Subgroups — #1771

Order 128.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#336	C₂	1	1	yes	maximal, characteristic
7	#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄	C₄	1	2	yes	—
13	C₂×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₄²	C₂²	1	1	yes	characteristic
16	C₄⋊C₈	C₄	1	4	yes	complemented
17	#97	—	2	2	no	—
18	#97	C₄	1	4	yes	complemented
19	C₄×C₈	—	2	2	no	—
20	#95	C₄	1	2	yes	—
21	C₂²×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄	C₂×C₄	1	2	yes	—
25	C₂²×C₄	C₂³	1	1	yes	characteristic
26	C₂×C₈	—	2	1	no	—
27	C₂×C₈	—	2	1	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₈	D₄	1	2	yes	—
37	C₂×C₈	Q₈	1	2	yes	—
38	C₄⋊C₄	—	2	4	no	—
39	C₄²	—	2	1	no	—
40	C₄²	—	2	1	no	—
41	C₄²	—	2	1	no	—
42	C₄²	—	2	1	no	—
43	C₄²	C₂×C₄	1	2	yes	—
44	C₄²	C₂×C₄	1	2	yes	—
45	C₂³	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	C₄²	1	2	yes	—
54	C₂×C₄	C₄²	1	2	yes	—
55	C₂×C₄	M₁₆	1	4	yes	—
56	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄	#41	1	1	yes	characteristic
61	C₂×C₄	#41	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	2	2	no	—
64	C₈	—	2	2	no	—
65	C₈	C₄⋊C₄	1	4	yes	complemented
66	C₂²	C₂×C₄²	1	1	yes	characteristic
67	C₂²	C₄×D₄	1	1	yes	characteristic
68	C₂²	C₄×Q₈	1	1	yes	characteristic
69	C₂²	C₂×M₁₆	1	2	yes	—
70	C₂²	#131	1	2	yes	—
71	C₄	—	2	2	no	—
72	C₄	—	2	2	no	—
73	C₄	—	2	4	no	—
74	C₄	—	4	2	no	—
75	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄	#117	1	1	yes	characteristic
77	C₄	C₄×D₄	1	1	yes	characteristic
78	C₄	C₄×Q₈	1	1	yes	characteristic
79	C₂	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂	C₄×M₁₆	1	1	yes	characteristic
81	C₂	#405	1	1	yes	characteristic
82	C₂	#435	1	2	yes	—
83	C₂	#446	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
16. C₄⋊C₈	C₄	semidirect
18. #97	C₄	semidirect
65. C₈	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈	2
C₂×#97	2
C₂×C₄×C₈	2
C₄×C₄⋊C₄	2
#336	2