Subgroups — C₂²×C₉×C₄⋊Q₈ #11768162

Order 1152.

2436 conjugacy classes of subgroups.

The subgroup lattice has 135 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₉	C₂	1	2	yes	maximal
2	C₂×C₉×C₄⋊Q₈	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₉×C₄⋊C₄	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₃×C₄⋊Q₈	C₃	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₉	—	2	16	no	—
7	C₂²×Q₈×C₉	C₂²	1	24	yes	—
8	C₂³×C₄×C₉	C₂²	1	4	yes	—
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₉	C₂²	1	2	yes	—
11	C₉×C₄⋊Q₈	C₂²	1	64	yes	direct product factor
12	C₂×C₉×C₄⋊C₄	C₂²	1	48	yes	—
13	C₂×C₄²×C₉	C₂²	1	12	yes	—
14	C₂³×C₃×Q₈	C₂×C₃	1	2	yes	—
15	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	24	yes	—
16	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	4	yes	—
17	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
18	C₂⁴×C₉	C₂³	1	1	yes	characteristic
19	C₂×Q₈×C₉	—	2	96	no	—
20	C₂×Q₈×C₉	C₂³	1	32	yes	—
21	C₂²×C₄×C₉	—	2	16	no	—
22	C₂²×C₄×C₉	D₄	1	8	yes	—
23	C₂²×C₄×C₉	Q₈	1	16	yes	complemented
24	C₂²×C₄×C₉	C₂³	1	24	yes	—
25	C₂²×C₄×C₉	C₂³	1	6	yes	—
26	C₂²×C₄×C₉	C₂³	1	12	yes	—
27	C₉×C₄⋊C₄	C₂³	1	64	yes	—
28	C₄²×C₉	C₂³	1	16	yes	—
29	C₂²×C₄⋊Q₈	C₉	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²×C₃×Q₈	—	2	16	no	—
31	C₂²×C₃×Q₈	C₂²×C₃	1	24	yes	—
32	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
33	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
34	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
35	C₃×C₄⋊Q₈	C₂²×C₃	1	64	yes	—
36	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	48	yes	—
37	C₂×C₃×C₄²	C₂²×C₃	1	12	yes	—
38	C₂³×C₉	C₂×D₄	1	4	yes	—
39	C₂³×C₉	C₂×Q₈	1	8	yes	—
40	C₂³×C₉	C₂⁴	1	3	yes	—
41	Q₈×C₉	—	2	64	no	—
42	C₂×C₄×C₉	—	2	48	no	—
43	C₂×C₄×C₉	C₂×D₄	1	24	yes	—
44	C₂×C₄×C₉	C₂×Q₈	1	48	yes	complemented
45	C₂×C₄×C₉	C₂⁴	1	16	yes	—
46	C₂×C₄×C₉	C₂⁴	1	4	yes	—
47	C₂×C₄×C₉	C₂⁴	1	8	yes	—
48	C₂³×Q₈	C₂×C₉	1	2	yes	—
49	C₂×C₄⋊Q₈	C₂×C₉	1	24	yes	direct product factor
50	C₂²×C₄⋊C₄	C₂×C₉	1	4	yes	—
51	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic
52	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
53	C₂×C₃×Q₈	—	2	96	no	—
54	C₂×C₃×Q₈	C₂³×C₃	1	32	yes	—
55	C₂²×C₃×C₄	—	2	16	no	—
56	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
57	C₂²×C₃×C₄	C₃×Q₈	1	16	yes	—
58	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
59	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
60	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
61	C₃×C₄⋊C₄	C₂³×C₃	1	64	yes	—
62	C₃×C₄²	C₂³×C₃	1	16	yes	—
63	C₂²×C₉	C₄⋊Q₈	1	16	yes	direct product factor
64	C₂²×C₉	C₂²×D₄	1	6	yes	—
65	C₂²×C₉	C₂²×Q₈	1	12	yes	—
66	C₂²×C₉	C₂⁵	1	1	yes	characteristic
67	C₄×C₉	—	2	16	no	—
68	C₄×C₉	C₂²×D₄	1	8	yes	—
69	C₄×C₉	C₂²×Q₈	1	16	yes	complemented
70	C₂²×Q₈	—	2	16	no	—
71	C₂²×Q₈	C₂²×C₉	1	24	yes	—
72	C₂³×C₄	C₂²×C₉	1	4	yes	—
73	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
74	C₂³×C₄	C₂²×C₉	1	2	yes	—
75	C₄⋊Q₈	C₂²×C₉	1	64	yes	direct product factor
76	C₂×C₄⋊C₄	C₂²×C₉	1	48	yes	—
77	C₂×C₄²	C₂²×C₉	1	12	yes	—
78	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
79	C₂³×C₃	C₂×C₃×Q₈	1	8	yes	—
80	C₂³×C₃	C₂⁴×C₃	1	3	yes	—
81	C₃×Q₈	—	2	64	no	—
82	C₂×C₃×C₄	—	2	48	no	—
83	C₂×C₃×C₄	C₂×C₃×D₄	1	24	yes	—
84	C₂×C₃×C₄	C₂×C₃×Q₈	1	48	yes	—
85	C₂×C₃×C₄	C₂⁴×C₃	1	16	yes	—
86	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
87	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
88	C₂×C₉	C₂×C₄⋊Q₈	1	12	yes	direct product factor
89	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
90	C₂×C₉	C₂³×Q₈	1	2	yes	—
91	C₂⁴	C₂³×C₉	1	1	yes	characteristic
92	C₂×Q₈	—	2	96	no	—
93	C₂×Q₈	C₂³×C₉	1	32	yes	—
94	C₂²×C₄	—	2	16	no	—
95	C₂²×C₄	D₄×C₉	1	8	yes	—
96	C₂²×C₄	Q₈×C₉	1	16	yes	complemented
97	C₂²×C₄	C₂³×C₉	1	24	yes	—
98	C₂²×C₄	C₂³×C₉	1	6	yes	—
99	C₂²×C₄	C₂³×C₉	1	12	yes	—
100	C₄⋊C₄	C₂³×C₉	1	64	yes	—
101	C₄²	C₂³×C₉	1	16	yes	—
102	C₂²×C₃	C₃×C₄⋊Q₈	1	16	yes	—
103	C₂²×C₃	C₂²×C₃×D₄	1	6	yes	—
104	C₂²×C₃	C₂²×C₃×Q₈	1	12	yes	—
105	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
106	C₃×C₄	—	2	16	no	—
107	C₃×C₄	C₂²×C₃×D₄	1	8	yes	—
108	C₃×C₄	C₂²×C₃×Q₈	1	16	yes	—
109	C₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
110	C₂³	C₂×D₄×C₉	1	4	yes	—
111	C₂³	C₂×Q₈×C₉	1	8	yes	—
112	C₂³	C₂⁴×C₉	1	3	yes	—
113	Q₈	—	2	64	no	—
114	C₂×C₄	—	2	48	no	—
115	C₂×C₄	C₂×D₄×C₉	1	24	yes	—
116	C₂×C₄	C₂×Q₈×C₉	1	48	yes	complemented
117	C₂×C₄	C₂⁴×C₉	1	16	yes	—
118	C₂×C₄	C₂⁴×C₉	1	4	yes	—
119	C₂×C₄	C₂⁴×C₉	1	8	yes	—
120	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	12	yes	—
121	C₂×C₃	C₂³×C₃×D₄	1	1	yes	characteristic
122	C₂×C₃	C₂³×C₃×Q₈	1	2	yes	—
123	C₂²	C₉×C₄⋊Q₈	1	16	yes	direct product factor
124	C₂²	C₂²×D₄×C₉	1	6	yes	—
125	C₂²	C₂²×Q₈×C₉	1	12	yes	—
126	C₂²	C₂⁵×C₉	1	1	yes	characteristic
127	C₄	—	2	16	no	—
128	C₄	C₂²×D₄×C₉	1	8	yes	—
129	C₄	C₂²×Q₈×C₉	1	16	yes	complemented
130	C₃	C₂²×C₃×C₄⋊Q₈	1	1	yes	characteristic
131	C₂	C₂×C₉×C₄⋊Q₈	1	12	yes	direct product factor
132	C₂	C₂³×D₄×C₉	1	1	yes	characteristic
133	C₂	C₂³×Q₈×C₉	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₉	C₂³
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle, prime radical	C₂⁴×C₃	C₂³×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₄⋊Q₈	C₂	direct (N × H)
11. C₉×C₄⋊Q₈	C₂²	direct (N × H)
23. C₂²×C₄×C₉	Q₈	semidirect
29. C₂²×C₄⋊Q₈	C₉	direct (N × H)
44. C₂×C₄×C₉	C₂×Q₈	semidirect
49. C₂×C₄⋊Q₈	C₂×C₉	direct (N × H)
63. C₂²×C₉	C₄⋊Q₈	direct (N × H)
69. C₄×C₉	C₂²×Q₈	semidirect
75. C₄⋊Q₈	C₂²×C₉	direct (N × H)
88. C₂×C₉	C₂×C₄⋊Q₈	direct (N × H)
96. C₂²×C₄	Q₈×C₉	semidirect
109. C₉	C₂²×C₄⋊Q₈	direct (N × H)
116. C₂×C₄	C₂×Q₈×C₉	semidirect
123. C₂²	C₉×C₄⋊Q₈	direct (N × H)
129. C₄	C₂²×Q₈×C₉	semidirect
131. C₂	C₂×C₉×C₄⋊Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₉	2
C₂×C₉×C₄⋊Q₈	2
C₂²×C₉×C₄⋊C₄	2
C₂²×C₄²×C₉	2
C₂²×C₃×C₄⋊Q₈	3

Subgroups — C22×C9×C4⋊Q8 #11768162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₉×C₄⋊Q₈ #11768162