Subgroups — #1846

Order 128.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107	C₂	1	2	yes	maximal
2	C₂×#106	C₂	1	2	yes	maximal
3	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	#382	C₂	1	2	yes	maximal
5	C₂²×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂×C₄²	C₂²	1	1	yes	characteristic
9	#107	—	2	4	no	—
10	#106	—	2	4	no	—
11	C₄×C₈	C₄	1	4	yes	complemented
12	#95	—	2	4	no	—
13	C₂²×C₄	—	2	4	no	—
14	C₂²×C₄	D₄	1	2	yes	—
15	C₂²×C₄	C₂³	1	1	yes	characteristic
16	C₂×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₈	D₄	1	2	yes	—
19	C₂×C₈	D₄	1	2	yes	—
20	C₂×C₈	Q₈	1	2	yes	—
21	C₂×C₈	Q₈	1	2	yes	—
22	C₄⋊C₄	—	2	4	no	—
23	C₄⋊C₄	—	2	4	no	—
24	C₄²	C₂×C₄	1	2	yes	—
25	C₄²	Q₈	1	2	yes	—
26	C₂³	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄	—	2	4	no	—
28	C₂×C₄	—	4	4	no	—
29	C₂×C₄	—	4	8	no	—
30	C₂×C₄	C₄⋊C₄	1	4	yes	—
31	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
37	C₈	C₄⋊C₄	1	4	yes	—
38	C₈	C₄⋊C₄	1	4	yes	—
39	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²	#127	1	1	yes	characteristic
41	C₂²	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²	#135	1	4	yes	—
43	C₄	—	2	4	no	—
44	C₄	—	8	4	no	—
45	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₄	C₄⋊Q₈	1	1	yes	characteristic
48	C₄	C₄⋊Q₈	1	1	yes	characteristic
49	C₂	#384	1	1	yes	characteristic
50	C₂	#427	1	2	yes	—
51	C₂	#495	1	2	yes	—
52	C₂	#499	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. C₄×C₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107	2
C₂×#106	2
C₂×C₄×C₈	2
#382	2