Subgroups — #1647

Order 128.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#547	C₂	1	1	yes	maximal, characteristic
2	#500	C₂	1	1	yes	maximal, characteristic, complemented
3	#474	C₂	1	1	yes	maximal, characteristic, complemented
4	#461	C₂	1	1	yes	maximal, characteristic, complemented
5	#332	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	—	2	1	no	—
9	C₂×SD₁₆	—	2	1	no	—
10	C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	#123	—	2	2	no	—
13	C₂²⋊Q₈	—	2	1	no	—
14	#121	—	2	1	no	—
15	C₄×D₄	C₂²	1	1	yes	characteristic
16	C₄×D₄	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄	—	2	1	no	—
18	#107	—	2	1	no	—
19	#107	—	2	2	no	—
20	C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊C₈	C₂²	1	1	yes	characteristic
22	#103	—	2	1	no	—
23	#102	—	2	1	no	—
24	C₄×C₈	C₂²	1	1	yes	characteristic
25	#41	—	4	2	no	—
26	#41	—	4	2	no	—
27	C₂×Q₈	—	2	1	no	—
28	C₂×Q₈	—	2	1	no	—
29	C₂×D₄	—	2	2	no	—
30	C₂×D₄	D₄	1	1	yes	characteristic
31	C₂×D₄	D₄	1	1	yes	characteristic
32	C₂²×C₄	—	2	2	no	—
33	C₂²×C₄	—	2	1	no	—
34	C₂²×C₄	—	2	1	no	—
35	SD₁₆	—	4	2	no	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₄⋊C₄	D₄	1	2	yes	—
46	C₂²⋊C₄	—	2	2	no	—
47	C₂²⋊C₄	—	2	1	no	—
48	C₂²⋊C₄	—	2	1	no	—
49	C₄²	C₂³	1	1	yes	characteristic
50	C₂³	—	2	1	no	—
51	C₂³	—	2	1	no	—
52	Q₈	—	4	1	no	—
53	Q₈	—	4	1	no	—
54	Q₈	—	4	1	no	—
55	Q₈	—	4	1	no	—
56	D₄	—	2	1	no	—
57	D₄	—	2	1	no	—
58	D₄	—	2	1	no	—
59	D₄	—	4	4	no	—
60	D₄	D₈	1	2	yes	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	4	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄	C₂×D₄	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	—	4	1	no	—
75	C₈	—	4	1	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	#120	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	2	no	—
86	C₄	—	4	1	no	—
87	C₄	—	8	1	no	—
88	C₄	C₂×D₈	1	1	yes	characteristic
89	C₄	#135	1	1	yes	characteristic
90	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₄	#137	1	1	yes	characteristic
92	C₂	—	4	2	no	—
93	C₂	—	8	1	no	—
94	C₂	#447	1	1	yes	characteristic
95	C₂	#452	1	1	yes	characteristic
96	C₂	#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#120
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#547	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #500	C₂	semidirect
3. #474	C₂	semidirect
4. #461	C₂	semidirect
5. #332	C₂	semidirect
6. D₄⋊C₈	C₂	semidirect
7. D₄⋊C₈	C₂	semidirect

Maximal subgroups

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

subgroup	index
#547	2
#500	2
#474	2
#461	2
#332	2
D₄⋊C₈	2