Subgroups — #1451

Order 128.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431	C₂	1	1	yes	maximal, characteristic
2	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	#322	C₂	1	2	yes	maximal, complemented
5	#322	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆	C₄	1	2	yes	—
7	C₂²×C₈	C₄	1	2	yes	—
8	C₂×C₄²	C₂²	1	1	yes	characteristic
9	C₄⋊C₈	C₄	1	2	yes	—
10	C₂²⋊C₈	C₄	1	2	yes	—
11	#97	C₄	1	2	yes	—
12	C₄×C₈	C₄	1	2	yes	—
13	C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₈	C₂²	1	2	yes	—
16	C₄×C₈	C₂²	1	2	yes	—
17	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
20	M₁₆	—	2	4	no	—
21	C₂×C₈	—	2	4	no	—
22	C₂×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₈	C₂×C₄	1	4	yes	—
24	C₂×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₈	C₂×C₄	1	2	yes	—
26	C₄²	C₂×C₄	1	1	yes	characteristic
27	C₄²	C₂×C₄	1	1	yes	characteristic
28	C₄²	C₂×C₄	1	1	yes	characteristic
29	C₄²	C₂³	1	1	yes	characteristic
30	C₂³	C₄²	1	1	yes	characteristic
31	C₂×C₄	—	2	2	no	—
32	C₂×C₄	—	2	2	no	—
33	C₂×C₄	C₄²	1	1	yes	characteristic
34	C₂×C₄	C₄²	1	1	yes	characteristic
35	C₂×C₄	C₄²	1	1	yes	characteristic
36	C₂×C₄	M₁₆	1	4	yes	—
37	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₈	—	2	2	no	—
41	C₈	—	2	4	no	—
42	C₈	—	2	4	no	—
43	C₈	M₁₆	1	4	yes	complemented
44	C₂²	—	2	1	no	—
45	C₂²	—	2	1	no	—
46	C₂²	#97	1	2	yes	—
47	C₂²	C₂×C₄²	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	2	1	no	—
50	C₄	#97	1	2	yes	—
51	C₄	C₂×M₁₆	1	2	yes	—
52	C₄	C₂×M₁₆	1	2	yes	—
53	C₄	#131	1	2	yes	—
54	C₂	—	2	2	no	—
55	C₂	C₂×#97	1	1	yes	characteristic
56	C₂	C₄×M₁₆	1	1	yes	characteristic
57	C₂	#405	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #322	C₂	semidirect
5. #322	C₂	semidirect
43. C₈	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#431	2
C₄×M₁₆	2
C₂×C₄×C₈	2
#322	2