Subgroups — #1652

Order 128.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556	C₂	1	1	yes	maximal, characteristic
2	#498	C₂	1	1	yes	maximal, characteristic, complemented
3	#474	C₂	1	1	yes	maximal, characteristic
4	#462	C₂	1	1	yes	maximal, characteristic, complemented
5	#332	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆	—	2	1	no	—
9	C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	#126	—	2	2	no	—
12	#124	—	2	1	no	—
13	C₂²⋊Q₈	—	2	1	no	—
14	C₄×Q₈	—	2	1	no	—
15	C₄×Q₈	C₂²	1	1	yes	characteristic
16	C₄×D₄	C₂²	1	1	yes	characteristic
17	#117	—	2	1	no	—
18	#107	—	2	1	no	—
19	#106	—	2	2	no	—
20	C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊C₈	C₂²	1	1	yes	characteristic
22	#103	—	2	1	no	—
23	#102	—	2	1	no	—
24	C₄×C₈	C₂²	1	1	yes	characteristic
25	C₂×Q₈	—	2	1	no	—
26	C₂×Q₈	—	2	1	no	—
27	C₂×Q₈	D₄	1	1	yes	characteristic
28	C₂×D₄	D₄	1	1	yes	characteristic
29	C₂²×C₄	—	2	1	no	—
30	Q₁₆	—	4	2	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₄⋊C₄	—	2	2	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	—	2	1	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₄⋊C₄	D₄	1	2	yes	—
41	C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₂²⋊C₄	—	2	2	no	—
43	C₂²⋊C₄	—	2	2	no	—
44	C₂²⋊C₄	—	2	1	no	—
45	C₄²	—	2	2	no	—
46	C₄²	—	2	1	no	—
47	C₄²	C₂³	1	1	yes	characteristic
48	C₂³	—	2	1	no	—
49	Q₈	—	2	1	no	—
50	Q₈	—	2	1	no	—
51	Q₈	—	4	1	no	—
52	Q₈	—	4	1	no	—
53	Q₈	—	4	1	no	—
54	Q₈	—	4	1	no	—
55	D₄	—	2	1	no	—
56	D₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄	C₂×D₄	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	4	1	no	—
69	C₈	—	4	1	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	#120	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	4	1	no	—
79	C₄	—	8	1	no	—
80	C₄	#135	1	1	yes	characteristic
81	C₄	#135	1	1	yes	characteristic
82	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₄	#137	1	1	yes	characteristic
84	C₂	—	8	1	no	—
85	C₂	#449	1	1	yes	characteristic
86	C₂	#452	1	1	yes	characteristic
87	C₂	#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#120
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	C₂×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #498	C₂	semidirect
4. #462	C₂	semidirect
5. #332	C₂	semidirect
6. Q₈⋊C₈	C₂	semidirect

Maximal subgroups

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

subgroup	index
#556	2
#498	2
#474	2
#462	2
#332	2
Q₈⋊C₈	2
D₄⋊C₈	2