Subgroups — C₂×C₉×#541 #11768174

Order 1152.

1506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#541	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#126	C₂	1	4	yes	maximal
4	C₂×C₉×#125	C₂	1	4	yes	maximal, complemented
5	C₂×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₉×C₂²⋊Q₈	C₂	1	2	yes	maximal
7	C₂×C₄×Q₈×C₉	C₂	1	2	yes	maximal, complemented
8	C₂×C₉×#117	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃×#541	C₃	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₉	C₂²	1	2	yes	—
11	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₉×C₄⋊Q₈	C₂²	1	8	yes	complemented
13	C₉×#126	C₂²	1	32	yes	—
14	C₉×#125	C₂²	1	8	yes	complemented
15	C₉×#125	C₂²	1	32	yes	complemented
16	C₉×C₂²⋊Q₈	C₂²	1	16	yes	—
17	C₄×Q₈×C₉	C₂²	1	16	yes	complemented
18	C₉×#117	C₂²	1	8	yes	—
19	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
22	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
24	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
25	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄²×C₉	C₂²	1	4	yes	—
28	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
30	C₃×#541	C₂×C₃	1	16	yes	—
31	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×#126	C₂×C₃	1	4	yes	—
33	C₂×C₃×#125	C₂×C₃	1	1	yes	characteristic
34	C₂×C₃×#125	C₂×C₃	1	4	yes	—
35	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
36	C₂×C₃×C₄×Q₈	C₂×C₃	1	2	yes	—
37	C₂×C₃×#117	C₂×C₃	1	1	yes	characteristic
38	C₂⁴×C₉	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₉	—	2	8	no	—
40	C₂×Q₈×C₉	C₂³	1	8	yes	—
41	C₂²×C₄×C₉	—	2	4	no	—
42	C₂²×C₄×C₉	C₂³	1	4	yes	—
43	C₂²×C₄×C₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₉	C₂³	1	4	yes	—
47	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₉×C₄⋊C₄	C₂³	1	8	yes	—
50	C₉×C₄⋊C₄	C₂³	1	8	yes	—
51	C₉×C₄⋊C₄	C₂³	1	16	yes	—
52	C₉×C₄⋊C₄	C₂³	1	16	yes	—
53	C₉×C₄⋊C₄	C₂³	1	16	yes	—
54	C₉×C₄⋊C₄	C₂³	1	16	yes	—
55	C₉×C₂²⋊C₄	C₂³	1	8	yes	—
56	C₉×C₂²⋊C₄	C₂³	1	16	yes	—
57	C₄²×C₉	C₂³	1	4	yes	—
58	C₄²×C₉	C₂³	1	4	yes	—
59	C₄²×C₉	C₂³	1	16	yes	—
60	C₂×#541	C₉	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
62	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
63	C₃×C₄⋊Q₈	C₂²×C₃	1	8	yes	—
64	C₃×#126	C₂²×C₃	1	32	yes	—
65	C₃×#125	C₂²×C₃	1	8	yes	—
66	C₃×#125	C₂²×C₃	1	32	yes	—
67	C₃×C₂²⋊Q₈	C₂²×C₃	1	16	yes	—
68	C₃×C₄×Q₈	C₂²×C₃	1	16	yes	—
69	C₃×#117	C₂²×C₃	1	8	yes	—
70	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
73	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
74	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
75	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
76	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
78	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
81	C₂³×C₉	—	2	2	no	—
82	C₂³×C₉	—	2	4	no	—
83	C₂³×C₉	C₂⁴	1	2	yes	—
84	C₂³×C₉	C₂⁴	1	1	yes	characteristic
85	Q₈×C₉	—	2	16	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	8	no	—
88	C₂×C₄×C₉	—	2	4	no	—
89	C₂×C₄×C₉	—	2	8	no	—
90	C₂×C₄×C₉	—	2	4	no	—
91	C₂×C₄×C₉	—	2	8	no	—
92	C₂×C₄×C₉	#41	1	4	yes	—
93	C₂×C₄×C₉	C₂⁴	1	4	yes	—
94	C₂×C₄×C₉	C₂⁴	1	4	yes	—
95	C₂×C₄×C₉	C₂⁴	1	8	yes	—
96	C₂×C₄×C₉	C₂⁴	1	8	yes	—
97	C₂×C₄×C₉	C₂⁴	1	2	yes	—
98	C₂×C₄×C₉	C₂⁴	1	2	yes	—
99	#541	C₂×C₉	1	16	yes	direct product factor
100	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
101	C₂×#126	C₂×C₉	1	4	yes	—
102	C₂×#125	C₂×C₉	1	1	yes	characteristic, complemented
103	C₂×#125	C₂×C₉	1	4	yes	complemented
104	C₂×C₂²⋊Q₈	C₂×C₉	1	2	yes	—
105	C₂×C₄×Q₈	C₂×C₉	1	2	yes	complemented
106	C₂×#117	C₂×C₉	1	1	yes	characteristic
107	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₃×Q₈	—	2	8	no	—
109	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
112	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
113	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
114	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
115	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
117	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
119	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
120	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
121	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
122	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
123	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
124	C₃×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
125	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
126	C₃×C₄²	C₂³×C₃	1	4	yes	—
127	C₃×C₄²	C₂³×C₃	1	4	yes	—
128	C₃×C₄²	C₂³×C₃	1	16	yes	—
129	C₂²×C₉	—	2	2	no	—
130	C₂²×C₉	—	2	4	no	—
131	C₂²×C₉	—	4	4	no	—
132	C₂²×C₉	C₂×#41	1	2	yes	—
133	C₂²×C₉	2₋¹⁺⁴	1	4	yes	—
134	C₂²×C₉	C₂⁵	1	1	yes	characteristic
135	C₄×C₉	—	2	2	no	—
136	C₄×C₉	—	2	8	no	—
137	C₄×C₉	—	2	4	no	—
138	C₄×C₉	—	2	4	no	—
139	C₄×C₉	—	2	8	no	—
140	C₄×C₉	C₂×#41	1	4	yes	—
141	C₂²×Q₈	C₂²×C₉	1	2	yes	—
142	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
143	C₄⋊Q₈	C₂²×C₉	1	8	yes	complemented
144	#126	C₂²×C₉	1	32	yes	—
145	#125	C₂²×C₉	1	8	yes	complemented
146	#125	C₂²×C₉	1	32	yes	complemented
147	C₂²⋊Q₈	C₂²×C₉	1	16	yes	—
148	C₄×Q₈	C₂²×C₉	1	16	yes	complemented
149	#117	C₂²×C₉	1	8	yes	—
150	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
153	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
154	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
155	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
158	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
159	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
160	C₂×C₄²	C₂²×C₉	1	4	yes	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	C₂⁴×C₃	1	2	yes	—
164	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
165	C₃×Q₈	—	2	16	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	8	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	8	no	—
171	C₂×C₃×C₄	—	2	8	no	—
172	C₂×C₃×C₄	C₃×#41	1	4	yes	—
173	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
174	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
175	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
176	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
177	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
178	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
179	C₂×C₉	—	4	2	no	—
180	C₂×C₉	#541	1	4	yes	direct product factor
181	C₂×C₉	C₂²×#41	1	1	yes	characteristic
182	C₂×C₉	C₂×2₋¹⁺⁴	1	2	yes	—
183	C₂⁴	C₂³×C₉	1	1	yes	characteristic
184	C₂×Q₈	—	2	8	no	—
185	C₂×Q₈	C₂³×C₉	1	8	yes	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	C₂³×C₉	1	2	yes	—
188	C₂²×C₄	C₂³×C₉	1	2	yes	—
189	C₂²×C₄	C₂³×C₉	1	4	yes	—
190	C₂²×C₄	C₂³×C₉	1	4	yes	—
191	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₉	1	4	yes	—
193	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₉	1	8	yes	—
195	C₄⋊C₄	C₂³×C₉	1	8	yes	—
196	C₄⋊C₄	C₂³×C₉	1	16	yes	—
197	C₄⋊C₄	C₂³×C₉	1	16	yes	—
198	C₄⋊C₄	C₂³×C₉	1	16	yes	—
199	C₄⋊C₄	C₂³×C₉	1	16	yes	—
200	C₂²⋊C₄	C₂³×C₉	1	8	yes	—
201	C₂²⋊C₄	C₂³×C₉	1	16	yes	—
202	C₄²	C₂³×C₉	1	4	yes	—
203	C₄²	C₂³×C₉	1	4	yes	—
204	C₄²	C₂³×C₉	1	16	yes	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	—	4	4	no	—
208	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
209	C₂²×C₃	C₃×2₋¹⁺⁴	1	4	yes	—
210	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	8	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	2	8	no	—
216	C₃×C₄	C₂×C₃×#41	1	4	yes	—
217	C₉	C₂×#541	1	1	yes	characteristic, direct product factor, Sylow
218	C₂³	—	2	2	no	—
219	C₂³	—	2	4	no	—
220	C₂³	C₂⁴×C₉	1	2	yes	—
221	C₂³	C₂⁴×C₉	1	1	yes	characteristic
222	Q₈	—	2	16	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	C₉×#41	1	4	yes	—
230	C₂×C₄	C₂⁴×C₉	1	4	yes	—
231	C₂×C₄	C₂⁴×C₉	1	4	yes	—
232	C₂×C₄	C₂⁴×C₉	1	8	yes	—
233	C₂×C₄	C₂⁴×C₉	1	8	yes	—
234	C₂×C₄	C₂⁴×C₉	1	2	yes	—
235	C₂×C₄	C₂⁴×C₉	1	2	yes	—
236	C₂×C₃	—	4	2	no	—
237	C₂×C₃	C₃×#541	1	4	yes	—
238	C₂×C₃	C₂²×C₃×#41	1	1	yes	characteristic
239	C₂×C₃	C₂×C₃×2₋¹⁺⁴	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	4	4	no	—
243	C₂²	C₂×C₉×#41	1	2	yes	—
244	C₂²	C₉×2₋¹⁺⁴	1	4	yes	—
245	C₂²	C₂⁵×C₉	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	8	no	—
248	C₄	—	2	4	no	—
249	C₄	—	2	4	no	—
250	C₄	—	2	8	no	—
251	C₄	C₂×C₉×#41	1	4	yes	—
252	C₃	C₂×C₃×#541	1	1	yes	characteristic
253	C₂	—	4	2	no	—
254	C₂	C₉×#541	1	4	yes	direct product factor
255	C₂	C₂²×C₉×#41	1	1	yes	characteristic
256	C₂	C₂×C₉×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle	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
1. C₉×#541	C₂	direct (N × H)
2. C₂×C₉×C₄⋊Q₈	C₂	semidirect
4. C₂×C₉×#125	C₂	semidirect
5. C₂×C₉×#125	C₂	semidirect
7. C₂×C₄×Q₈×C₉	C₂	semidirect
12. C₉×C₄⋊Q₈	C₂²	semidirect
14. C₉×#125	C₂²	semidirect
15. C₉×#125	C₂²	semidirect
17. C₄×Q₈×C₉	C₂²	semidirect
60. C₂×#541	C₉	direct (N × H)
99. #541	C₂×C₉	direct (N × H)
100. C₂×C₄⋊Q₈	C₂×C₉	semidirect
102. C₂×#125	C₂×C₉	semidirect
103. C₂×#125	C₂×C₉	semidirect
105. C₂×C₄×Q₈	C₂×C₉	semidirect
143. C₄⋊Q₈	C₂²×C₉	semidirect
145. #125	C₂²×C₉	semidirect
146. #125	C₂²×C₉	semidirect
148. C₄×Q₈	C₂²×C₉	semidirect
180. C₂×C₉	#541	direct (N × H)
217. C₉	C₂×#541	direct (N × H)
254. C₂	C₉×#541	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×#541	2
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×#126	2
C₂×C₉×#125	2
C₂×C₉×C₂²⋊Q₈	2
C₂×C₄×Q₈×C₉	2
C₂×C₉×#117	2
C₂×C₃×#541	3

Subgroups — C2×C9×#541 #11768174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#541 #11768174