Subgroups — C₂×C₃×#5085 #11874150

Order 1152.

1457 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#215	C₂	1	4	yes	maximal, complemented
3	C₂×C₃×#874	C₂	1	1	yes	maximal, characteristic
4	C₃×#5085	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#492	—	3	1	no	maximal
7	C₂×#5085	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
9	C₂²×C₃×D₁₂	—	2	2	no	—
10	C₂²×C₃×D₁₂	C₂²	1	2	yes	—
11	C₃²×C₄⋊Q₈	—	2	2	no	—
12	C₃²×C₄⋊Q₈	C₂²	1	4	yes	complemented
13	C₂×C₃²×C₄⋊C₄	—	2	2	no	—
14	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₃×#215	—	2	16	no	—
16	C₂×C₃×#215	C₂²	1	16	yes	complemented
17	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃×#874	—	2	2	no	—
19	C₃×#874	C₂²	1	4	yes	—
20	C₃×C₄×C₃⋊C₈	C₂²	1	4	yes	complemented
21	C₂²×C₃×SD₁₆	—	3	4	no	—
22	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
23	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×C₃×#127	—	3	1	no	—
26	C₂²×#215	C₂×C₃	1	4	yes	complemented
27	C₂×#874	C₂×C₃	1	1	yes	characteristic
28	C₃×#492	—	3	8	no	—
29	C₂×C₃×C₄×C₈	—	3	1	no	—
30	#5085	C₂×C₃	1	8	yes	direct product factor
31	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂³×C₃×D₃	—	2	2	no	—
33	C₂×C₃²×Q₈	—	2	2	no	—
34	C₂×C₃²×Q₈	—	2	8	no	—
35	C₂×C₃²×Q₈	C₂³	1	4	yes	—
36	C₂²×C₃²×C₄	—	2	2	no	—
37	C₂²×C₃²×C₄	D₄	1	2	yes	—
38	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
39	C₂×C₃×D₁₂	—	2	8	no	—
40	C₂×C₃×D₁₂	—	2	8	no	—
41	C₂×C₃×D₁₂	—	2	2	no	—
42	C₂×C₃×D₁₂	—	4	8	no	—
43	C₂×C₃×D₁₂	C₂³	1	4	yes	—
44	C₃²×C₄⋊C₄	—	2	8	no	—
45	C₃²×C₄²	D₄	1	2	yes	—
46	C₃²×C₄²	C₂³	1	2	yes	—
47	C₃×#215	—	2	32	no	—
48	C₂×C₃×C₃⋊C₈	D₄	1	8	yes	complemented
49	C₂×C₃×C₃⋊C₈	C₂³	1	4	yes	—
50	C₂×#492	—	3	1	no	Sylow
51	C₂²×C₃×Q₈	—	2	2	no	—
52	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
53	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
54	C₂²×C₃×D₄	—	3	2	no	—
55	C₂²×C₃×D₄	—	6	2	no	—
56	C₂²×D₁₂	—	2	2	no	—
57	C₂²×D₁₂	C₂²×C₃	1	2	yes	—
58	C₂×C₃×SD₁₆	—	3	16	no	—
59	C₂×C₃×SD₁₆	—	6	16	no	—
60	C₂²×C₃×C₈	—	3	2	no	—
61	C₃×C₄⋊Q₈	—	2	4	no	—
62	C₃×C₄⋊Q₈	—	2	2	no	—
63	C₃×C₄⋊Q₈	—	2	2	no	—
64	C₃×C₄⋊Q₈	—	2	4	no	—
65	C₃×C₄⋊Q₈	C₂×D₃	1	4	yes	complemented
66	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	complemented
67	C₃×#127	—	3	4	no	—
68	C₃×#127	—	6	2	no	—
69	C₂×C₃×C₄⋊C₄	—	2	4	no	—
70	C₂×C₃×C₄⋊C₄	—	2	2	no	—
71	C₂×C₃×C₄⋊C₄	—	2	2	no	—
72	C₂×C₃×C₄²	—	2	1	no	—
73	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
74	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
75	C₂×#215	—	2	16	no	—
76	C₂×#215	C₂²×C₃	1	16	yes	complemented
77	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
78	#874	—	2	2	no	—
79	#874	C₂²×C₃	1	4	yes	—
80	C₃×C₄×C₈	—	3	4	no	—
81	C₄×C₃⋊C₈	C₂²×C₃	1	4	yes	complemented
82	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₃×D₃	—	2	4	no	—
84	C₂²×C₃×D₃	—	4	4	no	—
85	C₂²×C₃×D₃	—	4	8	no	—
86	C₃²×Q₈	—	2	8	no	—
87	C₃²×Q₈	—	4	4	no	—
88	C₂×C₃²×C₄	—	2	4	no	—
89	C₂×C₃²×C₄	—	4	4	no	—
90	C₂×C₃²×C₄	SD₁₆	1	8	yes	complemented
91	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
92	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
94	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
95	C₃×D₁₂	—	2	8	no	—
96	C₃×D₁₂	—	4	4	no	—
97	C₃×D₁₂	—	4	16	no	—
98	C₃×C₃⋊C₈	C₂×D₄	1	8	yes	complemented
99	C₂²×SD₁₆	—	3	4	no	—
100	C₂×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic, complemented
101	C₂×#127	—	3	1	no	—
102	#492	—	3	8	no	—
103	C₂×C₄×C₈	—	3	1	no	—
104	C₂⁴×C₃	—	6	2	no	—
105	C₂³×D₃	—	2	2	no	—
106	C₂×C₃×Q₈	—	2	4	no	—
107	C₂×C₃×Q₈	—	2	2	no	—
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	—	2	8	no	—
110	C₂×C₃×Q₈	—	2	8	no	—
111	C₂×C₃×Q₈	—	2	4	no	—
112	C₂×C₃×Q₈	—	4	8	no	—
113	C₂×C₃×Q₈	C₂²×D₃	1	4	yes	—
114	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
115	C₂×C₃×D₄	—	3	4	no	—
116	C₂×C₃×D₄	—	6	8	no	—
117	C₂×C₃×D₄	—	6	8	no	—
118	C₂×C₃×D₄	—	6	2	no	—
119	C₂×C₃×D₄	—	12	8	no	—
120	C₂²×C₃×C₄	—	2	4	no	—
121	C₂²×C₃×C₄	—	2	2	no	—
122	C₂²×C₃×C₄	—	2	2	no	—
123	C₂²×C₃×C₄	—	2	2	no	—
124	C₂²×C₃×C₄	—	2	1	no	—
125	C₂²×C₃×C₄	#67	1	2	yes	—
126	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
127	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₂×D₁₂	—	2	8	no	—
130	C₂×D₁₂	—	2	2	no	—
131	C₂×D₁₂	—	2	8	no	—
132	C₂×D₁₂	—	4	8	no	—
133	C₂×D₁₂	C₂³×C₃	1	4	yes	—
134	C₃×SD₁₆	—	6	32	no	—
135	C₂×C₃×C₈	—	3	8	no	—
136	C₂×C₃×C₈	—	3	4	no	—
137	C₃×C₄⋊C₄	—	2	16	no	—
138	C₃×C₄⋊C₄	—	2	8	no	—
139	C₃×C₄⋊C₄	—	2	8	no	—
140	C₃×C₄²	—	2	2	no	—
141	C₃×C₄²	—	2	2	no	—
142	C₃×C₄²	#67	1	2	yes	—
143	C₃×C₄²	C₃×D₄	1	2	yes	—
144	C₃×C₄²	C₂²×D₃	1	2	yes	—
145	C₃×C₄²	C₂³×C₃	1	2	yes	—
146	#215	—	2	32	no	—
147	C₂×C₃⋊C₈	C₃×D₄	1	8	yes	complemented
148	C₂×C₃⋊C₈	C₂³×C₃	1	4	yes	—
149	C₂²×C₃²	#127	1	2	yes	—
150	C₂²×C₃²	C₂×SD₁₆	1	4	yes	—
151	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₃×D₃	—	4	4	no	—
153	C₂×C₃×D₃	—	4	8	no	—
154	C₂×C₃×D₃	—	8	8	no	—
155	C₃²×C₄	—	4	4	no	—
156	C₃²×C₄	#127	1	2	yes	—
157	C₃²×C₄	C₂×SD₁₆	1	8	yes	complemented
158	C₃²×C₄	C₂²×D₄	1	2	yes	—
159	C₂²×Q₈	C₂×C₃×D₃	1	2	yes	—
160	C₂²×D₄	—	3	2	no	—
161	C₂²×D₄	—	6	2	no	—
162	C₂×SD₁₆	—	3	16	no	—
163	C₂×SD₁₆	—	6	16	no	—
164	C₂²×C₈	—	3	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	C₂×C₃×D₃	1	4	yes	complemented
167	#127	—	3	4	no	—
168	#127	—	6	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
171	C₄×C₈	—	3	4	no	—
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	—	6	4	no	—
174	C₂³×C₃	—	12	4	no	—
175	C₂³×C₃	—	12	8	no	—
176	C₂³×C₃	C₂×#67	1	1	yes	characteristic
177	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂²×D₃	—	2	4	no	—
179	C₂²×D₃	—	4	4	no	—
180	C₂²×D₃	—	4	8	no	—
181	C₃×Q₈	—	2	8	no	—
182	C₃×Q₈	—	2	8	no	—
183	C₃×Q₈	—	4	8	no	—
184	C₃×Q₈	—	4	8	no	—
185	C₃×Q₈	—	4	4	no	—
186	C₃×Q₈	—	4	4	no	—
187	C₃×D₄	—	6	8	no	—
188	C₃×D₄	—	12	4	no	—
189	C₃×D₄	—	12	16	no	—
190	C₂×C₃×C₄	—	2	8	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	8	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	4	8	no	—
199	C₂×C₃×C₄	—	4	4	no	—
200	C₂×C₃×C₄	—	4	4	no	—
201	C₂×C₃×C₄	#215	1	8	yes	complemented
202	C₂×C₃×C₄	C₃×SD₁₆	1	8	yes	complemented
203	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
204	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×#67	1	4	yes	—
206	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
207	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
209	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
211	D₁₂	—	2	8	no	—
212	D₁₂	—	4	4	no	—
213	D₁₂	—	4	16	no	—
214	C₃×C₈	—	3	8	no	—
215	C₃⋊C₈	C₂×C₃×D₄	1	8	yes	complemented
216	C₂×C₃²	#492	1	4	yes	complemented
217	C₂×C₃²	C₂×#127	1	1	yes	characteristic
218	C₂×C₃²	C₂²×SD₁₆	1	2	yes	—
219	C₃×D₃	—	8	4	no	—
220	C₂⁴	—	6	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	8	no	—
223	C₂×Q₈	C₂²×C₃×D₃	1	4	yes	—
224	C₂×D₄	—	3	4	no	—
225	C₂×D₄	—	6	8	no	—
226	C₂×D₄	—	6	8	no	—
227	C₂×D₄	—	6	2	no	—
228	C₂×D₄	—	12	8	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	C₃×#67	1	2	yes	—
231	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
232	SD₁₆	—	6	32	no	—
233	C₂×C₈	—	3	4	no	—
234	C₂×C₈	—	3	8	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄²	C₃×#67	1	2	yes	—
237	C₄²	C₂²×C₃×D₃	1	2	yes	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	4	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	12	4	no	—
242	C₂²×C₃	—	12	8	no	—
243	C₂²×C₃	—	24	8	no	—
244	C₂²×C₃	#940	1	2	yes	—
245	C₂²×C₃	C₂×#215	1	4	yes	—
246	C₂²×C₃	C₃×#127	1	2	yes	—
247	C₂²×C₃	C₂×C₃×SD₁₆	1	4	yes	—
248	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
249	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
250	C₂×D₃	—	4	8	no	—
251	C₂×D₃	—	4	4	no	—
252	C₂×D₃	—	8	8	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	8	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	4	8	no	—
257	C₃×C₄	—	4	4	no	—
258	C₃×C₄	—	4	4	no	—
259	C₃×C₄	#940	1	2	yes	—
260	C₃×C₄	C₂×#215	1	8	yes	complemented
261	C₃×C₄	C₃×#127	1	2	yes	—
262	C₃×C₄	C₂×C₃×SD₁₆	1	8	yes	complemented
263	C₃×C₄	C₂×D₃×D₄	1	2	yes	—
264	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
265	C₃²	C₂×#492	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	6	4	no	—
267	C₂³	—	12	4	no	—
268	C₂³	—	12	8	no	—
269	C₂³	C₂×C₃×#67	1	1	yes	characteristic
270	Q₈	—	2	8	no	—
271	Q₈	—	4	4	no	—
272	D₄	—	6	8	no	—
273	D₄	—	12	4	no	—
274	D₄	—	12	16	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₃×#215	1	8	yes	complemented
278	C₂×C₄	C₃×D₃×D₄	1	4	yes	—
279	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
280	C₂×C₄	C₂×C₃×#67	1	4	yes	—
281	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
282	C₈	—	3	8	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	2	no	—
285	C₂×C₃	—	2	4	no	—
286	C₂×C₃	—	24	4	no	—
287	C₂×C₃	#5085	1	4	yes	direct product factor
288	C₂×C₃	C₃×#492	1	4	yes	complemented
289	C₂×C₃	C₂×#940	1	1	yes	characteristic
290	C₂×C₃	C₂²×#215	1	2	yes	—
291	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
292	C₂×C₃	C₂²×C₃×SD₁₆	1	2	yes	—
293	D₃	—	8	4	no	—
294	C₂²	—	12	4	no	—
295	C₂²	—	12	8	no	—
296	C₂²	—	24	8	no	—
297	C₂²	C₃×#940	1	2	yes	—
298	C₂²	C₂×C₃×#215	1	4	yes	—
299	C₂²	C₂²×C₃×#67	1	1	yes	characteristic
300	C₄	—	4	4	no	—
301	C₄	C₃×#940	1	2	yes	—
302	C₄	C₂×C₃×#215	1	8	yes	complemented
303	C₄	C₂×C₃×D₃×D₄	1	2	yes	—
304	C₃	—	2	1	no	—
305	C₃	C₂×#5085	1	1	yes	characteristic, direct product factor
306	C₃	C₂×C₃×#492	1	1	yes	characteristic, complemented
307	C₂	—	24	4	no	—
308	C₂	C₃×#5085	1	4	yes	direct product factor
309	C₂	C₂×C₃×#940	1	1	yes	characteristic
310	C₂	C₂²×C₃×#215	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄⋊Q₈	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂×C₃×#874	C₂
hypercenter	C₂×C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×C₃×#492
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₄⋊Q₈	C₂	semidirect
2. C₂²×C₃×#215	C₂	semidirect
4. C₃×#5085	C₂	direct (N × H)
5. C₂×C₃×C₄×C₃⋊C₈	C₂	semidirect
7. C₂×#5085	C₃	direct (N × H)
12. C₃²×C₄⋊Q₈	C₂²	semidirect
16. C₂×C₃×#215	C₂²	semidirect
20. C₃×C₄×C₃⋊C₈	C₂²	semidirect
23. C₂×C₃×C₄⋊Q₈	D₃	semidirect
24. C₂×C₃×C₄⋊Q₈	C₂×C₃	semidirect
26. C₂²×#215	C₂×C₃	semidirect
30. #5085	C₂×C₃	direct (N × H)
31. C₂×C₄×C₃⋊C₈	C₂×C₃	semidirect
48. C₂×C₃×C₃⋊C₈	D₄	semidirect
65. C₃×C₄⋊Q₈	C₂×D₃	semidirect
66. C₃×C₄⋊Q₈	C₂²×C₃	semidirect
76. C₂×#215	C₂²×C₃	semidirect
81. C₄×C₃⋊C₈	C₂²×C₃	semidirect
90. C₂×C₃²×C₄	SD₁₆	semidirect
98. C₃×C₃⋊C₈	C₂×D₄	semidirect
100. C₂×C₄⋊Q₈	C₃×D₃	semidirect
147. C₂×C₃⋊C₈	C₃×D₄	semidirect
157. C₃²×C₄	C₂×SD₁₆	semidirect
166. C₄⋊Q₈	C₂×C₃×D₃	semidirect
201. C₂×C₃×C₄	#215	semidirect
202. C₂×C₃×C₄	C₃×SD₁₆	semidirect
215. C₃⋊C₈	C₂×C₃×D₄	semidirect
216. C₂×C₃²	#492	semidirect
260. C₃×C₄	C₂×#215	semidirect
262. C₃×C₄	C₂×C₃×SD₁₆	semidirect
265. C₃²	C₂×#492	semidirect
277. C₂×C₄	C₃×#215	semidirect
287. C₂×C₃	#5085	direct (N × H)
288. C₂×C₃	C₃×#492	semidirect
302. C₄	C₂×C₃×#215	semidirect
305. C₃	C₂×#5085	direct (N × H)
306. C₃	C₂×C₃×#492	semidirect
308. C₂	C₃×#5085	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄⋊Q₈	2
C₂²×C₃×#215	2
C₂×C₃×#874	2
C₃×#5085	2
C₂×C₃×C₄×C₃⋊C₈	2
C₂×C₃×#492	3
C₂×#5085	3

Subgroups — C2×C3×#5085 #11874150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#5085 #11874150