Subgroups — #3390

Order 128.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557	C₂	1	2	yes	maximal
2	#501	C₂	1	2	yes	maximal
3	#499	C₂	1	1	yes	maximal, characteristic
4	#479	C₂	1	2	yes	maximal
5	#479	C₂	1	2	yes	maximal
6	#477	C₂	1	2	yes	maximal
7	#446	C₂	1	1	yes	maximal, characteristic
8	#441	C₂	1	2	yes	maximal
9	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈	—	2	2	no	—
12	C₄⋊Q₈	C₂²	1	2	yes	—
13	#125	—	2	2	no	—
14	#125	—	2	2	no	—
15	#125	C₂²	1	2	yes	—
16	#125	C₂²	1	2	yes	—
17	C₄×Q₈	—	2	2	no	—
18	C₄×Q₈	C₂²	1	2	yes	—
19	C₄×Q₈	C₂²	1	2	yes	—
20	C₄×Q₈	C₂²	1	2	yes	—
21	C₄×Q₈	C₂²	1	1	yes	characteristic
22	#107	C₂²	1	1	yes	characteristic
23	#107	C₂²	1	1	yes	characteristic
24	#107	C₂²	1	2	yes	—
25	#107	C₂²	1	1	yes	characteristic
26	#106	C₂²	1	2	yes	—
27	#106	C₂²	1	2	yes	—
28	C₄⋊C₈	C₂²	1	2	yes	—
29	C₄⋊C₈	C₂²	1	1	yes	characteristic
30	#103	C₂²	1	2	yes	—
31	#103	C₂²	1	2	yes	—
32	#103	C₂²	1	2	yes	—
33	#97	C₂²	1	2	yes	—
34	C₄×C₈	C₂²	1	1	yes	characteristic
35	C₂×Q₈	—	2	2	no	—
36	C₂×Q₈	D₄	1	1	yes	characteristic
37	C₂×Q₈	C₂³	1	2	yes	—
38	Q₁₆	Q₈	1	4	yes	complemented
39	C₂×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₈	C₂³	1	2	yes	—
41	C₂×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊C₄	—	2	2	no	—
43	C₄⋊C₄	—	2	2	no	—
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	—	2	2	no	—
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	D₄	1	2	yes	—
51	C₄⋊C₄	D₄	1	1	yes	characteristic
52	C₄⋊C₄	C₂³	1	2	yes	—
53	C₄⋊C₄	C₂³	1	2	yes	—
54	C₄⋊C₄	C₂³	1	2	yes	—
55	C₄²	—	2	2	no	—
56	C₄²	—	2	2	no	—
57	C₄²	—	2	2	no	—
58	C₄²	C₂³	1	1	yes	characteristic
59	C₄²	C₂³	1	2	yes	—
60	Q₈	—	2	2	no	—
61	Q₈	—	4	1	no	—
62	Q₈	—	4	4	no	—
63	Q₈	C₂×Q₈	1	4	yes	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄	C₂×D₄	1	2	yes	—
70	C₂×C₄	C₂×D₄	1	2	yes	—
71	C₂×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄	C₂⁴	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	2	2	no	—
75	C₈	C₂×Q₈	1	2	yes	—
76	C₂²	C₂²×D₄	1	1	yes	characteristic
77	C₄	—	2	4	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	2	2	no	—
82	C₄	—	4	2	no	—
83	C₄	—	4	2	no	—
84	C₄	—	4	2	no	—
85	C₄	C₂²×Q₈	1	1	yes	characteristic
86	C₄	2₋¹⁺⁴	1	1	yes	characteristic
87	C₂	D₄×Q₈	1	1	yes	characteristic
88	C₂	#575	1	1	yes	characteristic
89	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
38. Q₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#557	2
#501	2
#499	2
#479	2
#477	2
#446	2
#441	2
C₄×Q₁₆	2