Subgroups — C₂³×#66949 #15660415

Order 2816.

14078 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×D₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#66949	C₂	1	56	yes	maximal, direct product factor
4	C₂³×Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×(C₂×D₁₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂³×(C₂×D₁₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₁₁	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
13	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
15	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×D₁₁	—	2	16	no	—
17	C₂³×C₄×D₁₁	C₂²	1	28	yes	—
18	C₂³×Dic₂₂	—	2	16	no	—
19	C₂³×Dic₂₂	C₂²	1	28	yes	complemented
20	C₂²×C₁₁×C₄⋊C₄	C₂²	1	28	yes	complemented
21	C₂²×(C₂×D₁₁)⋊C₄	C₂²	1	28	yes	—
22	C₂²×(C₂×D₁₁)⋊C₄	C₂²	1	28	yes	—
23	C₂²×Dic₁₁⋊C₄	C₂²	1	28	yes	complemented
24	C₂²×Dic₁₁⋊C₄	C₂²	1	28	yes	complemented
25	C₂×#66949	C₂²	1	448	yes	direct product factor
26	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
27	C₂⁴×D₁₁	—	2	8	no	—
28	C₂⁴×D₁₁	—	2	8	no	—
29	C₂⁴×D₁₁	Q₈	1	16	yes	complemented
30	C₂⁴×D₁₁	C₂³	1	14	yes	—
31	C₂³×C₄×C₁₁	—	2	8	no	—
32	C₂³×C₄×C₁₁	—	2	8	no	—
33	C₂³×C₄×C₁₁	—	2	8	no	—
34	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
35	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
36	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
37	C₂³×Dic₁₁	—	2	8	no	—
38	C₂³×Dic₁₁	—	2	8	no	—
39	C₂³×Dic₁₁	D₄	1	16	yes	—
40	C₂³×Dic₁₁	C₂³	1	14	yes	—
41	C₂³×Dic₁₁	C₂³	1	14	yes	—
42	C₂³×Dic₁₁	C₂³	1	14	yes	—
43	C₂²×C₄×D₁₁	—	2	224	no	—
44	C₂²×C₄×D₁₁	C₂³	1	112	yes	—
45	C₂²×Dic₂₂	—	2	224	no	—
46	C₂²×Dic₂₂	C₂³	1	112	yes	complemented
47	C₂×C₁₁×C₄⋊C₄	C₂³	1	112	yes	complemented
48	C₂×(C₂×D₁₁)⋊C₄	C₂³	1	112	yes	—
49	C₂×(C₂×D₁₁)⋊C₄	C₂³	1	112	yes	—
50	C₂×Dic₁₁⋊C₄	C₂³	1	112	yes	complemented
51	C₂×Dic₁₁⋊C₄	C₂³	1	112	yes	complemented
52	#66949	C₂³	1	512	yes	direct product factor
53	C₂³×C₂²⋊Q₈	—	11	1	no	maximal, Sylow
54	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
55	C₂⁴×C₁₁	C₂×Q₈	1	8	yes	—
56	C₂⁴×C₁₁	#41	1	8	yes	—
57	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
58	C₂³×D₁₁	—	2	128	no	—
59	C₂³×D₁₁	—	2	56	no	—
60	C₂³×D₁₁	—	2	56	no	—
61	C₂³×D₁₁	C₂×Q₈	1	112	yes	complemented
62	C₂³×D₁₁	C₂⁴	1	28	yes	—
63	C₂²×C₄×C₁₁	—	2	56	no	—
64	C₂²×C₄×C₁₁	—	2	56	no	—
65	C₂²×C₄×C₁₁	—	2	56	no	—
66	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
67	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
68	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
69	C₂²×Dic₁₁	—	2	56	no	—
70	C₂²×Dic₁₁	—	2	56	no	—
71	C₂²×Dic₁₁	C₂×D₄	1	112	yes	—
72	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
73	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
74	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
75	C₂×C₄×D₁₁	—	2	448	no	—
76	C₂×C₄×D₁₁	C₂⁴	1	64	yes	—
77	C₂×Dic₂₂	—	2	448	no	—
78	C₂×Dic₂₂	C₂⁴	1	64	yes	complemented
79	C₁₁×C₄⋊C₄	C₂⁴	1	64	yes	complemented
80	(C₂×D₁₁)⋊C₄	C₂⁴	1	64	yes	—
81	(C₂×D₁₁)⋊C₄	C₂⁴	1	64	yes	—
82	Dic₁₁⋊C₄	C₂⁴	1	64	yes	complemented
83	Dic₁₁⋊C₄	C₂⁴	1	64	yes	complemented
84	C₂⁴×Q₈	—	11	1	no	—
85	C₂⁵×C₄	—	11	1	no	—
86	C₂²×C₂²⋊Q₈	—	11	56	no	—
87	C₂³×C₄⋊C₄	—	11	1	no	—
88	C₂³×C₄⋊C₄	—	11	1	no	—
89	C₂³×C₄⋊C₄	D₁₁	1	1	yes	characteristic, complemented
90	C₂³×C₂²⋊C₄	—	11	1	no	—
91	C₂³×C₂²⋊C₄	—	11	1	no	—
92	C₂³×C₁₁	C₂²⋊Q₈	1	64	yes	complemented
93	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
94	C₂³×C₁₁	C₂²×Q₈	1	28	yes	—
95	C₂³×C₁₁	C₂×#41	1	28	yes	—
96	C₂³×C₁₁	C₂⁵	1	7	yes	—
97	C₂²×D₁₁	—	2	448	no	—
98	C₂²×D₁₁	—	2	56	no	—
99	C₂²×D₁₁	—	2	56	no	—
100	C₂²×D₁₁	C₂²×Q₈	1	112	yes	complemented
101	C₂²×D₁₁	C₂⁵	1	8	yes	—
102	C₂×C₄×C₁₁	—	2	56	no	—
103	C₂×C₄×C₁₁	—	2	56	no	—
104	C₂×C₄×C₁₁	—	2	56	no	—
105	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
106	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
107	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
108	C₂×Dic₁₁	—	2	56	no	—
109	C₂×Dic₁₁	—	2	56	no	—
110	C₂×Dic₁₁	C₂²×D₄	1	112	yes	—
111	C₂×Dic₁₁	C₂⁵	1	8	yes	—
112	C₂×Dic₁₁	C₂⁵	1	8	yes	—
113	C₂×Dic₁₁	C₂⁵	1	8	yes	—
114	C₄×D₁₁	—	2	128	no	—
115	Dic₂₂	—	2	128	no	—
116	C₂⁶	—	11	1	no	—
117	C₂³×Q₈	—	11	28	no	—
118	C₂³×Q₈	—	22	16	no	—
119	C₂⁴×C₄	—	11	1	no	—
120	C₂⁴×C₄	—	11	1	no	—
121	C₂⁴×C₄	—	11	1	no	—
122	C₂⁴×C₄	—	11	28	no	—
123	C₂⁴×C₄	—	22	16	no	—
124	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
125	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
126	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
127	C₂×C₂²⋊Q₈	—	11	448	no	—
128	C₂²×C₄⋊C₄	—	11	28	no	—
129	C₂²×C₄⋊C₄	—	11	28	no	—
130	C₂²×C₄⋊C₄	C₂×D₁₁	1	28	yes	complemented
131	C₂²×C₂²⋊C₄	—	11	28	no	—
132	C₂²×C₂²⋊C₄	—	11	28	no	—
133	C₂²×C₁₁	C₂×C₂²⋊Q₈	1	112	yes	complemented
134	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
135	C₂²×C₁₁	C₂³×Q₈	1	14	yes	—
136	C₂²×C₁₁	C₂²×#41	1	14	yes	—
137	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
138	C₂×D₁₁	—	2	8	no	—
139	C₂×D₁₁	—	2	8	no	—
140	C₂×D₁₁	—	2	224	no	—
141	C₂×D₁₁	C₂³×Q₈	1	16	yes	complemented
142	C₄×C₁₁	—	2	8	no	—
143	C₄×C₁₁	—	2	8	no	—
144	C₄×C₁₁	—	2	8	no	—
145	Dic₁₁	—	2	8	no	—
146	Dic₁₁	—	2	8	no	—
147	Dic₁₁	C₂³×D₄	1	16	yes	—
148	C₂⁵	—	11	16	no	—
149	C₂⁵	—	11	14	no	—
150	C₂⁵	—	22	8	no	—
151	C₂⁵	—	22	8	no	—
152	C₂⁵	C₂²×D₁₁	1	1	yes	characteristic
153	C₂²×Q₈	—	11	112	no	—
154	C₂²×Q₈	—	22	224	no	—
155	C₂³×C₄	—	2	8	no	—
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	2	8	no	—
158	C₂³×C₄	—	11	16	no	—
159	C₂³×C₄	—	11	14	no	—
160	C₂³×C₄	—	11	14	no	—
161	C₂³×C₄	—	11	14	no	—
162	C₂³×C₄	—	11	112	no	—
163	C₂³×C₄	—	22	224	no	—
164	C₂³×C₄	—	22	8	no	—
165	C₂³×C₄	—	22	8	no	—
166	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
167	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
168	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
169	C₂²⋊Q₈	—	11	512	no	—
170	C₂×C₄⋊C₄	—	11	112	no	—
171	C₂×C₄⋊C₄	—	11	112	no	—
172	C₂×C₄⋊C₄	C₂²×D₁₁	1	112	yes	complemented
173	C₂×C₂²⋊C₄	—	11	112	no	—
174	C₂×C₂²⋊C₄	—	11	112	no	—
175	C₂×C₁₁	C₂²×C₂²⋊Q₈	1	28	yes	complemented
176	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
177	C₂×C₁₁	C₂⁴×Q₈	1	1	yes	characteristic
178	C₂×C₁₁	C₂³×#41	1	1	yes	characteristic
179	D₁₁	—	2	16	no	—
180	C₂⁴	—	11	112	no	—
181	C₂⁴	—	11	28	no	—
182	C₂⁴	—	22	128	no	—
183	C₂⁴	—	22	56	no	—
184	C₂⁴	—	22	56	no	—
185	C₂⁴	#4339	1	8	yes	—
186	C₂⁴	D₄×D₁₁	1	8	yes	—
187	C₂⁴	Q₈×D₁₁	1	8	yes	—
188	C₂⁴	C₂³×D₁₁	1	7	yes	—
189	C₂×Q₈	—	11	64	no	—
190	C₂×Q₈	—	22	448	no	—
191	C₂²×C₄	—	2	56	no	—
192	C₂²×C₄	—	2	56	no	—
193	C₂²×C₄	—	2	56	no	—
194	C₂²×C₄	—	11	112	no	—
195	C₂²×C₄	—	11	28	no	—
196	C₂²×C₄	—	11	28	no	—
197	C₂²×C₄	—	11	28	no	—
198	C₂²×C₄	—	11	64	no	—
199	C₂²×C₄	—	22	448	no	—
200	C₂²×C₄	—	22	56	no	—
201	C₂²×C₄	—	22	56	no	—
202	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
203	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
204	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
205	C₄⋊C₄	—	11	64	no	—
206	C₄⋊C₄	—	11	64	no	—
207	C₄⋊C₄	C₂³×D₁₁	1	64	yes	complemented
208	C₂²⋊C₄	—	11	64	no	—
209	C₂²⋊C₄	—	11	64	no	—
210	C₁₁	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	11	112	no	—
212	C₂³	—	11	8	no	—
213	C₂³	—	22	448	no	—
214	C₂³	—	22	56	no	—
215	C₂³	—	22	56	no	—
216	C₂³	#66949	1	64	yes	direct product factor
217	C₂³	C₂×#4339	1	28	yes	—
218	C₂³	C₂×D₄×D₁₁	1	28	yes	—
219	C₂³	C₂×Q₈×D₁₁	1	28	yes	—
220	C₂³	C₂⁴×D₁₁	1	7	yes	—
221	Q₈	—	22	128	no	—
222	C₂×C₄	—	2	56	no	—
223	C₂×C₄	—	2	56	no	—
224	C₂×C₄	—	2	56	no	—
225	C₂×C₄	—	11	112	no	—
226	C₂×C₄	—	11	8	no	—
227	C₂×C₄	—	11	8	no	—
228	C₂×C₄	—	11	8	no	—
229	C₂×C₄	—	22	128	no	—
230	C₂×C₄	—	22	56	no	—
231	C₂×C₄	—	22	56	no	—
232	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
233	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
234	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
235	C₂²	—	11	16	no	—
236	C₂²	—	22	8	no	—
237	C₂²	—	22	8	no	—
238	C₂²	—	22	224	no	—
239	C₂²	C₂×#66949	1	112	yes	direct product factor
240	C₂²	C₂²×#4339	1	14	yes	—
241	C₂²	C₂²×D₄×D₁₁	1	14	yes	—
242	C₂²	C₂²×Q₈×D₁₁	1	14	yes	—
243	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
244	C₄	—	2	8	no	—
245	C₄	—	2	8	no	—
246	C₄	—	2	8	no	—
247	C₄	—	11	16	no	—
248	C₄	—	22	8	no	—
249	C₄	—	22	8	no	—
250	C₂	—	22	16	no	—
251	C₂	C₂²×#66949	1	28	yes	direct product factor
252	C₂	C₂³×D₄×D₁₁	1	1	yes	characteristic
253	C₂	C₂³×Q₈×D₁₁	1	1	yes	characteristic
254	C₂	C₂³×#4339	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂³×C₁₁×C₄⋊C₄	C₂
socle	C₂⁵×C₁₁	C₂³
prime radical	C₂⁵×D₁₁	C₂²
hypercenter	C₂³×C₄⋊C₄	D₁₁
hypocenter	C₁₁	C₂³×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₂₂	C₂	semidirect
3. C₂²×#66949	C₂	direct (N × H)
4. C₂³×Dic₁₁⋊C₄	C₂	semidirect
5. C₂³×Dic₁₁⋊C₄	C₂	semidirect
8. C₂³×C₁₁×C₄⋊C₄	C₂	semidirect
19. C₂³×Dic₂₂	C₂²	semidirect
20. C₂²×C₁₁×C₄⋊C₄	C₂²	semidirect
23. C₂²×Dic₁₁⋊C₄	C₂²	semidirect
24. C₂²×Dic₁₁⋊C₄	C₂²	semidirect
25. C₂×#66949	C₂²	direct (N × H)
29. C₂⁴×D₁₁	Q₈	semidirect
46. C₂²×Dic₂₂	C₂³	semidirect
47. C₂×C₁₁×C₄⋊C₄	C₂³	semidirect
50. C₂×Dic₁₁⋊C₄	C₂³	semidirect
51. C₂×Dic₁₁⋊C₄	C₂³	semidirect
52. #66949	C₂³	direct (N × H)
61. C₂³×D₁₁	C₂×Q₈	semidirect
78. C₂×Dic₂₂	C₂⁴	semidirect
79. C₁₁×C₄⋊C₄	C₂⁴	semidirect
82. Dic₁₁⋊C₄	C₂⁴	semidirect
83. Dic₁₁⋊C₄	C₂⁴	semidirect
89. C₂³×C₄⋊C₄	D₁₁	semidirect
92. C₂³×C₁₁	C₂²⋊Q₈	semidirect
100. C₂²×D₁₁	C₂²×Q₈	semidirect
130. C₂²×C₄⋊C₄	C₂×D₁₁	semidirect
133. C₂²×C₁₁	C₂×C₂²⋊Q₈	semidirect
141. C₂×D₁₁	C₂³×Q₈	semidirect
172. C₂×C₄⋊C₄	C₂²×D₁₁	semidirect
175. C₂×C₁₁	C₂²×C₂²⋊Q₈	semidirect
207. C₄⋊C₄	C₂³×D₁₁	semidirect
210. C₁₁	C₂³×C₂²⋊Q₈	semidirect
216. C₂³	#66949	direct (N × H)
239. C₂²	C₂×#66949	direct (N × H)
251. C₂	C₂²×#66949	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₂₂	2
C₂⁴×C₄×D₁₁	2
C₂²×#66949	2
C₂³×Dic₁₁⋊C₄	2
C₂³×(C₂×D₁₁)⋊C₄	2
C₂³×C₁₁×C₄⋊C₄	2
C₂³×C₂²⋊Q₈	11

Subgroups — C23×#66949 #15660415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#66949 #15660415