Subgroups — #1677

Order 128.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#486	C₂	1	1	yes	maximal, characteristic, complemented
2	#476	C₂	1	1	yes	maximal, characteristic, complemented
3	#459	C₂	1	1	yes	maximal, characteristic
4	C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#334	C₂	1	1	yes	maximal, characteristic, complemented
6	#333	C₂	1	1	yes	maximal, characteristic, complemented
7	#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈	—	2	1	no	—
9	C₂²×C₈	—	2	1	no	—
10	C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	#127	C₂²	1	1	yes	characteristic
12	C₄×D₄	C₂²	1	1	yes	characteristic
13	#106	—	2	1	no	—
14	C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₈	C₂²	1	1	yes	characteristic
17	#102	—	2	1	no	—
18	#102	—	2	1	no	—
19	#102	—	2	1	no	—
20	#102	—	2	1	no	—
21	C₂²⋊C₈	—	2	1	no	—
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈	—	2	1	no	—
24	C₂×D₄	—	2	1	no	—
25	C₂×D₄	—	2	1	no	—
26	C₂×D₄	—	2	1	no	—
27	C₂×D₄	D₄	1	1	yes	characteristic
28	C₂²×C₄	—	2	1	no	—
29	D₈	—	4	2	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	4	2	no	—
34	C₂×C₈	D₄	1	1	yes	characteristic
35	C₂×C₈	D₄	1	1	yes	characteristic
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	—	2	1	no	—
38	C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂²⋊C₄	—	2	1	no	—
40	C₄²	C₂³	1	1	yes	characteristic
41	C₂³	—	2	1	no	—
42	C₂³	—	4	1	no	—
43	Q₈	—	8	1	no	—
44	D₄	—	2	1	no	—
45	D₄	—	4	1	no	—
46	D₄	—	4	1	no	—
47	D₄	—	4	1	no	—
48	D₄	—	4	1	no	—
49	D₄	—	8	1	no	—
50	D₄	SD₁₆	1	2	yes	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄	#41	1	1	yes	characteristic
58	C₈	—	2	1	no	—
59	C₈	—	2	1	no	—
60	C₈	—	4	1	no	—
61	C₈	—	4	1	no	—
62	C₈	—	4	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	8	1	no	—
67	C₂²	—	8	1	no	—
68	C₂²	—	8	1	no	—
69	C₂²	#121	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	8	1	no	—
73	C₄	C₂×SD₁₆	1	1	yes	characteristic
74	C₄	#135	1	1	yes	characteristic
75	C₄	#135	1	1	yes	characteristic
76	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	—	16	1	no	—
79	C₂	#463	1	1	yes	characteristic
80	C₂	#465	1	1	yes	characteristic
81	C₂	#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#459	C₂
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
1. #486	C₂	semidirect
2. #476	C₂	semidirect
4. C₈×D₄	C₂	semidirect
5. #334	C₂	semidirect
6. #333	C₂	semidirect
7. #331	C₂	semidirect

Maximal subgroups

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

subgroup	index
#486	2
#476	2
#459	2
C₈×D₄	2
#334	2
#333	2
#331	2