Subgroups — #3166

Order 128.

174 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic
2	C₂×#125	C₂	1	1	yes	maximal, characteristic
3	#501	C₂	1	4	yes	maximal, complemented
4	#499	C₂	1	4	yes	maximal, complemented
5	#428	C₂	1	4	yes	maximal
6	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆	C₂²	1	2	yes	—
8	C₄⋊Q₈	C₂²	1	2	yes	—
9	#125	—	2	2	no	—
10	#125	C₂²	1	2	yes	—
11	#125	C₂²	1	2	yes	—
12	#125	C₂²	1	2	yes	—
13	C₂²⋊Q₈	—	2	2	no	—
14	C₄×Q₈	—	2	2	no	—
15	#117	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄	—	2	2	no	—
17	C₂×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄²	C₂²	1	1	yes	characteristic
19	#107	C₂²	1	8	yes	—
20	#106	C₂²	1	8	yes	—
21	#97	C₂²	1	2	yes	—
22	C₄×C₈	C₂²	1	2	yes	—
23	C₂×Q₈	—	2	2	no	—
24	C₂²×C₄	—	2	2	no	—
25	C₂²×C₄	D₄	1	2	yes	—
26	C₂²×C₄	C₂³	1	1	yes	characteristic
27	M₁₆	Q₈	1	8	yes	complemented
28	C₂×C₈	C₂³	1	4	yes	—
29	C₄⋊C₄	—	2	4	no	—
30	C₄⋊C₄	—	2	4	no	—
31	C₄⋊C₄	—	2	2	no	—
32	C₄⋊C₄	—	2	4	no	—
33	C₄⋊C₄	—	2	2	no	—
34	C₄⋊C₄	C₂³	1	4	yes	—
35	C₄⋊C₄	C₂³	1	4	yes	—
36	C₂²⋊C₄	—	2	2	no	—
37	C₄²	—	2	2	no	—
38	C₄²	D₄	1	2	yes	—
39	C₄²	C₂³	1	1	yes	characteristic
40	C₄²	C₂³	1	1	yes	characteristic
41	C₂³	C₂×D₄	1	1	yes	characteristic
42	Q₈	—	4	4	no	—
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	—	4	2	no	—
46	C₂×C₄	—	4	4	no	—
47	C₂×C₄	C₂×D₄	1	2	yes	—
48	C₂×C₄	C₂×D₄	1	2	yes	—
49	C₂×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄	C₂×Q₈	1	4	yes	—
51	C₂×C₄	C₂⁴	1	1	yes	characteristic
52	C₈	C₂×Q₈	1	8	yes	—
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊Q₈	1	2	yes	—
55	C₂²	C₂²×D₄	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	2	2	no	—
58	C₄	—	4	4	no	—
59	C₄	—	4	4	no	—
60	C₄	C₄⋊Q₈	1	2	yes	—
61	C₄	C₂²×Q₈	1	2	yes	—
62	C₂	—	2	2	no	—
63	C₂	C₂×C₄⋊Q₈	1	1	yes	characteristic
64	C₂	#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
3. #501	C₂	semidirect
4. #499	C₂	semidirect
27. M₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#533	2
C₂×#125	2
#501	2
#499	2
#428	2
C₄×M₁₆	2