Subgroups — #1969

Order 128.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×#103	C₂	1	2	yes	maximal
4	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	#391	C₂	1	2	yes	maximal
6	C₂²×Q₈	C₂²	1	2	yes	—
7	C₂×Q₁₆	—	2	2	no	—
8	C₂×Q₁₆	—	2	2	no	—
9	C₂×Q₁₆	C₄	1	4	yes	complemented
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²	C₂²	1	1	yes	characteristic
14	#107	—	2	2	no	—
15	#103	—	2	4	no	—
16	C₄×C₈	—	2	2	no	—
17	#95	—	2	4	no	—
18	C₂×Q₈	—	2	2	no	—
19	C₂×Q₈	—	2	2	no	—
20	C₂×Q₈	—	2	2	no	—
21	C₂×Q₈	—	2	2	no	—
22	C₂×Q₈	—	2	2	no	—
23	C₂×Q₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄	—	2	2	no	—
25	C₂²×C₄	—	2	2	no	—
26	C₂²×C₄	D₄	1	2	yes	—
27	C₂²×C₄	C₂³	1	1	yes	characteristic
28	Q₁₆	—	2	8	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₈	D₄	1	2	yes	—
33	C₂×C₈	D₄	1	2	yes	—
34	C₂×C₈	D₄	1	2	yes	—
35	C₄⋊C₄	—	2	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄²	—	2	1	no	—
38	C₄²	—	2	1	no	—
39	C₂³	C₂×D₄	1	1	yes	characteristic
40	Q₈	—	2	4	no	—
41	Q₈	—	2	4	no	—
42	Q₈	—	4	2	no	—
43	Q₈	—	4	2	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	2	4	no	—
47	C₂×C₄	—	4	2	no	—
48	C₂×C₄	—	4	2	no	—
49	C₂×C₄	—	4	2	no	—
50	C₂×C₄	—	4	4	no	—
51	C₂×C₄	Q₁₆	1	4	yes	complemented
52	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄	#41	1	1	yes	characteristic
57	C₂×C₄	#41	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	C₂²⋊C₄	1	4	yes	—
60	C₂²	C₄×D₄	1	1	yes	characteristic
61	C₂²	#121	1	1	yes	characteristic
62	C₂²	#127	1	1	yes	characteristic
63	C₂²	C₂×Q₁₆	1	2	yes	—
64	C₂²	#135	1	2	yes	—
65	C₄	—	2	4	no	—
66	C₄	—	4	4	no	—
67	C₄	—	8	2	no	—
68	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₄	C₄×D₄	1	1	yes	characteristic
70	C₄	#121	1	1	yes	characteristic
71	C₄	#124	1	1	yes	characteristic
72	C₂	#390	1	1	yes	characteristic
73	C₂	C₄×Q₁₆	1	2	yes	—
74	C₂	#467	1	2	yes	—
75	C₂	#494	1	1	yes	characteristic
76	C₂	#495	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
9. C₂×Q₁₆	C₄	semidirect
51. C₂×C₄	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₁₆	2
C₂×#107	2
C₂×#103	2
C₂×C₄×C₈	2
#391	2