Subgroups — #2232

Order 128.

75 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#466	C₂	1	1	yes	maximal, characteristic
2	C₂×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	#366	C₂	1	2	yes	maximal, complemented
4	#357	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈	C₂²	1	1	yes	characteristic
7	C₂²×C₈	C₂²	1	1	yes	characteristic
8	#121	—	2	1	no	—
9	C₂×C₄⋊C₄	—	2	1	no	—
10	C₂×C₁₆	C₂²	1	2	yes	—
11	#107	—	2	1	no	—
12	#107	C₂²	1	2	yes	—
13	#107	C₂²	1	1	yes	characteristic
14	#102	—	2	1	no	—
15	C₂×D₄	—	2	1	no	—
16	C₂×D₄	—	4	1	no	—
17	C₂²×C₄	—	4	1	no	—
18	C₂²×C₄	D₄	1	1	yes	characteristic
19	D₈	—	2	2	no	—
20	C₂×C₈	—	2	2	no	—
21	C₂×C₈	D₄	1	1	yes	characteristic
22	C₂×C₈	C₂³	1	1	yes	characteristic
23	C₄⋊C₄	—	2	1	no	—
24	C₄⋊C₄	—	2	2	no	—
25	C₄⋊C₄	—	4	1	no	—
26	C₂²⋊C₄	—	4	1	no	—
27	C₁₆	—	2	2	no	—
28	C₂³	—	4	1	no	—
29	C₂³	D₈	1	1	yes	characteristic
30	D₄	—	4	1	no	—
31	D₄	—	4	1	no	—
32	D₄	—	8	2	no	—
33	C₂×C₄	—	2	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	4	1	no	—
36	C₂×C₄	—	4	2	no	—
37	C₂×C₄	—	8	2	no	—
38	C₂×C₄	D₈	1	1	yes	characteristic
39	C₂×C₄	C₂×D₄	1	1	yes	characteristic
40	C₈	—	2	1	no	—
41	C₈	#41	1	2	yes	—
42	C₂²	—	2	1	no	—
43	C₂²	—	2	1	no	—
44	C₂²	—	8	1	no	—
45	C₂²	—	8	1	no	—
46	C₂²	—	8	1	no	—
47	C₂²	D₁₆	1	2	yes	—
48	C₂²	C₂×D₈	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	8	2	no	—
51	C₄	—	8	1	no	—
52	C₄	#123	1	1	yes	characteristic
53	C₄	#137	1	1	yes	characteristic
54	C₂	—	2	2	no	—
55	C₂	—	16	1	no	—
56	C₂	#480	1	1	yes	characteristic
57	C₂	C₂×D₁₆	1	1	yes	characteristic
58	C₂	#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#107	C₂	semidirect
3. #366	C₂	semidirect
4. #357	C₂	semidirect
5. C₂²⋊C₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#466	2
C₂×#107	2
#366	2
#357	2
C₂²⋊C₁₆	2