Subgroups — #1552

Order 128.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic
2	#432	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#332	C₂	1	2	yes	maximal, complemented
5	#332	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈	—	2	1	no	—
7	C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#125	C₄	1	2	yes	—
10	C₂²⋊Q₈	—	2	1	no	—
11	C₂²⋊Q₈	—	2	1	no	—
12	C₄×Q₈	—	2	1	no	—
13	C₄×Q₈	—	2	1	no	—
14	#117	C₄	1	2	yes	—
15	C₂×C₄²	C₂²	1	1	yes	characteristic
16	C₄⋊C₈	—	2	1	no	—
17	C₄⋊C₈	C₂²	1	2	yes	—
18	C₄⋊C₈	C₂²	1	2	yes	—
19	C₂²⋊C₈	—	2	1	no	—
20	#97	—	2	1	no	—
21	C₂×Q₈	—	2	1	no	—
22	C₂×Q₈	—	2	1	no	—
23	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₄	D₄	1	1	yes	characteristic
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	—	4	2	no	—
29	C₄⋊C₄	—	2	2	no	—
30	C₄⋊C₄	—	2	2	no	—
31	C₄⋊C₄	—	2	1	no	—
32	C₄⋊C₄	—	2	1	no	—
33	C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂²⋊C₄	—	2	2	no	—
36	C₄²	—	2	2	no	—
37	C₄²	C₂×C₄	1	1	yes	characteristic
38	C₄²	D₄	1	1	yes	characteristic
39	C₄²	D₄	1	1	yes	characteristic
40	C₄²	C₂³	1	1	yes	characteristic
41	C₂³	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈	—	4	2	no	—
43	Q₈	—	4	1	no	—
44	Q₈	—	4	1	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄	C₂×D₄	1	1	yes	characteristic
57	C₈	—	4	2	no	—
58	C₈	—	4	2	no	—
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	#101	1	2	yes	—
62	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	2	1	no	—
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	4	2	no	—
68	C₄	—	4	2	no	—
69	C₄	#135	1	2	yes	—
70	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄	#137	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	C₂×#101	1	1	yes	characteristic
74	C₂	#416	1	1	yes	characteristic
75	C₂	#417	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #332	C₂	semidirect
5. #332	C₂	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#432	2
C₂×C₄⋊C₈	2
#332	2