Subgroups — #2228

Order 128.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₈	C₂	1	1	yes	maximal, characteristic
3	#367	C₂	1	1	yes	maximal, characteristic, complemented
4	#366	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#357	C₂	1	1	yes	maximal, characteristic
7	#357	C₂	1	1	yes	maximal, characteristic
8	C₂×D₈	C₂²	1	1	yes	characteristic
9	#125	—	2	1	no	—
10	C₄×D₄	—	2	1	no	—
11	C₂×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₆	C₂²	1	1	yes	characteristic
13	#107	C₂²	1	1	yes	characteristic
14	#107	C₂²	1	1	yes	characteristic
15	#107	C₂²	1	1	yes	characteristic
16	#106	—	2	1	no	—
17	#102	—	2	1	no	—
18	C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×D₄	—	2	1	no	—
20	C₂²×C₄	—	4	1	no	—
21	D₈	—	2	1	no	—
22	D₈	Q₈	1	2	yes	—
23	C₂×C₈	D₄	1	1	yes	characteristic
24	C₂×C₈	C₂³	1	1	yes	characteristic
25	C₄⋊C₄	—	2	1	no	—
26	C₄⋊C₄	—	2	1	no	—
27	C₄⋊C₄	—	2	1	no	—
28	C₄⋊C₄	—	4	1	no	—
29	C₄⋊C₄	—	4	1	no	—
30	C₂²⋊C₄	—	4	1	no	—
31	C₄²	D₄	1	1	yes	characteristic
32	C₁₆	—	2	1	no	—
33	C₁₆	—	2	1	no	—
34	C₂³	—	4	1	no	—
35	D₄	—	2	2	no	—
36	D₄	—	4	1	no	—
37	C₂×C₄	—	4	1	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	4	1	no	—
40	C₂×C₄	—	8	2	no	—
41	C₂×C₄	D₈	1	1	yes	characteristic
42	C₂×C₄	D₈	1	1	yes	characteristic
43	C₂×C₄	C₂×D₄	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	C₂×Q₈	1	1	yes	characteristic
46	C₈	#41	1	1	yes	characteristic
47	C₂²	—	4	2	no	—
48	C₂²	—	8	1	no	—
49	C₂²	—	8	1	no	—
50	C₂²	C₂×D₈	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	8	1	no	—
54	C₄	—	8	1	no	—
55	C₄	—	8	1	no	—
56	C₄	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄	#137	1	1	yes	characteristic
58	C₂	—	8	2	no	—
59	C₂	#474	1	1	yes	characteristic
60	C₂	#508	1	1	yes	characteristic
61	C₂	#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #499	C₂	semidirect
3. #367	C₂	semidirect
4. #366	C₂	semidirect
5. C₄⋊C₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#499	2
C₄×D₈	2
#367	2
#366	2
C₄⋊C₁₆	2
#357	2