Subgroups — C₉×#2978 #11767699

Order 1152.

603 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#525	C₂	1	1	yes	maximal, characteristic
2	C₉×#436	C₂	1	4	yes	maximal, complemented
3	C₉×#435	C₂	1	4	yes	maximal, complemented
4	C₉×#433	C₂	1	2	yes	maximal, complemented
5	C₉×#408	C₂	1	2	yes	maximal, complemented
6	C₉×#407	C₂	1	2	yes	maximal, complemented
7	C₃×#2978	C₃	1	1	yes	maximal, characteristic
8	C₂×C₉×#41	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×M₁₆	C₂²	1	4	yes	—
11	C₂×C₉×M₁₆	C₂²	1	2	yes	complemented
12	C₂²×C₈×C₉	C₂²	1	2	yes	complemented
13	C₉×#123	C₄	1	2	yes	—
14	C₉×C₂²⋊Q₈	C₄	1	2	yes	—
15	C₉×#121	C₄	1	2	yes	—
16	C₉×#120	C₄	1	2	yes	—
17	C₄×D₄×C₉	C₂²	1	4	yes	—
18	C₉×#117	C₂²	1	1	yes	characteristic
19	C₉×C₄⋊C₈	C₂²	1	4	yes	complemented
20	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
21	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₉×C₂²⋊C₈	C₂²	1	4	yes	complemented
24	C₉×#97	C₂²	1	2	yes	complemented
25	C₄×C₈×C₉	C₂²	1	2	yes	complemented
26	C₃×#525	C₂×C₃	1	1	yes	characteristic
27	C₃×#436	C₂×C₃	1	4	yes	—
28	C₃×#435	C₂×C₃	1	4	yes	—
29	C₃×#433	C₂×C₃	1	2	yes	—
30	C₃×#408	C₂×C₃	1	2	yes	—
31	C₃×#407	C₂×C₃	1	2	yes	—
32	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
33	C₉×#41	—	4	2	no	—
34	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
36	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	—	4	2	no	—
39	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₉	C₂³	1	2	yes	—
44	C₉×M₁₆	—	4	4	no	—
45	C₉×M₁₆	—	4	2	no	—
46	C₂×C₈×C₉	—	4	2	no	—
47	C₂×C₈×C₉	C₂³	1	4	yes	—
48	C₂×C₈×C₉	C₂³	1	4	yes	—
49	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
51	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
53	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
54	C₄²×C₉	C₂³	1	2	yes	—
55	#2978	C₉	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
57	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
59	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
60	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
61	C₃×#123	C₃×C₄	1	2	yes	—
62	C₃×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
63	C₃×#121	C₃×C₄	1	2	yes	—
64	C₃×#120	C₃×C₄	1	2	yes	—
65	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
66	C₃×#117	C₂²×C₃	1	1	yes	characteristic
67	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
68	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
69	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
70	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
71	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
72	C₃×#97	C₂²×C₃	1	2	yes	—
73	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
74	C₂³×C₉	—	4	1	no	—
75	C₂³×C₉	—	4	2	no	—
76	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
77	C₂³×C₉	C₂²×C₄	1	2	yes	—
78	C₂³×C₉	C₂²×C₄	1	2	yes	—
79	Q₈×C₉	—	4	1	no	—
80	D₄×C₉	—	4	4	no	—
81	D₄×C₉	—	4	1	no	—
82	D₄×C₉	—	4	2	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	4	1	no	—
88	C₂×C₄×C₉	—	4	2	no	—
89	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
94	C₈×C₉	—	2	4	no	—
95	C₈×C₉	—	2	4	no	—
96	#525	C₂×C₉	1	1	yes	characteristic
97	#436	C₂×C₉	1	4	yes	complemented
98	#435	C₂×C₉	1	4	yes	complemented
99	#433	C₂×C₉	1	2	yes	complemented
100	#408	C₂×C₉	1	2	yes	complemented
101	#407	C₂×C₉	1	2	yes	complemented
102	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
103	C₃×#41	—	4	2	no	—
104	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
106	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
107	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	4	2	no	—
109	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
110	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
113	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
114	C₃×M₁₆	—	4	4	no	—
115	C₃×M₁₆	—	4	2	no	—
116	C₂×C₃×C₈	—	4	2	no	—
117	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
118	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
119	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
120	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
121	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
122	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
123	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
124	C₃×C₄²	C₂³×C₃	1	2	yes	—
125	C₂²×C₉	—	2	1	no	—
126	C₂²×C₉	—	2	2	no	—
127	C₂²×C₉	—	2	2	no	—
128	C₂²×C₉	—	2	2	no	—
129	C₂²×C₉	—	2	4	no	—
130	C₂²×C₉	—	2	2	no	—
131	C₂²×C₉	—	2	2	no	—
132	C₂²×C₉	—	4	4	no	—
133	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₉	—	2	2	no	—
135	C₄×C₉	—	2	1	no	—
136	C₄×C₉	—	2	2	no	—
137	C₄×C₉	—	2	4	no	—
138	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
139	C₂×#41	C₂²×C₉	1	1	yes	characteristic
140	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
141	C₂×M₁₆	C₂²×C₉	1	4	yes	—
142	C₂×M₁₆	C₂²×C₉	1	2	yes	complemented
143	C₂²×C₈	C₂²×C₉	1	2	yes	complemented
144	#123	C₄×C₉	1	2	yes	—
145	C₂²⋊Q₈	C₄×C₉	1	2	yes	—
146	#121	C₄×C₉	1	2	yes	—
147	#120	C₄×C₉	1	2	yes	—
148	C₄×D₄	C₂²×C₉	1	4	yes	—
149	#117	C₂²×C₉	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₉	1	4	yes	complemented
151	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
152	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
153	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
154	C₂²⋊C₈	C₂²×C₉	1	4	yes	complemented
155	#97	C₂²×C₉	1	2	yes	complemented
156	C₄×C₈	C₂²×C₉	1	2	yes	complemented
157	C₂³×C₃	—	4	1	no	—
158	C₂³×C₃	—	4	2	no	—
159	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
160	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
161	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
162	C₃×Q₈	—	4	1	no	—
163	C₃×D₄	—	4	4	no	—
164	C₃×D₄	—	4	1	no	—
165	C₃×D₄	—	4	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	—	4	1	no	—
172	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
173	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
174	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
175	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
177	C₃×C₈	—	2	4	no	—
178	C₃×C₈	—	2	4	no	—
179	C₂×C₉	—	4	2	no	—
180	C₂×C₉	—	4	1	no	—
181	C₂×C₉	—	4	2	no	—
182	C₂×C₉	#518	1	1	yes	characteristic
183	C₂×C₉	#568	1	2	yes	—
184	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
185	#41	—	4	2	no	—
186	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
187	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
188	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
189	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
192	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₉	1	2	yes	—
195	C₂²×C₄	C₂³×C₉	1	2	yes	—
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	C₂³×C₉	1	4	yes	—
200	C₂×C₈	C₂³×C₉	1	4	yes	—
201	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
202	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
203	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
204	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
205	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
206	C₄²	C₂³×C₉	1	2	yes	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	4	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	4	4	no	—
215	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
221	C₉	#2978	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³	—	4	1	no	—
223	C₂³	—	4	2	no	—
224	C₂³	C₂²×C₄×C₉	1	2	yes	—
225	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
226	C₂³	C₂²×C₄×C₉	1	2	yes	—
227	Q₈	—	4	1	no	—
228	D₄	—	4	1	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
238	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
239	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
240	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
241	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	2	4	no	—
244	C₂×C₃	—	4	2	no	—
245	C₂×C₃	—	4	1	no	—
246	C₂×C₃	—	4	2	no	—
247	C₂×C₃	C₃×#518	1	1	yes	characteristic
248	C₂×C₃	C₃×#568	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
263	C₃	C₃×#2978	1	1	yes	characteristic
264	C₂	—	4	2	no	—
265	C₂	—	4	1	no	—
266	C₂	—	4	2	no	—
267	C₂	C₉×#518	1	1	yes	characteristic
268	C₂	C₉×#568	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₃×C₄
prime radical	C₃×#525	C₂×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
2. C₉×#436	C₂	semidirect
3. C₉×#435	C₂	semidirect
4. C₉×#433	C₂	semidirect
5. C₉×#408	C₂	semidirect
6. C₉×#407	C₂	semidirect
11. C₂×C₉×M₁₆	C₂²	semidirect
12. C₂²×C₈×C₉	C₂²	semidirect
19. C₉×C₄⋊C₈	C₂²	semidirect
23. C₉×C₂²⋊C₈	C₂²	semidirect
24. C₉×#97	C₂²	semidirect
25. C₄×C₈×C₉	C₂²	semidirect
55. #2978	C₉	direct (N × H)
97. #436	C₂×C₉	semidirect
98. #435	C₂×C₉	semidirect
99. #433	C₂×C₉	semidirect
100. #408	C₂×C₉	semidirect
101. #407	C₂×C₉	semidirect
142. C₂×M₁₆	C₂²×C₉	semidirect
143. C₂²×C₈	C₂²×C₉	semidirect
150. C₄⋊C₈	C₂²×C₉	semidirect
154. C₂²⋊C₈	C₂²×C₉	semidirect
155. #97	C₂²×C₉	semidirect
156. C₄×C₈	C₂²×C₉	semidirect
221. C₉	#2978	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×#525	2
C₉×#436	2
C₉×#435	2
C₉×#433	2
C₉×#408	2
C₉×#407	2
C₃×#2978	3

Subgroups — C9×#2978 #11767699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2978 #11767699