Subgroups — #8780

Order 256.

155 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1726	C₂	1	1	yes	maximal, characteristic
2	#1564	C₂	1	2	yes	maximal
3	#1281	C₂	1	2	yes	maximal
4	#1274	C₂	1	2	yes	maximal
5	#431	C₂²	1	4	yes	—
6	C₂×#97	C₄	1	1	yes	characteristic
7	C₂×#97	C₄	1	1	yes	characteristic
8	C₂×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#376	C₂²	1	1	yes	characteristic
13	#335	—	2	2	no	—
14	#334	—	2	2	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²	C₂³	1	1	yes	characteristic
22	C₄⋊C₈	—	2	4	no	—
23	C₂²⋊C₈	—	2	4	no	—
24	#97	—	2	2	no	—
25	#97	—	2	2	no	—
26	#97	—	2	2	no	—
27	C₄×C₈	—	2	2	no	—
28	C₄×C₈	—	2	4	no	—
29	C₄×C₈	C₈	1	4	yes	complemented
30	C₄×C₈	D₄	1	1	yes	characteristic
31	C₄×C₈	D₄	1	1	yes	characteristic
32	C₄×C₈	Q₈	1	1	yes	characteristic
33	C₄×C₈	Q₈	1	1	yes	characteristic
34	#95	—	2	2	no	—
35	C₂²×C₄	—	2	1	no	—
36	C₂²×C₄	—	2	1	no	—
37	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄	#41	1	2	yes	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	4	4	no	—
46	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈	C₂×C₈	1	2	yes	—
51	C₂×C₈	C₂×C₈	1	2	yes	—
52	C₂×C₈	M₁₆	1	2	yes	—
53	C₂×C₈	M₁₆	1	2	yes	—
54	C₄²	—	2	1	no	—
55	C₄²	—	2	1	no	—
56	C₄²	C₂×C₈	1	2	yes	—
57	C₄²	M₁₆	1	2	yes	—
58	C₄²	C₂×D₄	1	1	yes	characteristic
59	C₄²	C₂×Q₈	1	1	yes	characteristic
60	C₄²	#41	1	2	yes	—
61	C₂³	#117	1	1	yes	characteristic
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	2	no	—
67	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄	#117	1	1	yes	characteristic
69	C₂×C₄	#124	1	2	yes	—
70	C₂×C₄	#125	1	2	yes	—
71	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	2	1	no	—
77	C₈	—	2	2	no	—
78	C₈	—	2	2	no	—
79	C₈	—	8	4	no	—
80	C₈	C₄⋊C₈	1	2	yes	—
81	C₈	C₄⋊C₈	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	#382	1	1	yes	characteristic
84	C₂²	#431	1	1	yes	characteristic
85	C₂²	#432	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	4	2	no	—
89	C₄	—	4	2	no	—
90	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄	#423	1	1	yes	characteristic
92	C₄	#431	1	1	yes	characteristic
93	C₄	#432	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	2	1	no	—
96	C₂	#1831	1	1	yes	characteristic
97	C₂	#1833	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#117
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#382
prime radical	C₂³	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
29. C₄×C₈	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1726	2
#1564	2
#1281	2
#1274	2