Subgroups — #2931

Order 128.

250 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	2	yes	maximal, complemented
2	#532	C₂	1	1	yes	maximal, characteristic
3	#436	C₂	1	4	yes	maximal, complemented
4	C₈×D₄	C₂	1	4	yes	maximal, complemented
5	#423	C₂	1	1	yes	maximal, characteristic, complemented
6	#408	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	2	yes	—
9	#131	—	2	8	no	—
10	C₂×M₁₆	C₂²	1	2	yes	—
11	C₂×M₁₆	C₂²	1	4	yes	—
12	C₂²×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₈	C₂²	1	4	yes	—
14	C₂²×C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈	C₄	1	1	yes	characteristic
16	#127	C₄	1	1	yes	characteristic
17	#124	C₄	1	2	yes	—
18	#121	C₄	1	4	yes	—
19	C₄×D₄	C₂²	1	4	yes	—
20	C₂×C₄²	C₂²	1	1	yes	characteristic
21	C₄⋊C₈	C₂²	1	4	yes	—
22	C₂²⋊C₈	C₂²	1	8	yes	—
23	C₄×C₈	C₂²	1	2	yes	—
24	C₄×C₈	C₂²	1	2	yes	—
25	#41	—	2	4	no	—
26	#41	—	2	4	no	—
27	C₂×Q₈	C₂×C₄	1	2	yes	—
28	C₂×D₄	C₂×C₄	1	4	yes	—
29	C₂×D₄	C₂×C₄	1	4	yes	—
30	C₂×D₄	C₂×C₄	1	2	yes	—
31	C₂²×C₄	C₂×C₄	1	2	yes	—
32	C₂²×C₄	C₂³	1	4	yes	—
33	C₂²×C₄	C₂³	1	1	yes	characteristic
34	M₁₆	—	2	4	no	—
35	M₁₆	—	2	8	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	8	no	—
38	C₂×C₈	D₄	1	4	yes	complemented
39	C₂×C₈	C₂³	1	2	yes	—
40	C₂×C₈	C₂³	1	4	yes	—
41	C₂×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₈	C₂³	1	1	yes	characteristic
43	C₄⋊C₄	C₂×C₄	1	4	yes	—
44	C₂²⋊C₄	C₂×C₄	1	8	yes	—
45	C₄²	C₂×C₄	1	1	yes	characteristic
46	C₄²	C₂×C₄	1	1	yes	characteristic
47	C₄²	C₂³	1	2	yes	—
48	C₂³	C₂²×C₄	1	4	yes	—
49	C₂³	C₂²×C₄	1	1	yes	characteristic
50	Q₈	—	2	4	no	—
51	D₄	—	2	8	no	—
52	D₄	—	2	4	no	—
53	D₄	—	2	8	no	—
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	C₂²×C₄	1	2	yes	—
58	C₂×C₄	C₂²×C₄	1	4	yes	—
59	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄	C₂²×C₄	1	2	yes	—
61	C₂×C₄	C₂×D₄	1	2	yes	—
62	C₂×C₄	#41	1	2	yes	—
63	C₂×C₄	C₂⁴	1	1	yes	characteristic
64	C₈	—	2	4	no	—
65	C₈	—	2	2	no	—
66	C₈	C₂×D₄	1	2	yes	—
67	C₈	C₂×D₄	1	2	yes	—
68	C₂²	—	2	2	no	—
69	C₂²	—	2	4	no	—
70	C₂²	—	2	8	no	—
71	C₂²	C₄×D₄	1	2	yes	—
72	C₂²	C₂³×C₄	1	1	yes	characteristic
73	C₄	—	2	4	no	—
74	C₄	—	2	2	no	—
75	C₄	C₄×D₄	1	2	yes	—
76	C₄	#131	1	4	yes	complemented
77	C₄	C₂²×D₄	1	1	yes	characteristic
78	C₄	C₂×#41	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	—	4	4	no	—
81	C₂	C₂×C₄×D₄	1	1	yes	characteristic
82	C₂	C₂×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂³
commutator, socle	C₂²	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#131	C₂	semidirect
3. #436	C₂	semidirect
4. C₈×D₄	C₂	semidirect
5. #423	C₂	semidirect
6. #408	C₂	semidirect
7. C₂×C₄×C₈	C₂	semidirect
38. C₂×C₈	D₄	semidirect
76. C₄	#131	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131	2
#532	2
#436	2
C₈×D₄	2
#423	2
#408	2
C₂×C₄×C₈	2