Subgroups — #3170

Order 128.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	2	yes	maximal
2	#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#494	C₂	1	1	yes	maximal, characteristic, complemented
4	#487	C₂	1	2	yes	maximal, complemented
5	#467	C₂	1	4	yes	maximal
6	C₄×Q₁₆	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆	C₂²	1	4	yes	—
9	C₂²×C₈	C₂²	1	2	yes	—
10	C₄⋊Q₈	C₂²	1	2	yes	—
11	C₄⋊Q₈	C₂²	1	2	yes	—
12	#125	—	2	2	no	—
13	C₂²⋊Q₈	—	2	2	no	—
14	C₂²⋊Q₈	C₂²	1	4	yes	—
15	C₄×Q₈	—	2	2	no	—
16	C₄×Q₈	C₂²	1	4	yes	—
17	#117	—	2	2	no	—
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	#107	C₂²	1	4	yes	—
20	#103	C₂²	1	8	yes	—
21	C₄×C₈	C₂²	1	2	yes	—
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₄×C₈	C₂²	1	1	yes	characteristic
24	C₂×Q₈	—	2	2	no	—
25	C₂×Q₈	C₂³	1	4	yes	—
26	C₂²×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₄	C₂³	1	1	yes	characteristic
29	Q₁₆	—	2	8	no	—
30	C₂×C₈	—	2	4	no	—
31	C₂×C₈	C₂³	1	2	yes	—
32	C₂×C₈	C₂³	1	2	yes	—
33	C₄⋊C₄	—	2	2	no	—
34	C₄⋊C₄	—	2	2	no	—
35	C₄⋊C₄	—	2	2	no	—
36	C₄⋊C₄	—	2	4	no	—
37	C₄⋊C₄	—	2	4	no	—
38	C₄⋊C₄	C₂³	1	4	yes	—
39	C₂²⋊C₄	—	2	4	no	—
40	C₄²	—	2	4	no	—
41	C₄²	D₄	1	1	yes	characteristic
42	C₄²	D₄	1	1	yes	characteristic
43	C₄²	C₂³	1	1	yes	characteristic
44	C₄²	C₂³	1	1	yes	characteristic
45	C₂³	C₂×D₄	1	1	yes	characteristic
46	Q₈	—	2	8	no	—
47	Q₈	—	4	4	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	Q₁₆	1	4	yes	complemented
53	C₂×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄	C₂⁴	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	#41	1	4	yes	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×Q₁₆	1	2	yes	—
64	C₂²	C₂²×D₄	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	4	4	no	—
68	C₄	—	4	4	no	—
69	C₄	C₂×Q₁₆	1	2	yes	—
70	C₄	C₂×Q₁₆	1	2	yes	—
71	C₄	#135	1	2	yes	—
72	C₄	C₂×#41	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#532	1	1	yes	characteristic
75	C₂	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂	C₂×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂²	C₂²×D₄
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #500	C₂	semidirect
3. #494	C₂	semidirect
4. #487	C₂	semidirect
6. C₄×Q₁₆	C₂	semidirect
52. C₂×C₄	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#500	2
#494	2
#487	2
#467	2
C₄×Q₁₆	2
C₂×C₄×C₈	2