Subgroups — #13707

Order 256.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic
3	#1424	C₂	1	1	yes	maximal, characteristic, complemented
4	#1423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#1415	C₂	1	1	yes	maximal, characteristic
7	#1415	C₂	1	1	yes	maximal, characteristic
8	C₂×D₁₆	C₂²	1	1	yes	characteristic
9	#499	—	2	1	no	—
10	C₄×D₈	—	2	1	no	—
11	C₂×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₃₂	C₂²	1	1	yes	characteristic
13	#367	—	2	1	no	—
14	#366	C₂²	1	1	yes	characteristic
15	#366	C₂²	1	1	yes	characteristic
16	#366	C₂²	1	1	yes	characteristic
17	#357	—	2	1	no	—
18	C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×D₈	—	2	1	no	—
20	#125	—	4	1	no	—
21	C₄×D₄	—	4	1	no	—
22	D₁₆	—	2	1	no	—
23	D₁₆	Q₈	1	2	yes	—
24	C₂×C₁₆	D₄	1	1	yes	characteristic
25	C₂×C₁₆	C₂³	1	1	yes	characteristic
26	#107	—	2	1	no	—
27	#107	—	2	1	no	—
28	#107	—	2	1	no	—
29	#106	—	4	1	no	—
30	#102	—	4	1	no	—
31	C₄×C₈	D₄	1	1	yes	characteristic
32	C₃₂	—	2	1	no	—
33	C₃₂	—	2	1	no	—
34	C₂×D₄	—	4	1	no	—
35	C₂²×C₄	—	8	1	no	—
36	D₈	—	2	2	no	—
37	D₈	—	4	1	no	—
38	C₂×C₈	D₈	1	1	yes	characteristic
39	C₂×C₈	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄	—	4	1	no	—
41	C₄⋊C₄	—	4	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	8	1	no	—
44	C₄⋊C₄	—	8	1	no	—
45	C₂²⋊C₄	—	8	1	no	—
46	C₄²	D₈	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	C₂×Q₈	1	1	yes	characteristic
49	C₁₆	#41	1	1	yes	characteristic
50	C₂³	—	8	1	no	—
51	D₄	—	4	2	no	—
52	D₄	—	8	1	no	—
53	C₂×C₄	—	8	1	no	—
54	C₂×C₄	—	8	1	no	—
55	C₂×C₄	—	8	1	no	—
56	C₂×C₄	—	16	2	no	—
57	C₂×C₄	D₁₆	1	1	yes	characteristic
58	C₂×C₄	D₁₆	1	1	yes	characteristic
59	C₂×C₄	C₂×D₈	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈	#137	1	1	yes	characteristic
63	C₂²	—	8	2	no	—
64	C₂²	—	16	1	no	—
65	C₂²	—	16	1	no	—
66	C₂²	C₂×D₁₆	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	—	16	1	no	—
70	C₄	—	16	1	no	—
71	C₄	—	16	1	no	—
72	C₄	#474	1	1	yes	characteristic
73	C₄	#510	1	1	yes	characteristic
74	C₂	—	16	2	no	—
75	C₂	#2224	1	1	yes	characteristic
76	C₂	#2262	1	1	yes	characteristic
77	C₂	#2263	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2253	C₂	semidirect
3. #1424	C₂	semidirect
4. #1423	C₂	semidirect
5. C₄⋊C₃₂	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2253	2
C₄×D₁₆	2
#1424	2
#1423	2
C₄⋊C₃₂	2
#1415	2