Subgroups — #2950

Order 128.

228 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	#519	C₂	1	2	yes	maximal
3	#441	C₂	1	4	yes	maximal, complemented
4	C₄×Q₁₆	C₂	1	4	yes	maximal, complemented
5	#427	C₂	1	1	yes	maximal, characteristic
6	#419	C₂	1	2	yes	maximal
7	#405	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈	C₂²	1	2	yes	—
9	C₂×Q₁₆	C₄	1	8	yes	complemented
10	C₂×Q₁₆	C₂²	1	2	yes	—
11	C₂×Q₁₆	C₂²	1	2	yes	—
12	C₂×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	C₄×Q₈	—	2	4	no	—
15	C₄×Q₈	C₂²	1	8	yes	—
16	#117	—	2	4	no	—
17	#117	C₂²	1	2	yes	—
18	#117	C₂²	1	1	yes	characteristic
19	#107	C₂²	1	2	yes	—
20	#106	C₂²	1	2	yes	—
21	#103	C₂²	1	8	yes	—
22	#97	C₂²	1	2	yes	—
23	C₄×C₈	C₂²	1	2	yes	—
24	C₂×Q₈	—	2	2	no	—
25	C₂×Q₈	—	2	4	no	—
26	C₂×Q₈	C₂×C₄	1	8	yes	—
27	C₂×Q₈	C₂³	1	4	yes	—
28	C₂²×C₄	—	2	2	no	—
29	C₂²×C₄	C₂³	1	1	yes	characteristic
30	Q₁₆	C₂×C₄	1	8	yes	—
31	Q₁₆	C₂×C₄	1	8	yes	—
32	M₁₆	—	2	2	no	—
33	C₂×C₈	C₂×C₄	1	4	yes	—
34	C₂×C₈	C₂³	1	2	yes	—
35	C₂×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈	C₂³	1	1	yes	characteristic
37	C₄⋊C₄	—	2	8	no	—
38	C₄⋊C₄	D₄	1	2	yes	—
39	C₄⋊C₄	C₂³	1	4	yes	—
40	C₂²⋊C₄	—	2	2	no	—
41	C₂²⋊C₄	D₄	1	2	yes	—
42	C₄²	—	2	8	no	—
43	C₄²	—	2	2	no	—
44	C₄²	C₂³	1	2	yes	—
45	C₂³	C₂×D₄	1	1	yes	characteristic
46	Q₈	—	2	8	no	—
47	Q₈	—	2	4	no	—
48	Q₈	C₂²×C₄	1	8	yes	—
49	C₂×C₄	—	2	8	no	—
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	C₂²×C₄	1	2	yes	—
53	C₂×C₄	C₂×D₄	1	4	yes	—
54	C₂×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄	#41	1	2	yes	—
56	C₂×C₄	C₂⁴	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₈	C₂²×C₄	1	2	yes	—
59	C₈	C₂²×C₄	1	2	yes	—
60	C₂²	—	2	2	no	—
61	C₂²	C₄×D₄	1	2	yes	—
62	C₂²	C₂²×D₄	1	1	yes	characteristic
63	C₄	—	2	8	no	—
64	C₄	—	2	4	no	—
65	C₄	—	4	4	no	—
66	C₄	C₄×D₄	1	2	yes	—
67	C₄	C₂³×C₄	1	1	yes	characteristic
68	C₄	C₂×#41	1	1	yes	characteristic
69	C₂	—	2	2	no	—
70	C₂	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂	#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator	C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #441	C₂	semidirect
4. C₄×Q₁₆	C₂	semidirect
9. C₂×Q₁₆	C₄	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
#519	2
#441	2
C₄×Q₁₆	2
#427	2
#419	2
#405	2