Subgroups — #7271

Order 256.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	1	yes	maximal, characteristic
2	C₂×#364	C₂	1	1	yes	maximal, characteristic
3	#1846	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	C₂×M₃₂	—	2	1	no	—
6	C₂×#107	—	2	1	no	—
7	C₂×#106	—	2	1	no	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	#382	C₄	1	2	yes	—
10	#364	—	2	1	no	—
11	#364	—	2	1	no	—
12	#364	C₄	1	2	yes	complemented
13	#364	C₄	1	2	yes	complemented
14	C₂²×C₈	D₄	1	1	yes	characteristic
15	C₂²×C₈	D₄	1	1	yes	characteristic
16	C₂×C₄⋊C₄	—	2	2	no	—
17	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
18	M₃₂	—	2	2	no	—
19	M₃₂	—	2	2	no	—
20	M₃₂	—	4	1	no	—
21	M₃₂	—	4	1	no	—
22	C₂×C₁₆	—	4	1	no	—
23	C₂×C₁₆	—	4	1	no	—
24	#107	—	4	1	no	—
25	#107	—	4	1	no	—
26	#106	—	4	1	no	—
27	#106	—	4	1	no	—
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈	D₄	1	1	yes	characteristic
31	C₄×C₈	Q₈	1	1	yes	characteristic
32	#95	—	4	2	no	—
33	C₂²×C₄	—	2	1	no	—
34	C₂²×C₄	—	4	2	no	—
35	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈	D₈	1	1	yes	characteristic
41	C₂×C₈	D₈	1	1	yes	characteristic
42	C₂×C₈	SD₁₆	1	1	yes	characteristic
43	C₂×C₈	SD₁₆	1	1	yes	characteristic
44	C₂×C₈	SD₁₆	1	1	yes	characteristic
45	C₂×C₈	SD₁₆	1	1	yes	characteristic
46	C₂×C₈	Q₁₆	1	1	yes	characteristic
47	C₂×C₈	Q₁₆	1	1	yes	characteristic
48	C₄⋊C₄	—	4	2	no	—
49	C₄⋊C₄	—	4	2	no	—
50	C₄²	—	2	1	no	—
51	C₄²	C₄²	1	1	yes	characteristic
52	C₄²	C₄⋊C₄	1	1	yes	characteristic
53	C₁₆	—	4	2	no	—
54	C₁₆	—	4	2	no	—
55	C₂³	#100	1	1	yes	characteristic
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	—	8	4	no	—
60	C₂×C₄	—	8	2	no	—
61	C₂×C₄	#95	1	1	yes	characteristic
62	C₂×C₄	#101	1	1	yes	characteristic
63	C₂×C₄	#102	1	1	yes	characteristic
64	C₂×C₄	#102	1	1	yes	characteristic
65	C₂×C₄	#103	1	1	yes	characteristic
66	C₂×C₄	#103	1	1	yes	characteristic
67	C₈	#102	1	1	yes	characteristic
68	C₈	#102	1	1	yes	characteristic
69	C₈	#103	1	1	yes	characteristic
70	C₈	#103	1	1	yes	characteristic
71	C₈	#106	1	1	yes	characteristic
72	C₈	#106	1	1	yes	characteristic
73	C₈	#107	1	1	yes	characteristic
74	C₈	#107	1	1	yes	characteristic
75	C₂²	—	2	2	no	—
76	C₂²	#331	1	1	yes	characteristic
77	C₂²	#333	1	1	yes	characteristic
78	C₂²	#343	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	16	2	no	—
82	C₄	#332	1	1	yes	characteristic
83	C₄	#332	1	1	yes	characteristic
84	C₄	#340	1	1	yes	characteristic
85	C₄	#340	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#1296	1	1	yes	characteristic
89	C₂	#1366	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#333
commutator	C₄²	C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂²
prime radical	C₂³	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #364	C₄	semidirect
13. #364	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#364	2
#1846	2