Subgroups — C₂²×#64146 #11900242

Order 1152.

13632 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#638	C₂	1	2	yes	maximal, complemented
4	C₂²×#3757	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64146	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#3752	C₂	1	2	yes	maximal, complemented
7	C₂²×#937	—	3	4	no	maximal
8	C₂⁵×C₃²	C₂²	1	1	yes	characteristic
9	C₂⁵×C₃²	C₂²	1	1	yes	characteristic, complemented
10	C₂⁴×C₃⋊D₃	—	2	8	no	—
11	C₂⁴×C₃⋊D₃	—	2	16	no	—
12	C₂⁴×C₃⋊D₃	C₂²	1	2	yes	—
13	C₂⁴×C₃⋊D₃	C₂²	1	12	yes	complemented
14	C₂²×C₃²×D₄	—	2	8	no	—
15	C₂²×C₃²×D₄	C₂²	1	12	yes	complemented
16	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic, complemented
17	C₂²×#638	—	2	16	no	—
18	C₂²×#638	C₂²	1	24	yes	complemented
19	C₂³×C₃⋊Dic₃	C₂²	1	2	yes	complemented
20	C₂×#3757	C₂²	1	12	yes	complemented
21	#64146	C₂²	1	64	yes	direct product factor
22	C₂×#3752	C₂²	1	24	yes	complemented
23	C₂⁵×D₃	—	3	4	no	—
24	C₂³×C₃×D₄	D₃	1	4	yes	complemented
25	C₂³×#67	—	3	8	no	—
26	C₂²×#217	—	3	4	no	—
27	C₂×#937	—	3	96	no	—
28	C₂²×(C₂×D₃)⋊C₄	—	3	8	no	—
29	C₂⁴×C₃²	—	2	8	no	—
30	C₂⁴×C₃²	—	2	8	no	—
31	C₂⁴×C₃²	—	2	4	no	—
32	C₂⁴×C₃²	D₄	1	8	yes	complemented
33	C₂⁴×C₃²	C₂³	1	6	yes	—
34	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
35	C₂⁴×C₃²	C₂³	1	6	yes	complemented
36	C₂³×C₃⋊D₃	—	2	128	no	—
37	C₂³×C₃⋊D₃	—	2	8	no	—
38	C₂³×C₃⋊D₃	—	2	8	no	—
39	C₂³×C₃⋊D₃	—	2	48	no	—
40	C₂³×C₃⋊D₃	—	2	96	no	—
41	C₂³×C₃⋊D₃	D₄	1	16	yes	complemented
42	C₂³×C₃⋊D₃	C₂³	1	12	yes	—
43	C₂³×C₃⋊D₃	C₂³	1	16	yes	complemented
44	C₂×C₃²×D₄	—	2	48	no	—
45	C₂×C₃²×D₄	C₂³	1	16	yes	complemented
46	C₂²×C₃²×C₄	—	2	4	no	—
47	C₂²×C₃²×C₄	C₂³	1	6	yes	complemented
48	C₂×#638	—	2	96	no	—
49	C₂×#638	C₂³	1	32	yes	complemented
50	C₂²×C₃⋊Dic₃	—	2	8	no	—
51	C₂²×C₃⋊Dic₃	C₂³	1	12	yes	complemented
52	#3757	C₂³	1	16	yes	complemented
53	#3752	C₂³	1	32	yes	complemented
54	C₂²×#120	—	9	1	no	Sylow
55	C₂⁵×C₃	C₂×D₃	1	4	yes	—
56	C₂⁵×C₃	C₂×D₃	1	4	yes	complemented
57	C₂⁴×D₃	—	3	8	no	—
58	C₂⁴×D₃	—	3	48	no	—
59	C₂⁴×D₃	—	6	64	no	—
60	C₂⁴×D₃	—	6	32	no	—
61	C₂²×C₃×D₄	—	2	32	no	—
62	C₂²×C₃×D₄	C₂×D₃	1	48	yes	complemented
63	C₂³×C₃×C₄	C₂×D₃	1	4	yes	complemented
64	C₂²×#67	—	3	96	no	—
65	C₂²×#67	—	6	64	no	—
66	C₂³×Dic₃	—	3	8	no	—
67	C₂×#217	—	3	48	no	—
68	#937	—	3	256	no	—
69	C₂×(C₂×D₃)⋊C₄	—	3	96	no	—
70	C₂³×C₃²	—	2	32	no	—
71	C₂³×C₃²	—	2	24	no	—
72	C₂³×C₃²	—	2	24	no	—
73	C₂³×C₃²	—	2	12	no	—
74	C₂³×C₃²	—	4	16	no	—
75	C₂³×C₃²	C₂×D₄	1	8	yes	—
76	C₂³×C₃²	C₂×D₄	1	4	yes	—
77	C₂³×C₃²	C₂×D₄	1	24	yes	complemented
78	C₂³×C₃²	C₂⁴	1	4	yes	—
79	C₂³×C₃²	C₂⁴	1	3	yes	—
80	C₂³×C₃²	C₂⁴	1	4	yes	complemented
81	C₂²×C₃⋊D₃	—	2	384	no	—
82	C₂²×C₃⋊D₃	—	2	64	no	—
83	C₂²×C₃⋊D₃	—	2	24	no	—
84	C₂²×C₃⋊D₃	—	2	24	no	—
85	C₂²×C₃⋊D₃	—	2	32	no	—
86	C₂²×C₃⋊D₃	—	2	64	no	—
87	C₂²×C₃⋊D₃	C₂×D₄	1	48	yes	complemented
88	C₂²×C₃⋊D₃	C₂⁴	1	8	yes	—
89	C₃²×D₄	—	2	32	no	—
90	C₂×C₃²×C₄	—	2	12	no	—
91	C₂×C₃²×C₄	C₂⁴	1	4	yes	complemented
92	#638	—	2	64	no	—
93	C₂×C₃⋊Dic₃	—	2	24	no	—
94	C₂×C₃⋊Dic₃	C₂⁴	1	8	yes	complemented
95	C₂⁶	—	9	1	no	—
96	C₂³×D₄	—	9	2	no	—
97	C₂³×D₄	C₃⋊D₃	1	1	yes	characteristic, complemented
98	C₂×#120	—	9	24	no	—
99	C₂²×C₂²⋊C₄	—	9	1	no	—
100	C₂²×C₂²⋊C₄	—	9	2	no	—
101	C₂⁴×C₃	—	2	32	no	—
102	C₂⁴×C₃	—	2	32	no	—
103	C₂⁴×C₃	—	2	16	no	—
104	C₂⁴×C₃	#67	1	32	yes	complemented
105	C₂⁴×C₃	C₂²×D₃	1	24	yes	—
106	C₂⁴×C₃	C₂²×D₃	1	4	yes	—
107	C₂⁴×C₃	C₂²×D₃	1	24	yes	complemented
108	C₂³×D₃	—	3	64	no	—
109	C₂³×D₃	—	3	48	no	—
110	C₂³×D₃	—	3	64	no	—
111	C₂³×D₃	—	6	384	no	—
112	C₂³×D₃	—	6	512	no	—
113	C₂³×D₃	—	6	32	no	—
114	C₂³×D₃	—	6	32	no	—
115	C₂³×D₃	—	6	192	no	—
116	C₂×C₃×D₄	—	2	192	no	—
117	C₂×C₃×D₄	C₂²×D₃	1	64	yes	complemented
118	C₂²×C₃×C₄	—	2	16	no	—
119	C₂²×C₃×C₄	C₂²×D₃	1	24	yes	complemented
120	C₂×#67	—	3	128	no	—
121	C₂×#67	—	6	384	no	—
122	C₂²×Dic₃	—	3	48	no	—
123	C₂²×Dic₃	—	6	32	no	—
124	#217	—	3	64	no	—
125	(C₂×D₃)⋊C₄	—	3	128	no	—
126	C₂²×C₃²	—	2	48	no	—
127	C₂²×C₃²	—	2	8	no	—
128	C₂²×C₃²	—	2	8	no	—
129	C₂²×C₃²	—	2	4	no	—
130	C₂²×C₃²	—	4	24	no	—
131	C₂²×C₃²	#120	1	16	yes	complemented
132	C₂²×C₃²	C₂²×D₄	1	12	yes	—
133	C₂²×C₃²	C₂²×D₄	1	6	yes	—
134	C₂²×C₃²	C₂²×D₄	1	8	yes	complemented
135	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
136	C₂×C₃⋊D₃	—	2	128	no	—
137	C₂×C₃⋊D₃	—	2	96	no	—
138	C₂×C₃⋊D₃	—	2	8	no	—
139	C₂×C₃⋊D₃	—	2	8	no	—
140	C₂×C₃⋊D₃	C₂²×D₄	1	16	yes	complemented
141	C₃²×C₄	—	2	4	no	—
142	C₃⋊Dic₃	—	2	8	no	—
143	C₂⁵	—	9	2	no	—
144	C₂⁵	—	9	12	no	—
145	C₂⁵	—	18	16	no	—
146	C₂⁵	—	18	8	no	—
147	C₂⁵	C₂×C₃⋊D₃	1	1	yes	characteristic
148	C₂⁵	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
149	C₂²×D₄	—	2	8	no	—
150	C₂²×D₄	—	9	24	no	—
151	C₂²×D₄	—	18	16	no	—
152	C₂²×D₄	C₂×C₃⋊D₃	1	12	yes	complemented
153	C₂³×C₄	—	9	2	no	—
154	C₂³×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
155	#120	—	9	64	no	—
156	C₂×C₂²⋊C₄	—	9	12	no	—
157	C₂×C₂²⋊C₄	—	9	24	no	—
158	C₂³×C₃	—	2	128	no	—
159	C₂³×C₃	—	2	96	no	—
160	C₂³×C₃	—	2	96	no	—
161	C₂³×C₃	—	2	48	no	—
162	C₂³×C₃	—	4	64	no	—
163	C₂³×C₃	D₃×D₄	1	32	yes	—
164	C₂³×C₃	C₂×#67	1	16	yes	—
165	C₂³×C₃	C₂×#67	1	96	yes	complemented
166	C₂³×C₃	C₂³×D₃	1	16	yes	—
167	C₂³×C₃	C₂³×D₃	1	12	yes	—
168	C₂³×C₃	C₂³×D₃	1	16	yes	complemented
169	C₂²×D₃	—	3	192	no	—
170	C₂²×D₃	—	3	32	no	—
171	C₂²×D₃	—	6	256	no	—
172	C₂²×D₃	—	6	1536	no	—
173	C₂²×D₃	—	6	256	no	—
174	C₂²×D₃	—	6	96	no	—
175	C₂²×D₃	—	6	96	no	—
176	C₂²×D₃	—	6	128	no	—
177	C₃×D₄	—	2	128	no	—
178	C₂×C₃×C₄	—	2	48	no	—
179	C₂×C₃×C₄	C₂³×D₃	1	16	yes	complemented
180	#67	—	6	256	no	—
181	C₂×Dic₃	—	3	32	no	—
182	C₂×Dic₃	—	6	96	no	—
183	C₂×C₃²	—	2	8	no	—
184	C₂×C₃²	—	4	4	no	—
185	C₂×C₃²	C₂×#120	1	12	yes	complemented
186	C₂×C₃²	C₂³×D₄	1	2	yes	—
187	C₂×C₃²	C₂³×D₄	1	1	yes	characteristic
188	C₃⋊D₃	—	2	16	no	—
189	C₂⁴	—	2	8	no	—
190	C₂⁴	—	2	8	no	—
191	C₂⁴	—	2	4	no	—
192	C₂⁴	—	9	12	no	—
193	C₂⁴	—	9	16	no	—
194	C₂⁴	—	9	16	no	—
195	C₂⁴	—	18	96	no	—
196	C₂⁴	—	18	128	no	—
197	C₂⁴	—	18	8	no	—
198	C₂⁴	—	18	8	no	—
199	C₂⁴	—	18	48	no	—
200	C₂⁴	#638	1	8	yes	complemented
201	C₂⁴	C₂²×C₃⋊D₃	1	6	yes	—
202	C₂⁴	C₂²×C₃⋊D₃	1	1	yes	characteristic
203	C₂⁴	C₂²×C₃⋊D₃	1	6	yes	complemented
204	C₂×D₄	—	2	48	no	—
205	C₂×D₄	—	9	32	no	—
206	C₂×D₄	—	18	96	no	—
207	C₂×D₄	C₂²×C₃⋊D₃	1	16	yes	complemented
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	9	12	no	—
210	C₂²×C₄	—	18	8	no	—
211	C₂²×C₄	C₂²×C₃⋊D₃	1	6	yes	complemented
212	C₂²⋊C₄	—	9	16	no	—
213	C₂²⋊C₄	—	9	32	no	—
214	C₂²×C₃	—	2	32	no	—
215	C₂²×C₃	—	2	192	no	—
216	C₂²×C₃	—	2	32	no	—
217	C₂²×C₃	—	2	16	no	—
218	C₂²×C₃	—	4	96	no	—
219	C₂²×C₃	#937	1	64	yes	complemented
220	C₂²×C₃	C₂×D₃×D₄	1	48	yes	—
221	C₂²×C₃	C₂²×#67	1	24	yes	—
222	C₂²×C₃	C₂²×#67	1	32	yes	complemented
223	C₂²×C₃	C₂⁴×D₃	1	4	yes	—
224	C₂×D₃	—	3	64	no	—
225	C₂×D₃	—	6	512	no	—
226	C₂×D₃	—	6	32	no	—
227	C₂×D₃	—	6	32	no	—
228	C₂×D₃	—	6	384	no	—
229	C₃×C₄	—	2	16	no	—
230	Dic₃	—	6	32	no	—
231	C₃²	C₂²×#120	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	2	24	no	—
233	C₂³	—	2	32	no	—
234	C₂³	—	2	24	no	—
235	C₂³	—	2	12	no	—
236	C₂³	—	4	16	no	—
237	C₂³	—	9	48	no	—
238	C₂³	—	9	8	no	—
239	C₂³	—	18	64	no	—
240	C₂³	—	18	384	no	—
241	C₂³	—	18	24	no	—
242	C₂³	—	18	24	no	—
243	C₂³	—	18	32	no	—
244	C₂³	—	18	64	no	—
245	C₂³	D₄×C₃⋊D₃	1	8	yes	—
246	C₂³	C₂×#638	1	4	yes	—
247	C₂³	C₂×#638	1	24	yes	complemented
248	C₂³	C₂³×C₃⋊D₃	1	4	yes	—
249	C₂³	C₂³×C₃⋊D₃	1	3	yes	—
250	C₂³	C₂³×C₃⋊D₃	1	4	yes	complemented
251	D₄	—	2	32	no	—
252	D₄	—	18	64	no	—
253	C₂×C₄	—	2	12	no	—
254	C₂×C₄	—	9	8	no	—
255	C₂×C₄	—	18	24	no	—
256	C₂×C₄	C₂³×C₃⋊D₃	1	4	yes	complemented
257	C₂×C₃	—	2	32	no	—
258	C₂×C₃	—	4	16	no	—
259	C₂×C₃	C₂×#937	1	48	yes	complemented
260	C₂×C₃	C₂²×D₃×D₄	1	8	yes	—
261	C₂×C₃	C₂³×#67	1	4	yes	—
262	D₃	—	6	64	no	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	48	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	24	no	—
268	C₂²	—	9	16	no	—
269	C₂²	—	18	128	no	—
270	C₂²	—	18	8	no	—
271	C₂²	—	18	8	no	—
272	C₂²	—	18	96	no	—
273	C₂²	#64146	1	16	yes	direct product factor
274	C₂²	C₂×D₄×C₃⋊D₃	1	12	yes	—
275	C₂²	C₂²×#638	1	6	yes	—
276	C₂²	C₂²×#638	1	8	yes	complemented
277	C₂²	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	18	8	no	—
280	C₃	C₂²×#937	1	4	yes	complemented
281	C₂	—	2	8	no	—
282	C₂	—	4	4	no	—
283	C₂	—	18	16	no	—
284	C₂	C₂×#64146	1	12	yes	direct product factor
285	C₂	C₂²×D₄×C₃⋊D₃	1	2	yes	—
286	C₂	C₂³×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁵
Frattini	C₂²	C₂⁴×C₃⋊D₃
Fitting	C₂³×C₃²×D₄	C₂
socle	C₂⁴×C₃²	C₂³
hypercenter	C₂³×D₄	C₃⋊D₃
hypocenter	C₃²	C₂²×#120
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₃⋊D₃	C₂	semidirect
2. C₂³×C₃²×D₄	C₂	semidirect
3. C₂³×#638	C₂	semidirect
4. C₂²×#3757	C₂	semidirect
5. C₂×#64146	C₂	direct (N × H)
6. C₂²×#3752	C₂	semidirect
9. C₂⁵×C₃²	C₂²	semidirect
13. C₂⁴×C₃⋊D₃	C₂²	semidirect
15. C₂²×C₃²×D₄	C₂²	semidirect
16. C₂³×C₃²×C₄	C₂²	semidirect
18. C₂²×#638	C₂²	semidirect
19. C₂³×C₃⋊Dic₃	C₂²	semidirect
20. C₂×#3757	C₂²	semidirect
21. #64146	C₂²	direct (N × H)
22. C₂×#3752	C₂²	semidirect
24. C₂³×C₃×D₄	D₃	semidirect
32. C₂⁴×C₃²	D₄	semidirect
35. C₂⁴×C₃²	C₂³	semidirect
41. C₂³×C₃⋊D₃	D₄	semidirect
43. C₂³×C₃⋊D₃	C₂³	semidirect
45. C₂×C₃²×D₄	C₂³	semidirect
47. C₂²×C₃²×C₄	C₂³	semidirect
49. C₂×#638	C₂³	semidirect
51. C₂²×C₃⋊Dic₃	C₂³	semidirect
52. #3757	C₂³	semidirect
53. #3752	C₂³	semidirect
56. C₂⁵×C₃	C₂×D₃	semidirect
62. C₂²×C₃×D₄	C₂×D₃	semidirect
63. C₂³×C₃×C₄	C₂×D₃	semidirect
77. C₂³×C₃²	C₂×D₄	semidirect
80. C₂³×C₃²	C₂⁴	semidirect
87. C₂²×C₃⋊D₃	C₂×D₄	semidirect
91. C₂×C₃²×C₄	C₂⁴	semidirect
94. C₂×C₃⋊Dic₃	C₂⁴	semidirect
97. C₂³×D₄	C₃⋊D₃	semidirect
104. C₂⁴×C₃	#67	semidirect
107. C₂⁴×C₃	C₂²×D₃	semidirect
117. C₂×C₃×D₄	C₂²×D₃	semidirect
119. C₂²×C₃×C₄	C₂²×D₃	semidirect
131. C₂²×C₃²	#120	semidirect
134. C₂²×C₃²	C₂²×D₄	semidirect
140. C₂×C₃⋊D₃	C₂²×D₄	semidirect
148. C₂⁵	C₂×C₃⋊D₃	semidirect
152. C₂²×D₄	C₂×C₃⋊D₃	semidirect
154. C₂³×C₄	C₂×C₃⋊D₃	semidirect
165. C₂³×C₃	C₂×#67	semidirect
168. C₂³×C₃	C₂³×D₃	semidirect
179. C₂×C₃×C₄	C₂³×D₃	semidirect
185. C₂×C₃²	C₂×#120	semidirect
200. C₂⁴	#638	semidirect
203. C₂⁴	C₂²×C₃⋊D₃	semidirect
207. C₂×D₄	C₂²×C₃⋊D₃	semidirect
211. C₂²×C₄	C₂²×C₃⋊D₃	semidirect
219. C₂²×C₃	#937	semidirect
222. C₂²×C₃	C₂²×#67	semidirect
231. C₃²	C₂²×#120	semidirect
247. C₂³	C₂×#638	semidirect
250. C₂³	C₂³×C₃⋊D₃	semidirect
256. C₂×C₄	C₂³×C₃⋊D₃	semidirect
259. C₂×C₃	C₂×#937	semidirect
273. C₂²	#64146	direct (N × H)
276. C₂²	C₂²×#638	semidirect
280. C₃	C₂²×#937	semidirect
284. C₂	C₂×#64146	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃⋊D₃	2
C₂³×C₃²×D₄	2
C₂³×#638	2
C₂²×#3757	2
C₂×#64146	2
C₂²×#3752	2
C₂²×#937	3

Subgroups — C22×#64146 #11900242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64146 #11900242