Subgroups — C₉×#2331 #11767052

Order 1152.

690 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₉×#391	C₂	1	4	yes	maximal
3	C₉×#389	C₂	1	2	yes	maximal
4	C₉×#387	C₂	1	4	yes	maximal
5	C₉×#382	C₂	1	1	yes	maximal, characteristic
6	C₄×C₉×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₉×#376	C₂	1	1	yes	maximal, characteristic
8	C₃×#2331	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	2	yes	—
10	C₉×C₄⋊Q₈	C₄	1	8	yes	complemented
11	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₉	C₂²	1	4	yes	—
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₉×#95	C₂²	1	2	yes	—
20	C₉×#95	C₂²	1	2	yes	—
21	C₉×#95	C₂²	1	4	yes	—
22	C₉×#95	C₂²	1	2	yes	—
23	C₉×#95	C₂²	1	2	yes	—
24	C₉×#95	C₂²	1	4	yes	—
25	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
26	C₃×#391	C₂×C₃	1	4	yes	—
27	C₃×#389	C₂×C₃	1	2	yes	—
28	C₃×#387	C₂×C₃	1	4	yes	—
29	C₃×#382	C₂×C₃	1	1	yes	characteristic
30	C₃×C₄×C₄⋊C₄	C₂×C₃	1	2	yes	—
31	C₃×#376	C₂×C₃	1	1	yes	characteristic
32	C₂×Q₈×C₉	—	4	4	no	—
33	C₂×Q₈×C₉	C₂×C₄	1	8	yes	—
34	C₂²×C₄×C₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₉	C₂³	1	4	yes	—
36	C₂²×C₄×C₉	C₂³	1	4	yes	—
37	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₉×C₄⋊C₄	—	2	4	no	—
42	C₉×C₄⋊C₄	—	4	4	no	—
43	C₉×C₄⋊C₄	C₂×C₄	1	8	yes	—
44	C₉×C₄⋊C₄	C₂×C₄	1	8	yes	—
45	C₄²×C₉	—	2	2	no	—
46	C₄²×C₉	—	4	4	no	—
47	C₄²×C₉	C₂×C₄	1	4	yes	—
48	C₄²×C₉	Q₈	1	4	yes	complemented
49	#2331	C₉	1	1	yes	characteristic, direct product factor, Sylow
50	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
51	C₃×C₄⋊Q₈	C₃×C₄	1	8	yes	—
52	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
53	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
54	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
58	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
60	C₃×#95	C₂²×C₃	1	2	yes	—
61	C₃×#95	C₂²×C₃	1	2	yes	—
62	C₃×#95	C₂²×C₃	1	4	yes	—
63	C₃×#95	C₂²×C₃	1	2	yes	—
64	C₃×#95	C₂²×C₃	1	2	yes	—
65	C₃×#95	C₂²×C₃	1	4	yes	—
66	C₂³×C₉	C₂⁴	1	1	yes	characteristic
67	Q₈×C₉	—	4	8	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	4	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	4	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₉	C₂²×C₄	1	8	yes	—
83	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
85	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
86	C₂×C₄×C₉	#41	1	2	yes	—
87	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
88	#391	C₂×C₉	1	4	yes	—
89	#389	C₂×C₉	1	2	yes	—
90	#387	C₂×C₉	1	4	yes	—
91	#382	C₂×C₉	1	1	yes	characteristic
92	C₄×C₄⋊C₄	C₂×C₉	1	2	yes	—
93	#376	C₂×C₉	1	1	yes	characteristic
94	C₂×C₃×Q₈	—	4	4	no	—
95	C₂×C₃×Q₈	C₂×C₃×C₄	1	8	yes	—
96	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
97	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
98	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
101	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
102	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₃×C₄⋊C₄	—	2	4	no	—
104	C₃×C₄⋊C₄	—	4	4	no	—
105	C₃×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
106	C₃×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
107	C₃×C₄²	—	2	2	no	—
108	C₃×C₄²	—	4	4	no	—
109	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
110	C₃×C₄²	C₃×Q₈	1	4	yes	—
111	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
112	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
113	C₂²×C₉	C₂×#41	1	1	yes	characteristic
114	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
115	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
116	C₄×C₉	—	2	8	no	—
117	C₄×C₉	—	2	2	no	—
118	C₄×C₉	—	2	2	no	—
119	C₄×C₉	—	2	4	no	—
120	C₄×C₉	—	4	2	no	—
121	C₄×C₉	—	4	4	no	—
122	C₄×C₉	C₄×Q₈	1	4	yes	—
123	C₂²×Q₈	C₂²×C₉	1	2	yes	—
124	C₄⋊Q₈	C₄×C₉	1	8	yes	complemented
125	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
127	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
129	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
130	C₂×C₄²	C₂²×C₉	1	4	yes	—
131	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
133	#95	C₂²×C₉	1	2	yes	—
134	#95	C₂²×C₉	1	2	yes	—
135	#95	C₂²×C₉	1	4	yes	—
136	#95	C₂²×C₉	1	2	yes	—
137	#95	C₂²×C₉	1	2	yes	—
138	#95	C₂²×C₉	1	4	yes	—
139	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
140	C₃×Q₈	—	4	8	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
154	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
155	C₂×C₃×C₄	C₂²×C₃×C₄	1	8	yes	—
156	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
158	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
159	C₂×C₃×C₄	C₃×#41	1	2	yes	—
160	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₂×C₉	#518	1	1	yes	characteristic
162	C₂×C₉	#519	1	1	yes	characteristic
163	C₂×C₉	#542	1	2	yes	—
164	C₂×C₉	#544	1	2	yes	—
165	C₂×Q₈	—	4	4	no	—
166	C₂×Q₈	C₂×C₄×C₉	1	8	yes	—
167	C₂²×C₄	C₂³×C₉	1	2	yes	—
168	C₂²×C₄	C₂³×C₉	1	4	yes	—
169	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₉	1	4	yes	—
172	C₂²×C₄	C₂³×C₉	1	2	yes	—
173	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	C₂×C₄×C₉	1	8	yes	—
177	C₄⋊C₄	C₂×C₄×C₉	1	8	yes	—
178	C₄²	—	2	2	no	—
179	C₄²	—	4	4	no	—
180	C₄²	C₂×C₄×C₉	1	4	yes	—
181	C₄²	Q₈×C₉	1	4	yes	complemented
182	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
183	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
184	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
185	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
186	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	8	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	4	2	no	—
192	C₃×C₄	—	4	4	no	—
193	C₃×C₄	C₃×C₄×Q₈	1	4	yes	—
194	C₉	#2331	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³	C₂⁴×C₉	1	1	yes	characteristic
196	Q₈	—	4	8	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
210	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
211	C₂×C₄	C₂²×C₄×C₉	1	8	yes	—
212	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
213	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
214	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
215	C₂×C₄	C₉×#41	1	2	yes	—
216	C₂×C₃	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
217	C₂×C₃	C₃×#518	1	1	yes	characteristic
218	C₂×C₃	C₃×#519	1	1	yes	characteristic
219	C₂×C₃	C₃×#542	1	2	yes	—
220	C₂×C₃	C₃×#544	1	2	yes	—
221	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
222	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
223	C₂²	C₂×C₉×#41	1	1	yes	characteristic
224	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
225	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
226	C₄	—	2	4	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	8	no	—
229	C₄	—	2	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	4	no	—
232	C₄	C₄×Q₈×C₉	1	4	yes	—
233	C₃	C₃×#2331	1	1	yes	characteristic
234	C₂	C₂×C₄×Q₈×C₉	1	1	yes	characteristic
235	C₂	C₉×#518	1	1	yes	characteristic
236	C₂	C₉×#519	1	1	yes	characteristic
237	C₂	C₉×#542	1	2	yes	—
238	C₂	C₉×#544	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, prime radical, cosocle	C₂³×C₃	C₂⁴×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
10. C₉×C₄⋊Q₈	C₄	semidirect
48. C₄²×C₉	Q₈	semidirect
49. #2331	C₉	direct (N × H)
124. C₄⋊Q₈	C₄×C₉	semidirect
181. C₄²	Q₈×C₉	semidirect
194. C₉	#2331	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊Q₈	2
C₉×#391	2
C₉×#389	2
C₉×#387	2
C₉×#382	2
C₄×C₉×C₄⋊C₄	2
C₉×#376	2
C₃×#2331	3

Subgroups — C9×#2331 #11767052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2331 #11767052