Subgroups — #8294

Order 256.

179 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#108	C₂	1	1	yes	maximal, characteristic
2	#1564	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆	C₂	1	2	yes	maximal
4	C₂×#322	C₂	1	1	yes	maximal, characteristic
5	#1278	C₂	1	2	yes	maximal
6	#431	C₂²	1	2	yes	—
7	C₂×#108	C₄	1	2	yes	—
8	C₄×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#341	C₄	1	2	yes	—
13	#335	C₄	1	2	yes	—
14	#334	C₄	1	2	yes	—
15	#322	—	2	2	no	—
16	#322	C₄	1	2	yes	—
17	#322	C₄	1	2	yes	—
18	C₈²	—	2	2	no	—
19	C₂×M₁₆	C₂×C₄	1	4	yes	—
20	C₂²×C₈	—	2	1	no	—
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈	Q₈	1	2	yes	—
24	C₂×C₄²	C₂³	1	1	yes	characteristic
25	#108	C₈	1	8	yes	complemented
26	C₄⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²⋊C₈	—	2	2	no	—
28	#97	—	2	2	no	—
29	C₄×C₈	—	2	2	no	—
30	C₄×C₈	—	2	2	no	—
31	C₄×C₈	—	2	1	no	—
32	C₄×C₈	C₂×C₄	1	2	yes	—
33	C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈	D₄	1	2	yes	—
39	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄	#41	1	1	yes	characteristic
42	M₁₆	—	2	4	no	—
43	M₁₆	C₂×C₈	1	8	yes	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	C₄²	1	1	yes	characteristic
51	C₂×C₈	C₄²	1	1	yes	characteristic
52	C₂×C₈	C₄²	1	1	yes	characteristic
53	C₂×C₈	C₄²	1	1	yes	characteristic
54	C₂×C₈	C₄⋊C₄	1	4	yes	—
55	C₂×C₈	C₂×C₈	1	4	yes	—
56	C₄²	C₂²×C₄	1	1	yes	characteristic
57	C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₄²	C₂×D₄	1	1	yes	characteristic
59	C₄²	#41	1	1	yes	characteristic
60	C₂³	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄	#117	1	1	yes	characteristic
66	C₂×C₄	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄	C₂²×C₈	1	2	yes	—
70	C₂×C₄	#131	1	2	yes	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	—	2	4	no	—
74	C₈	—	2	8	no	—
75	C₈	—	4	2	no	—
76	C₈	—	4	4	no	—
77	C₈	C₄×C₈	1	2	yes	—
78	C₈	C₄×C₈	1	2	yes	—
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²	C₈×Q₈	1	2	yes	—
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄	#405	1	1	yes	characteristic
87	C₄	#430	1	2	yes	—
88	C₄	C₈×D₄	1	2	yes	—
89	C₄	#444	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	C₈×C₄⋊C₄	1	1	yes	characteristic
92	C₂	#1773	1	1	yes	characteristic
93	C₂	#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₄×C₄⋊C₄
prime radical	C₂³	C₄×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
25. #108	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#108	2
#1564	2
C₈×M₁₆	2
C₂×#322	2
#1278	2