Subgroups — #1810

Order 128.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#108	C₂	1	2	yes	maximal, complemented
3	#407	C₂	1	2	yes	maximal
4	#341	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄	C₂²	1	1	yes	characteristic
6	C₂×M₁₆	C₂²	1	4	yes	—
7	C₂²×C₈	—	2	4	no	—
8	C₂²×C₈	C₄	1	4	yes	—
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	#108	—	2	4	no	—
12	C₂²⋊C₈	—	2	4	no	—
13	C₂⁴	Q₈	1	1	yes	characteristic
14	C₂²×C₄	—	2	2	no	—
15	C₂²×C₄	—	2	2	no	—
16	C₂²×C₄	C₂×C₄	1	2	yes	—
17	C₂²×C₄	D₄	1	2	yes	—
18	C₂²×C₄	Q₈	1	1	yes	characteristic
19	C₂²×C₄	C₂³	1	1	yes	characteristic
20	M₁₆	—	2	4	no	—
21	M₁₆	—	2	4	no	—
22	C₂×C₈	—	2	2	no	—
23	C₂×C₈	—	2	8	no	—
24	C₂×C₈	—	2	4	no	—
25	C₂×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₈	D₄	1	4	yes	—
28	C₂³	—	2	4	no	—
29	C₂³	—	2	2	no	—
30	C₂³	C₄⋊C₄	1	2	yes	—
31	C₂³	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄	—	2	2	no	—
33	C₂×C₄	—	2	2	no	—
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	—	2	4	no	—
36	C₂×C₄	C₄⋊C₄	1	2	yes	—
37	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄	C₂×D₄	1	2	yes	—
39	C₂×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄	#41	1	2	yes	—
41	C₈	—	2	2	no	—
42	C₈	—	4	4	no	—
43	C₈	C₂²⋊C₄	1	4	yes	—
44	C₂²	—	2	8	no	—
45	C₂²	—	2	2	no	—
46	C₂²	—	2	2	no	—
47	C₂²	—	2	2	no	—
48	C₂²	#108	1	4	yes	complemented
49	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²	C₂²⋊Q₈	1	2	yes	—
51	C₄	—	2	2	no	—
52	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄	#117	1	1	yes	characteristic
54	C₄	#121	1	2	yes	—
55	C₂	—	2	4	no	—
56	C₂	—	2	2	no	—
57	C₂	#380	1	1	yes	characteristic
58	C₂	C₂×#108	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₂×C₄⋊C₄
prime radical	C₂⁴	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#108	C₂	semidirect
4. #341	C₂	semidirect
48. C₂²	#108	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈	2
C₂×#108	2
#407	2
#341	2