Subgroups — #72255643

Order 2048.

Only the 2332 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 276 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251779	C₂	1	8	yes	maximal, complemented
2	#72251662	C₂	1	4	yes	maximal, complemented
3	#72249344	C₂	1	1	yes	maximal, characteristic, complemented
4	#72237402	C₂	1	4	yes	maximal, complemented
5	#72236667	C₂	1	1	yes	maximal, characteristic, complemented
6	#72235216	C₂	1	1	yes	maximal, characteristic, complemented
7	#72226123	C₂	1	2	yes	maximal, complemented
8	#72225712	C₂	1	8	yes	maximal, complemented
9	#72225690	C₂	1	2	yes	maximal, complemented
10	#10586422	C₂²	1	4	yes	—
11	#10586404	C₂²	1	8	yes	—
12	C₂×#62852	C₂²	1	1	yes	characteristic
13	C₂×#62842	C₂²	1	2	yes	—
14	C₂×#35346	C₂²	1	1	yes	characteristic, complemented
15	#6499867	C₂²	1	16	yes	complemented
16	C₂×#23900	C₂²	1	4	yes	complemented
17	C₂×#23900	C₂²	1	4	yes	complemented
18	C₂×#23900	C₂²	1	4	yes	complemented
19	#616961	C₂²	1	2	yes	complemented
20	#616960	C₂²	1	4	yes	complemented
21	#392111	C₂²	1	1	yes	characteristic, complemented
22	#392111	C₂²	1	2	yes	complemented
23	#359554	C₂²	1	4	yes	complemented
24	#359554	C₂²	1	4	yes	complemented
25	#359554	C₂²	1	4	yes	complemented
26	#359434	C₂²	1	4	yes	complemented
27	#359433	C₂²	1	2	yes	complemented
28	#188050	C₂²	1	4	yes	complemented
29	#188049	C₂²	1	8	yes	complemented
30	#187944	C₂²	1	8	yes	complemented
31	#187944	C₂²	1	8	yes	complemented
32	#187944	C₂²	1	8	yes	complemented
33	#187934	C₂²	1	32	yes	complemented
34	#187934	C₂²	1	16	yes	complemented
35	C₂²×#585	D₄	1	1	yes	characteristic, complemented
36	C₂³×H(5,𝔽₂)	D₄	1	2	yes	complemented
37	C₂³×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
38	#62832	D₄	1	8	yes	complemented
39	#62824	D₄	1	8	yes	complemented
40	#62820	D₄	1	16	yes	complemented
41	#62818	D₄	1	32	yes	complemented
42	#62818	D₄	1	16	yes	complemented
43	#62817	D₄	1	16	yes	complemented
44	#62817	C₂³	1	16	yes	—
45	C₂×#3485	D₄	1	4	yes	complemented
46	C₂×#3485	D₄	1	4	yes	complemented
47	C₂×#3484	D₄	1	4	yes	complemented
48	C₂×#3484	C₂³	1	4	yes	—
49	C₂×#3484	C₂³	1	4	yes	—
50	C₂×#3484	C₂³	1	4	yes	—
51	C₂×#3449	D₄	1	2	yes	complemented
52	C₂²×#535	C₂³	1	2	yes	—
53	C₂²×#534	D₄	1	4	yes	complemented
54	C₂²×#534	C₂³	1	4	yes	—
55	C₂³×#120	C₂³	1	2	yes	complemented
56	C₂³×#120	C₂³	1	1	yes	characteristic, complemented
57	C₂×#3428	D₄	1	1	yes	characteristic, complemented
58	C₂²×#518	D₄	1	2	yes	complemented
59	C₂×#2846	C₂³	1	8	yes	complemented
60	C₂×#2846	C₂³	1	4	yes	complemented
61	C₂×#2308	C₂³	1	1	yes	characteristic, complemented
62	C₂×#2308	C₂³	1	2	yes	complemented
63	#21746	C₂³	1	32	yes	complemented
64	#21746	C₂³	1	16	yes	complemented
65	C₂×#1898	C₂³	1	8	yes	complemented
66	C₂×#1898	C₂³	1	8	yes	complemented
67	C₂×#1898	C₂³	1	8	yes	complemented
68	C₂×#1781	C₂³	1	4	yes	complemented
69	C₂×#1781	C₂³	1	4	yes	complemented
70	C₂×#1781	C₂³	1	4	yes	complemented
71	#14335	C₂³	1	4	yes	complemented
72	#14335	C₂³	1	2	yes	complemented
73	#14255	C₂³	1	8	yes	complemented
74	#14255	C₂³	1	4	yes	complemented
75	C₂⁷	C₂⁴	1	1	yes	characteristic
76	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
77	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
78	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
79	C₂³×#41	C₂×D₄	1	2	yes	—
80	C₂³×#41	C₂×D₄	1	2	yes	—
81	C₂³×#41	C₂×D₄	1	1	yes	characteristic
82	C₂³×#41	C₂×D₄	1	1	yes	characteristic
83	C₂³×#41	C₂×D₄	1	2	yes	—
84	C₂⁴×D₄	C₂×D₄	1	2	yes	—
85	C₂⁴×D₄	C₂×D₄	1	2	yes	—
86	C₂⁴×D₄	C₂×D₄	1	2	yes	—
87	C₂⁴×D₄	C₂×D₄	1	4	yes	—
88	C₂⁴×D₄	C₂⁴	1	2	yes	—
89	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
90	#3490	C₂×D₄	1	64	yes	—
91	#3490	C₂×D₄	1	32	yes	—
92	#3484	C₂×D₄	1	16	yes	—
93	#3484	C₂×D₄	1	16	yes	—
94	#3484	C₂×D₄	1	16	yes	—
95	#3484	C₂×D₄	1	16	yes	—
96	#3484	C₂×D₄	1	16	yes	—
97	#3484	C₂×D₄	1	16	yes	—
98	#3484	C₂×D₄	1	64	yes	—
99	#3484	C₂×D₄	1	32	yes	—
100	C₂×#534	C₂×D₄	1	8	yes	—
101	C₂×#534	C₂×D₄	1	8	yes	—
102	C₂×#534	C₂×D₄	1	8	yes	—
103	C₂²×#127	C₂×D₄	1	4	yes	—
104	C₂²×#124	C₂×D₄	1	4	yes	—
105	C₂²×#123	C₂×D₄	1	8	yes	—
106	C₂²×#123	C₂×D₄	1	8	yes	—
107	C₂²×#121	C₂×D₄	1	16	yes	—
108	C₂²×#121	C₂×D₄	1	8	yes	—
109	C₂²×#120	C₂×D₄	1	8	yes	—
110	C₂²×#120	C₂×D₄	1	4	yes	—
111	C₂²×#120	C₂⁴	1	8	yes	—
112	C₂²×#120	C₂⁴	1	4	yes	—
113	C₂×#518	C₂×D₄	1	4	yes	—
114	C₂×#518	C₂×D₄	1	4	yes	—
115	C₂×#518	C₂×D₄	1	4	yes	—
116	C₂²×#117	C₂×D₄	1	2	yes	—
117	C₂³×C₂²⋊C₄	C₂×D₄	1	4	yes	—
118	C₂³×C₂²⋊C₄	C₂⁴	1	2	yes	—
119	C₂³×C₂²⋊C₄	C₂⁴	1	1	yes	characteristic
120	C₂×#379	C₂⁴	1	8	yes	—
121	C₂×#379	C₂⁴	1	4	yes	—
122	C₂⁶	#120	1	2	yes	—
123	C₂⁶	#120	1	4	yes	complemented
124	C₂⁶	C₂²×D₄	1	2	yes	—
125	C₂⁶	C₂²×D₄	1	1	yes	characteristic
126	C₂⁶	C₂⁵	1	1	yes	characteristic
127	C₂×H(5,𝔽₂)	#120	1	16	yes	—
128	C₂×H(5,𝔽₂)	#120	1	16	yes	complemented
129	C₂×H(5,𝔽₂)	#120	1	16	yes	complemented
130	C₂×H(5,𝔽₂)	#120	1	32	yes	complemented
131	C₂³×Q₈	#120	1	1	yes	characteristic
132	C₂³×Q₈	#120	1	1	yes	characteristic
133	C₂³×Q₈	#120	1	2	yes	complemented
134	C₂³×D₄	#120	1	1	yes	characteristic
135	C₂³×D₄	#120	1	2	yes	—
136	C₂³×D₄	#120	1	1	yes	characteristic
137	C₂³×D₄	#120	1	2	yes	—
138	C₂³×D₄	#120	1	8	yes	complemented
139	C₂³×D₄	#120	1	2	yes	complemented
140	C₂³×D₄	#120	1	8	yes	complemented
141	C₂³×D₄	C₂²×D₄	1	2	yes	—
142	C₂³×D₄	C₂²×D₄	1	2	yes	—
143	C₂³×D₄	C₂²×D₄	1	2	yes	—
144	C₂³×D₄	C₂²×D₄	1	2	yes	—
145	C₂³×D₄	C₂²×D₄	1	2	yes	—
146	C₂³×D₄	C₂²×D₄	1	2	yes	—
147	C₂³×D₄	C₂²×D₄	1	8	yes	—
148	C₂³×D₄	C₂²×D₄	1	8	yes	—
149	C₂³×D₄	C₂²×D₄	1	8	yes	—
150	C₂³×D₄	C₂²×D₄	1	4	yes	—
151	C₂³×D₄	C₂²×D₄	1	4	yes	—
152	C₂³×D₄	C₂²×D₄	1	4	yes	—
153	C₂³×D₄	C₂²×D₄	1	16	yes	—
154	C₂³×D₄	C₂²×D₄	1	8	yes	—
155	C₂⁴×C₄	#120	1	2	yes	—
156	C₂⁴×C₄	#120	1	2	yes	—
157	C₂⁴×C₄	#120	1	2	yes	—
158	C₂⁴×C₄	#120	1	4	yes	complemented
159	C₂⁴×C₄	#120	1	4	yes	complemented
160	C₂⁴×C₄	#120	1	8	yes	complemented
161	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
162	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
163	#543	#120	1	32	yes	complemented
164	#534	#120	1	16	yes	—
165	#534	#120	1	16	yes	—
166	#534	#120	1	16	yes	complemented
167	#534	#120	1	16	yes	complemented
168	#534	#120	1	16	yes	complemented
169	#534	#120	1	16	yes	complemented
170	#534	#120	1	32	yes	complemented
171	#534	#120	1	64	yes	complemented
172	C₂×#120	C₂²×D₄	1	32	yes	—
173	C₂×#120	C₂²×D₄	1	32	yes	—
174	C₂×#120	C₂²×D₄	1	32	yes	—
175	#518	#120	1	16	yes	—
176	#518	#120	1	16	yes	complemented
177	#518	#120	1	16	yes	complemented
178	#518	#120	1	64	yes	complemented
179	#518	#120	1	32	yes	complemented
180	C₂²×C₄⋊C₄	#120	1	4	yes	complemented
181	C₂²×C₄⋊C₄	#120	1	16	yes	complemented
182	C₂²×C₂²⋊C₄	#120	1	8	yes	complemented
183	C₂²×C₂²⋊C₄	#120	1	16	yes	complemented
184	C₂²×C₂²⋊C₄	#120	1	8	yes	complemented
185	C₂²×C₂²⋊C₄	C₂²×D₄	1	8	yes	—
186	C₂²×C₂²⋊C₄	C₂²×D₄	1	4	yes	—
187	C₂²×C₂²⋊C₄	C₂²×D₄	1	16	yes	—
188	C₂²×C₂²⋊C₄	C₂²×D₄	1	8	yes	—
189	C₂²×C₄²	#120	1	4	yes	complemented
190	C₂⁵	C₂×#120	1	4	yes	—
191	C₂⁵	C₂×#120	1	4	yes	—
192	C₂⁵	C₂×#120	1	8	yes	—
193	C₂⁵	C₂×#120	1	2	yes	—
194	C₂⁵	C₂×#120	1	2	yes	—
195	C₂⁵	C₂×#120	1	4	yes	—
196	C₂⁵	C₂×#120	1	4	yes	—
197	C₂⁵	C₂×#120	1	2	yes	—
198	C₂⁵	C₂³×D₄	1	2	yes	—
199	C₂⁵	C₂³×D₄	1	1	yes	characteristic
200	C₂⁵	C₂³×D₄	1	8	yes	—
201	C₂⁵	C₂³×D₄	1	4	yes	—
202	C₂²×D₄	B(4,𝔽₂)	1	4	yes	complemented
203	C₂²×D₄	C₂×#120	1	8	yes	—
204	C₂²×D₄	C₂×#120	1	8	yes	—
205	C₂²×D₄	C₂×#120	1	8	yes	—
206	C₂²×D₄	C₂×#120	1	64	yes	—
207	C₂²×D₄	C₂×#120	1	32	yes	—
208	C₂²×D₄	C₂×#120	1	16	yes	—
209	C₂²×D₄	C₂×#120	1	16	yes	—
210	C₂²×D₄	C₂×#120	1	16	yes	—
211	C₂²×D₄	C₂×#120	1	4	yes	—
212	C₂²×D₄	C₂×#120	1	4	yes	—
213	C₂²×D₄	C₂×#120	1	4	yes	—
214	C₂²×D₄	C₂×#120	1	4	yes	—
215	C₂²×D₄	C₂×#120	1	4	yes	—
216	C₂²×D₄	C₂×#120	1	4	yes	—
217	C₂³×C₄	C₂×#120	1	1	yes	characteristic
218	C₂³×C₄	C₂×#120	1	16	yes	—
219	C₂³×C₄	C₂×#120	1	4	yes	—
220	C₂³×C₄	C₂×#120	1	8	yes	—
221	C₂³×C₄	C₂×#120	1	8	yes	—
222	C₂³×C₄	C₂×#120	1	8	yes	—
223	C₂³×C₄	C₂×#120	1	2	yes	—
224	C₂³×C₄	C₂×#120	1	1	yes	characteristic
225	C₂³×C₄	C₂×#120	1	2	yes	—
226	C₂³×C₄	C₂×#120	1	2	yes	—
227	C₂³×C₄	C₂×#120	1	2	yes	—
228	C₂×C₂²⋊C₄	C₂×#120	1	16	yes	—
229	C₂×C₂²⋊C₄	C₂×#120	1	16	yes	—
230	C₂×C₂²⋊C₄	C₂×#120	1	16	yes	—
231	C₂×C₂²⋊C₄	C₂×#120	1	64	yes	—
232	C₂×C₂²⋊C₄	C₂×#120	1	32	yes	—
233	C₂⁴	#2846	1	16	yes	—
234	C₂⁴	#2846	1	8	yes	—
235	C₂⁴	#2846	1	4	yes	—
236	C₂⁴	#2846	1	2	yes	—
237	C₂⁴	#2846	1	2	yes	—
238	C₂⁴	#2846	1	4	yes	—
239	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
240	C₂⁴	#3026	1	4	yes	—
241	C₂⁴	C₂²×#120	1	16	yes	—
242	C₂⁴	C₂²×#120	1	4	yes	—
243	C₂⁴	C₂²×#120	1	4	yes	—
244	C₂⁴	C₂²×#120	1	4	yes	—
245	C₂⁴	C₂²×#120	1	4	yes	—
246	C₂⁴	C₂²×#120	1	2	yes	—
247	C₂⁴	C₂²×#120	1	1	yes	characteristic
248	C₂²×C₄	#2846	1	2	yes	—
249	C₂²×C₄	#2846	1	32	yes	—
250	C₂²×C₄	#2846	1	16	yes	—
251	C₂²×C₄	#2846	1	8	yes	—
252	C₂²×C₄	#2846	1	16	yes	—
253	C₂²×C₄	#2846	1	4	yes	—
254	C₂²×C₄	#2846	1	4	yes	—
255	C₂²×C₄	#2846	1	1	yes	characteristic
256	C₂²×C₄	#2846	1	1	yes	characteristic
257	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
258	C₂²×C₄	#3026	1	2	yes	—
259	C₂³	#32898	1	8	yes	—
260	C₂³	C₂×#2846	1	1	yes	characteristic
261	C₂³	C₂×#2846	1	2	yes	—
262	C₂³	C₂×#2846	1	8	yes	—
263	C₂³	C₂×#2846	1	4	yes	—
264	C₂³	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
265	C₂³	C₂×#3026	1	1	yes	characteristic
266	C₂³	C₂×#3026	1	2	yes	—
267	C₂×C₄	#32898	1	4	yes	—
268	C₂²	#6342371	1	1	yes	characteristic
269	C₂²	C₂×#32898	1	1	yes	characteristic
270	C₂²	C₂×#32898	1	2	yes	—
271	C₂²	#6628892	1	2	yes	—
272	C₂²	#6628892	1	1	yes	characteristic
273	C₂	#72210025	1	2	yes	—
274	C₂	#72219118	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#6342371
commutator, Frattini, cosocle	C₂⁶	C₂⁵
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72251779	C₂	semidirect
2. #72251662	C₂	semidirect
3. #72249344	C₂	semidirect
4. #72237402	C₂	semidirect
5. #72236667	C₂	semidirect
6. #72235216	C₂	semidirect
7. #72226123	C₂	semidirect
8. #72225712	C₂	semidirect
9. #72225690	C₂	semidirect
14. C₂×#35346	C₂²	semidirect
15. #6499867	C₂²	semidirect
16. C₂×#23900	C₂²	semidirect
17. C₂×#23900	C₂²	semidirect
18. C₂×#23900	C₂²	semidirect
19. #616961	C₂²	semidirect
20. #616960	C₂²	semidirect
21. #392111	C₂²	semidirect
22. #392111	C₂²	semidirect
23. #359554	C₂²	semidirect
24. #359554	C₂²	semidirect
25. #359554	C₂²	semidirect
26. #359434	C₂²	semidirect
27. #359433	C₂²	semidirect
28. #188050	C₂²	semidirect
29. #188049	C₂²	semidirect
30. #187944	C₂²	semidirect
31. #187944	C₂²	semidirect
32. #187944	C₂²	semidirect
33. #187934	C₂²	semidirect
34. #187934	C₂²	semidirect
35. C₂²×#585	D₄	semidirect
36. C₂³×H(5,𝔽₂)	D₄	semidirect
38. #62832	D₄	semidirect
39. #62824	D₄	semidirect
40. #62820	D₄	semidirect
41. #62818	D₄	semidirect
42. #62818	D₄	semidirect
43. #62817	D₄	semidirect
45. C₂×#3485	D₄	semidirect
46. C₂×#3485	D₄	semidirect
47. C₂×#3484	D₄	semidirect
51. C₂×#3449	D₄	semidirect
53. C₂²×#534	D₄	semidirect
55. C₂³×#120	C₂³	semidirect
56. C₂³×#120	C₂³	semidirect
57. C₂×#3428	D₄	semidirect
58. C₂²×#518	D₄	semidirect
59. C₂×#2846	C₂³	semidirect
60. C₂×#2846	C₂³	semidirect
61. C₂×#2308	C₂³	semidirect
62. C₂×#2308	C₂³	semidirect
63. #21746	C₂³	semidirect
64. #21746	C₂³	semidirect
65. C₂×#1898	C₂³	semidirect
66. C₂×#1898	C₂³	semidirect
67. C₂×#1898	C₂³	semidirect
68. C₂×#1781	C₂³	semidirect
69. C₂×#1781	C₂³	semidirect
70. C₂×#1781	C₂³	semidirect
71. #14335	C₂³	semidirect
72. #14335	C₂³	semidirect
73. #14255	C₂³	semidirect
74. #14255	C₂³	semidirect
123. C₂⁶	#120	semidirect
128. C₂×H(5,𝔽₂)	#120	semidirect
129. C₂×H(5,𝔽₂)	#120	semidirect
130. C₂×H(5,𝔽₂)	#120	semidirect
133. C₂³×Q₈	#120	semidirect
138. C₂³×D₄	#120	semidirect
139. C₂³×D₄	#120	semidirect
140. C₂³×D₄	#120	semidirect
158. C₂⁴×C₄	#120	semidirect
159. C₂⁴×C₄	#120	semidirect
160. C₂⁴×C₄	#120	semidirect
163. #543	#120	semidirect
166. #534	#120	semidirect
167. #534	#120	semidirect
168. #534	#120	semidirect
169. #534	#120	semidirect
170. #534	#120	semidirect
171. #534	#120	semidirect
176. #518	#120	semidirect
177. #518	#120	semidirect
178. #518	#120	semidirect
179. #518	#120	semidirect
180. C₂²×C₄⋊C₄	#120	semidirect
181. C₂²×C₄⋊C₄	#120	semidirect
182. C₂²×C₂²⋊C₄	#120	semidirect
183. C₂²×C₂²⋊C₄	#120	semidirect
184. C₂²×C₂²⋊C₄	#120	semidirect
189. C₂²×C₄²	#120	semidirect
202. C₂²×D₄	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72251779	2
#72251662	2
#72249344	2
#72237402	2
#72236667	2
#72235216	2
#72226123	2
#72225712	2
#72225690	2