Subgroups — #1833

Order 128.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108	C₂	1	1	yes	maximal, characteristic
2	#427	C₂	1	1	yes	maximal, characteristic
3	#405	C₂	1	1	yes	maximal, characteristic
4	#344	C₂	1	2	yes	maximal
5	H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂×M₁₆	C₂²	1	2	yes	—
7	C₂×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	#117	C₂²	1	2	yes	—
10	#117	C₂²	1	1	yes	characteristic
11	#108	—	2	1	no	—
12	#108	—	2	1	no	—
13	#107	—	2	1	no	—
14	#106	—	2	1	no	—
15	#97	C₄	1	2	yes	complemented
16	C₄×C₈	C₄	1	2	yes	complemented
17	C₂²×C₄	C₂³	1	1	yes	characteristic
18	M₁₆	—	2	1	no	—
19	M₁₆	—	2	1	no	—
20	M₁₆	—	2	2	no	—
21	M₁₆	—	2	2	no	—
22	C₂×C₈	—	2	2	no	—
23	C₂×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈	D₄	1	1	yes	characteristic
27	C₂×C₈	D₄	1	1	yes	characteristic
28	C₂×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₈	Q₈	1	1	yes	characteristic
30	C₄⋊C₄	—	2	1	no	—
31	C₄⋊C₄	—	2	2	no	—
32	C₂²⋊C₄	—	2	1	no	—
33	C₂²⋊C₄	—	2	2	no	—
34	C₄²	—	2	2	no	—
35	C₄²	C₂×C₄	1	2	yes	—
36	C₂³	#41	1	1	yes	characteristic
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	4	2	no	—
39	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄	#41	1	1	yes	characteristic
43	C₂×C₄	#41	1	1	yes	characteristic
44	C₂×C₄	#41	1	1	yes	characteristic
45	C₈	—	2	2	no	—
46	C₈	—	4	2	no	—
47	C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂²	—	4	1	no	—
52	C₂²	#117	1	1	yes	characteristic
53	C₂²	#124	1	1	yes	characteristic
54	C₂²	#125	1	1	yes	characteristic
55	C₄	—	4	2	no	—
56	C₄	—	8	2	no	—
57	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄	#117	1	1	yes	characteristic
59	C₄	#124	1	1	yes	characteristic
60	C₄	#125	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	—	2	1	no	—
63	C₂	—	2	1	no	—
64	C₂	#382	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#382
prime radical	C₂³	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
15. #97	C₄	semidirect
16. 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×#108	2
#427	2
#405	2
#344	2
H(3,ℤ/4)	2