Subgroups — C₂×C₃×#4491 #11796586

Order 1152.

609 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4491	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×#357	—	3	1	no	maximal
6	C₂×#4491	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×D₁₂	—	2	1	no	—
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
10	C₂×C₃×D₂₄	—	2	1	no	—
11	C₂×C₃×D₂₄	—	2	2	no	—
12	C₂×C₃×D₂₄	C₄	1	4	yes	complemented
13	C₂×C₃×D₂₄	C₂²	1	2	yes	—
14	C₃²×#107	—	2	1	no	—
15	C₃²×#107	C₂²	1	2	yes	complemented
16	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	complemented
18	C₂²×C₃×D₈	—	3	1	no	—
19	C₂²×D₂₄	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₃×C₁₆	—	3	1	no	—
21	C₂×C₃×#107	—	2	1	no	—
22	C₂×C₃×#107	D₃	1	1	yes	characteristic, complemented
23	C₂×C₃×#107	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×#357	—	3	4	no	—
26	#4491	C₂×C₃	1	4	yes	direct product factor
27	C₂³×C₃×D₃	—	4	1	no	—
28	C₂²×C₃²×C₄	—	4	1	no	—
29	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃×D₁₂	—	2	4	no	—
31	C₂×C₃×D₁₂	—	2	2	no	—
32	C₂×C₃×D₁₂	—	4	1	no	—
33	C₂×C₃×D₁₂	—	4	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	D₄	1	2	yes	—
37	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
38	C₃²×C₄⋊C₄	—	2	2	no	—
39	C₃²×C₄⋊C₄	—	4	1	no	—
40	C₃×D₂₄	—	2	4	no	—
41	C₃×D₂₄	—	2	2	no	—
42	C₃×D₂₄	C₂×C₄	1	4	yes	complemented
43	C₃×C₃⋊C₁₆	—	2	2	no	—
44	C₂×#357	—	3	1	no	Sylow
45	C₂²×C₃×D₄	—	6	1	no	—
46	C₂²×D₁₂	—	2	1	no	—
47	C₂×C₃×D₈	—	3	4	no	—
48	C₂×C₃×D₈	—	3	2	no	—
49	C₂×C₃×D₈	—	6	2	no	—
50	C₂×C₃×D₈	—	6	1	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	—	2	2	no	—
55	C₂×C₃×C₄⋊C₄	—	2	1	no	—
56	C₂×C₃×C₄⋊C₄	—	2	1	no	—
57	C₂×D₂₄	—	2	2	no	—
58	C₂×D₂₄	—	2	1	no	—
59	C₂×D₂₄	C₃×C₄	1	4	yes	complemented
60	C₂×D₂₄	C₂²×C₃	1	2	yes	—
61	C₂×C₃×C₁₆	—	3	2	no	—
62	C₂×C₃×C₁₆	—	6	2	no	—
63	C₃×#107	—	2	2	no	—
64	C₃×#107	—	2	1	no	—
65	C₃×#107	—	2	1	no	—
66	C₃×#107	—	2	2	no	—
67	C₃×#107	C₂×D₃	1	2	yes	complemented
68	C₃×#107	C₂²×C₃	1	2	yes	complemented
69	C₂×C₃⋊C₁₆	—	2	2	no	—
70	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	complemented
71	C₂³×C₃²	D₈	1	1	yes	characteristic
72	C₂²×C₃×D₃	—	4	4	no	—
73	C₂²×C₃×D₃	—	4	2	no	—
74	C₂²×C₃×D₃	—	8	2	no	—
75	C₂²×C₃×D₃	—	8	2	no	—
76	C₂×C₃²×C₄	—	4	2	no	—
77	C₂×C₃²×C₄	—	8	2	no	—
78	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
79	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
80	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
81	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₃×D₁₂	—	2	4	no	—
83	C₃×D₁₂	—	4	4	no	—
84	C₃×D₁₂	—	4	2	no	—
85	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
86	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₈	—	3	1	no	—
89	C₂²×C₁₆	—	3	1	no	—
90	C₂×#107	C₃×D₃	1	1	yes	characteristic, complemented
91	#357	—	3	4	no	—
92	C₂⁴×C₃	—	12	1	no	—
93	C₂³×D₃	—	4	1	no	—
94	C₂×C₃×D₄	—	6	4	no	—
95	C₂×C₃×D₄	—	6	2	no	—
96	C₂×C₃×D₄	—	12	1	no	—
97	C₂×C₃×D₄	—	12	2	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	—	4	2	no	—
100	C₂²×C₃×C₄	—	4	1	no	—
101	C₂²×C₃×C₄	—	4	1	no	—
102	C₂²×C₃×C₄	#67	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
104	C₂×D₁₂	—	2	4	no	—
105	C₂×D₁₂	—	2	2	no	—
106	C₂×D₁₂	—	4	1	no	—
107	C₂×D₁₂	—	4	2	no	—
108	C₃×D₈	—	3	4	no	—
109	C₃×D₈	—	6	4	no	—
110	C₃×D₈	—	6	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
116	C₂×C₃×C₈	D₁₂	1	2	yes	—
117	C₂×C₃×C₈	#67	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
119	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
120	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
123	C₃×C₄⋊C₄	—	2	4	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	—	4	2	no	—
127	C₃×C₄⋊C₄	—	4	1	no	—
128	C₃×C₄⋊C₄	—	4	1	no	—
129	D₂₄	—	2	4	no	—
130	D₂₄	—	2	2	no	—
131	D₂₄	C₂×C₃×C₄	1	4	yes	complemented
132	C₃×C₁₆	—	6	2	no	—
133	C₃⋊C₁₆	—	2	2	no	—
134	C₂²×C₃²	#102	1	2	yes	—
135	C₂²×C₃²	D₁₆	1	2	yes	—
136	C₂²×C₃²	SD₃₂	1	2	yes	—
137	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
138	C₂×C₃×D₃	—	4	4	no	—
139	C₂×C₃×D₃	—	8	8	no	—
140	C₂×C₃×D₃	—	8	2	no	—
141	C₂×C₃×D₃	—	8	2	no	—
142	C₃²×C₄	—	8	2	no	—
143	C₃²×C₄	#102	1	2	yes	—
144	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
146	C₂²×D₄	—	6	1	no	—
147	C₂×D₈	—	3	4	no	—
148	C₂×D₈	—	3	2	no	—
149	C₂×D₈	—	6	2	no	—
150	C₂×D₈	—	6	1	no	—
151	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₁₆	—	3	2	no	—
154	C₂×C₁₆	—	6	2	no	—
155	#107	—	2	1	no	—
156	#107	C₂×C₃×D₃	1	2	yes	complemented
157	C₂³×C₃	—	2	1	no	—
158	C₂³×C₃	—	12	4	no	—
159	C₂³×C₃	—	12	2	no	—
160	C₂³×C₃	—	24	2	no	—
161	C₂³×C₃	—	24	2	no	—
162	C₂³×C₃	#213	1	1	yes	characteristic
163	C₂³×C₃	C₃×D₈	1	1	yes	characteristic
164	C₂²×D₃	—	4	4	no	—
165	C₂²×D₃	—	4	2	no	—
166	C₂²×D₃	—	8	2	no	—
167	C₂²×D₃	—	8	2	no	—
168	C₃×D₄	—	6	4	no	—
169	C₃×D₄	—	12	4	no	—
170	C₃×D₄	—	12	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	4	4	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	—	8	4	no	—
179	C₂×C₃×C₄	—	8	2	no	—
180	C₂×C₃×C₄	—	8	2	no	—
181	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
182	C₂×C₃×C₄	#213	1	1	yes	characteristic
183	C₂×C₃×C₄	#215	1	2	yes	—
184	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
185	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
186	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
187	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
189	D₁₂	—	2	4	no	—
190	D₁₂	—	4	4	no	—
191	D₁₂	—	4	2	no	—
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	—	2	1	no	—
194	C₃×C₈	—	2	1	no	—
195	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
196	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
197	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
198	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
199	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
200	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂×C₃²	#357	1	4	yes	complemented
202	C₂×C₃²	C₂×#102	1	1	yes	characteristic
203	C₂×C₃²	C₂×D₁₆	1	1	yes	characteristic
204	C₂×C₃²	C₂×SD₃₂	1	1	yes	characteristic
205	C₃×D₃	—	8	4	no	—
206	C₂⁴	—	12	1	no	—
207	C₂×D₄	—	6	4	no	—
208	C₂×D₄	—	6	2	no	—
209	C₂×D₄	—	12	1	no	—
210	C₂×D₄	—	12	2	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	C₃×#67	1	1	yes	characteristic
213	D₈	—	3	4	no	—
214	D₈	—	6	4	no	—
215	D₈	—	6	2	no	—
216	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
217	C₂×C₈	C₃×D₁₂	1	2	yes	—
218	C₂×C₈	C₃×#67	1	1	yes	characteristic
219	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₁₆	—	6	2	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	12	4	no	—
228	C₂²×C₃	—	24	2	no	—
229	C₂²×C₃	—	24	2	no	—
230	C₂²×C₃	—	24	8	no	—
231	C₂²×C₃	#809	1	2	yes	—
232	C₂²×C₃	#826	1	2	yes	—
233	C₂²×C₃	#828	1	2	yes	—
234	C₂²×C₃	C₃×#102	1	2	yes	—
235	C₂²×C₃	C₃×D₁₆	1	2	yes	—
236	C₂²×C₃	C₃×SD₃₂	1	2	yes	—
237	C₂²×C₃	C₂×#213	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
239	C₂×D₃	—	4	4	no	—
240	C₂×D₃	—	8	2	no	—
241	C₂×D₃	—	8	2	no	—
242	C₂×D₃	—	8	8	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	8	4	no	—
247	C₃×C₄	—	8	2	no	—
248	C₃×C₄	—	8	2	no	—
249	C₃×C₄	#809	1	2	yes	—
250	C₃×C₄	C₃×#102	1	2	yes	—
251	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
252	C₃×C₄	C₂×#215	1	1	yes	characteristic
253	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₃×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
255	C₃²	C₂×#357	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	12	2	no	—
257	C₂³	—	12	4	no	—
258	C₂³	—	24	2	no	—
259	C₂³	—	24	2	no	—
260	C₂³	C₃×#213	1	1	yes	characteristic
261	D₄	—	6	4	no	—
262	D₄	—	12	2	no	—
263	D₄	—	12	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
267	C₂×C₄	C₃×#213	1	1	yes	characteristic
268	C₂×C₄	C₃×#215	1	2	yes	—
269	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
270	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
271	C₈	C₂×C₃×C₄×D₃	1	1	yes	characteristic
272	C₈	C₂×C₃×D₁₂	1	1	yes	characteristic
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	24	4	no	—
278	C₂×C₃	#4491	1	4	yes	direct product factor
279	C₂×C₃	C₃×#357	1	4	yes	complemented
280	C₂×C₃	C₂×#809	1	1	yes	characteristic
281	C₂×C₃	C₂×#826	1	1	yes	characteristic
282	C₂×C₃	C₂×#828	1	1	yes	characteristic
283	C₂×C₃	C₂×C₃×#102	1	1	yes	characteristic
284	C₂×C₃	C₂×C₃×D₁₆	1	1	yes	characteristic
285	C₂×C₃	C₂×C₃×SD₃₂	1	1	yes	characteristic
286	D₃	—	8	4	no	—
287	C₂²	—	12	4	no	—
288	C₂²	—	24	2	no	—
289	C₂²	—	24	8	no	—
290	C₂²	—	24	2	no	—
291	C₂²	C₃×#809	1	2	yes	—
292	C₂²	C₃×#826	1	2	yes	—
293	C₂²	C₃×#828	1	2	yes	—
294	C₂²	C₂×C₃×#213	1	1	yes	characteristic
295	C₄	—	8	2	no	—
296	C₄	C₃×#809	1	2	yes	—
297	C₄	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
298	C₄	C₂×C₃×#215	1	1	yes	characteristic
299	C₃	—	2	1	no	—
300	C₃	C₂×#4491	1	1	yes	characteristic, direct product factor
301	C₃	C₂×C₃×#357	1	1	yes	characteristic, complemented
302	C₂	—	24	4	no	—
303	C₂	C₃×#4491	1	4	yes	direct product factor
304	C₂	C₂×C₃×#809	1	1	yes	characteristic
305	C₂	C₂×C₃×#826	1	1	yes	characteristic
306	C₂	C₂×C₃×#828	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	#213
commutator	C₃×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₂×C₃²×#107	C₂
socle	C₂³×C₃²	D₈
prime radical	C₂²×C₃×D₂₄	C₂
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₂×C₃×#107	D₃
hypocenter	C₃	C₂×C₃×#357
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×#107

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#107	C₂	semidirect
3. C₂²×C₃×C₃⋊C₁₆	C₂	semidirect
4. C₃×#4491	C₂	direct (N × H)
6. C₂×#4491	C₃	direct (N × H)
12. C₂×C₃×D₂₄	C₄	semidirect
15. C₃²×#107	C₂²	semidirect
17. C₂×C₃×C₃⋊C₁₆	C₂²	semidirect
22. C₂×C₃×#107	D₃	semidirect
23. C₂×C₃×#107	C₂×C₃	semidirect
24. C₂²×C₃⋊C₁₆	C₂×C₃	semidirect
26. #4491	C₂×C₃	direct (N × H)
42. C₃×D₂₄	C₂×C₄	semidirect
59. C₂×D₂₄	C₃×C₄	semidirect
67. C₃×#107	C₂×D₃	semidirect
68. C₃×#107	C₂²×C₃	semidirect
70. C₂×C₃⋊C₁₆	C₂²×C₃	semidirect
90. C₂×#107	C₃×D₃	semidirect
131. D₂₄	C₂×C₃×C₄	semidirect
156. #107	C₂×C₃×D₃	semidirect
201. C₂×C₃²	#357	semidirect
255. C₃²	C₂×#357	semidirect
278. C₂×C₃	#4491	direct (N × H)
279. C₂×C₃	C₃×#357	semidirect
300. C₃	C₂×#4491	direct (N × H)
301. C₃	C₂×C₃×#357	semidirect
303. C₂	C₃×#4491	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×C₃×D₂₄	2
C₂×C₃²×#107	2
C₂²×C₃×C₃⋊C₁₆	2
C₃×#4491	2
C₂×C₃×#357	3
C₂×#4491	3

Subgroups — C2×C3×#4491 #11796586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4491 #11796586