Subgroups — #2985

Order 128.

161 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	#433	C₂	1	4	yes	maximal
6	#432	C₂	1	1	yes	maximal, characteristic
7	#431	C₂	1	1	yes	maximal, characteristic
8	C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₄⋊Q₈	C₄	1	1	yes	characteristic
10	#125	C₄	1	2	yes	—
11	C₂²⋊Q₈	C₄	1	2	yes	—
12	C₂²⋊Q₈	C₄	1	2	yes	—
13	C₄×Q₈	C₂²	1	2	yes	—
14	C₄×Q₈	C₂²	1	2	yes	—
15	#117	C₂²	1	2	yes	—
16	C₂×C₄²	C₂²	1	1	yes	characteristic
17	C₄⋊C₈	C₂²	1	2	yes	—
18	C₄⋊C₈	C₂²	1	2	yes	—
19	C₄⋊C₈	C₂²	1	4	yes	—
20	C₄⋊C₈	C₂²	1	4	yes	—
21	C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂²⋊C₈	C₂²	1	2	yes	—
23	#97	C₂²	1	2	yes	—
24	#97	C₂²	1	4	yes	—
25	C₄×C₈	C₂²	1	1	yes	characteristic
26	C₄×C₈	C₂²	1	1	yes	characteristic
27	C₄×C₈	C₂²	1	4	yes	—
28	C₂×Q₈	C₂×C₄	1	2	yes	—
29	C₂×Q₈	C₂×C₄	1	2	yes	—
30	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₈	C₂³	1	2	yes	—
34	C₂×C₈	C₂³	1	2	yes	—
35	C₂×C₈	C₂³	1	4	yes	—
36	C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₄⋊C₄	C₂×C₄	1	4	yes	—
40	C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂²⋊C₄	C₂×C₄	1	4	yes	—
43	C₄²	C₂×C₄	1	1	yes	characteristic
44	C₄²	C₂×C₄	1	1	yes	characteristic
45	C₄²	C₂³	1	1	yes	characteristic
46	C₄²	C₂³	1	4	yes	—
47	C₄²	C₂³	1	1	yes	characteristic
48	C₂³	C₂²×C₄	1	1	yes	characteristic
49	Q₈	—	2	4	no	—
50	Q₈	—	4	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄	C₂²×C₄	1	4	yes	—
59	C₂×C₄	C₂²×C₄	1	4	yes	—
60	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄	C₂⁴	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	2	2	no	—
64	C₈	—	2	4	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	C₂³×C₄	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	—	2	4	no	—
72	C₄	—	2	1	no	—
73	C₄	—	2	1	no	—
74	C₄	—	2	4	no	—
75	C₄	#131	1	2	yes	—
76	C₄	2₋¹⁺⁴	1	2	yes	—
77	C₂	—	4	1	no	—
78	C₂	#519	1	1	yes	characteristic
79	C₂	C₂×#131	1	1	yes	characteristic
80	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

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
#433	2
#432	2
#431	2