Subgroups — #7062

Order 256.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1831	C₂	1	1	yes	maximal, characteristic
2	#1322	C₂	1	2	yes	maximal
3	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
4	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
5	C₄³	C₄	1	1	yes	characteristic
6	C₂²⋊C₁₆	—	2	2	no	—
7	#336	—	2	2	no	—
8	C₂²×C₈	—	2	1	no	—
9	C₂²×C₈	D₄	1	2	yes	—
10	C₂×C₄²	—	2	1	no	—
11	C₂×C₄²	—	2	1	no	—
12	C₂×C₄²	C₈	1	1	yes	characteristic
13	C₂×C₄²	C₈	1	1	yes	characteristic
14	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆	—	4	2	no	—
16	C₄⋊C₈	—	4	1	no	—
17	C₄⋊C₈	—	4	1	no	—
18	C₂²×C₄	—	2	1	no	—
19	C₂²×C₄	—	2	1	no	—
20	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
21	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
22	C₂²×C₄	M₁₆	1	1	yes	characteristic
23	C₂×C₈	—	2	2	no	—
24	C₂×C₈	—	4	2	no	—
25	C₂×C₈	—	4	2	no	—
26	C₂×C₈	—	4	1	no	—
27	C₄²	—	2	1	no	—
28	C₄²	—	2	1	no	—
29	C₄²	—	2	1	no	—
30	C₄²	—	4	1	no	—
31	C₄²	—	4	1	no	—
32	C₄²	—	4	1	no	—
33	C₄²	—	4	2	no	—
34	C₄²	C₁₆	1	2	yes	complemented
35	C₁₆	—	8	2	no	—
36	C₂³	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	4	1	no	—
45	C₂×C₄	—	4	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	#99	1	1	yes	characteristic
50	C₂×C₄	#101	1	1	yes	characteristic
51	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄	M₃₂	1	1	yes	characteristic
53	C₈	—	4	2	no	—
54	C₈	—	8	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	#324	1	1	yes	characteristic
58	C₂²	C₂²⋊C₁₆	1	1	yes	characteristic
59	C₂²	#349	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	2	1	no	—
62	C₄	—	2	1	no	—
63	C₄	—	4	2	no	—
64	C₄	—	4	2	no	—
65	C₄	—	4	1	no	—
66	C₄	#354	1	1	yes	characteristic
67	C₄	#355	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	2	1	no	—
70	C₂	#1322	1	1	yes	characteristic
71	C₂	#1324	1	1	yes	characteristic
72	C₂	#1327	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#99
commutator	C₂×C₄	C₂×C₁₆
Frattini, cosocle	C₂×C₄⋊C₈	C₂²
socle	C₂²	#324
prime radical	C₂³	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
34. C₄²	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#1831	2
#1322	2