Subgroups — C₉×#2310 #11767031

Order 1152.

1176 conjugacy classes of subgroups.

The subgroup lattice has 174 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	1	yes	maximal, characteristic
2	C₂×C₉×#117	C₂	1	2	yes	maximal
3	C₉×#391	C₂	1	8	yes	maximal, complemented
4	C₉×#381	C₂	1	4	yes	maximal
5	C₃×#2310	C₃	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₉	—	2	8	no	—
7	C₂²×Q₈×C₉	C₄	1	8	yes	complemented
8	C₂²×Q₈×C₉	C₂²	1	4	yes	—
9	C₂³×C₄×C₉	C₂²	1	2	yes	—
10	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₉×#117	—	2	8	no	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄²×C₉	C₂²	1	4	yes	—
15	C₉×#95	C₂²	1	16	yes	—
16	C₂³×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×#117	C₂×C₃	1	2	yes	—
18	C₃×#391	C₂×C₃	1	8	yes	—
19	C₃×#381	C₂×C₃	1	4	yes	—
20	C₂⁴×C₉	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₉	—	2	32	no	—
22	C₂×Q₈×C₉	—	2	8	no	—
23	C₂×Q₈×C₉	—	2	8	no	—
24	C₂×Q₈×C₉	C₂×C₄	1	16	yes	—
25	C₂²×C₄×C₉	—	2	8	no	—
26	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
27	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₉	D₄	1	8	yes	—
29	C₂²×C₄×C₉	C₂³	1	8	yes	—
30	C₂²×C₄×C₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₉	C₂³	1	2	yes	—
32	C₉×C₄⋊C₄	—	2	8	no	—
33	C₉×C₂²⋊C₄	—	2	8	no	—
34	C₄²×C₉	—	2	8	no	—
35	#2310	C₉	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₃×Q₈	—	2	8	no	—
37	C₂²×C₃×Q₈	C₃×C₄	1	8	yes	—
38	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	—
39	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
40	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
41	C₃×#117	—	2	8	no	—
42	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
43	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
44	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
45	C₃×#95	C₂²×C₃	1	16	yes	—
46	C₂³×C₉	—	2	4	no	—
47	C₂³×C₉	C₂²×C₄	1	2	yes	—
48	C₂³×C₉	C₂×D₄	1	4	yes	—
49	C₂³×C₉	C₂⁴	1	1	yes	characteristic
50	Q₈×C₉	—	2	32	no	—
51	C₂×C₄×C₉	—	2	16	no	—
52	C₂×C₄×C₉	—	2	8	no	—
53	C₂×C₄×C₉	—	2	8	no	—
54	C₂×C₄×C₉	—	2	16	no	—
55	C₂×C₄×C₉	C₂²⋊C₄	1	16	yes	—
56	C₂×C₄×C₉	C₂²×C₄	1	8	yes	—
57	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
59	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
60	C₂³×Q₈	C₂×C₉	1	1	yes	characteristic
61	C₂×#117	C₂×C₉	1	2	yes	—
62	#391	C₂×C₉	1	8	yes	complemented
63	#381	C₂×C₉	1	4	yes	—
64	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
65	C₂×C₃×Q₈	—	2	32	no	—
66	C₂×C₃×Q₈	—	2	8	no	—
67	C₂×C₃×Q₈	—	2	8	no	—
68	C₂×C₃×Q₈	C₂×C₃×C₄	1	16	yes	—
69	C₂²×C₃×C₄	—	2	8	no	—
70	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
71	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
72	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
73	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
74	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
75	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
76	C₃×C₄⋊C₄	—	2	8	no	—
77	C₃×C₂²⋊C₄	—	2	8	no	—
78	C₃×C₄²	—	2	8	no	—
79	C₂²×C₉	—	2	4	no	—
80	C₂²×C₉	—	2	8	no	—
81	C₂²×C₉	C₂×C₂²⋊C₄	1	4	yes	—
82	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
83	C₂²×C₉	C₂²×D₄	1	2	yes	—
84	C₂²×C₉	2₋¹⁺⁴	1	4	yes	—
85	C₄×C₉	—	2	8	no	—
86	C₄×C₉	—	4	8	no	—
87	C₄×C₉	C₂×C₂²⋊C₄	1	8	yes	—
88	C₂²×Q₈	—	2	8	no	—
89	C₂²×Q₈	C₄×C₉	1	8	yes	complemented
90	C₂²×Q₈	C₂²×C₉	1	4	yes	—
91	C₂³×C₄	C₂²×C₉	1	2	yes	—
92	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
93	#117	—	2	8	no	—
94	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
95	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
96	C₂×C₄²	C₂²×C₉	1	4	yes	—
97	#95	C₂²×C₉	1	16	yes	—
98	C₂³×C₃	—	2	4	no	—
99	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
100	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
101	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
102	C₃×Q₈	—	2	32	no	—
103	C₂×C₃×C₄	—	2	16	no	—
104	C₂×C₃×C₄	—	2	8	no	—
105	C₂×C₃×C₄	—	2	8	no	—
106	C₂×C₃×C₄	—	2	16	no	—
107	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	16	yes	—
108	C₂×C₃×C₄	C₂²×C₃×C₄	1	8	yes	—
109	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
110	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
111	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
112	C₂×C₉	—	2	4	no	—
113	C₂×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₉	#519	1	2	yes	—
115	C₂×C₉	#536	1	4	yes	—
116	C₂⁴	C₂³×C₉	1	1	yes	characteristic
117	C₂×Q₈	—	2	32	no	—
118	C₂×Q₈	—	2	8	no	—
119	C₂×Q₈	—	2	8	no	—
120	C₂×Q₈	C₂×C₄×C₉	1	16	yes	—
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	C₂×C₄×C₉	1	8	yes	—
123	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
124	C₂²×C₄	D₄×C₉	1	8	yes	—
125	C₂²×C₄	C₂³×C₉	1	8	yes	—
126	C₂²×C₄	C₂³×C₉	1	4	yes	—
127	C₂²×C₄	C₂³×C₉	1	2	yes	—
128	C₄⋊C₄	—	2	8	no	—
129	C₂²⋊C₄	—	2	8	no	—
130	C₄²	—	2	8	no	—
131	C₂²×C₃	—	2	4	no	—
132	C₂²×C₃	—	2	8	no	—
133	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	4	yes	—
134	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
135	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
136	C₂²×C₃	C₃×2₋¹⁺⁴	1	4	yes	—
137	C₃×C₄	—	2	8	no	—
138	C₃×C₄	—	4	8	no	—
139	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	8	yes	—
140	C₉	#2310	1	1	yes	characteristic, direct product factor, Sylow
141	C₂³	—	2	4	no	—
142	C₂³	C₂²×C₄×C₉	1	2	yes	—
143	C₂³	C₂×D₄×C₉	1	4	yes	—
144	C₂³	C₂⁴×C₉	1	1	yes	characteristic
145	Q₈	—	2	32	no	—
146	C₂×C₄	—	2	16	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	2	16	no	—
150	C₂×C₄	C₉×C₂²⋊C₄	1	16	yes	—
151	C₂×C₄	C₂²×C₄×C₉	1	8	yes	—
152	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
153	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
154	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
155	C₂×C₃	—	2	4	no	—
156	C₂×C₃	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₃	C₃×#519	1	2	yes	—
158	C₂×C₃	C₃×#536	1	4	yes	—
159	C₂²	—	2	8	no	—
160	C₂²	—	2	4	no	—
161	C₂²	C₂×C₉×C₂²⋊C₄	1	4	yes	—
162	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
163	C₂²	C₂²×D₄×C₉	1	2	yes	—
164	C₂²	C₉×2₋¹⁺⁴	1	4	yes	—
165	C₄	—	2	8	no	—
166	C₄	—	4	8	no	—
167	C₄	C₂×C₉×C₂²⋊C₄	1	8	yes	—
168	C₃	C₃×#2310	1	1	yes	characteristic
169	C₂	—	2	4	no	—
170	C₂	C₂²×C₉×C₂²⋊C₄	1	1	yes	characteristic
171	C₂	C₉×#519	1	2	yes	—
172	C₂	C₉×#536	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₉×#391	C₂	semidirect
7. C₂²×Q₈×C₉	C₄	semidirect
35. #2310	C₉	direct (N × H)
62. #391	C₂×C₉	semidirect
89. C₂²×Q₈	C₄×C₉	semidirect
140. C₉	#2310	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×#117	2
C₉×#391	2
C₉×#381	2
C₃×#2310	3

Subgroups — C9×#2310 #11767031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2310 #11767031