Finite Group Databasebeta

Subgroups — #62637

Order 256.

1013 conjugacy classes of subgroups; only the 477 normal subgroups are shown.

normal subgroups only

This group has 1013 conjugacy classes of subgroups — too many to draw. Only its 477 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#578	C₂	1	1	yes	maximal, characteristic, complemented
3	#3417	C₂	1	8	yes	maximal, complemented
4	#3416	C₂	1	8	yes	maximal, complemented
5	#3414	C₂	1	6	yes	maximal, complemented
6	C₂×#510	C₂	1	6	yes	maximal, complemented
7	#3407	C₂	1	1	yes	maximal, characteristic, complemented
8	#585	—	2	1	no	—
9	C₂×2₋¹⁺⁴	—	2	1	no	—
10	#578	—	2	4	no	—
11	#578	C₂²	1	4	yes	complemented
12	#578	C₂²	1	8	yes	complemented
13	#577	—	2	4	no	—
14	#576	C₂²	1	4	yes	complemented
15	#575	—	2	3	no	—
16	C₂×#137	—	2	3	no	—
17	C₂×#135	C₂²	1	6	yes	—
18	C₂×#135	C₂²	1	3	yes	complemented
19	C₂²×Q₁₆	C₂²	1	3	yes	complemented
20	C₂×#131	C₂²	1	1	yes	characteristic
21	#510	C₂²	1	12	yes	complemented
22	#510	C₂²	1	24	yes	complemented
23	#510	C₂²	1	12	yes	complemented
24	#509	C₂²	1	4	yes	complemented
25	#509	C₂²	1	12	yes	complemented
26	#508	C₂²	1	24	yes	complemented
27	C₂×Q₃₂	C₂²	1	12	yes	complemented
28	C₂×SD₃₂	C₂²	1	12	yes	complemented
29	#504	C₂²	1	8	yes	complemented
30	C₂×M₃₂	C₂²	1	6	yes	complemented
31	2₋¹⁺⁴	—	2	4	no	—
32	2₋¹⁺⁴	—	2	8	no	—
33	2₋¹⁺⁴	—	4	4	no	—
34	2₋¹⁺⁴	—	4	1	no	—
35	H(5,𝔽₂)	—	2	4	no	—
36	H(5,𝔽₂)	—	4	3	no	—
37	C₂×#41	—	2	6	no	—
38	C₂×#41	—	2	3	no	—
39	C₂×#41	—	4	4	no	—
40	C₂×#41	—	4	3	no	—
41	C₂×#41	D₄	1	1	yes	characteristic
42	C₂²×Q₈	—	2	3	no	—
43	C₂²×Q₈	—	4	1	no	—
44	#137	—	2	24	no	—
45	#137	—	2	12	no	—
46	AGL(1,ℤ/8)	—	2	12	no	—
47	#135	—	2	6	no	—
48	#135	—	2	6	no	—
49	#135	—	2	6	no	—
50	#135	C₂³	1	4	yes	—
51	#135	C₂³	1	12	yes	—
52	#135	C₂³	1	12	yes	—
53	C₂×Q₁₆	—	2	3	no	—
54	C₂×Q₁₆	—	2	6	no	—
55	C₂×Q₁₆	C₂³	1	3	yes	—
56	C₂×Q₁₆	C₂³	1	12	yes	—
57	C₂×Q₁₆	C₂³	1	6	yes	—
58	C₂×SD₁₆	—	2	6	no	—
59	C₂×SD₁₆	—	2	3	no	—
60	C₂×D₈	C₂³	1	6	yes	—
61	C₂×D₈	C₂³	1	1	yes	characteristic
62	#131	D₄	1	4	yes	—
63	#131	C₂³	1	4	yes	—
64	C₂×M₁₆	D₄	1	3	yes	—
65	C₂²×C₈	C₂³	1	3	yes	—
66	Q₃₂	C₂³	1	24	yes	—
67	SD₃₂	C₂³	1	24	yes	—
68	SD₃₂	C₂³	1	8	yes	—
69	D₁₆	C₂³	1	8	yes	—
70	M₃₂	C₂³	1	4	yes	—
71	M₃₂	C₂³	1	12	yes	—
72	C₂×C₁₆	C₂³	1	12	yes	—
73	#41	—	2	4	no	—
74	#41	—	2	12	no	—
75	#41	—	2	12	no	—
76	#41	—	4	4	no	—
77	#41	—	4	12	no	—
78	#41	—	4	24	no	—
79	#41	—	4	6	no	—
80	#41	—	4	6	no	—
81	#41	—	4	6	no	—
82	#41	D₈	1	4	yes	—
83	#41	C₂×D₄	1	4	yes	—
84	C₂×Q₈	—	2	3	no	—
85	C₂×Q₈	—	2	6	no	—
86	C₂×Q₈	—	2	12	no	—
87	C₂×Q₈	—	4	8	no	—
88	C₂×Q₈	—	4	4	no	—
89	C₂×Q₈	—	4	3	no	—
90	C₂×Q₈	—	4	6	no	—
91	C₂×Q₈	D₈	1	1	yes	characteristic
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	6	no	—
94	C₂×D₄	—	4	3	no	—
95	C₂×D₄	—	4	6	no	—
96	C₂×D₄	—	4	12	no	—
97	C₂×D₄	D₈	1	3	yes	—
98	C₂²×C₄	—	4	3	no	—
99	C₂²×C₄	—	4	3	no	—
100	C₂²×C₄	—	4	6	no	—
101	C₂²×C₄	C₂×D₄	1	3	yes	—
102	Q₁₆	—	2	3	no	—
103	Q₁₆	—	2	12	no	—
104	Q₁₆	—	2	6	no	—
105	Q₁₆	C₂⁴	1	6	yes	—
106	Q₁₆	C₂⁴	1	4	yes	—
107	SD₁₆	—	2	4	no	—
108	SD₁₆	—	2	12	no	—
109	SD₁₆	—	2	12	no	—
110	D₈	—	2	1	no	—
111	D₈	—	2	6	no	—
112	D₈	C₂⁴	1	2	yes	—
113	D₈	C₂⁴	1	4	yes	—
114	M₁₆	C₂×D₄	1	12	yes	—
115	C₂×C₈	C₂×D₄	1	3	yes	—
116	C₂×C₈	C₂×D₄	1	6	yes	—
117	C₂×C₈	C₂⁴	1	6	yes	—
118	C₂×C₈	C₂⁴	1	1	yes	characteristic
119	C₁₆	C₂⁴	1	8	yes	—
120	C₂³	—	4	1	no	—
121	C₂³	—	4	6	no	—
122	C₂³	C₂×D₈	1	3	yes	—
123	Q₈	—	2	4	no	—
124	Q₈	—	2	6	no	—
125	Q₈	—	4	3	no	—
126	Q₈	—	4	12	no	—
127	Q₈	—	4	6	no	—
128	Q₈	C₂×D₈	1	4	yes	—
129	D₄	—	2	4	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	4	no	—
132	D₄	—	4	12	no	—
133	D₄	—	4	12	no	—
134	D₄	—	4	6	no	—
135	D₄	—	4	1	no	—
136	D₄	C₂×D₈	1	12	yes	—
137	C₂×C₄	—	4	3	no	—
138	C₂×C₄	—	4	6	no	—
139	C₂×C₄	—	4	12	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	—	4	12	no	—
142	C₂×C₄	—	4	12	no	—
143	C₂×C₄	C₂×D₈	1	3	yes	—
144	C₂×C₄	C₂×D₈	1	6	yes	—
145	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄	C₂²×D₄	1	6	yes	—
147	C₈	C₂²×D₄	1	1	yes	characteristic
148	C₈	C₂²×D₄	1	6	yes	—
149	C₈	C₂⁵	1	1	yes	characteristic
150	C₂²	—	4	3	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	4	no	—
153	C₂²	—	8	2	no	—
154	C₂²	—	16	6	no	—
155	C₂²	C₂²×D₈	1	1	yes	characteristic
156	C₂²	C₂²×D₈	1	6	yes	—
157	C₄	—	4	4	no	—
158	C₄	—	4	6	no	—
159	C₄	C₂²×D₈	1	1	yes	characteristic
160	C₄	C₂²×D₈	1	6	yes	—
161	C₄	C₂³×D₄	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	6	no	—
164	C₂	—	8	2	no	—
165	C₂	—	8	4	no	—
166	C₂	C₂³×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂³×D₈
commutator, Frattini, cosocle	C₈	C₂⁵
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3584	C₂	semidirect
2. C₂×#578	C₂	semidirect
3. #3417	C₂	semidirect
4. #3416	C₂	semidirect
5. #3414	C₂	semidirect
6. C₂×#510	C₂	semidirect
7. #3407	C₂	semidirect
11. #578	C₂²	semidirect
12. #578	C₂²	semidirect
14. #576	C₂²	semidirect
18. C₂×#135	C₂²	semidirect
19. C₂²×Q₁₆	C₂²	semidirect
21. #510	C₂²	semidirect
22. #510	C₂²	semidirect
23. #510	C₂²	semidirect
24. #509	C₂²	semidirect
25. #509	C₂²	semidirect
26. #508	C₂²	semidirect
27. C₂×Q₃₂	C₂²	semidirect
28. C₂×SD₃₂	C₂²	semidirect
29. #504	C₂²	semidirect
30. C₂×M₃₂	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3584	2
C₂×#578	2
#3417	2
#3416	2
#3414	2
C₂×#510	2
#3407	2