Subgroups — C₉×#3458 #11768179

Order 1152.

1494 conjugacy classes of subgroups.

The subgroup lattice has 141 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#544	C₂	1	12	yes	maximal, complemented
2	C₉×#543	C₂	1	4	yes	maximal
3	C₉×#533	C₂	1	12	yes	maximal, complemented
4	C₂×C₉×#117	C₂	1	3	yes	maximal
5	C₃×#3458	C₃	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₉×C₄⋊Q₈	C₂²	1	24	yes	complemented
8	C₉×#125	C₂²	1	24	yes	complemented
9	C₉×C₂²⋊Q₈	C₂²	1	48	yes	—
10	C₄×Q₈×C₉	C₂²	1	16	yes	complemented
11	C₉×#117	C₂²	1	12	yes	—
12	C₉×#117	C₂²	1	12	yes	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	6	yes	—
14	C₂×C₉×C₂²⋊C₄	C₂²	1	6	yes	—
15	C₂×C₄²×C₉	C₂²	1	6	yes	—
16	C₃×#544	C₂×C₃	1	12	yes	—
17	C₃×#543	C₂×C₃	1	4	yes	—
18	C₃×#533	C₂×C₃	1	12	yes	—
19	C₂×C₃×#117	C₂×C₃	1	3	yes	—
20	C₂⁴×C₉	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₉	C₂³	1	16	yes	—
22	C₂²×C₄×C₉	Q₈	1	8	yes	complemented
23	C₂²×C₄×C₉	C₂³	1	12	yes	—
24	C₂²×C₄×C₉	C₂³	1	3	yes	—
25	C₂²×C₄×C₉	C₂³	1	3	yes	—
26	C₉×C₄⋊C₄	C₂³	1	24	yes	—
27	C₉×C₄⋊C₄	C₂³	1	48	yes	—
28	C₉×C₂²⋊C₄	C₂³	1	24	yes	—
29	C₄²×C₉	C₂³	1	12	yes	—
30	C₄²×C₉	C₂³	1	12	yes	—
31	#3458	C₉	1	1	yes	characteristic, direct product factor, Sylow
32	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
33	C₃×C₄⋊Q₈	C₂²×C₃	1	24	yes	—
34	C₃×#125	C₂²×C₃	1	24	yes	—
35	C₃×C₂²⋊Q₈	C₂²×C₃	1	48	yes	—
36	C₃×C₄×Q₈	C₂²×C₃	1	16	yes	—
37	C₃×#117	C₂²×C₃	1	12	yes	—
38	C₃×#117	C₂²×C₃	1	12	yes	—
39	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	6	yes	—
40	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
41	C₂×C₃×C₄²	C₂²×C₃	1	6	yes	—
42	C₂³×C₉	—	4	2	no	—
43	C₂³×C₉	C₂×Q₈	1	4	yes	—
44	C₂³×C₉	C₂⁴	1	3	yes	—
45	Q₈×C₉	—	4	16	no	—
46	C₂×C₄×C₉	—	4	12	no	—
47	C₂×C₄×C₉	C₂×Q₈	1	12	yes	—
48	C₂×C₄×C₉	C₂×Q₈	1	12	yes	—
49	C₂×C₄×C₉	C₂⁴	1	24	yes	—
50	C₂×C₄×C₉	C₂⁴	1	3	yes	—
51	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
52	#544	C₂×C₉	1	12	yes	complemented
53	#543	C₂×C₉	1	4	yes	—
54	#533	C₂×C₉	1	12	yes	complemented
55	C₂×#117	C₂×C₉	1	3	yes	—
56	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³×C₃	1	16	yes	—
58	C₂²×C₃×C₄	C₃×Q₈	1	8	yes	—
59	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
60	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
61	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
62	C₃×C₄⋊C₄	C₂³×C₃	1	24	yes	—
63	C₃×C₄⋊C₄	C₂³×C₃	1	48	yes	—
64	C₃×C₂²⋊C₄	C₂³×C₃	1	24	yes	—
65	C₃×C₄²	C₂³×C₃	1	12	yes	—
66	C₃×C₄²	C₂³×C₃	1	12	yes	—
67	C₂²×C₉	—	2	6	no	—
68	C₂²×C₉	—	4	4	no	—
69	C₂²×C₉	C₂²×Q₈	1	6	yes	—
70	C₂²×C₉	C₂⁵	1	1	yes	characteristic
71	C₄×C₉	—	2	24	no	—
72	C₄×C₉	C₂²×Q₈	1	6	yes	—
73	C₄×C₉	C₂²×Q₈	1	2	yes	—
74	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
75	C₄⋊Q₈	C₂²×C₉	1	24	yes	complemented
76	#125	C₂²×C₉	1	24	yes	complemented
77	C₂²⋊Q₈	C₂²×C₉	1	48	yes	—
78	C₄×Q₈	C₂²×C₉	1	16	yes	complemented
79	#117	C₂²×C₉	1	12	yes	—
80	#117	C₂²×C₉	1	12	yes	—
81	C₂×C₄⋊C₄	C₂²×C₉	1	6	yes	—
82	C₂×C₂²⋊C₄	C₂²×C₉	1	6	yes	—
83	C₂×C₄²	C₂²×C₉	1	6	yes	—
84	C₂³×C₃	—	4	2	no	—
85	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
86	C₂³×C₃	C₂⁴×C₃	1	3	yes	—
87	C₃×Q₈	—	4	16	no	—
88	C₂×C₃×C₄	—	4	12	no	—
89	C₂×C₃×C₄	C₂×C₃×Q₈	1	12	yes	—
90	C₂×C₃×C₄	C₂×C₃×Q₈	1	12	yes	—
91	C₂×C₃×C₄	C₂⁴×C₃	1	24	yes	—
92	C₂×C₃×C₄	C₂⁴×C₃	1	3	yes	—
93	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
94	C₂×C₉	—	2	6	no	—
95	C₂×C₉	C₂³×Q₈	1	1	yes	characteristic
96	C₂×C₉	#585	1	2	yes	—
97	C₂⁴	C₂³×C₉	1	1	yes	characteristic
98	C₂×Q₈	C₂³×C₉	1	16	yes	—
99	C₂²×C₄	Q₈×C₉	1	8	yes	complemented
100	C₂²×C₄	C₂³×C₉	1	12	yes	—
101	C₂²×C₄	C₂³×C₉	1	3	yes	—
102	C₂²×C₄	C₂³×C₉	1	3	yes	—
103	C₄⋊C₄	C₂³×C₉	1	24	yes	—
104	C₄⋊C₄	C₂³×C₉	1	48	yes	—
105	C₂²⋊C₄	C₂³×C₉	1	24	yes	—
106	C₄²	C₂³×C₉	1	12	yes	—
107	C₄²	C₂³×C₉	1	12	yes	—
108	C₂²×C₃	—	2	6	no	—
109	C₂²×C₃	—	4	4	no	—
110	C₂²×C₃	C₂²×C₃×Q₈	1	6	yes	—
111	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
112	C₃×C₄	—	2	24	no	—
113	C₃×C₄	C₂²×C₃×Q₈	1	6	yes	—
114	C₃×C₄	C₂²×C₃×Q₈	1	2	yes	—
115	C₉	#3458	1	1	yes	characteristic, direct product factor, Sylow
116	C₂³	—	4	2	no	—
117	C₂³	C₂×Q₈×C₉	1	4	yes	—
118	C₂³	C₂⁴×C₉	1	3	yes	—
119	Q₈	—	4	16	no	—
120	C₂×C₄	—	4	12	no	—
121	C₂×C₄	C₂×Q₈×C₉	1	12	yes	—
122	C₂×C₄	C₂×Q₈×C₉	1	12	yes	—
123	C₂×C₄	C₂⁴×C₉	1	24	yes	—
124	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
125	C₂×C₄	C₂⁴×C₉	1	3	yes	—
126	C₂×C₃	—	2	6	no	—
127	C₂×C₃	C₂³×C₃×Q₈	1	1	yes	characteristic
128	C₂×C₃	C₃×#585	1	2	yes	—
129	C₂²	—	2	6	no	—
130	C₂²	—	4	4	no	—
131	C₂²	C₂²×Q₈×C₉	1	6	yes	—
132	C₂²	C₂⁵×C₉	1	1	yes	characteristic
133	C₄	—	2	24	no	—
134	C₄	C₂²×Q₈×C₉	1	2	yes	—
135	C₄	C₂²×Q₈×C₉	1	6	yes	—
136	C₃	C₃×#3458	1	1	yes	characteristic
137	C₂	—	2	6	no	—
138	C₂	C₂³×Q₈×C₉	1	1	yes	characteristic
139	C₂	C₉×#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂²	C₂⁵×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂⁵×C₃
prime radical	C₂⁴×C₃	C₂³×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#544	C₂	semidirect
3. C₉×#533	C₂	semidirect
7. C₉×C₄⋊Q₈	C₂²	semidirect
8. C₉×#125	C₂²	semidirect
10. C₄×Q₈×C₉	C₂²	semidirect
22. C₂²×C₄×C₉	Q₈	semidirect
31. #3458	C₉	direct (N × H)
52. #544	C₂×C₉	semidirect
54. #533	C₂×C₉	semidirect
75. C₄⋊Q₈	C₂²×C₉	semidirect
76. #125	C₂²×C₉	semidirect
78. C₄×Q₈	C₂²×C₉	semidirect
99. C₂²×C₄	Q₈×C₉	semidirect
115. C₉	#3458	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#544	2
C₉×#543	2
C₉×#533	2
C₂×C₉×#117	2
C₃×#3458	3

Subgroups — C9×#3458 #11768179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3458 #11768179