Subgroups — C₉×#3486 #11768207

Order 1152.

2238 conjugacy classes of subgroups.

The subgroup lattice has 267 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₈×C₉	C₂	1	8	yes	maximal, complemented
4	C₉×#548	C₂	1	8	yes	maximal, complemented
5	C₉×#536	C₂	1	6	yes	maximal, complemented
6	C₉×#525	C₂	1	6	yes	maximal, complemented
7	C₉×#519	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3486	C₃	1	1	yes	maximal, characteristic
9	C₉×2₋¹⁺⁴	—	2	8	no	—
10	C₂×C₉×#41	C₂²	1	6	yes	—
11	C₂×C₉×#41	C₂²	1	4	yes	complemented
12	C₂²×Q₈×C₉	—	2	8	no	—
13	C₂²×Q₈×C₉	C₂²	1	3	yes	—
14	C₂²×Q₈×C₉	C₂²	1	8	yes	—
15	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
16	C₂²×Q₈×C₉	C₂²	1	4	yes	complemented
17	C₂³×C₄×C₉	C₂²	1	3	yes	—
18	C₉×C₄⋊Q₈	C₂²	1	12	yes	complemented
19	C₉×#124	C₂²	1	12	yes	complemented
20	C₉×#123	C₂²	1	12	yes	—
21	C₉×C₂²⋊Q₈	C₂²	1	12	yes	—
22	C₉×C₂²⋊Q₈	C₂²	1	24	yes	complemented
23	C₉×#121	C₂²	1	12	yes	—
24	C₉×#120	C₂²	1	4	yes	—
25	C₄×Q₈×C₉	C₂²	1	8	yes	complemented
26	C₄×D₄×C₉	C₂²	1	24	yes	—
27	C₉×#117	C₂²	1	6	yes	—
28	C₂×C₃×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic
29	C₂³×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
30	C₃×D₄×Q₈	C₂×C₃	1	8	yes	—
31	C₃×#548	C₂×C₃	1	8	yes	—
32	C₃×#536	C₂×C₃	1	6	yes	—
33	C₃×#525	C₂×C₃	1	6	yes	—
34	C₃×#519	C₂×C₃	1	1	yes	characteristic
35	C₂⁴×C₉	C₂³	1	1	yes	characteristic
36	C₉×#41	—	2	16	no	—
37	C₉×#41	—	2	24	no	—
38	C₂×Q₈×C₉	—	2	32	no	—
39	C₂×Q₈×C₉	—	2	16	no	—
40	C₂×Q₈×C₉	—	2	12	no	—
41	C₂×Q₈×C₉	D₄	1	8	yes	complemented
42	C₂×Q₈×C₉	C₂³	1	6	yes	—
43	C₂×Q₈×C₉	C₂³	1	12	yes	—
44	C₂×Q₈×C₉	C₂³	1	4	yes	—
45	C₂×D₄×C₉	C₂³	1	8	yes	—
46	C₂×D₄×C₉	C₂³	1	4	yes	—
47	C₂×D₄×C₉	C₂³	1	6	yes	—
48	C₂²×C₄×C₉	—	2	12	no	—
49	C₂²×C₄×C₉	C₂³	1	12	yes	—
50	C₂²×C₄×C₉	C₂³	1	12	yes	—
51	C₂²×C₄×C₉	C₂³	1	3	yes	—
52	C₂²×C₄×C₉	C₂³	1	3	yes	—
53	C₉×C₄⋊C₄	C₂³	1	12	yes	—
54	C₉×C₄⋊C₄	C₂³	1	24	yes	—
55	C₉×C₂²⋊C₄	C₂³	1	4	yes	—
56	C₉×C₂²⋊C₄	C₂³	1	8	yes	—
57	C₉×C₂²⋊C₄	C₂³	1	24	yes	—
58	C₄²×C₉	C₂³	1	12	yes	—
59	#3486	C₉	1	1	yes	characteristic, direct product factor, Sylow
60	C₃×2₋¹⁺⁴	—	2	8	no	—
61	C₂×C₃×#41	C₂²×C₃	1	6	yes	—
62	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
63	C₂²×C₃×Q₈	—	2	8	no	—
64	C₂²×C₃×Q₈	C₂²×C₃	1	3	yes	—
65	C₂²×C₃×Q₈	C₂²×C₃	1	8	yes	—
66	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	—
68	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
69	C₃×C₄⋊Q₈	C₂²×C₃	1	12	yes	—
70	C₃×#124	C₂²×C₃	1	12	yes	—
71	C₃×#123	C₂²×C₃	1	12	yes	—
72	C₃×C₂²⋊Q₈	C₂²×C₃	1	12	yes	—
73	C₃×C₂²⋊Q₈	C₂²×C₃	1	24	yes	—
74	C₃×#121	C₂²×C₃	1	12	yes	—
75	C₃×#120	C₂²×C₃	1	4	yes	—
76	C₃×C₄×Q₈	C₂²×C₃	1	8	yes	—
77	C₃×C₄×D₄	C₂²×C₃	1	24	yes	—
78	C₃×#117	C₂²×C₃	1	6	yes	—
79	C₂³×C₉	—	2	2	no	—
80	C₂³×C₉	—	2	4	no	—
81	C₂³×C₉	C₂⁴	1	4	yes	—
82	C₂³×C₉	C₂⁴	1	1	yes	characteristic
83	C₂³×C₉	C₂⁴	1	2	yes	—
84	Q₈×C₉	—	2	24	no	—
85	Q₈×C₉	—	2	12	no	—
86	Q₈×C₉	C₂×D₄	1	16	yes	—
87	D₄×C₉	—	2	8	no	—
88	D₄×C₉	—	2	16	no	—
89	D₄×C₉	—	2	12	no	—
90	C₂×C₄×C₉	—	2	6	no	—
91	C₂×C₄×C₉	—	2	24	no	—
92	C₂×C₄×C₉	—	2	24	no	—
93	C₂×C₄×C₉	C₂×D₄	1	12	yes	—
94	C₂×C₄×C₉	C₂⁴	1	8	yes	—
95	C₂×C₄×C₉	C₂⁴	1	6	yes	—
96	C₂×C₄×C₉	C₂⁴	1	4	yes	—
97	C₂×C₄×C₉	C₂⁴	1	6	yes	—
98	C₂×2₋¹⁺⁴	C₂×C₉	1	1	yes	characteristic, complemented
99	C₂³×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
100	D₄×Q₈	C₂×C₉	1	8	yes	complemented
101	#548	C₂×C₉	1	8	yes	complemented
102	#536	C₂×C₉	1	6	yes	complemented
103	#525	C₂×C₉	1	6	yes	complemented
104	#519	C₂×C₉	1	1	yes	characteristic, complemented
105	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
106	C₃×#41	—	2	16	no	—
107	C₃×#41	—	2	24	no	—
108	C₂×C₃×Q₈	—	2	32	no	—
109	C₂×C₃×Q₈	—	2	16	no	—
110	C₂×C₃×Q₈	—	2	12	no	—
111	C₂×C₃×Q₈	C₃×D₄	1	8	yes	—
112	C₂×C₃×Q₈	C₂³×C₃	1	12	yes	—
113	C₂×C₃×Q₈	C₂³×C₃	1	6	yes	—
114	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
115	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
116	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
117	C₂×C₃×D₄	C₂³×C₃	1	6	yes	—
118	C₂²×C₃×C₄	—	2	12	no	—
119	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
120	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
121	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
122	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
123	C₃×C₄⋊C₄	C₂³×C₃	1	12	yes	—
124	C₃×C₄⋊C₄	C₂³×C₃	1	24	yes	—
125	C₃×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
126	C₃×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
127	C₃×C₂²⋊C₄	C₂³×C₃	1	24	yes	—
128	C₃×C₄²	C₂³×C₃	1	12	yes	—
129	C₂²×C₉	—	2	8	no	—
130	C₂²×C₉	—	2	2	no	—
131	C₂²×C₉	—	2	2	no	—
132	C₂²×C₉	—	2	2	no	—
133	C₂²×C₉	—	2	4	no	—
134	C₂²×C₉	—	2	8	no	—
135	C₂²×C₉	C₂²×D₄	1	2	yes	—
136	C₂²×C₉	2₋¹⁺⁴	1	4	yes	complemented
137	C₂²×C₉	C₂⁵	1	1	yes	characteristic
138	C₄×C₉	—	2	4	no	—
139	C₄×C₉	—	2	6	no	—
140	C₄×C₉	—	2	8	no	—
141	C₄×C₉	C₂²×D₄	1	12	yes	—
142	2₋¹⁺⁴	—	2	8	no	—
143	C₂×#41	C₂²×C₉	1	6	yes	—
144	C₂×#41	C₂²×C₉	1	4	yes	complemented
145	C₂²×Q₈	—	2	8	no	—
146	C₂²×Q₈	C₂²×C₉	1	3	yes	—
147	C₂²×Q₈	C₂²×C₉	1	8	yes	—
148	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
149	C₂²×Q₈	C₂²×C₉	1	4	yes	complemented
150	C₂³×C₄	C₂²×C₉	1	3	yes	—
151	C₄⋊Q₈	C₂²×C₉	1	12	yes	complemented
152	#124	C₂²×C₉	1	12	yes	complemented
153	#123	C₂²×C₉	1	12	yes	—
154	C₂²⋊Q₈	C₂²×C₉	1	12	yes	—
155	C₂²⋊Q₈	C₂²×C₉	1	24	yes	complemented
156	#121	C₂²×C₉	1	12	yes	—
157	#120	C₂²×C₉	1	4	yes	—
158	C₄×Q₈	C₂²×C₉	1	8	yes	complemented
159	C₄×D₄	C₂²×C₉	1	24	yes	—
160	#117	C₂²×C₉	1	6	yes	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
164	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
165	C₂³×C₃	C₂⁴×C₃	1	2	yes	—
166	C₃×Q₈	—	2	24	no	—
167	C₃×Q₈	—	2	12	no	—
168	C₃×Q₈	C₂×C₃×D₄	1	16	yes	—
169	C₃×D₄	—	2	8	no	—
170	C₃×D₄	—	2	16	no	—
171	C₃×D₄	—	2	12	no	—
172	C₂×C₃×C₄	—	2	6	no	—
173	C₂×C₃×C₄	—	2	24	no	—
174	C₂×C₃×C₄	—	2	24	no	—
175	C₂×C₃×C₄	C₂×C₃×D₄	1	12	yes	—
176	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
177	C₂×C₃×C₄	C₂⁴×C₃	1	6	yes	—
178	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
179	C₂×C₃×C₄	C₂⁴×C₃	1	6	yes	—
180	C₂×C₉	—	2	2	no	—
181	C₂×C₉	—	2	4	no	—
182	C₂×C₉	—	4	4	no	—
183	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
184	C₂×C₉	C₂×2₋¹⁺⁴	1	2	yes	—
185	C₂⁴	C₂³×C₉	1	1	yes	characteristic
186	#41	—	2	16	no	—
187	#41	—	2	24	no	—
188	C₂×Q₈	—	2	32	no	—
189	C₂×Q₈	—	2	16	no	—
190	C₂×Q₈	—	2	12	no	—
191	C₂×Q₈	D₄×C₉	1	8	yes	complemented
192	C₂×Q₈	C₂³×C₉	1	6	yes	—
193	C₂×Q₈	C₂³×C₉	1	12	yes	—
194	C₂×Q₈	C₂³×C₉	1	4	yes	—
195	C₂×D₄	C₂³×C₉	1	8	yes	—
196	C₂×D₄	C₂³×C₉	1	4	yes	—
197	C₂×D₄	C₂³×C₉	1	6	yes	—
198	C₂²×C₄	—	2	12	no	—
199	C₂²×C₄	C₂³×C₉	1	12	yes	—
200	C₂²×C₄	C₂³×C₉	1	12	yes	—
201	C₂²×C₄	C₂³×C₉	1	3	yes	—
202	C₂²×C₄	C₂³×C₉	1	3	yes	—
203	C₄⋊C₄	C₂³×C₉	1	12	yes	—
204	C₄⋊C₄	C₂³×C₉	1	24	yes	—
205	C₂²⋊C₄	C₂³×C₉	1	4	yes	—
206	C₂²⋊C₄	C₂³×C₉	1	8	yes	—
207	C₂²⋊C₄	C₂³×C₉	1	24	yes	—
208	C₄²	C₂³×C₉	1	12	yes	—
209	C₂²×C₃	—	2	8	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	—	2	8	no	—
215	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
216	C₂²×C₃	C₃×2₋¹⁺⁴	1	4	yes	—
217	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	2	6	no	—
220	C₃×C₄	—	2	8	no	—
221	C₃×C₄	C₂²×C₃×D₄	1	12	yes	—
222	C₉	#3486	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³	—	2	2	no	—
224	C₂³	—	2	4	no	—
225	C₂³	C₂⁴×C₉	1	4	yes	—
226	C₂³	C₂⁴×C₉	1	1	yes	characteristic
227	C₂³	C₂⁴×C₉	1	2	yes	—
228	Q₈	—	2	24	no	—
229	Q₈	—	2	12	no	—
230	Q₈	C₂×D₄×C₉	1	16	yes	—
231	D₄	—	2	8	no	—
232	D₄	—	2	16	no	—
233	D₄	—	2	12	no	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	24	no	—
236	C₂×C₄	—	2	24	no	—
237	C₂×C₄	C₂×D₄×C₉	1	12	yes	—
238	C₂×C₄	C₂⁴×C₉	1	8	yes	—
239	C₂×C₄	C₂⁴×C₉	1	6	yes	—
240	C₂×C₄	C₂⁴×C₉	1	4	yes	—
241	C₂×C₄	C₂⁴×C₉	1	6	yes	—
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	2	4	no	—
244	C₂×C₃	—	4	4	no	—
245	C₂×C₃	C₂³×C₃×D₄	1	1	yes	characteristic
246	C₂×C₃	C₂×C₃×2₋¹⁺⁴	1	2	yes	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	8	no	—
253	C₂²	C₂²×D₄×C₉	1	2	yes	—
254	C₂²	C₉×2₋¹⁺⁴	1	4	yes	complemented
255	C₂²	C₂⁵×C₉	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	2	6	no	—
258	C₄	—	2	8	no	—
259	C₄	C₂²×D₄×C₉	1	12	yes	—
260	C₃	C₃×#3486	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	2	4	no	—
263	C₂	—	4	4	no	—
264	C₂	C₂³×D₄×C₉	1	1	yes	characteristic
265	C₂	C₂×C₉×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂⁵
commutator	C₂²	C₂⁵×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂⁵×C₃
prime radical	C₃×#3486	C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×2₋¹⁺⁴	C₂	semidirect
2. C₂³×Q₈×C₉	C₂	semidirect
3. D₄×Q₈×C₉	C₂	semidirect
4. C₉×#548	C₂	semidirect
5. C₉×#536	C₂	semidirect
6. C₉×#525	C₂	semidirect
7. C₉×#519	C₂	semidirect
11. C₂×C₉×#41	C₂²	semidirect
16. C₂²×Q₈×C₉	C₂²	semidirect
18. C₉×C₄⋊Q₈	C₂²	semidirect
19. C₉×#124	C₂²	semidirect
22. C₉×C₂²⋊Q₈	C₂²	semidirect
25. C₄×Q₈×C₉	C₂²	semidirect
41. C₂×Q₈×C₉	D₄	semidirect
59. #3486	C₉	direct (N × H)
98. C₂×2₋¹⁺⁴	C₂×C₉	semidirect
99. C₂³×Q₈	C₂×C₉	semidirect
100. D₄×Q₈	C₂×C₉	semidirect
101. #548	C₂×C₉	semidirect
102. #536	C₂×C₉	semidirect
103. #525	C₂×C₉	semidirect
104. #519	C₂×C₉	semidirect
136. C₂²×C₉	2₋¹⁺⁴	semidirect
144. C₂×#41	C₂²×C₉	semidirect
149. C₂²×Q₈	C₂²×C₉	semidirect
151. C₄⋊Q₈	C₂²×C₉	semidirect
152. #124	C₂²×C₉	semidirect
155. C₂²⋊Q₈	C₂²×C₉	semidirect
158. C₄×Q₈	C₂²×C₉	semidirect
191. C₂×Q₈	D₄×C₉	semidirect
222. C₉	#3486	direct (N × H)
254. C₂²	C₉×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×2₋¹⁺⁴	2
C₂³×Q₈×C₉	2
D₄×Q₈×C₉	2
C₉×#548	2
C₉×#536	2
C₉×#525	2
C₉×#519	2
C₃×#3486	3

Subgroups — C9×#3486 #11768207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3486 #11768207