Subgroups — #8274

Order 256.

219 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1767	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	2	yes	maximal
4	#1277	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×#97	—	2	2	no	—
7	C₂×C₄×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄³	C₂²	1	1	yes	characteristic
12	#336	—	2	2	no	—
13	#335	C₄	1	8	yes	complemented
14	C₂²×C₈	—	2	4	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²	C₂×C₄	1	2	yes	—
19	C₂×C₄²	D₄	1	1	yes	characteristic
20	C₂×C₄²	D₄	1	1	yes	characteristic
21	C₂×C₄²	Q₈	1	2	yes	—
22	C₂×C₄²	C₂³	1	1	yes	characteristic
23	C₄⋊C₈	—	2	4	no	—
24	C₄⋊C₈	C₂×C₄	1	8	yes	—
25	#97	—	4	4	no	—
26	C₄×C₈	—	2	4	no	—
27	C₄×C₈	—	2	4	no	—
28	C₄×C₈	—	2	2	no	—
29	C₄×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	—	2	8	no	—
39	C₂×C₈	—	4	4	no	—
40	C₂×C₈	—	4	4	no	—
41	C₂×C₈	C₄²	1	2	yes	—
42	C₂×C₈	C₄²	1	2	yes	—
43	C₂×C₈	M₁₆	1	4	yes	—
44	C₂×C₈	M₁₆	1	4	yes	—
45	C₄²	—	2	4	no	—
46	C₄²	—	2	2	no	—
47	C₄²	—	2	2	no	—
48	C₄²	—	2	2	no	—
49	C₄²	D₈	1	2	yes	—
50	C₄²	Q₁₆	1	2	yes	—
51	C₄²	C₂²×C₄	1	2	yes	—
52	C₄²	#41	1	2	yes	—
53	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	#107	1	4	yes	—
60	C₂×C₄	#108	1	4	yes	—
61	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄	#117	1	1	yes	characteristic
63	C₂×C₄	C₄×D₄	1	2	yes	—
64	C₂×C₄	C₄×Q₈	1	2	yes	—
65	C₂×C₄	C₂×M₁₆	1	2	yes	—
66	C₂×C₄	C₂×M₁₆	1	2	yes	—
67	C₂×C₄	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄	#135	1	2	yes	—
70	C₈	—	2	2	no	—
71	C₈	—	2	4	no	—
72	C₈	—	4	8	no	—
73	C₈	#97	1	4	yes	complemented
74	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
75	C₂²	#423	1	2	yes	—
76	C₂²	C₂×#107	1	1	yes	characteristic
77	C₂²	#427	1	1	yes	characteristic
78	C₂²	C₂×#108	1	2	yes	—
79	C₄	—	2	4	no	—
80	C₄	—	2	4	no	—
81	C₄	—	2	2	no	—
82	C₄	C₂×#97	1	1	yes	characteristic
83	C₄	C₄×M₁₆	1	1	yes	characteristic
84	C₄	#432	1	2	yes	—
85	C₄	C₄×D₈	1	2	yes	—
86	C₄	C₄×Q₁₆	1	2	yes	—
87	C₂	#1567	1	2	yes	—
88	C₂	#1568	1	2	yes	—
89	C₂	#1767	1	1	yes	characteristic
90	C₂	C₄×#107	1	1	yes	characteristic
91	C₂	C₄×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
13. #335	C₄	semidirect
73. C₈	#97	semidirect

Maximal subgroups

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

subgroup	index
#1767	2
C₄²×C₈	2
C₂×#335	2
#1277	2