Subgroups — C₂²×C₉×#124 #11768157

Order 1152.

2820 conjugacy classes of subgroups.

The subgroup lattice has 189 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, complemented
2	C₂³×D₄×C₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#124	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₉×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×#124	C₃	1	1	yes	maximal, characteristic
7	C₂⁵×C₉	C₂²	1	2	yes	—
8	C₂²×Q₈×C₉	—	2	8	no	—
9	C₂²×Q₈×C₉	C₂²	1	12	yes	complemented
10	C₂²×D₄×C₉	—	2	8	no	—
11	C₂²×D₄×C₉	C₂²	1	12	yes	—
12	C₂³×C₄×C₉	C₂²	1	2	yes	—
13	C₂³×C₄×C₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
15	C₉×#124	C₂²	1	64	yes	direct product factor
16	C₂×C₉×C₂²⋊C₄	C₂²	1	48	yes	complemented
17	C₂×C₄²×C₉	C₂²	1	12	yes	complemented
18	C₂³×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂³×C₃×D₄	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#124	C₂×C₃	1	24	yes	—
21	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	4	yes	—
22	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
23	C₂⁴×C₉	—	2	8	no	—
24	C₂⁴×C₉	—	2	16	no	—
25	C₂⁴×C₉	C₂³	1	12	yes	—
26	C₂⁴×C₉	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₉	—	2	48	no	—
28	C₂×Q₈×C₉	C₂³	1	16	yes	complemented
29	C₂×D₄×C₉	—	2	48	no	—
30	C₂×D₄×C₉	C₂³	1	16	yes	—
31	C₂²×C₄×C₉	—	2	8	no	—
32	C₂²×C₄×C₉	—	2	8	no	—
33	C₂²×C₄×C₉	D₄	1	8	yes	—
34	C₂²×C₄×C₉	C₂³	1	12	yes	—
35	C₂²×C₄×C₉	C₂³	1	12	yes	—
36	C₂²×C₄×C₉	C₂³	1	6	yes	—
37	C₉×C₂²⋊C₄	C₂³	1	64	yes	complemented
38	C₄²×C₉	C₂³	1	16	yes	complemented
39	C₂²×#124	C₉	1	1	yes	characteristic, direct product factor, Sylow
40	C₂⁵×C₃	C₂²×C₃	1	2	yes	—
41	C₂²×C₃×Q₈	—	2	8	no	—
42	C₂²×C₃×Q₈	C₂²×C₃	1	12	yes	—
43	C₂²×C₃×D₄	—	2	8	no	—
44	C₂²×C₃×D₄	C₂²×C₃	1	12	yes	—
45	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
46	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
47	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
48	C₃×#124	C₂²×C₃	1	64	yes	—
49	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	48	yes	—
50	C₂×C₃×C₄²	C₂²×C₃	1	12	yes	—
51	C₂³×C₉	—	2	24	no	—
52	C₂³×C₉	—	2	48	no	—
53	C₂³×C₉	—	4	32	no	—
54	C₂³×C₉	C₂×D₄	1	4	yes	—
55	C₂³×C₉	#41	1	8	yes	—
56	C₂³×C₉	C₂⁴	1	8	yes	—
57	C₂³×C₉	C₂⁴	1	3	yes	—
58	Q₈×C₉	—	2	32	no	—
59	D₄×C₉	—	2	32	no	—
60	C₂×C₄×C₉	—	2	24	no	—
61	C₂×C₄×C₉	—	2	24	no	—
62	C₂×C₄×C₉	C₂×D₄	1	24	yes	—
63	C₂×C₄×C₉	C₂⁴	1	8	yes	—
64	C₂×C₄×C₉	C₂⁴	1	8	yes	—
65	C₂×C₄×C₉	C₂⁴	1	4	yes	—
66	C₂³×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
67	C₂³×D₄	C₂×C₉	1	1	yes	characteristic
68	C₂×#124	C₂×C₉	1	24	yes	direct product factor
69	C₂²×C₂²⋊C₄	C₂×C₉	1	4	yes	complemented
70	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic, complemented
71	C₂⁴×C₃	—	2	8	no	—
72	C₂⁴×C₃	—	2	16	no	—
73	C₂⁴×C₃	C₂³×C₃	1	12	yes	—
74	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₃×Q₈	—	2	48	no	—
76	C₂×C₃×Q₈	C₂³×C₃	1	16	yes	—
77	C₂×C₃×D₄	—	2	48	no	—
78	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
79	C₂²×C₃×C₄	—	2	8	no	—
80	C₂²×C₃×C₄	—	2	8	no	—
81	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
82	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
83	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
84	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
85	C₃×C₂²⋊C₄	C₂³×C₃	1	64	yes	—
86	C₃×C₄²	C₂³×C₃	1	16	yes	—
87	C₂²×C₉	—	2	8	no	—
88	C₂²×C₉	—	2	16	no	—
89	C₂²×C₉	—	4	48	no	—
90	C₂²×C₉	#124	1	16	yes	direct product factor
91	C₂²×C₉	C₂²×D₄	1	6	yes	—
92	C₂²×C₉	C₂×#41	1	12	yes	—
93	C₂²×C₉	C₂⁵	1	1	yes	characteristic
94	C₄×C₉	—	2	8	no	—
95	C₄×C₉	—	2	8	no	—
96	C₄×C₉	C₂²×D₄	1	8	yes	—
97	C₂⁵	C₂²×C₉	1	2	yes	—
98	C₂²×Q₈	—	2	8	no	—
99	C₂²×Q₈	C₂²×C₉	1	12	yes	complemented
100	C₂²×D₄	—	2	8	no	—
101	C₂²×D₄	C₂²×C₉	1	12	yes	—
102	C₂³×C₄	C₂²×C₉	1	2	yes	—
103	C₂³×C₄	C₂²×C₉	1	2	yes	—
104	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
105	#124	C₂²×C₉	1	64	yes	direct product factor
106	C₂×C₂²⋊C₄	C₂²×C₉	1	48	yes	complemented
107	C₂×C₄²	C₂²×C₉	1	12	yes	complemented
108	C₂³×C₃	—	2	24	no	—
109	C₂³×C₃	—	2	48	no	—
110	C₂³×C₃	—	4	32	no	—
111	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
112	C₂³×C₃	C₃×#41	1	8	yes	—
113	C₂³×C₃	C₂⁴×C₃	1	8	yes	—
114	C₂³×C₃	C₂⁴×C₃	1	3	yes	—
115	C₃×Q₈	—	2	32	no	—
116	C₃×D₄	—	2	32	no	—
117	C₂×C₃×C₄	—	2	24	no	—
118	C₂×C₃×C₄	—	2	24	no	—
119	C₂×C₃×C₄	C₂×C₃×D₄	1	24	yes	—
120	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
121	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
122	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
123	C₂×C₉	—	4	8	no	—
124	C₂×C₉	C₂×#124	1	12	yes	direct product factor
125	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
126	C₂×C₉	C₂²×#41	1	2	yes	—
127	C₂⁴	—	2	8	no	—
128	C₂⁴	—	2	16	no	—
129	C₂⁴	C₂³×C₉	1	12	yes	—
130	C₂⁴	C₂³×C₉	1	1	yes	characteristic
131	C₂×Q₈	—	2	48	no	—
132	C₂×Q₈	C₂³×C₉	1	16	yes	complemented
133	C₂×D₄	—	2	48	no	—
134	C₂×D₄	C₂³×C₉	1	16	yes	—
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	D₄×C₉	1	8	yes	—
138	C₂²×C₄	C₂³×C₉	1	12	yes	—
139	C₂²×C₄	C₂³×C₉	1	12	yes	—
140	C₂²×C₄	C₂³×C₉	1	6	yes	—
141	C₂²⋊C₄	C₂³×C₉	1	64	yes	complemented
142	C₄²	C₂³×C₉	1	16	yes	complemented
143	C₂²×C₃	—	2	8	no	—
144	C₂²×C₃	—	2	16	no	—
145	C₂²×C₃	—	4	48	no	—
146	C₂²×C₃	C₃×#124	1	16	yes	—
147	C₂²×C₃	C₂²×C₃×D₄	1	6	yes	—
148	C₂²×C₃	C₂×C₃×#41	1	12	yes	—
149	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
150	C₃×C₄	—	2	8	no	—
151	C₃×C₄	—	2	8	no	—
152	C₃×C₄	C₂²×C₃×D₄	1	8	yes	—
153	C₉	C₂²×#124	1	1	yes	characteristic, direct product factor, Sylow
154	C₂³	—	2	24	no	—
155	C₂³	—	2	48	no	—
156	C₂³	—	4	32	no	—
157	C₂³	C₂×D₄×C₉	1	4	yes	—
158	C₂³	C₉×#41	1	8	yes	—
159	C₂³	C₂⁴×C₉	1	8	yes	—
160	C₂³	C₂⁴×C₉	1	3	yes	—
161	Q₈	—	2	32	no	—
162	D₄	—	2	32	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	2	24	no	—
165	C₂×C₄	C₂×D₄×C₉	1	24	yes	—
166	C₂×C₄	C₂⁴×C₉	1	8	yes	—
167	C₂×C₄	C₂⁴×C₉	1	8	yes	—
168	C₂×C₄	C₂⁴×C₉	1	4	yes	—
169	C₂×C₃	—	4	8	no	—
170	C₂×C₃	C₂×C₃×#124	1	12	yes	—
171	C₂×C₃	C₂³×C₃×D₄	1	1	yes	characteristic
172	C₂×C₃	C₂²×C₃×#41	1	2	yes	—
173	C₂²	—	2	8	no	—
174	C₂²	—	2	16	no	—
175	C₂²	—	4	48	no	—
176	C₂²	C₉×#124	1	16	yes	direct product factor
177	C₂²	C₂²×D₄×C₉	1	6	yes	—
178	C₂²	C₂×C₉×#41	1	12	yes	—
179	C₂²	C₂⁵×C₉	1	1	yes	characteristic
180	C₄	—	2	8	no	—
181	C₄	—	2	8	no	—
182	C₄	C₂²×D₄×C₉	1	8	yes	—
183	C₃	C₂²×C₃×#124	1	1	yes	characteristic
184	C₂	—	4	8	no	—
185	C₂	C₂×C₉×#124	1	12	yes	direct product factor
186	C₂	C₂³×D₄×C₉	1	1	yes	characteristic
187	C₂	C₂²×C₉×#41	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₉	C₂³
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle	C₂⁴×C₃	C₂³×C₃
prime radical	C₂³×C₃×D₄	C₂×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
1. C₂³×Q₈×C₉	C₂	semidirect
3. C₂×C₉×#124	C₂	direct (N × H)
4. C₂²×C₉×C₂²⋊C₄	C₂	semidirect
5. C₂²×C₄²×C₉	C₂	semidirect
9. C₂²×Q₈×C₉	C₂²	semidirect
15. C₉×#124	C₂²	direct (N × H)
16. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
17. C₂×C₄²×C₉	C₂²	semidirect
28. C₂×Q₈×C₉	C₂³	semidirect
37. C₉×C₂²⋊C₄	C₂³	semidirect
38. C₄²×C₉	C₂³	semidirect
39. C₂²×#124	C₉	direct (N × H)
66. C₂³×Q₈	C₂×C₉	semidirect
68. C₂×#124	C₂×C₉	direct (N × H)
69. C₂²×C₂²⋊C₄	C₂×C₉	semidirect
70. C₂²×C₄²	C₂×C₉	semidirect
90. C₂²×C₉	#124	direct (N × H)
99. C₂²×Q₈	C₂²×C₉	semidirect
105. #124	C₂²×C₉	direct (N × H)
106. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
107. C₂×C₄²	C₂²×C₉	semidirect
124. C₂×C₉	C₂×#124	direct (N × H)
132. C₂×Q₈	C₂³×C₉	semidirect
141. C₂²⋊C₄	C₂³×C₉	semidirect
142. C₄²	C₂³×C₉	semidirect
153. C₉	C₂²×#124	direct (N × H)
176. C₂²	C₉×#124	direct (N × H)
185. C₂	C₂×C₉×#124	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
C₂³×D₄×C₉	2
C₂×C₉×#124	2
C₂²×C₉×C₂²⋊C₄	2
C₂²×C₄²×C₉	2
C₂²×C₃×#124	3

Subgroups — C22×C9×#124 #11768157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₉×#124 #11768157