Subgroups — #2229

Order 128.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#367	C₂	1	1	yes	maximal, characteristic
4	#366	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#358	C₂	1	1	yes	maximal, characteristic
7	#358	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆	C₂²	1	1	yes	characteristic
9	#125	—	2	1	no	—
10	C₄×Q₈	—	2	1	no	—
11	C₂×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₆	C₂²	1	1	yes	characteristic
13	#107	C₂²	1	1	yes	characteristic
14	#107	C₂²	1	1	yes	characteristic
15	#107	C₂²	1	1	yes	characteristic
16	#106	—	2	1	no	—
17	#103	—	2	1	no	—
18	C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×Q₈	—	2	1	no	—
20	Q₁₆	—	2	1	no	—
21	Q₁₆	Q₈	1	2	yes	—
22	C₂×C₈	D₄	1	1	yes	characteristic
23	C₂×C₈	C₂³	1	1	yes	characteristic
24	C₄⋊C₄	—	2	1	no	—
25	C₄⋊C₄	—	2	1	no	—
26	C₄⋊C₄	—	2	1	no	—
27	C₄⋊C₄	—	4	1	no	—
28	C₄⋊C₄	—	4	1	no	—
29	C₄⋊C₄	—	4	1	no	—
30	C₄²	—	4	1	no	—
31	C₄²	D₄	1	1	yes	characteristic
32	C₁₆	—	2	1	no	—
33	C₁₆	—	2	1	no	—
34	Q₈	—	2	2	no	—
35	Q₈	—	4	1	no	—
36	C₂×C₄	—	4	1	no	—
37	C₂×C₄	—	4	1	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	4	1	no	—
40	C₂×C₄	D₈	1	1	yes	characteristic
41	C₂×C₄	D₈	1	1	yes	characteristic
42	C₂×C₄	C₂×D₄	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₈	C₂×Q₈	1	1	yes	characteristic
45	C₈	#41	1	1	yes	characteristic
46	C₂²	C₂×D₈	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	2	1	no	—
49	C₄	—	4	2	no	—
50	C₄	—	8	1	no	—
51	C₄	—	8	1	no	—
52	C₄	—	8	1	no	—
53	C₄	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄	#137	1	1	yes	characteristic
55	C₂	#474	1	1	yes	characteristic
56	C₂	#508	1	1	yes	characteristic
57	C₂	#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#499	2
C₄×Q₁₆	2
#367	2
#366	2
C₄⋊C₁₆	2
#358	2