	1	yes	characteristic, complemented
58	C₂×#2697	D₄	1	1	yes	characteristic, complemented
59	C₂×#2679	D₄	1	1	yes	characteristic, complemented
60	C₂×#2669	D₄	1	1	yes	characteristic, complemented
61	C₂×#2433	C₂³	1	1	yes	characteristic
62	C₂×#2403	D₄	1	1	yes	characteristic, complemented
63	C₂×#2403	C₂³	1	1	yes	characteristic
64	C₂×#2403	C₂³	1	1	yes	characteristic
65	C₂×#2377	D₄	1	1	yes	characteristic, complemented
66	C₂×#2375	D₄	1	1	yes	characteristic, complemented
67	#34171	C₂³	1	1	yes	characteristic
68	C₂×#2358	D₄	1	1	yes	characteristic, complemented
69	C₂×D₄×C₂²⋊C₄	C₂³	1	1	yes	characteristic
70	#32990	D₄	1	2	yes	complemented
71	#32888	D₄	1	2	yes	complemented
72	C₂×#1783	C₂³	1	1	yes	characteristic
73	C₂×#1783	C₂³	1	1	yes	characteristic
74	C₂×#1781	C₂³	1	1	yes	characteristic
75	C₂×#1781	C₂³	1	1	yes	characteristic
76	C₂×#1781	C₂³	1	1	yes	characteristic
77	C₂×#1781	C₂³	1	1	yes	characteristic
78	#16014	D₄	1	1	yes	characteristic, complemented
79	#15948	D₄	1	1	yes	characteristic, complemented
80	#15948	D₄	1	1	yes	characteristic, complemented
81	#15947	D₄	1	1	yes	characteristic, complemented
82	#15947	D₄	1	1	yes	characteristic, complemented
83	#15947	D₄	1	1	yes	characteristic, complemented
84	#15266	D₄	1	1	yes	characteristic, complemented
85	#15266	D₄	1	1	yes	characteristic, complemented
86	#14679	D₄	1	1	yes	characteristic, complemented
87	#14335	C₂³	1	1	yes	characteristic
88	#14290	D₄	1	1	yes	characteristic, complemented
89	#14262	D₄	1	1	yes	characteristic, complemented
90	#13953	D₄	1	1	yes	characteristic, complemented
91	#13877	D₄	1	1	yes	characteristic, complemented
92	#13867	D₄	1	1	yes	characteristic, complemented
93	#13867	D₄	1	1	yes	characteristic, complemented
94	C₂×#1438	D₄	1	1	yes	characteristic, complemented
95	#13780	D₄	1	1	yes	characteristic, complemented
96	#13088	D₄	1	4	yes	complemented
97	#13088	D₄	1	4	yes	complemented
98	C₂⁷	C₂×D₄	1	1	yes	characteristic
99	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
100	C₂⁴×D₄	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	C₂⁴	1	1	yes	characteristic
104	C₂²×#127	C₂×D₄	1	1	yes	characteristic
105	C₂²×#127	C₂×D₄	1	1	yes	characteristic
106	C₂²×#123	C₂×D₄	1	1	yes	characteristic
107	C₂²×#123	C₂×D₄	1	1	yes	characteristic
108	C₂²×#121	C₂×D₄	1	1	yes	characteristic
109	C₂²×#120	C₂×D₄	1	1	yes	characteristic
110	C₂²×#120	C₂×D₄	1	1	yes	characteristic
111	C₂²×#120	C₂×D₄	1	1	yes	characteristic
112	C₂²×#120	C₂×D₄	1	1	yes	characteristic
113	C₂²×#120	C₂×D₄	1	1	yes	characteristic
114	C₂²×#120	C₂×D₄	1	1	yes	characteristic
115	C₂²×#120	C₂×D₄	1	1	yes	characteristic
116	C₂²×#120	C₂×D₄	1	1	yes	characteristic
117	C₂²×#120	C₂×D₄	1	1	yes	characteristic
118	C₂²×#120	C₂×D₄	1	1	yes	characteristic
119	C₂×#518	C₂×D₄	1	1	yes	characteristic, complemented
120	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
122	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
123	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
124	C₂²×#99	C₂×D₄	1	2	yes	complemented
125	C₂²×#99	C₂×D₄	1	2	yes	complemented
126	#2433	C₂×D₄	1	2	yes	—
127	#2433	C₂×D₄	1	2	yes	—
128	#2433	C₂×D₄	1	2	yes	—
129	#2433	#41	1	2	yes	—
130	C₂×#390	C₂×D₄	1	1	yes	characteristic
131	C₂×#390	C₂×D₄	1	1	yes	characteristic
132	C₂×#386	C₂×D₄	1	1	yes	characteristic
133	C₂×#386	C₂×D₄	1	1	yes	characteristic
134	C₂×#386	C₂×D₄	1	1	yes	characteristic
135	C₂×#386	C₂×D₄	1	1	yes	characteristic
136	C₂×#381	C₂×D₄	1	1	yes	characteristic
137	C₂×#381	C₂×D₄	1	1	yes	characteristic
138	C₂×#380	C₂×D₄	1	1	yes	characteristic
139	C₂×#379	C₂×D₄	1	1	yes	characteristic
140	C₂×#379	C₂×D₄	1	1	yes	characteristic
141	C₂×#379	C₂×D₄	1	1	yes	characteristic
142	C₂×#379	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
145	#1783	C₂×D₄	1	2	yes	complemented
146	#1781	C₂×D₄	1	2	yes	complemented
147	#1781	C₂×D₄	1	2	yes	complemented
148	#1438	C₂×D₄	1	1	yes	characteristic
149	#1438	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	C₂⁶	#120	1	1	yes	characteristic
156	C₂⁶	#120	1	1	yes	characteristic
157	C₂⁶	C₂²×D₄	1	1	yes	characteristic
158	C₂⁶	C₂²×D₄	1	1	yes	characteristic
159	C₂³×D₄	#120	1	1	yes	characteristic
160	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
161	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
162	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
163	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
164	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
165	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
166	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
167	C₂³×D₄	C₂×#41	1	1	yes	characteristic
168	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂⁴×C₄	#120	1	1	yes	characteristic
173	C₂⁴×C₄	#120	1	1	yes	characteristic
174	C₂⁴×C₄	#120	1	1	yes	characteristic
175	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
176	C₂×#127	#120	1	2	yes	—
177	C₂×#127	#120	1	2	yes	—
178	C₂×#127	#121	1	2	yes	—
179	C₂×#127	#121	1	2	yes	—
180	C₂×#121	#121	1	2	yes	—
181	C₂×#121	#127	1	2	yes	—
182	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
183	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
184	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
185	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
186	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
187	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
188	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
189	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
190	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
191	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
192	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
193	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
194	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
195	C₂²×C₄²	#120	1	1	yes	characteristic
196	C₂²×C₄²	#120	1	1	yes	characteristic
197	#390	#120	1	2	yes	—
198	#390	#120	1	2	yes	—
199	#390	#121	1	2	yes	—
200	#390	#121	1	2	yes	—
201	C₄×C₂²⋊C₄	#121	1	2	yes	—
202	C₄×C₂²⋊C₄	#127	1	2	yes	—
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	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	D₄²	1	1	yes	characteristic
216	C₂⁵	D₄²	1	1	yes	characteristic
217	C₂⁵	D₄²	1	1	yes	characteristic
218	C₂⁵	#546	1	1	yes	characteristic
219	C₂⁵	#546	1	1	yes	characteristic
220	C₂²×D₄	#392	1	2	yes	—
221	C₂²×D₄	C₂×#120	1	1	yes	characteristic
222	C₂²×D₄	C₂×#120	1	1	yes	characteristic
223	C₂²×D₄	C₂×#121	1	1	yes	characteristic
224	C₂²×D₄	C₂×#121	1	1	yes	characteristic
225	C₂²×D₄	#534	1	1	yes	characteristic
226	C₂²×D₄	#534	1	1	yes	characteristic
227	C₂²×D₄	#534	1	1	yes	characteristic
228	C₂²×D₄	#534	1	1	yes	characteristic
229	C₂²×D₄	D₄²	1	1	yes	characteristic
230	C₂²×D₄	D₄²	1	1	yes	characteristic
231	C₂²×D₄	D₄²	1	1	yes	characteristic
232	C₂²×D₄	D₄²	1	1	yes	characteristic
233	C₂²×D₄	D₄²	1	1	yes	characteristic
234	C₂²×D₄	D₄²	1	1	yes	characteristic
235	C₂²×D₄	#546	1	1	yes	characteristic
236	C₂²×D₄	#546	1	1	yes	characteristic
237	C₂³×C₄	C₂×#120	1	1	yes	characteristic
238	C₂³×C₄	C₂×#120	1	1	yes	characteristic
239	C₂³×C₄	C₂×#121	1	1	yes	characteristic
240	C₂³×C₄	C₂×#127	1	1	yes	characteristic
241	C₂³×C₄	#535	1	1	yes	characteristic
242	C₂³×C₄	#535	1	1	yes	characteristic
243	C₂×C₄⋊C₄	#392	1	2	yes	—
244	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
245	C₂×C₄²	#392	1	2	yes	—
246	C₂×C₄²	#392	1	2	yes	—
247	C₂⁴	#2403	1	1	yes	characteristic
248	C₂⁴	#2403	1	1	yes	characteristic
249	C₂⁴	#2403	1	1	yes	characteristic
250	C₂⁴	#2403	1	1	yes	characteristic
251	C₂⁴	#2408	1	1	yes	characteristic
252	C₂⁴	#2410	1	1	yes	characteristic
253	C₂⁴	#2410	1	1	yes	characteristic
254	C₂⁴	#2410	1	1	yes	characteristic
255	C₂⁴	#2410	1	1	yes	characteristic
256	C₂⁴	#2433	1	1	yes	characteristic
257	C₂⁴	#2846	1	1	yes	characteristic
258	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
259	C₂⁴	#3025	1	1	yes	characteristic
260	C₂⁴	#3026	1	1	yes	characteristic
261	C₂⁴	#3027	1	1	yes	characteristic
262	C₂²×C₄	C₂×#392	1	1	yes	characteristic
263	C₂²×C₄	#2408	1	1	yes	characteristic
264	C₂²×C₄	#2433	1	1	yes	characteristic
265	C₂²×C₄	#2613	1	1	yes	characteristic
266	C₂³	#14715	1	1	yes	characteristic
267	C₂³	#23895	1	2	yes	—
268	C₂³	#23895	1	2	yes	—
269	C₂³	#24260	1	1	yes	characteristic
270	C₂³	#24262	1	1	yes	characteristic
271	C₂³	#25771	1	1	yes	characteristic
272	C₂³	#25773	1	1	yes	characteristic
273	C₂³	#32898	1	1	yes	characteristic
274	C₂³	#32899	1	1	yes	characteristic
275	C₂²	#393934	1	1	yes	characteristic
276	C₂²	#394064	1	1	yes	characteristic
277	C₂²	#394064	1	1	yes	characteristic
278	C₂²	#511790	1	1	yes	characteristic
279	C₂²	#511794	1	1	yes	characteristic
280	C₂²	#511853	1	1	yes	characteristic
281	C₂²	#511853	1	1	yes	characteristic
282	C₂	#72233127	1	1	yes	characteristic
283	C₂	#72233152	1	1	yes	characteristic
284	C₂	#72233154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#393934
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂×D₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72233145	C₂	semidirect
2. #72233140	C₂	semidirect
3. #72233085	C₂	semidirect
4. #72233080	C₂	semidirect
5. #72233056	C₂	semidirect
6. #72233037	C₂	semidirect
7. #72233004	C₂	semidirect
8. #72232999	C₂	semidirect
9. #72232754	C₂	semidirect
10. #72232637	C₂	semidirect
11. #72231473	C₂	semidirect
12. #72231451	C₂	semidirect
13. #72218840	C₂	semidirect
14. #72218838	C₂	semidirect
15. #72216587	C₂	semidirect
18. #6470225	C₂²	semidirect
19. #6470225	C₂²	semidirect
21. #6437621	C₂²	semidirect
22. C₂×#23900	C₂²	semidirect
23. #6393523	C₂²	semidirect
29. #496225	C₂²	semidirect
30. #496225	C₂²	semidirect
31. #496225	C₂²	semidirect
32. #496225	C₂²	semidirect
35. #493357	C₂²	semidirect
36. #493357	C₂²	semidirect
37. C₂×#12687	C₂²	semidirect
38. C₂×#12687	C₂²	semidirect
39. #359501	C₂²	semidirect
40. #359442	C₂²	semidirect
41. #201664	C₂²	semidirect
42. #201664	C₂²	semidirect
44. #186835	C₂²	semidirect
45. #128009	C₂²	semidirect
46. #128009	C₂²	semidirect
47. #108238	C₂²	semidirect
48. #106660	C₂²	semidirect
49. #106654	C₂²	semidirect
50. #106654	C₂²	semidirect
52. C₂²×D₄²	D₄	semidirect
53. C₂²×D₄²	D₄	semidirect
55. C₂³×#120	D₄	semidirect
57. C₂×#2709	D₄	semidirect
58. C₂×#2697	D₄	semidirect
59. C₂×#2679	D₄	semidirect
60. C₂×#2669	D₄	semidirect
62. C₂×#2403	D₄	semidirect
65. C₂×#2377	D₄	semidirect
66. C₂×#2375	D₄	semidirect
68. C₂×#2358	D₄	semidirect
70. #32990	D₄	semidirect
71. #32888	D₄	semidirect
78. #16014	D₄	semidirect
79. #15948	D₄	semidirect
80. #15948	D₄	semidirect
81. #15947	D₄	semidirect
82. #15947	D₄	semidirect
83. #15947	D₄	semidirect
84. #15266	D₄	semidirect
85. #15266	D₄	semidirect
86. #14679	D₄	semidirect
88. #14290	D₄	semidirect
89. #14262	D₄	semidirect
90. #13953	D₄	semidirect
91. #13877	D₄	semidirect
92. #13867	D₄	semidirect
93. #13867	D₄	semidirect
94. C₂×#1438	D₄	semidirect
95. #13780	D₄	semidirect
96. #13088	D₄	semidirect
97. #13088	D₄	semidirect
99. C₂²×H(5,𝔽₂)	C₂×D₄	semidirect
119. C₂×#518	C₂×D₄	semidirect
124. C₂²×#99	C₂×D₄	semidirect
125. C₂²×#99	C₂×D₄	semidirect
145. #1783	C₂×D₄	semidirect
146. #1781	C₂×D₄	semidirect
147. #1781	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72233145	2
#72233140	2
#72233085	2
#72233080	2
#72233056	2
#72233037	2
#72233004	2
#72232999	2
#72232754	2
#72232637	2
#72231473	2
#72231451	2
#72218840	2
#72218838	2
#72216587	2