Subgroups — #3209

Order 128.

192 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	#494	C₂	1	2	yes	maximal, complemented
4	#487	C₂	1	2	yes	maximal, complemented
5	#462	C₂	1	4	yes	maximal, complemented
6	#451	C₂	1	4	yes	maximal
7	#431	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈	C₂²	1	2	yes	—
9	C₂×Q₁₆	C₂²	1	8	yes	—
10	C₄⋊Q₈	—	2	1	no	—
11	C₄⋊Q₈	—	2	1	no	—
12	C₄⋊Q₈	C₂²	1	2	yes	—
13	C₄⋊Q₈	C₂²	1	2	yes	—
14	C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	#125	—	2	1	no	—
17	C₂²⋊Q₈	—	2	1	no	—
18	C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₄×Q₈	—	2	1	no	—
20	C₄×Q₈	C₂²	1	2	yes	—
21	#117	—	2	1	no	—
22	C₂×C₄⋊C₄	—	2	1	no	—
23	C₂×C₄⋊C₄	—	2	1	no	—
24	C₂×C₄²	C₂²	1	1	yes	characteristic
25	C₄⋊C₈	C₂²	1	2	yes	—
26	#103	C₂²	1	8	yes	—
27	C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈	C₂²	1	2	yes	—
29	C₂×Q₈	—	2	1	no	—
30	C₂×Q₈	—	2	2	no	—
31	C₂×Q₈	—	4	4	no	—
32	C₂×Q₈	C₂³	1	4	yes	—
33	C₂×Q₈	C₂³	1	2	yes	—
34	C₂²×C₄	—	2	2	no	—
35	C₂²×C₄	D₄	1	1	yes	characteristic
36	C₂²×C₄	D₄	1	1	yes	characteristic
37	C₂²×C₄	C₂³	1	1	yes	characteristic
38	Q₁₆	—	4	8	no	—
39	C₂×C₈	C₂³	1	4	yes	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	—	2	2	no	—
43	C₄⋊C₄	—	2	2	no	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	C₂³	1	2	yes	—
50	C₂²⋊C₄	—	2	2	no	—
51	C₄²	—	2	2	no	—
52	C₄²	D₄	1	1	yes	characteristic
53	C₄²	D₄	1	1	yes	characteristic
54	C₄²	C₂³	1	1	yes	characteristic
55	C₄²	C₂³	1	1	yes	characteristic
56	C₂³	C₂×D₄	1	1	yes	characteristic
57	Q₈	—	2	4	no	—
58	Q₈	—	2	8	no	—
59	Q₈	—	4	4	no	—
60	Q₈	—	4	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	Q₁₆	1	4	yes	complemented
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄	C₂⁴	1	1	yes	characteristic
74	C₈	—	2	4	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	C₂×Q₁₆	1	2	yes	—
78	C₂²	C₂²×D₄	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	4	4	no	—
82	C₄	—	4	2	no	—
83	C₄	—	4	2	no	—
84	C₄	C₂×Q₁₆	1	2	yes	—
85	C₄	C₂×Q₁₆	1	2	yes	—
86	C₄	#137	1	2	yes	—
87	C₄	H(5,𝔽₂)	1	2	yes	—
88	C₂	—	2	2	no	—
89	C₂	#535	1	1	yes	characteristic
90	C₂	C₂²×Q₁₆	1	1	yes	characteristic
91	C₂	C₂×#137	1	1	yes	characteristic

	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. #494	C₂	semidirect
4. #487	C₂	semidirect
5. #462	C₂	semidirect
67. 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
C₂×C₄⋊Q₈	2
#494	2
#487	2
#462	2
#451	2
#431	2