Subgroups — #7119

Order 256.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1945	C₂	1	1	yes	maximal, characteristic
2	#1376	C₂	1	1	yes	maximal, characteristic
3	#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	C₂×#108	C₄	1	1	yes	characteristic
6	#427	C₄	1	1	yes	characteristic
7	#424	—	2	1	no	—
8	C₂×C₄×C₈	—	2	1	no	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	#339	—	2	1	no	—
11	C₂×M₁₆	—	2	1	no	—
12	C₂²×C₈	—	2	2	no	—
13	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈	D₄	1	1	yes	characteristic
15	#117	—	2	1	no	—
16	C₂×C₄²	D₄	1	1	yes	characteristic
17	M₃₂	—	2	2	no	—
18	M₃₂	—	4	1	no	—
19	C₂×C₁₆	—	4	1	no	—
20	#108	—	4	1	no	—
21	#107	C₈	1	1	yes	characteristic, complemented
22	#107	C₈	1	1	yes	characteristic, complemented
23	#106	—	2	1	no	—
24	C₄⋊C₈	—	4	1	no	—
25	C₄⋊C₈	—	4	1	no	—
26	C₄×C₈	—	4	1	no	—
27	C₄×C₈	—	4	1	no	—
28	C₄×C₈	—	4	1	no	—
29	C₂²×C₄	—	2	1	no	—
30	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	M₁₆	—	4	1	no	—
32	M₁₆	—	4	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	—	4	2	no	—
39	C₂×C₈	—	4	2	no	—
40	C₂×C₈	—	4	1	no	—
41	C₂×C₈	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈	M₁₆	1	1	yes	characteristic
43	C₂×C₈	D₈	1	1	yes	characteristic
44	C₂×C₈	SD₁₆	1	1	yes	characteristic
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₂²⋊C₄	—	4	1	no	—
48	C₄²	—	2	1	no	—
49	C₄²	—	4	1	no	—
50	C₄²	SD₁₆	1	1	yes	characteristic
51	C₄²	Q₁₆	1	1	yes	characteristic
52	C₁₆	—	4	2	no	—
53	C₂³	#104	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₄	#99	1	1	yes	characteristic
62	C₂×C₄	#101	1	1	yes	characteristic
63	C₂×C₄	#102	1	1	yes	characteristic
64	C₂×C₄	#103	1	1	yes	characteristic
65	C₂×C₄	#104	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₈	—	2	1	no	—
70	C₈	—	4	4	no	—
71	C₈	—	8	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	D₄⋊C₈	1	1	yes	characteristic
75	C₂²	#328	1	1	yes	characteristic
76	C₂²	#330	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	2	no	—
79	C₄	—	8	1	no	—
80	C₄	—	8	1	no	—
81	C₄	#324	1	1	yes	characteristic
82	C₄	Q₈⋊C₈	1	1	yes	characteristic
83	C₄	#327	1	1	yes	characteristic
84	C₄	#332	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	—	2	1	no	—
87	C₂	#1271	1	1	yes	characteristic
88	C₂	#1335	1	1	yes	characteristic
89	C₂	#1336	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#102
commutator	C₂×C₈	C₂×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	D₄⋊C₈
prime radical	C₂³	#104
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
21. #107	C₈	semidirect
22. #107	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1945	2
#1376	2
#1273	2