Subgroups — Q₈×#131 #60194

Order 256.

941 conjugacy classes of subgroups.

The subgroup lattice has 131 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆	C₂	1	3	yes	maximal, complemented
3	#2962	C₂	1	9	yes	maximal, complemented
4	#2960	C₂	1	9	yes	maximal, complemented
5	C₂×C₈×Q₈	C₂	1	3	yes	maximal, complemented
6	#2906	C₂	1	3	yes	maximal
7	C₄×#131	C₂	1	3	yes	maximal
8	C₂×#131	C₂²	1	3	yes	—
9	C₂×#131	C₂²	1	1	yes	characteristic
10	Q₈²	C₄	1	1	yes	characteristic
11	#557	C₄	1	3	yes	—
12	#554	C₄	1	9	yes	—
13	D₄×Q₈	C₄	1	3	yes	—
14	#533	C₂²	1	9	yes	—
15	C₄×#41	C₂²	1	3	yes	—
16	C₂×C₄×Q₈	C₂²	1	3	yes	—
17	#446	C₂²	1	18	yes	—
18	#446	C₂²	1	18	yes	—
19	C₈×Q₈	C₂²	1	1	yes	characteristic
20	C₈×Q₈	C₂²	1	9	yes	—
21	C₈×Q₈	C₂²	1	9	yes	—
22	C₈×Q₈	C₂²	1	3	yes	—
23	C₈×Q₈	C₂²	1	6	yes	—
24	#424	C₂²	1	18	yes	—
25	#423	C₂²	1	9	yes	—
26	C₂×C₄⋊C₈	C₂²	1	9	yes	—
27	#405	C₂²	1	9	yes	—
28	#405	C₂²	1	9	yes	—
29	C₄×M₁₆	C₂²	1	9	yes	—
30	C₂×C₄×C₈	C₂²	1	9	yes	—
31	C₂×#41	C₂³	1	1	yes	characteristic
32	C₂²×Q₈	C₂×C₄	1	3	yes	—
33	#131	—	2	12	no	—
34	#131	Q₈	1	8	yes	direct product factor
35	C₂×M₁₆	C₂³	1	9	yes	—
36	C₂×M₁₆	C₂³	1	3	yes	—
37	C₂²×C₈	C₂³	1	9	yes	—
38	C₂²×C₈	C₂³	1	3	yes	—
39	C₄⋊Q₈	C₂×C₄	1	9	yes	—
40	C₄⋊Q₈	C₂×C₄	1	9	yes	—
41	#125	C₂×C₄	1	18	yes	—
42	C₂²⋊Q₈	C₂×C₄	1	18	yes	—
43	C₂²⋊Q₈	C₂×C₄	1	18	yes	—
44	C₄×Q₈	C₂×C₄	1	3	yes	—
45	C₄×Q₈	C₂×C₄	1	9	yes	—
46	C₄×Q₈	C₂×C₄	1	3	yes	—
47	C₄×Q₈	C₂×C₄	1	3	yes	—
48	C₄×Q₈	C₂³	1	1	yes	characteristic
49	C₄×Q₈	C₂³	1	9	yes	—
50	C₄×Q₈	C₂³	1	6	yes	—
51	C₄×D₄	C₂×C₄	1	9	yes	—
52	C₄×D₄	C₂×C₄	1	9	yes	—
53	#117	C₂³	1	9	yes	—
54	C₂×C₄⋊C₄	C₂×C₄	1	9	yes	—
55	C₂×C₄²	C₂³	1	9	yes	—
56	C₄⋊C₈	C₂³	1	3	yes	—
57	C₄⋊C₈	C₂³	1	9	yes	—
58	C₄⋊C₈	C₂³	1	18	yes	—
59	C₄⋊C₈	C₂³	1	18	yes	—
60	#97	C₂³	1	18	yes	—
61	C₄×C₈	C₂³	1	9	yes	—
62	C₄×C₈	C₂³	1	9	yes	—
63	C₄×C₈	C₂³	1	3	yes	—
64	C₄×C₈	C₂³	1	9	yes	—
65	#41	C₂×Q₈	1	4	yes	—
66	#41	#41	1	4	yes	—
67	C₂×Q₈	—	2	12	no	—
68	C₂×Q₈	C₂²×C₄	1	9	yes	—
69	C₂×Q₈	C₂²×C₄	1	6	yes	—
70	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
71	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×D₄	C₂²×C₄	1	3	yes	—
73	C₂²×C₄	C₂²×C₄	1	9	yes	—
74	C₂²×C₄	C₂²×C₄	1	9	yes	—
75	C₂²×C₄	C₂⁴	1	3	yes	—
76	M₁₆	—	2	18	no	—
77	M₁₆	C₂×Q₈	1	12	yes	complemented
78	C₂×C₈	—	2	18	no	—
79	C₂×C₈	C₂×Q₈	1	12	yes	complemented
80	C₂×C₈	C₂⁴	1	3	yes	—
81	C₂×C₈	C₂⁴	1	9	yes	—
82	C₂×C₈	C₂⁴	1	3	yes	—
83	C₂×C₈	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄	C₂²×C₄	1	3	yes	—
85	C₄⋊C₄	C₂²×C₄	1	18	yes	—
86	C₄⋊C₄	C₂²×C₄	1	9	yes	—
87	C₄⋊C₄	C₂²×C₄	1	9	yes	—
88	C₄⋊C₄	C₂²×C₄	1	9	yes	—
89	C₄⋊C₄	C₂²×C₄	1	18	yes	—
90	C₂²⋊C₄	C₂²×C₄	1	18	yes	—
91	C₄²	C₂²×C₄	1	9	yes	—
92	C₄²	C₂²×C₄	1	9	yes	—
93	C₄²	C₂⁴	1	3	yes	—
94	C₄²	C₂⁴	1	9	yes	—
95	C₂³	C₂³×C₄	1	3	yes	—
96	Q₈	—	2	18	no	—
97	Q₈	—	4	6	no	—
98	Q₈	C₄×Q₈	1	4	yes	—
99	Q₈	#131	1	4	yes	direct product factor
100	D₄	C₄×Q₈	1	12	yes	—
101	C₂×C₄	—	2	18	no	—
102	C₂×C₄	—	4	9	no	—
103	C₂×C₄	C₂³×C₄	1	3	yes	—
104	C₂×C₄	C₂³×C₄	1	9	yes	—
105	C₂×C₄	C₂³×C₄	1	9	yes	—
106	C₂×C₄	C₂³×C₄	1	3	yes	—
107	C₂×C₄	C₂³×C₄	1	3	yes	—
108	C₂×C₄	C₂²×Q₈	1	6	yes	—
109	C₂×C₄	C₂×#41	1	6	yes	—
110	C₂×C₄	C₂⁵	1	1	yes	characteristic
111	C₈	—	2	3	no	—
112	C₈	—	2	9	no	—
113	C₈	C₂²×Q₈	1	2	yes	—
114	C₈	C₂²×Q₈	1	6	yes	—
115	C₂²	—	2	3	no	—
116	C₂²	—	2	3	no	—
117	C₂²	C₂×C₄×Q₈	1	6	yes	—
118	C₂²	C₂⁴×C₄	1	1	yes	characteristic
119	C₄	—	2	9	no	—
120	C₄	—	2	3	no	—
121	C₄	—	2	9	no	—
122	C₄	C₂×C₄×Q₈	1	6	yes	—
123	C₄	C₂×#131	1	6	yes	—
124	C₄	C₂³×Q₈	1	1	yes	characteristic
125	C₄	C₂²×#41	1	1	yes	characteristic
126	C₂	—	2	6	no	—
127	C₂	C₂²×C₄×Q₈	1	1	yes	characteristic
128	C₂	C₂²×#131	1	1	yes	characteristic
129	C₂	#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	C₂×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆	C₂	semidirect
3. #2962	C₂	semidirect
4. #2960	C₂	semidirect
5. C₂×C₈×Q₈	C₂	semidirect
34. #131	Q₈	direct (N × H)
77. M₁₆	C₂×Q₈	semidirect
79. C₂×C₈	C₂×Q₈	semidirect
99. Q₈	#131	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#41	2
Q₈×M₁₆	2
#2962	2
#2960	2
C₂×C₈×Q₈	2
#2906	2
C₄×#131	2

Subgroups — Q8×#131 #60194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#131 #60194