Subgroups — #7217

Order 256.

113 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2162	C₂	1	1	yes	maximal, characteristic
2	#1834	C₂	1	1	yes	maximal, characteristic
3	#1567	C₂	1	1	yes	maximal, characteristic
4	#423	—	2	1	no	—
5	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
6	#384	—	2	1	no	—
7	C₄⋊C₁₆	—	2	1	no	—
8	C₂²⋊C₁₆	—	2	1	no	—
9	C₄×C₁₆	C₄	1	2	yes	complemented
10	#340	—	2	1	no	—
11	#340	C₄	1	2	yes	—
12	#335	—	2	1	no	—
13	#335	C₄	1	2	yes	complemented
14	C₂×M₁₆	—	4	1	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄⋊C₄	—	4	1	no	—
19	C₂×C₄⋊C₄	—	4	1	no	—
20	C₂×C₄²	D₄	1	1	yes	characteristic
21	C₂×C₁₆	—	2	2	no	—
22	C₄⋊C₈	—	2	2	no	—
23	C₄⋊C₈	—	4	1	no	—
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈	D₄	1	1	yes	characteristic
27	C₄×C₈	Q₈	1	1	yes	characteristic
28	C₂²×C₄	—	4	2	no	—
29	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄	D₈	1	1	yes	characteristic
31	C₂²×C₄	SD₁₆	1	1	yes	characteristic
32	M₁₆	—	8	1	no	—
33	M₁₆	—	8	1	no	—
34	C₂×C₈	—	4	1	no	—
35	C₂×C₈	—	4	1	no	—
36	C₂×C₈	—	4	2	no	—
37	C₂×C₈	C₄²	1	1	yes	characteristic
38	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₄⋊C₄	—	4	2	no	—
42	C₄⋊C₄	—	4	2	no	—
43	C₄⋊C₄	—	8	2	no	—
44	C₄⋊C₄	—	8	1	no	—
45	C₄⋊C₄	—	8	1	no	—
46	C₄²	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²	C₄⋊C₄	1	1	yes	characteristic
48	C₄²	SD₁₆	1	1	yes	characteristic
49	C₄²	Q₁₆	1	1	yes	characteristic
50	C₁₆	—	4	2	no	—
51	C₂³	#102	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	8	2	no	—
56	C₂×C₄	—	8	2	no	—
57	C₂×C₄	—	8	2	no	—
58	C₂×C₄	#95	1	1	yes	characteristic
59	C₂×C₄	#102	1	1	yes	characteristic
60	C₂×C₄	#103	1	1	yes	characteristic
61	C₂×C₄	#103	1	1	yes	characteristic
62	C₂×C₄	#106	1	1	yes	characteristic
63	C₂×C₄	#107	1	1	yes	characteristic
64	C₂×C₄	D₁₆	1	1	yes	characteristic
65	C₂×C₄	SD₃₂	1	1	yes	characteristic
66	C₂×C₄	SD₃₂	1	1	yes	characteristic
67	C₂×C₄	Q₃₂	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	2	1	no	—
70	C₈	—	2	1	no	—
71	C₈	—	2	1	no	—
72	C₈	—	8	2	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	#340	1	1	yes	characteristic
76	C₂²	#357	1	1	yes	characteristic
77	C₂²	#358	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	16	2	no	—
81	C₄	H(3,ℤ/4)	1	1	yes	characteristic
82	C₄	#344	1	1	yes	characteristic
83	C₄	#357	1	1	yes	characteristic
84	C₄	#358	1	1	yes	characteristic
85	C₄	#365	1	2	yes	—
86	C₄	#366	1	1	yes	characteristic
87	C₄	#367	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	2	1	no	—
90	C₂	#1286	1	1	yes	characteristic
91	C₂	#1380	1	1	yes	characteristic
92	C₂	#1383	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#340
commutator	C₂×C₈	C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂²
prime radical	C₂³	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
9. C₄×C₁₆	C₄	semidirect
13. #335	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2162	2
#1834	2
#1567	2