Subgroups — Q₈×#106 #22724

Order 256.

419 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄	C₂	1	2	yes	maximal
3	#1948	C₂	1	6	yes	maximal
4	#1844	C₂	1	3	yes	maximal
5	C₄×#106	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈	—	2	2	no	—
7	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
8	C₈×Q₈	C₄	1	8	yes	complemented
9	C₂×#106	C₂²	1	3	yes	—
10	C₂×#106	C₂²	1	6	yes	—
11	C₂×#106	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	3	yes	—
13	C₂×C₄×C₈	C₂²	1	3	yes	—
14	#389	—	2	6	no	—
15	#384	C₂²	1	6	yes	—
16	C₄×C₄⋊C₄	C₂²	1	6	yes	—
17	#340	C₂²	1	6	yes	—
18	C₂²×Q₈	D₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	3	yes	—
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	C₄×Q₈	—	4	8	no	—
22	C₄×Q₈	C₂×C₄	1	4	yes	—
23	C₄×Q₈	Q₈	1	4	yes	—
24	C₂×C₄⋊C₄	—	2	6	no	—
25	C₂×C₄⋊C₄	—	2	6	no	—
26	C₂×C₄⋊C₄	D₄	1	3	yes	—
27	C₂×C₄⋊C₄	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄	C₂³	1	2	yes	—
29	C₂×C₄²	—	2	6	no	—
30	C₂×C₄²	C₂³	1	3	yes	—
31	#106	—	2	6	no	—
32	#106	—	4	6	no	—
33	#106	Q₈	1	4	yes	direct product factor
34	C₄⋊C₈	C₂×C₄	1	12	yes	—
35	C₄×C₈	C₂×C₄	1	12	yes	—
36	#95	—	2	6	no	—
37	C₂×Q₈	C₄⋊C₄	1	4	yes	—
38	C₂×Q₈	SD₁₆	1	8	yes	—
39	C₂²×C₄	—	2	2	no	—
40	C₂²×C₄	—	2	6	no	—
41	C₂²×C₄	C₂×D₄	1	3	yes	—
42	C₂²×C₄	C₂×D₄	1	3	yes	—
43	C₂²×C₄	C₂⁴	1	1	yes	characteristic
44	C₂×C₈	—	2	3	no	—
45	C₂×C₈	—	2	3	no	—
46	C₂×C₈	C₂²×C₄	1	6	yes	—
47	C₂×C₈	C₂²×C₄	1	2	yes	—
48	C₂×C₈	C₂×Q₈	1	2	yes	—
49	C₂×C₈	#41	1	2	yes	—
50	C₄⋊C₄	—	2	2	no	—
51	C₄⋊C₄	—	2	6	no	—
52	C₄⋊C₄	—	2	6	no	—
53	C₄⋊C₄	—	4	12	no	—
54	C₄⋊C₄	—	8	6	no	—
55	C₄⋊C₄	C₄⋊C₄	1	12	yes	—
56	C₄⋊C₄	C₂×Q₈	1	4	yes	—
57	C₄²	—	4	12	no	—
58	C₄²	C₂²×C₄	1	6	yes	—
59	C₄²	C₂×Q₈	1	6	yes	—
60	C₂³	C₂²×D₄	1	1	yes	characteristic
61	Q₈	#106	1	16	yes	direct product factor
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	6	no	—
67	C₂×C₄	—	4	6	no	—
68	C₂×C₄	—	4	6	no	—
69	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
70	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
71	C₂×C₄	C₂×SD₁₆	1	12	yes	—
72	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
74	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
75	C₂×C₄	C₂×#41	1	1	yes	characteristic
76	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
77	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
78	C₈	—	2	6	no	—
79	C₈	C₄×Q₈	1	4	yes	complemented
80	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
81	C₂²	D₄×Q₈	1	1	yes	characteristic
82	C₂²	#550	1	1	yes	characteristic
83	C₂²	C₂²×SD₁₆	1	2	yes	—
84	C₂²	#577	1	2	yes	—
85	C₄	—	2	6	no	—
86	C₄	—	4	4	no	—
87	C₄	—	8	6	no	—
88	C₄	C₂×#106	1	12	yes	—
89	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
90	C₄	#520	1	1	yes	characteristic
91	C₄	#557	1	1	yes	characteristic
92	C₄	Q₈²	1	1	yes	characteristic
93	C₂	Q₈×C₄⋊C₄	1	1	yes	characteristic
94	C₂	C₂²×#106	1	1	yes	characteristic
95	C₂	#2912	1	1	yes	characteristic
96	C₂	Q₈×SD₁₆	1	2	yes	—
97	C₂	#3392	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈	C₄	semidirect
33. #106	Q₈	direct (N × H)
61. Q₈	#106	direct (N × H)
79. C₈	C₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈	2
Q₈×C₄⋊C₄	2
#1948	2
#1844	2
C₄×#106	2

Subgroups — Q8×#106 #22724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#106 #22724