Subgroups — C₂²×C₁₁×#486 #14637795

Order 2816.

3086 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₁×#127	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₁×#486	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₁×#102	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₁₁	—	2	1	no	—
7	C₂³×D₄×C₁₁	—	2	1	no	—
8	C₂³×D₄×C₁₁	—	2	1	no	—
9	C₂³×D₄×C₁₁	C₂²	1	2	yes	—
10	C₂³×C₈×C₁₁	C₂²	1	2	yes	—
11	C₂×C₁₁×C₄⋊Q₈	—	2	6	no	—
12	C₂×C₁₁×C₄⋊Q₈	C₂²	1	12	yes	complemented
13	C₂×C₁₁×#127	—	2	6	no	—
14	C₂×C₁₁×#127	C₂²	1	12	yes	—
15	C₂²×C₁₁×C₄⋊C₄	—	2	1	no	—
16	C₂²×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
18	C₁₁×#486	C₂²	1	64	yes	direct product factor
19	C₂×C₁₁×#102	C₂²	1	48	yes	complemented
20	C₂×C₄×C₈×C₁₁	C₂²	1	12	yes	complemented
21	C₂⁵×C₁₁	—	2	2	no	—
22	C₂²×Q₈×C₁₁	—	2	12	no	—
23	C₂²×Q₈×C₁₁	—	4	8	no	—
24	C₂²×D₄×C₁₁	—	2	6	no	—
25	C₂²×D₄×C₁₁	—	2	12	no	—
26	C₂²×D₄×C₁₁	—	2	12	no	—
27	C₂²×D₄×C₁₁	—	2	16	no	—
28	C₂²×D₄×C₁₁	—	4	8	no	—
29	C₂²×D₄×C₁₁	—	4	8	no	—
30	C₂²×D₄×C₁₁	C₂³	1	12	yes	—
31	C₂³×C₄×C₁₁	—	2	2	no	—
32	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₁	—	2	8	no	—
36	C₂²×C₈×C₁₁	C₂³	1	12	yes	—
37	C₁₁×C₄⋊Q₈	—	2	24	no	—
38	C₁₁×C₄⋊Q₈	C₂³	1	16	yes	complemented
39	C₁₁×#127	—	2	24	no	—
40	C₁₁×#127	C₂³	1	16	yes	—
41	C₂×C₁₁×C₄⋊C₄	—	2	12	no	—
42	C₂×C₁₁×C₄⋊C₄	—	2	6	no	—
43	C₂×C₁₁×C₄⋊C₄	C₂³	1	12	yes	—
44	C₂×C₄²×C₁₁	D₄	1	6	yes	—
45	C₂×C₄²×C₁₁	C₂³	1	6	yes	—
46	C₁₁×#102	C₂³	1	64	yes	complemented
47	C₄×C₈×C₁₁	C₂³	1	16	yes	complemented
48	C₂²×#486	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₁	—	2	12	no	—
50	C₂⁴×C₁₁	—	4	8	no	—
51	C₂⁴×C₁₁	—	4	8	no	—
52	C₂⁴×C₁₁	—	4	8	no	—
53	C₂⁴×C₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂×Q₈×C₁₁	—	2	16	no	—
55	C₂×Q₈×C₁₁	—	4	48	no	—
56	C₂×D₄×C₁₁	—	2	12	no	—
57	C₂×D₄×C₁₁	—	2	16	no	—
58	C₂×D₄×C₁₁	—	2	16	no	—
59	C₂×D₄×C₁₁	—	2	48	no	—
60	C₂×D₄×C₁₁	—	4	24	no	—
61	C₂×D₄×C₁₁	—	4	48	no	—
62	C₂×D₄×C₁₁	—	4	48	no	—
63	C₂×D₄×C₁₁	C₂⁴	1	8	yes	—
64	C₂²×C₄×C₁₁	—	2	12	no	—
65	C₂²×C₄×C₁₁	—	4	8	no	—
66	C₂²×C₄×C₁₁	D₈	1	8	yes	—
67	C₂²×C₄×C₁₁	SD₁₆	1	8	yes	—
68	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₁₁	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₁₁	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₁₁	#41	1	8	yes	—
72	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
73	C₂×C₈×C₁₁	—	2	24	no	—
74	C₂×C₈×C₁₁	C₂⁴	1	8	yes	—
75	C₁₁×C₄⋊C₄	—	2	16	no	—
76	C₁₁×C₄⋊C₄	—	2	12	no	—
77	C₁₁×C₄⋊C₄	C₂⁴	1	8	yes	—
78	C₄²×C₁₁	C₂×D₄	1	12	yes	—
79	C₄²×C₁₁	C₂⁴	1	4	yes	—
80	C₂²×C₄⋊Q₈	C₂×C₁₁	1	1	yes	characteristic, complemented
81	C₂²×#127	C₂×C₁₁	1	1	yes	characteristic
82	C₂×#486	C₂×C₁₁	1	24	yes	direct product factor
83	C₂²×#102	C₂×C₁₁	1	4	yes	complemented
84	C₂²×C₄×C₈	C₂×C₁₁	1	1	yes	characteristic, complemented
85	C₂³×C₁₁	—	2	8	no	—
86	C₂³×C₁₁	—	4	24	no	—
87	C₂³×C₁₁	—	4	24	no	—
88	C₂³×C₁₁	—	4	24	no	—
89	C₂³×C₁₁	—	8	32	no	—
90	C₂³×C₁₁	#124	1	4	yes	—
91	C₂³×C₁₁	C₂×D₈	1	4	yes	—
92	C₂³×C₁₁	C₂×SD₁₆	1	4	yes	—
93	C₂³×C₁₁	C₂²×D₄	1	3	yes	—
94	Q₈×C₁₁	—	4	32	no	—
95	D₄×C₁₁	—	2	16	no	—
96	D₄×C₁₁	—	4	24	no	—
97	D₄×C₁₁	—	4	32	no	—
98	D₄×C₁₁	—	4	32	no	—
99	C₂×C₄×C₁₁	—	2	8	no	—
100	C₂×C₄×C₁₁	—	4	24	no	—
101	C₂×C₄×C₁₁	#124	1	12	yes	—
102	C₂×C₄×C₁₁	C₂×D₈	1	24	yes	—
103	C₂×C₄×C₁₁	C₂×SD₁₆	1	24	yes	—
104	C₂×C₄×C₁₁	C₂²×D₄	1	3	yes	—
105	C₂×C₄×C₁₁	C₂²×D₄	1	4	yes	—
106	C₂×C₄×C₁₁	C₂²×D₄	1	4	yes	—
107	C₂×C₄×C₁₁	C₂×#41	1	12	yes	—
108	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
109	C₈×C₁₁	—	2	8	no	—
110	C₂³×Q₈	—	2	1	no	—
111	C₂³×D₄	—	2	1	no	—
112	C₂³×D₄	—	2	1	no	—
113	C₂³×D₄	C₂²×C₁₁	1	2	yes	—
114	C₂³×C₈	C₂²×C₁₁	1	2	yes	—
115	C₂×C₄⋊Q₈	—	2	6	no	—
116	C₂×C₄⋊Q₈	C₂²×C₁₁	1	12	yes	complemented
117	C₂×#127	—	2	6	no	—
118	C₂×#127	C₂²×C₁₁	1	12	yes	—
119	C₂²×C₄⋊C₄	—	2	1	no	—
120	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
121	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
122	#486	C₂²×C₁₁	1	64	yes	direct product factor
123	C₂×#102	C₂²×C₁₁	1	48	yes	complemented
124	C₂×C₄×C₈	C₂²×C₁₁	1	12	yes	complemented
125	C₂²×C₁₁	—	4	8	no	—
126	C₂²×C₁₁	—	4	8	no	—
127	C₂²×C₁₁	—	4	8	no	—
128	C₂²×C₁₁	—	8	48	no	—
129	C₂²×C₁₁	#486	1	16	yes	direct product factor
130	C₂²×C₁₁	C₂×#124	1	6	yes	—
131	C₂²×C₁₁	C₂²×D₈	1	6	yes	—
132	C₂²×C₁₁	C₂²×SD₁₆	1	6	yes	—
133	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
134	C₄×C₁₁	—	4	8	no	—
135	C₄×C₁₁	C₂×#124	1	6	yes	—
136	C₄×C₁₁	C₂²×D₈	1	8	yes	—
137	C₄×C₁₁	C₂²×SD₁₆	1	8	yes	—
138	C₄×C₁₁	C₂²×#41	1	2	yes	—
139	C₂⁵	—	2	2	no	—
140	C₂²×Q₈	—	2	12	no	—
141	C₂²×Q₈	—	4	8	no	—
142	C₂²×D₄	—	2	6	no	—
143	C₂²×D₄	—	2	12	no	—
144	C₂²×D₄	—	2	12	no	—
145	C₂²×D₄	—	2	16	no	—
146	C₂²×D₄	—	4	8	no	—
147	C₂²×D₄	—	4	8	no	—
148	C₂²×D₄	C₂³×C₁₁	1	12	yes	—
149	C₂³×C₄	—	2	2	no	—
150	C₂³×C₄	D₄×C₁₁	1	1	yes	characteristic
151	C₂³×C₄	D₄×C₁₁	1	1	yes	characteristic
152	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
153	C₂²×C₈	—	2	8	no	—
154	C₂²×C₈	C₂³×C₁₁	1	12	yes	—
155	C₄⋊Q₈	—	2	24	no	—
156	C₄⋊Q₈	C₂³×C₁₁	1	16	yes	complemented
157	#127	—	2	24	no	—
158	#127	C₂³×C₁₁	1	16	yes	—
159	C₂×C₄⋊C₄	—	2	12	no	—
160	C₂×C₄⋊C₄	—	2	6	no	—
161	C₂×C₄⋊C₄	C₂³×C₁₁	1	12	yes	—
162	C₂×C₄²	D₄×C₁₁	1	6	yes	—
163	C₂×C₄²	C₂³×C₁₁	1	6	yes	—
164	#102	C₂³×C₁₁	1	64	yes	complemented
165	C₄×C₈	C₂³×C₁₁	1	16	yes	complemented
166	C₂×C₁₁	—	8	8	no	—
167	C₂×C₁₁	C₂×#486	1	12	yes	direct product factor
168	C₂×C₁₁	C₂²×#124	1	1	yes	characteristic
169	C₂×C₁₁	C₂³×D₈	1	1	yes	characteristic
170	C₂×C₁₁	C₂³×SD₁₆	1	1	yes	characteristic
171	C₂⁴	—	2	12	no	—
172	C₂⁴	—	4	8	no	—
173	C₂⁴	—	4	8	no	—
174	C₂⁴	—	4	8	no	—
175	C₂⁴	C₂×D₄×C₁₁	1	1	yes	characteristic
176	C₂×Q₈	—	2	16	no	—
177	C₂×Q₈	—	4	48	no	—
178	C₂×D₄	—	2	12	no	—
179	C₂×D₄	—	2	16	no	—
180	C₂×D₄	—	2	16	no	—
181	C₂×D₄	—	2	48	no	—
182	C₂×D₄	—	4	24	no	—
183	C₂×D₄	—	4	48	no	—
184	C₂×D₄	—	4	48	no	—
185	C₂×D₄	C₂⁴×C₁₁	1	8	yes	—
186	C₂²×C₄	—	2	12	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	C₁₁×D₈	1	8	yes	—
189	C₂²×C₄	C₁₁×SD₁₆	1	8	yes	—
190	C₂²×C₄	C₂×D₄×C₁₁	1	3	yes	—
191	C₂²×C₄	C₂×D₄×C₁₁	1	6	yes	—
192	C₂²×C₄	C₂×D₄×C₁₁	1	6	yes	—
193	C₂²×C₄	C₁₁×#41	1	8	yes	—
194	C₂²×C₄	C₂⁴×C₁₁	1	3	yes	—
195	C₂×C₈	—	2	24	no	—
196	C₂×C₈	C₂⁴×C₁₁	1	8	yes	—
197	C₄⋊C₄	—	2	16	no	—
198	C₄⋊C₄	—	2	12	no	—
199	C₄⋊C₄	C₂⁴×C₁₁	1	8	yes	—
200	C₄²	C₂×D₄×C₁₁	1	12	yes	—
201	C₄²	C₂⁴×C₁₁	1	4	yes	—
202	C₁₁	C₂²×#486	1	1	yes	characteristic, direct product factor, Sylow
203	C₂³	—	2	8	no	—
204	C₂³	—	4	24	no	—
205	C₂³	—	4	24	no	—
206	C₂³	—	4	24	no	—
207	C₂³	—	8	32	no	—
208	C₂³	C₁₁×#124	1	4	yes	—
209	C₂³	C₂×C₁₁×D₈	1	4	yes	—
210	C₂³	C₂×C₁₁×SD₁₆	1	4	yes	—
211	C₂³	C₂²×D₄×C₁₁	1	3	yes	—
212	Q₈	—	4	32	no	—
213	D₄	—	2	16	no	—
214	D₄	—	4	24	no	—
215	D₄	—	4	32	no	—
216	D₄	—	4	32	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	4	24	no	—
219	C₂×C₄	C₁₁×#124	1	12	yes	—
220	C₂×C₄	C₂×C₁₁×D₈	1	24	yes	—
221	C₂×C₄	C₂×C₁₁×SD₁₆	1	24	yes	—
222	C₂×C₄	C₂²×D₄×C₁₁	1	3	yes	—
223	C₂×C₄	C₂²×D₄×C₁₁	1	4	yes	—
224	C₂×C₄	C₂²×D₄×C₁₁	1	4	yes	—
225	C₂×C₄	C₂×C₁₁×#41	1	12	yes	—
226	C₂×C₄	C₂⁵×C₁₁	1	1	yes	characteristic
227	C₈	—	2	8	no	—
228	C₂²	—	4	8	no	—
229	C₂²	—	4	8	no	—
230	C₂²	—	4	8	no	—
231	C₂²	—	8	48	no	—
232	C₂²	C₁₁×#486	1	16	yes	direct product factor
233	C₂²	C₂×C₁₁×#124	1	6	yes	—
234	C₂²	C₂²×C₁₁×D₈	1	6	yes	—
235	C₂²	C₂²×C₁₁×SD₁₆	1	6	yes	—
236	C₂²	C₂³×D₄×C₁₁	1	1	yes	characteristic
237	C₄	—	4	8	no	—
238	C₄	C₂×C₁₁×#124	1	6	yes	—
239	C₄	C₂²×C₁₁×D₈	1	8	yes	—
240	C₄	C₂²×C₁₁×SD₁₆	1	8	yes	—
241	C₄	C₂²×C₁₁×#41	1	2	yes	—
242	C₂	—	8	8	no	—
243	C₂	C₂×C₁₁×#486	1	12	yes	direct product factor
244	C₂	C₂²×C₁₁×#124	1	1	yes	characteristic
245	C₂	C₂³×C₁₁×D₈	1	1	yes	characteristic
246	C₂	C₂³×C₁₁×SD₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₁
prime radical	C₂²×C₁₁×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁ ◁ C₂²×C₄²×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₁×C₄⋊Q₈	C₂	semidirect
3. C₂×C₁₁×#486	C₂	direct (N × H)
4. C₂²×C₁₁×#102	C₂	semidirect
5. C₂²×C₄×C₈×C₁₁	C₂	semidirect
12. C₂×C₁₁×C₄⋊Q₈	C₂²	semidirect
18. C₁₁×#486	C₂²	direct (N × H)
19. C₂×C₁₁×#102	C₂²	semidirect
20. C₂×C₄×C₈×C₁₁	C₂²	semidirect
38. C₁₁×C₄⋊Q₈	C₂³	semidirect
46. C₁₁×#102	C₂³	semidirect
47. C₄×C₈×C₁₁	C₂³	semidirect
48. C₂²×#486	C₁₁	direct (N × H)
80. C₂²×C₄⋊Q₈	C₂×C₁₁	semidirect
82. C₂×#486	C₂×C₁₁	direct (N × H)
83. C₂²×#102	C₂×C₁₁	semidirect
84. C₂²×C₄×C₈	C₂×C₁₁	semidirect
116. C₂×C₄⋊Q₈	C₂²×C₁₁	semidirect
122. #486	C₂²×C₁₁	direct (N × H)
123. C₂×#102	C₂²×C₁₁	semidirect
124. C₂×C₄×C₈	C₂²×C₁₁	semidirect
129. C₂²×C₁₁	#486	direct (N × H)
156. C₄⋊Q₈	C₂³×C₁₁	semidirect
164. #102	C₂³×C₁₁	semidirect
165. C₄×C₈	C₂³×C₁₁	semidirect
167. C₂×C₁₁	C₂×#486	direct (N × H)
202. C₁₁	C₂²×#486	direct (N × H)
232. C₂²	C₁₁×#486	direct (N × H)
243. C₂	C₂×C₁₁×#486	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₁×C₄⋊Q₈	2
C₂²×C₁₁×#127	2
C₂×C₁₁×#486	2
C₂²×C₁₁×#102	2
C₂²×C₄×C₈×C₁₁	2
C₂²×#486	11

Subgroups — C22×C11×#486 #14637795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#486 #14637795