Subgroups — #2906

Order 128.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	2	yes	maximal
2	C₄×#41	C₂	1	1	yes	maximal, characteristic
3	#424	C₂	1	6	yes	maximal, complemented
4	#423	C₂	1	3	yes	maximal, complemented
5	C₂×C₄⋊C₈	C₂	1	3	yes	maximal, complemented
6	C₂×#41	C₂²	1	1	yes	characteristic
7	#131	—	2	8	no	—
8	C₂×M₁₆	C₂²	1	6	yes	—
9	C₂²×C₈	C₂²	1	6	yes	—
10	C₄×Q₈	C₄	1	1	yes	characteristic
11	C₄×Q₈	C₄	1	1	yes	characteristic
12	C₄×D₄	C₄	1	3	yes	—
13	C₄×D₄	C₄	1	3	yes	—
14	#117	C₂²	1	3	yes	—
15	C₂×C₄²	C₂²	1	3	yes	—
16	C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₈	C₂²	1	3	yes	—
18	C₄⋊C₈	C₂²	1	6	yes	—
19	C₄⋊C₈	C₂²	1	6	yes	—
20	#41	D₄	1	4	yes	—
21	#41	Q₈	1	4	yes	—
22	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄	C₂×C₄	1	3	yes	—
24	C₂²×C₄	C₂×C₄	1	3	yes	—
25	C₂²×C₄	C₂×C₄	1	3	yes	—
26	C₂²×C₄	C₂³	1	3	yes	—
27	M₁₆	—	2	12	no	—
28	C₂×C₈	—	2	12	no	—
29	C₂×C₈	C₂³	1	2	yes	—
30	C₂×C₈	C₂³	1	6	yes	—
31	C₄⋊C₄	C₂×C₄	1	3	yes	—
32	C₄⋊C₄	C₂×C₄	1	3	yes	—
33	C₂²⋊C₄	C₂×C₄	1	6	yes	—
34	C₄²	C₂×C₄	1	3	yes	—
35	C₄²	C₂×C₄	1	3	yes	—
36	C₄²	C₂³	1	1	yes	characteristic
37	C₄²	C₂³	1	3	yes	—
38	C₂³	C₂²×C₄	1	3	yes	—
39	Q₈	C₄⋊C₄	1	4	yes	—
40	D₄	C₄⋊C₄	1	12	yes	—
41	C₂×C₄	—	2	6	no	—
42	C₂×C₄	—	4	3	no	—
43	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄	C₂²×C₄	1	3	yes	—
45	C₂×C₄	C₂²×C₄	1	3	yes	—
46	C₂×C₄	C₂²×C₄	1	3	yes	—
47	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄	C₂×D₄	1	6	yes	—
49	C₂×C₄	C₂×Q₈	1	6	yes	—
50	C₂×C₄	C₂⁴	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	2	6	no	—
53	C₂²	—	2	3	no	—
54	C₂²	—	2	3	no	—
55	C₂²	C₂×C₄⋊C₄	1	6	yes	—
56	C₂²	C₂³×C₄	1	1	yes	characteristic
57	C₄	—	2	3	no	—
58	C₄	—	2	1	no	—
59	C₄	—	2	3	no	—
60	C₄	C₂×C₄⋊C₄	1	6	yes	—
61	C₄	#131	1	2	yes	complemented
62	C₄	C₂²×D₄	1	1	yes	characteristic
63	C₄	C₂²×Q₈	1	1	yes	characteristic
64	C₂	—	2	6	no	—
65	C₂	C₂²×C₄⋊C₄	1	1	yes	characteristic
66	C₂	C₂×#131	1	1	yes	characteristic
67	C₂	#568	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator, socle	C₂²	C₂³×C₄
prime radical	C₂×#41	C₂²
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
3. #424	C₂	semidirect
4. #423	C₂	semidirect
5. C₂×C₄⋊C₈	C₂	semidirect
61. 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
C₄×#41	2
#424	2
#423	2
C₂×C₄⋊C₈	2