Subgroups — #2922

Order 128.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41	C₂	1	1	yes	maximal, characteristic
2	#433	C₂	1	4	yes	maximal, complemented
3	#431	C₂	1	4	yes	maximal, complemented
4	#424	C₂	1	2	yes	maximal, complemented
5	#408	C₂	1	2	yes	maximal
6	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	1	yes	characteristic
9	C₂×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₈	C₂²	1	2	yes	—
11	C₄×Q₈	C₄	1	2	yes	—
12	C₄×D₄	C₄	1	4	yes	—
13	C₄×D₄	C₄	1	2	yes	—
14	#117	C₂²	1	1	yes	characteristic
15	#117	C₂²	1	2	yes	—
16	C₂×C₄²	C₂²	1	1	yes	characteristic
17	C₂×C₄²	C₂²	1	2	yes	—
18	C₄⋊C₈	C₂²	1	8	yes	—
19	C₂²⋊C₈	C₂²	1	8	yes	—
20	#97	C₂²	1	2	yes	—
21	C₄×C₈	C₂²	1	2	yes	—
22	C₄×C₈	C₂²	1	2	yes	—
23	C₄×C₈	C₂²	1	2	yes	—
24	#41	—	2	4	no	—
25	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄	C₂×C₄	1	2	yes	—
27	C₂×D₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄	C₂×C₄	1	4	yes	—
29	C₂²×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₄	C₂³	1	2	yes	—
31	C₂²×C₄	C₂³	1	1	yes	characteristic
32	M₁₆	—	2	4	no	—
33	C₂×C₈	—	2	4	no	—
34	C₂×C₈	C₂³	1	4	yes	—
35	C₂×C₈	C₂³	1	4	yes	—
36	C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂²⋊C₄	C₂×C₄	1	4	yes	—
39	C₂²⋊C₄	C₂×C₄	1	2	yes	—
40	C₄²	C₂×C₄	1	4	yes	—
41	C₄²	C₂×C₄	1	2	yes	—
42	C₄²	C₂³	1	1	yes	characteristic
43	C₄²	C₂³	1	2	yes	—
44	C₄²	C₂³	1	1	yes	characteristic
45	C₂³	C₂²×C₄	1	2	yes	—
46	C₂³	C₂²×C₄	1	1	yes	characteristic
47	Q₈	—	2	2	no	—
48	D₄	—	2	4	no	—
49	D₄	—	2	2	no	—
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	8	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	C₂²×C₄	1	2	yes	—
54	C₂×C₄	C₂²×C₄	1	2	yes	—
55	C₂×C₄	C₂²×C₄	1	4	yes	—
56	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄	C₂²×C₄	1	2	yes	—
58	C₂×C₄	#41	1	4	yes	—
59	C₂×C₄	C₂⁴	1	1	yes	characteristic
60	C₈	—	2	4	no	—
61	C₈	—	2	4	no	—
62	C₂²	—	2	2	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	4	no	—
65	C₂²	#117	1	2	yes	—
66	C₂²	C₂³×C₄	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	2	no	—
69	C₄	—	2	4	no	—
70	C₄	#117	1	2	yes	—
71	C₄	#131	1	4	yes	—
72	C₄	C₂×#41	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	—	4	2	no	—
75	C₂	C₂×#117	1	1	yes	characteristic
76	C₂	C₂×#131	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂³
commutator, socle	C₂²	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #433	C₂	semidirect
3. #431	C₂	semidirect
4. #424	C₂	semidirect
6. C₄×M₁₆	C₂	semidirect
7. C₂×C₄×C₈	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#41	2
#433	2
#431	2
#424	2
#408	2
C₄×M₁₆	2
C₂×C₄×C₈	2