Subgroups — #1816

Order 128.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	#417	C₂	1	2	yes	maximal
5	#343	C₂	1	2	yes	maximal, complemented
6	C₂×#41	C₂²	1	1	yes	characteristic
7	#131	—	2	2	no	—
8	#131	C₄	1	4	yes	complemented
9	C₂×M₁₆	—	2	1	no	—
10	C₂×M₁₆	C₂²	1	2	yes	—
11	C₂×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₈	—	2	1	no	—
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄	C₂²	1	2	yes	—
15	#108	—	2	2	no	—
16	#107	—	2	1	no	—
17	#107	—	2	1	no	—
18	#103	—	2	2	no	—
19	#102	—	2	2	no	—
20	#41	—	2	1	no	—
21	#41	—	2	1	no	—
22	#41	C₂×C₄	1	2	yes	—
23	#41	Q₈	1	2	yes	—
24	C₂×Q₈	D₄	1	1	yes	characteristic
25	C₂×D₄	—	2	1	no	—
26	C₂×D₄	D₄	1	1	yes	characteristic
27	C₂²×C₄	—	2	1	no	—
28	C₂²×C₄	—	2	2	no	—
29	C₂²×C₄	C₂³	1	1	yes	characteristic
30	M₁₆	—	2	1	no	—
31	M₁₆	—	2	4	no	—
32	M₁₆	—	2	2	no	—
33	M₁₆	—	2	2	no	—
34	M₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	2	4	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₈	D₄	1	1	yes	characteristic
40	C₂×C₈	D₄	1	1	yes	characteristic
41	C₂×C₈	D₄	1	2	yes	—
42	C₄⋊C₄	—	2	2	no	—
43	C₄⋊C₄	—	2	2	no	—
44	C₂³	—	2	1	no	—
45	C₂³	#41	1	1	yes	characteristic
46	Q₈	—	2	1	no	—
47	Q₈	C₄⋊C₄	1	2	yes	—
48	D₄	—	2	2	no	—
49	D₄	—	2	2	no	—
50	D₄	—	2	1	no	—
51	D₄	C₄⋊C₄	1	2	yes	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	8	2	no	—
57	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄	#41	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	4	2	no	—
65	C₈	C₂²⋊C₄	1	2	yes	—
66	C₈	C₂²⋊C₄	1	2	yes	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	2	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	#117	1	1	yes	characteristic
73	C₂²	#121	1	1	yes	characteristic
74	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	8	2	no	—
77	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₄	#121	1	1	yes	characteristic
80	C₄	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	2	1	no	—
83	C₂	—	4	2	no	—
84	C₂	#380	1	1	yes	characteristic
85	C₂	#472	1	1	yes	characteristic
86	C₂	#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#108	C₂	semidirect
3. C₂×#107	C₂	semidirect
5. #343	C₂	semidirect
8. #131	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131	2
C₂×#108	2
C₂×#107	2
#417	2
#343	2