Subgroups — #13180

Order 256.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2159	C₂	1	2	yes	maximal
2	C₂×#364	C₂	1	1	yes	maximal, characteristic
3	#1846	C₂	1	1	yes	maximal, characteristic
4	#1772	C₂	1	1	yes	maximal, characteristic
5	#1387	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂×#108	C₂²	1	1	yes	characteristic
8	#427	C₂²	1	1	yes	characteristic
9	C₂×#107	C₂²	1	2	yes	—
10	C₂×#106	—	2	1	no	—
11	#405	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#382	—	2	1	no	—
14	#367	—	2	2	no	—
15	#366	—	2	2	no	—
16	#364	C₄	1	4	yes	complemented
17	#339	—	2	1	no	—
18	C₂×M₁₆	—	2	1	no	—
19	C₂²×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	#117	—	2	1	no	—
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄²	D₄	1	1	yes	characteristic
24	M₃₂	—	2	2	no	—
25	M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₆	—	2	2	no	—
27	#108	—	2	1	no	—
28	#108	Q₈	1	2	yes	—
29	#107	—	2	2	no	—
30	#107	—	2	2	no	—
31	#107	D₄	1	2	yes	—
32	#106	—	2	1	no	—
33	#106	—	4	1	no	—
34	#106	—	4	1	no	—
35	#97	—	4	1	no	—
36	C₄×C₈	—	4	1	no	—
37	C₄×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈	Q₈	1	2	yes	—
39	#95	—	4	2	no	—
40	C₂²×C₄	—	2	1	no	—
41	C₂²×C₄	—	4	2	no	—
42	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
43	M₁₆	—	2	2	no	—
44	M₁₆	—	4	1	no	—
45	C₂×C₈	—	4	1	no	—
46	C₂×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₈	D₈	1	1	yes	characteristic
48	C₂×C₈	D₈	1	1	yes	characteristic
49	C₂×C₈	Q₁₆	1	2	yes	—
50	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈	#41	1	1	yes	characteristic
55	C₂×C₈	#41	1	1	yes	characteristic
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₄²	—	2	1	no	—
61	C₄²	—	4	1	no	—
62	C₄²	C₄⋊C₄	1	2	yes	—
63	C₁₆	—	2	4	no	—
64	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	4	2	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	—	8	4	no	—
69	C₂×C₄	—	8	2	no	—
70	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄	#121	1	1	yes	characteristic
72	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄	#137	1	1	yes	characteristic
76	C₈	—	4	2	no	—
77	C₈	#107	1	2	yes	—
78	C₈	#107	1	2	yes	—
79	C₈	C₄×D₄	1	1	yes	characteristic
80	C₈	C₄×Q₈	1	1	yes	characteristic
81	C₈	#125	1	1	yes	characteristic
82	C₈	C₄⋊Q₈	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	#428	1	1	yes	characteristic
85	C₂²	#459	1	1	yes	characteristic
86	C₂²	#477	1	1	yes	characteristic
87	C₄	—	4	2	no	—
88	C₄	—	8	2	no	—
89	C₄	—	16	2	no	—
90	C₄	#389	1	1	yes	characteristic
91	C₄	C₂×#107	1	1	yes	characteristic
92	C₄	#462	1	1	yes	characteristic
93	C₄	#474	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	2	1	no	—
96	C₂	#1947	1	1	yes	characteristic
97	C₂	#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#459
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #364	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2159	2
C₂×#364	2
#1846	2
#1772	2
#1387	2