Subgroups — #2962

Order 128.

178 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	#446	C₂	1	2	yes	maximal, complemented
3	#446	C₂	1	4	yes	maximal, complemented
4	C₈×Q₈	C₂	1	2	yes	maximal, complemented
5	#424	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#405	C₂	1	2	yes	maximal
8	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈	C₄	1	1	yes	characteristic
13	C₄⋊Q₈	C₄	1	1	yes	characteristic
14	#125	C₄	1	2	yes	—
15	C₂²⋊Q₈	C₄	1	2	yes	—
16	C₂²⋊Q₈	C₄	1	2	yes	—
17	C₄×Q₈	C₂²	1	2	yes	—
18	C₄×Q₈	C₂²	1	2	yes	—
19	#117	C₂²	1	2	yes	—
20	C₂×C₄²	C₂²	1	1	yes	characteristic
21	C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₄⋊C₈	C₂²	1	2	yes	—
24	C₄⋊C₈	C₂²	1	4	yes	—
25	C₄⋊C₈	C₂²	1	4	yes	—
26	#97	C₂²	1	2	yes	—
27	#97	C₂²	1	4	yes	—
28	C₄×C₈	C₂²	1	2	yes	—
29	C₄×C₈	C₂²	1	4	yes	—
30	C₂×Q₈	C₂×C₄	1	2	yes	—
31	C₂×Q₈	C₂×C₄	1	2	yes	—
32	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄	C₂³	1	1	yes	characteristic
35	M₁₆	—	2	2	no	—
36	M₁₆	Q₈	1	4	yes	complemented
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	C₂³	1	2	yes	—
39	C₂×C₈	C₂³	1	2	yes	—
40	C₂×C₈	C₂³	1	2	yes	—
41	C₂×C₈	C₂³	1	2	yes	—
42	C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₄⋊C₄	C₂×C₄	1	4	yes	—
45	C₄⋊C₄	C₂×C₄	1	4	yes	—
46	C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄	C₂×C₄	1	4	yes	—
49	C₄²	C₂×C₄	1	1	yes	characteristic
50	C₄²	C₂×C₄	1	1	yes	characteristic
51	C₄²	C₂³	1	1	yes	characteristic
52	C₄²	C₂³	1	4	yes	—
53	C₄²	C₂³	1	1	yes	characteristic
54	C₂³	C₂²×C₄	1	1	yes	characteristic
55	Q₈	—	2	4	no	—
56	Q₈	—	4	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄	C₂²×C₄	1	4	yes	—
63	C₂×C₄	C₂²×C₄	1	4	yes	—
64	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄	C₂×Q₈	1	2	yes	—
67	C₂×C₄	#41	1	2	yes	—
68	C₂×C₄	C₂⁴	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₈	—	2	2	no	—
71	C₈	—	2	2	no	—
72	C₈	C₂×Q₈	1	4	yes	—
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	C₄×Q₈	1	2	yes	—
76	C₂²	C₂³×C₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	4	no	—
79	C₄	—	2	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	2	4	no	—
82	C₄	C₄×Q₈	1	2	yes	—
83	C₄	#131	1	2	yes	—
84	C₄	C₂²×Q₈	1	1	yes	characteristic
85	C₄	C₂×#41	1	1	yes	characteristic
86	C₂	—	2	2	no	—
87	C₂	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂	C₂×#131	1	1	yes	characteristic
89	C₂	#568	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. #446	C₂	semidirect
3. #446	C₂	semidirect
4. C₈×Q₈	C₂	semidirect
36. 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
#446	2
C₈×Q₈	2
#424	2
C₂×C₄⋊C₈	2
#405	2
C₄×M₁₆	2