Subgroups — #1271

Order 128.

79 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#389	C₂	1	1	yes	maximal, characteristic
2	#336	C₂	1	1	yes	maximal, characteristic
3	#336	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈	—	2	1	no	—
5	C₂²×C₈	—	2	1	no	—
6	C₂×C₄⋊C₄	—	2	1	no	—
7	C₂×C₄⋊C₄	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₄	C₄	1	1	yes	characteristic
9	C₂×C₄²	C₂²	1	1	yes	characteristic
10	#95	—	2	1	no	—
11	C₂²×C₄	—	2	1	no	—
12	C₂²×C₄	—	2	1	no	—
13	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄	D₄	1	1	yes	characteristic
15	C₂²×C₄	D₄	1	1	yes	characteristic
16	C₂×C₈	—	2	2	no	—
17	C₂×C₈	—	2	2	no	—
18	C₂×C₈	—	4	1	no	—
19	C₂×C₈	—	4	1	no	—
20	C₂×C₈	—	4	1	no	—
21	C₂×C₈	—	4	1	no	—
22	C₄⋊C₄	—	2	1	no	—
23	C₄⋊C₄	—	4	1	no	—
24	C₄⋊C₄	—	4	1	no	—
25	C₄⋊C₄	—	4	1	no	—
26	C₄⋊C₄	C₈	1	2	yes	complemented
27	C₄²	—	4	1	no	—
28	C₂³	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	—	2	1	no	—
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	—	2	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	—	4	1	no	—
37	C₂×C₄	—	4	1	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	4	1	no	—
40	C₂×C₄	—	4	1	no	—
41	C₂×C₄	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄	M₁₆	1	1	yes	characteristic
43	C₂×C₄	D₈	1	1	yes	characteristic
44	C₂×C₄	SD₁₆	1	1	yes	characteristic
45	C₂×C₄	SD₁₆	1	1	yes	characteristic
46	C₂×C₄	Q₁₆	1	1	yes	characteristic
47	C₈	—	4	2	no	—
48	C₈	—	4	2	no	—
49	C₂²	C₂²⋊C₈	1	1	yes	characteristic
50	C₂²	#99	1	1	yes	characteristic
51	C₂²	#101	1	1	yes	characteristic
52	C₂²	#102	1	1	yes	characteristic
53	C₂²	#103	1	1	yes	characteristic
54	C₂²	#104	1	1	yes	characteristic
55	C₂²	#104	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	2	1	no	—
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	2	1	no	—
62	C₄	—	4	2	no	—
63	C₄	—	8	1	no	—
64	C₂	#324	1	1	yes	characteristic
65	C₂	D₄⋊C₈	1	1	yes	characteristic
66	C₂	Q₈⋊C₈	1	1	yes	characteristic
67	C₂	#327	1	1	yes	characteristic
68	C₂	#328	1	1	yes	characteristic
69	C₂	#330	1	1	yes	characteristic
70	C₂	#332	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
26. C₄⋊C₄	C₈	semidirect

Maximal subgroups

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

subgroup	index
#389	2
#336	2