Subgroups — #62844

Order 256.

3737 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3505	C₂	1	24	yes	maximal, complemented
2	#3491	C₂	1	12	yes	maximal, complemented
3	#3490	C₂	1	12	yes	maximal, complemented
4	#3459	C₂	1	8	yes	maximal, complemented
5	C₂×#543	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#534	C₂	1	3	yes	maximal, complemented
7	C₂×#518	C₂	1	3	yes	maximal, complemented
8	C₂³×D₄	C₂²	1	3	yes	—
9	#553	C₂²	1	48	yes	complemented
10	#552	C₂²	1	48	yes	complemented
11	#551	C₂²	1	96	yes	complemented
12	#548	C₂²	1	16	yes	complemented
13	#546	C₂²	1	48	yes	complemented
14	#543	C₂²	1	4	yes	complemented
15	#543	C₂²	1	12	yes	complemented
16	#539	C₂²	1	48	yes	complemented
17	#538	C₂²	1	48	yes	complemented
18	#534	C₂²	1	12	yes	—
19	#534	C₂²	1	24	yes	complemented
20	#534	C₂²	1	12	yes	complemented
21	#529	C₂²	1	48	yes	complemented
22	#525	C₂²	1	48	yes	complemented
23	C₂×#123	C₂²	1	12	yes	complemented
24	C₂×C₂²⋊Q₈	C₂²	1	12	yes	complemented
25	C₂×#121	C₂²	1	12	yes	complemented
26	C₂×#120	C₂²	1	12	yes	complemented
27	#518	C₂²	1	24	yes	complemented
28	#518	C₂²	1	24	yes	complemented
29	C₂×C₄×D₄	C₂²	1	12	yes	complemented
30	C₂×C₄×D₄	C₂²	1	12	yes	complemented
31	C₂×#117	C₂²	1	6	yes	complemented
32	C₂²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
33	C₂²×C₂²⋊C₄	C₂²	1	3	yes	—
34	C₂²×C₂²⋊C₄	C₂²	1	6	yes	complemented
35	C₂⁵	C₂³	1	3	yes	—
36	C₂⁵	C₂³	1	1	yes	characteristic
37	C₂×#41	C₂³	1	16	yes	complemented
38	C₂²×Q₈	C₂³	1	4	yes	complemented
39	C₂²×D₄	—	2	24	no	—
40	C₂²×D₄	C₂³	1	6	yes	—
41	C₂²×D₄	C₂³	1	6	yes	—
42	C₂²×D₄	C₂³	1	12	yes	—
43	C₂²×D₄	C₂³	1	12	yes	—
44	C₂²×D₄	C₂³	1	12	yes	complemented
45	C₂³×C₄	C₂³	1	3	yes	—
46	C₂³×C₄	C₂³	1	2	yes	—
47	C₂³×C₄	C₂³	1	6	yes	—
48	C₂³×C₄	C₂³	1	12	yes	complemented
49	#126	C₂³	1	96	yes	complemented
50	#125	C₂³	1	48	yes	complemented
51	#124	C₂³	1	48	yes	complemented
52	#123	C₂³	1	48	yes	—
53	#123	C₂³	1	48	yes	complemented
54	#123	C₂³	1	96	yes	complemented
55	C₂²⋊Q₈	C₂³	1	48	yes	complemented
56	C₂²⋊Q₈	C₂³	1	48	yes	complemented
57	C₂²⋊Q₈	C₂³	1	48	yes	complemented
58	#121	C₂³	1	48	yes	—
59	#121	C₂³	1	48	yes	complemented
60	#121	C₂³	1	48	yes	complemented
61	#120	C₂³	1	48	yes	—
62	#120	C₂³	1	48	yes	complemented
63	C₄×Q₈	C₂³	1	16	yes	complemented
64	C₄×D₄	C₂³	1	48	yes	—
65	C₄×D₄	C₂³	1	96	yes	complemented
66	C₄×D₄	C₂³	1	48	yes	complemented
67	C₄×D₄	C₂³	1	48	yes	complemented
68	#117	C₂³	1	48	yes	complemented
69	#117	C₂³	1	48	yes	complemented
70	C₂×C₄⋊C₄	C₂³	1	12	yes	complemented
71	C₂×C₄⋊C₄	C₂³	1	12	yes	complemented
72	C₂×C₄⋊C₄	C₂³	1	12	yes	complemented
73	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
74	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
75	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
76	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
77	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
78	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
79	C₂×C₂²⋊C₄	C₂³	1	12	yes	complemented
80	C₂×C₂²⋊C₄	C₂³	1	48	yes	complemented
81	C₂×C₂²⋊C₄	C₂³	1	24	yes	complemented
82	C₂×C₄²	C₂³	1	12	yes	complemented
83	C₂⁴	—	2	12	no	—
84	C₂⁴	—	8	2	no	—
85	C₂⁴	—	8	6	no	—
86	C₂⁴	#41	1	8	yes	complemented
87	C₂⁴	C₂⁴	1	12	yes	—
88	C₂⁴	C₂⁴	1	6	yes	—
89	C₂⁴	C₂⁴	1	3	yes	—
90	C₂⁴	C₂⁴	1	3	yes	—
91	C₂⁴	C₂⁴	1	1	yes	characteristic
92	#41	—	8	16	no	—
93	C₂×Q₈	—	8	4	no	—
94	C₂×Q₈	C₂⁴	1	16	yes	—
95	C₂×D₄	—	2	24	no	—
96	C₂×D₄	—	2	24	no	—
97	C₂×D₄	—	2	48	no	—
98	C₂×D₄	—	2	48	no	—
99	C₂×D₄	—	8	12	no	—
100	C₂×D₄	C₂⁴	1	12	yes	—
101	C₂×D₄	C₂⁴	1	24	yes	—
102	C₂×D₄	C₂⁴	1	12	yes	—
103	C₂×D₄	C₂⁴	1	48	yes	—
104	C₂²×C₄	—	2	12	no	—
105	C₂²×C₄	—	8	2	no	—
106	C₂²×C₄	—	8	6	no	—
107	C₂²×C₄	—	8	12	no	—
108	C₂²×C₄	C₂⁴	1	6	yes	—
109	C₂²×C₄	C₂⁴	1	12	yes	—
110	C₂²×C₄	C₂⁴	1	12	yes	—
111	C₂²×C₄	C₂⁴	1	12	yes	—
112	C₂²×C₄	C₂⁴	1	12	yes	—
113	C₂²×C₄	C₂⁴	1	12	yes	—
114	C₂²×C₄	C₂⁴	1	48	yes	—
115	C₄⋊C₄	C₂⁴	1	48	yes	—
116	C₄⋊C₄	C₂⁴	1	48	yes	—
117	C₄⋊C₄	C₂⁴	1	48	yes	—
118	C₂²⋊C₄	C₂⁴	1	12	yes	—
119	C₂²⋊C₄	C₂⁴	1	24	yes	—
120	C₂²⋊C₄	C₂⁴	1	4	yes	—
121	C₂²⋊C₄	C₂⁴	1	24	yes	—
122	C₂²⋊C₄	C₂⁴	1	48	yes	—
123	C₂²⋊C₄	C₂⁴	1	96	yes	—
124	C₄²	C₂⁴	1	48	yes	—
125	C₂³	—	2	12	no	—
126	C₂³	—	2	24	no	—
127	C₂³	—	4	6	no	—
128	C₂³	—	4	2	no	—
129	C₂³	—	4	12	no	—
130	C₂³	—	4	6	no	—
131	C₂³	—	4	24	no	—
132	C₂³	—	8	8	no	—
133	C₂³	—	8	24	no	—
134	C₂³	C₂×#41	1	12	yes	—
135	C₂³	C₂×#41	1	12	yes	—
136	C₂³	C₂×#41	1	4	yes	—
137	C₂³	C₂⁵	1	12	yes	—
138	C₂³	C₂⁵	1	3	yes	—
139	C₂³	C₂⁵	1	3	yes	—
140	C₂³	C₂⁵	1	1	yes	characteristic
141	Q₈	—	4	16	no	—
142	D₄	—	2	24	no	—
143	D₄	—	2	48	no	—
144	D₄	—	2	24	no	—
145	D₄	—	4	48	no	—
146	C₂×C₄	—	2	24	no	—
147	C₂×C₄	—	2	12	no	—
148	C₂×C₄	—	4	12	no	—
149	C₂×C₄	—	4	12	no	—
150	C₂×C₄	—	4	12	no	—
151	C₂×C₄	—	4	12	no	—
152	C₂×C₄	—	4	48	no	—
153	C₂×C₄	C₂⁵	1	8	yes	—
154	C₂×C₄	C₂⁵	1	12	yes	—
155	C₂×C₄	C₂⁵	1	24	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	6	no	—
158	C₂²	—	2	6	no	—
159	C₂²	—	2	12	no	—
160	C₂²	—	2	24	no	—
161	C₂²	—	4	12	no	—
162	C₂²	—	4	4	no	—
163	C₂²	—	4	12	no	—
164	C₂²	—	4	24	no	—
165	C₂²	—	8	24	no	—
166	C₂²	C₂²×#41	1	6	yes	—
167	C₂²	C₂²×#41	1	6	yes	—
168	C₂²	C₂²×#41	1	2	yes	—
169	C₂²	C₂⁶	1	1	yes	characteristic
170	C₄	—	2	12	no	—
171	C₄	—	2	8	no	—
172	C₄	—	2	24	no	—
173	C₂	—	2	6	no	—
174	C₂	—	2	6	no	—
175	C₂	—	2	2	no	—
176	C₂	—	4	12	no	—
177	C₂	C₂³×#41	1	1	yes	characteristic
178	C₂	H(7,𝔽₂)	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. #3505	C₂	semidirect
2. #3491	C₂	semidirect
3. #3490	C₂	semidirect
4. #3459	C₂	semidirect
5. C₂×#543	C₂	semidirect
6. C₂×#534	C₂	semidirect
7. C₂×#518	C₂	semidirect
9. #553	C₂²	semidirect
10. #552	C₂²	semidirect
11. #551	C₂²	semidirect
12. #548	C₂²	semidirect
13. #546	C₂²	semidirect
14. #543	C₂²	semidirect
15. #543	C₂²	semidirect
16. #539	C₂²	semidirect
17. #538	C₂²	semidirect
19. #534	C₂²	semidirect
20. #534	C₂²	semidirect
21. #529	C₂²	semidirect
22. #525	C₂²	semidirect
23. C₂×#123	C₂²	semidirect
24. C₂×C₂²⋊Q₈	C₂²	semidirect
25. C₂×#121	C₂²	semidirect
26. C₂×#120	C₂²	semidirect
27. #518	C₂²	semidirect
28. #518	C₂²	semidirect
29. C₂×C₄×D₄	C₂²	semidirect
30. C₂×C₄×D₄	C₂²	semidirect
31. C₂×#117	C₂²	semidirect
34. C₂²×C₂²⋊C₄	C₂²	semidirect
37. C₂×#41	C₂³	semidirect
38. C₂²×Q₈	C₂³	semidirect
44. C₂²×D₄	C₂³	semidirect
48. C₂³×C₄	C₂³	semidirect
49. #126	C₂³	semidirect
50. #125	C₂³	semidirect
51. #124	C₂³	semidirect
53. #123	C₂³	semidirect
54. #123	C₂³	semidirect
55. C₂²⋊Q₈	C₂³	semidirect
56. C₂²⋊Q₈	C₂³	semidirect
57. C₂²⋊Q₈	C₂³	semidirect
59. #121	C₂³	semidirect
60. #121	C₂³	semidirect
62. #120	C₂³	semidirect
63. C₄×Q₈	C₂³	semidirect
65. C₄×D₄	C₂³	semidirect
66. C₄×D₄	C₂³	semidirect
67. C₄×D₄	C₂³	semidirect
68. #117	C₂³	semidirect
69. #117	C₂³	semidirect
70. C₂×C₄⋊C₄	C₂³	semidirect
71. C₂×C₄⋊C₄	C₂³	semidirect
72. C₂×C₄⋊C₄	C₂³	semidirect
79. C₂×C₂²⋊C₄	C₂³	semidirect
80. C₂×C₂²⋊C₄	C₂³	semidirect
81. C₂×C₂²⋊C₄	C₂³	semidirect
82. C₂×C₄²	C₂³	semidirect
86. C₂⁴	#41	semidirect

Maximal subgroups

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

subgroup	index
#3505	2
#3491	2
#3490	2
#3459	2
C₂×#543	2
C₂×#534	2
C₂×#518	2