Subgroups — C₂×C₃×#5076 #11874144

Order 1152.

1561 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×#214	C₂	1	4	yes	maximal, complemented
3	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#5076	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×#5076	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×D₄	—	2	2	no	—
9	C₂²×C₃²×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
11	C₃²×#127	—	2	2	no	—
12	C₃²×#127	C₂²	1	4	yes	—
13	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₃×#214	—	2	16	no	—
15	C₂×C₃×#214	C₂²	1	16	yes	complemented
16	C₂×C₃×C₄⋊Dic₃	—	2	2	no	—
17	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃×#869	—	2	2	no	—
19	C₃×#869	C₂²	1	4	yes	complemented
20	C₃×C₄×C₃⋊C₈	C₂²	1	4	yes	complemented
21	C₂²×C₃×SD₁₆	—	3	4	no	—
22	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
23	C₂×C₃×#127	—	2	1	no	—
24	C₂×C₃×#127	D₃	1	1	yes	characteristic
25	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
26	C₂²×#214	C₂×C₃	1	4	yes	complemented
27	C₂×#869	C₂×C₃	1	1	yes	characteristic, complemented
28	C₃×#492	—	3	8	no	—
29	C₂×C₃×C₄×C₈	—	3	1	no	—
30	#5076	C₂×C₃	1	8	yes	direct product factor
31	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂⁴×C₃²	—	2	2	no	—
33	C₂×C₃²×D₄	—	2	8	no	—
34	C₂×C₃²×D₄	—	2	8	no	—
35	C₂×C₃²×D₄	—	2	2	no	—
36	C₂×C₃²×D₄	—	4	8	no	—
37	C₂×C₃²×D₄	C₂³	1	4	yes	—
38	C₂²×C₃²×C₄	D₄	1	2	yes	—
39	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₃×Dic₃	—	2	2	no	—
41	C₂×C₃×Dic₆	—	2	8	no	—
42	C₂×C₃×Dic₆	—	2	2	no	—
43	C₂×C₃×Dic₆	C₂³	1	4	yes	—
44	C₃²×C₄²	D₄	1	2	yes	—
45	C₃²×C₄²	C₂³	1	2	yes	—
46	C₃×#214	—	2	32	no	—
47	C₃×C₄⋊Dic₃	—	2	8	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	—	3	2	no	—
52	C₂²×C₃×D₄	—	2	4	no	—
53	C₂²×C₃×D₄	—	2	2	no	—
54	C₂²×C₃×D₄	—	2	2	no	—
55	C₂²×C₃×D₄	—	2	2	no	—
56	C₂²×C₃×D₄	C₂×D₃	1	2	yes	—
57	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
58	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
59	C₂×C₃×SD₁₆	—	3	16	no	—
60	C₂×C₃×SD₁₆	—	6	16	no	—
61	C₂²×C₃×C₈	—	3	2	no	—
62	C₃×C₄⋊Q₈	—	3	4	no	—
63	C₃×C₄⋊Q₈	—	6	2	no	—
64	C₃×#127	—	2	4	no	—
65	C₃×#127	—	2	2	no	—
66	C₃×#127	—	2	2	no	—
67	C₃×#127	—	2	4	no	—
68	C₃×#127	C₂×D₃	1	4	yes	—
69	C₃×#127	C₂²×C₃	1	4	yes	—
70	C₂×C₃×C₄⋊C₄	—	6	2	no	—
71	C₂×C₃×C₄²	—	2	1	no	—
72	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
73	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
74	C₂×#214	—	2	16	no	—
75	C₂×#214	C₂²×C₃	1	16	yes	complemented
76	C₂×C₄⋊Dic₃	—	2	2	no	—
77	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
78	#869	—	2	2	no	—
79	#869	C₂²×C₃	1	4	yes	complemented
80	C₃×C₄×C₈	—	3	4	no	—
81	C₄×C₃⋊C₈	C₂²×C₃	1	4	yes	complemented
82	C₂³×C₃²	—	2	4	no	—
83	C₂³×C₃²	—	4	4	no	—
84	C₂³×C₃²	—	4	8	no	—
85	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
86	C₃²×D₄	—	2	8	no	—
87	C₃²×D₄	—	4	4	no	—
88	C₃²×D₄	—	4	16	no	—
89	C₂×C₃²×C₄	SD₁₆	1	8	yes	complemented
90	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
91	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
93	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
94	C₂×C₃×Dic₃	—	2	4	no	—
95	C₂×C₃×Dic₃	—	4	4	no	—
96	C₃×Dic₆	—	2	8	no	—
97	C₃×Dic₆	—	4	4	no	—
98	C₃×C₃⋊C₈	C₂×D₄	1	8	yes	complemented
99	C₂²×SD₁₆	—	3	4	no	—
100	C₂×C₄⋊Q₈	—	3	1	no	—
101	C₂×#127	C₃×D₃	1	1	yes	characteristic
102	#492	—	3	8	no	—
103	C₂×C₄×C₈	—	3	1	no	—
104	C₂⁴×C₃	—	2	4	no	—
105	C₂⁴×C₃	—	2	2	no	—
106	C₂⁴×C₃	—	2	2	no	—
107	C₂×C₃×Q₈	—	3	4	no	—
108	C₂×C₃×Q₈	—	6	2	no	—
109	C₂×C₃×Q₈	—	6	8	no	—
110	C₂×C₃×D₄	—	2	4	no	—
111	C₂×C₃×D₄	—	2	16	no	—
112	C₂×C₃×D₄	—	2	8	no	—
113	C₂×C₃×D₄	—	2	2	no	—
114	C₂×C₃×D₄	—	2	8	no	—
115	C₂×C₃×D₄	—	2	8	no	—
116	C₂×C₃×D₄	—	2	2	no	—
117	C₂×C₃×D₄	—	2	8	no	—
118	C₂×C₃×D₄	—	2	4	no	—
119	C₂×C₃×D₄	—	4	8	no	—
120	C₂×C₃×D₄	—	4	16	no	—
121	C₂×C₃×D₄	—	4	8	no	—
122	C₂×C₃×D₄	—	4	8	no	—
123	C₂×C₃×D₄	C₂²×D₃	1	4	yes	—
124	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
125	C₂²×C₃×C₄	—	2	2	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	6	2	no	—
128	C₂²×C₃×C₄	#67	1	2	yes	—
129	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
130	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂²×Dic₃	—	2	2	no	—
133	C₂×Dic₆	—	2	8	no	—
134	C₂×Dic₆	—	2	2	no	—
135	C₂×Dic₆	C₂³×C₃	1	4	yes	—
136	C₃×SD₁₆	—	6	32	no	—
137	C₂×C₃×C₈	—	3	8	no	—
138	C₂×C₃×C₈	—	3	4	no	—
139	C₃×C₄⋊C₄	—	6	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	#214	—	2	32	no	—
147	C₄⋊Dic₃	—	2	8	no	—
148	C₂×C₃⋊C₈	C₃×D₄	1	8	yes	complemented
149	C₂×C₃⋊C₈	C₂³×C₃	1	4	yes	—
150	C₂²×C₃²	—	4	4	no	—
151	C₂²×C₃²	—	4	8	no	—
152	C₂²×C₃²	—	8	8	no	—
153	C₂²×C₃²	#127	1	2	yes	—
154	C₂²×C₃²	C₂×SD₁₆	1	4	yes	—
155	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
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	C₃×Dic₃	—	4	4	no	—
160	C₂²×Q₈	—	3	2	no	—
161	C₂²×D₄	—	2	2	no	—
162	C₂²×D₄	C₂×C₃×D₃	1	2	yes	—
163	C₂×SD₁₆	—	3	16	no	—
164	C₂×SD₁₆	—	6	16	no	—
165	C₂²×C₈	—	3	2	no	—
166	C₄⋊Q₈	—	3	4	no	—
167	C₄⋊Q₈	—	6	2	no	—
168	#127	—	2	2	no	—
169	#127	C₂×C₃×D₃	1	4	yes	—
170	C₂×C₄⋊C₄	—	6	2	no	—
171	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
172	C₄×C₈	—	3	4	no	—
173	C₂³×C₃	—	2	8	no	—
174	C₂³×C₃	—	2	4	no	—
175	C₂³×C₃	—	2	4	no	—
176	C₂³×C₃	—	2	1	no	—
177	C₂³×C₃	—	4	16	no	—
178	C₂³×C₃	—	4	8	no	—
179	C₂³×C₃	—	4	8	no	—
180	C₂³×C₃	—	4	4	no	—
181	C₂³×C₃	—	4	4	no	—
182	C₂³×C₃	—	4	8	no	—
183	C₂³×C₃	C₂×#67	1	1	yes	characteristic
184	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
185	C₃×Q₈	—	6	8	no	—
186	C₃×Q₈	—	12	4	no	—
187	C₃×D₄	—	2	8	no	—
188	C₃×D₄	—	2	8	no	—
189	C₃×D₄	—	4	8	no	—
190	C₃×D₄	—	4	32	no	—
191	C₃×D₄	—	4	8	no	—
192	C₃×D₄	—	4	16	no	—
193	C₃×D₄	—	4	4	no	—
194	C₃×D₄	—	4	4	no	—
195	C₃×D₄	—	4	16	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	8	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	6	4	no	—
202	C₂×C₃×C₄	—	12	4	no	—
203	C₂×C₃×C₄	#214	1	8	yes	complemented
204	C₂×C₃×C₄	C₃×SD₁₆	1	8	yes	complemented
205	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
206	C₂×C₃×C₄	C₂×#67	1	4	yes	—
207	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
209	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
211	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
213	C₂×Dic₃	—	2	4	no	—
214	C₂×Dic₃	—	4	4	no	—
215	Dic₆	—	2	8	no	—
216	Dic₆	—	4	4	no	—
217	C₃×C₈	—	3	8	no	—
218	C₃⋊C₈	C₂×C₃×D₄	1	8	yes	complemented
219	C₂×C₃²	—	8	4	no	—
220	C₂×C₃²	#492	1	4	yes	complemented
221	C₂×C₃²	C₂×#127	1	1	yes	characteristic
222	C₂×C₃²	C₂²×SD₁₆	1	2	yes	—
223	C₂⁴	—	2	2	no	—
224	C₂×Q₈	—	3	4	no	—
225	C₂×Q₈	—	6	2	no	—
226	C₂×Q₈	—	6	8	no	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	8	no	—
230	C₂×D₄	—	4	8	no	—
231	C₂×D₄	C₂²×C₃×D₃	1	4	yes	—
232	C₂²×C₄	—	6	2	no	—
233	C₂²×C₄	C₃×#67	1	2	yes	—
234	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
235	SD₁₆	—	6	32	no	—
236	C₂×C₈	—	3	4	no	—
237	C₂×C₈	—	3	8	no	—
238	C₄⋊C₄	—	6	8	no	—
239	C₄²	C₃×#67	1	2	yes	—
240	C₄²	C₂²×C₃×D₃	1	2	yes	—
241	C₂²×C₃	—	2	4	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	4	16	no	—
245	C₂²×C₃	—	4	8	no	—
246	C₂²×C₃	—	4	8	no	—
247	C₂²×C₃	—	4	8	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	—	4	4	no	—
250	C₂²×C₃	—	8	16	no	—
251	C₂²×C₃	—	8	8	no	—
252	C₂²×C₃	—	8	8	no	—
253	C₂²×C₃	C₂×#214	1	4	yes	—
254	C₂²×C₃	#940	1	2	yes	—
255	C₂²×C₃	C₃×#127	1	2	yes	—
256	C₂²×C₃	C₂×C₃×SD₁₆	1	4	yes	—
257	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
258	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
259	C₃×C₄	—	2	8	no	—
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	12	4	no	—
263	C₃×C₄	C₂×#214	1	8	yes	complemented
264	C₃×C₄	#940	1	2	yes	—
265	C₃×C₄	C₃×#127	1	2	yes	—
266	C₃×C₄	C₂×C₃×SD₁₆	1	8	yes	complemented
267	C₃×C₄	C₂×D₃×D₄	1	2	yes	—
268	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
269	Dic₃	—	4	4	no	—
270	C₃²	C₂×#492	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	2	4	no	—
272	C₂³	—	4	8	no	—
273	C₂³	—	4	4	no	—
274	C₂³	C₂×C₃×#67	1	1	yes	characteristic
275	Q₈	—	6	8	no	—
276	Q₈	—	12	4	no	—
277	D₄	—	2	8	no	—
278	D₄	—	4	16	no	—
279	D₄	—	4	4	no	—
280	C₂×C₄	—	6	4	no	—
281	C₂×C₄	—	12	4	no	—
282	C₂×C₄	C₃×#214	1	8	yes	complemented
283	C₂×C₄	C₃×D₃×D₄	1	4	yes	—
284	C₂×C₄	C₂×C₃×#67	1	4	yes	—
285	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
286	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
287	C₈	—	3	8	no	—
288	C₂×C₃	—	2	2	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	4	no	—
291	C₂×C₃	—	8	8	no	—
292	C₂×C₃	—	8	4	no	—
293	C₂×C₃	—	8	4	no	—
294	C₂×C₃	#5076	1	4	yes	direct product factor
295	C₂×C₃	C₃×#492	1	4	yes	complemented
296	C₂×C₃	C₂²×#214	1	2	yes	—
297	C₂×C₃	C₂×#940	1	1	yes	characteristic
298	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
299	C₂×C₃	C₂²×C₃×SD₁₆	1	2	yes	—
300	C₂²	—	4	8	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	8	8	no	—
303	C₂²	C₂×C₃×#214	1	4	yes	—
304	C₂²	C₃×#940	1	2	yes	—
305	C₂²	C₂²×C₃×#67	1	1	yes	characteristic
306	C₄	—	12	4	no	—
307	C₄	C₂×C₃×#214	1	8	yes	complemented
308	C₄	C₃×#940	1	2	yes	—
309	C₄	C₂×C₃×D₃×D₄	1	2	yes	—
310	C₃	—	2	1	no	—
311	C₃	C₂×#5076	1	1	yes	characteristic, direct product factor
312	C₃	C₂×C₃×#492	1	1	yes	characteristic, complemented
313	C₂	—	8	4	no	—
314	C₂	C₃×#5076	1	4	yes	direct product factor
315	C₂	C₂²×C₃×#214	1	2	yes	—
316	C₂	C₂×C₃×#940	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting, prime radical	C₂×C₃²×#127	C₂
socle	C₂³×C₃²	C₂×D₄
hypercenter	C₂×C₃×#127	D₃
hypocenter	C₃	C₂×C₃×#492
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×#214	C₂	semidirect
3. C₂×C₃×#869	C₂	semidirect
4. C₃×#5076	C₂	direct (N × H)
5. C₂×C₃×C₄×C₃⋊C₈	C₂	semidirect
7. C₂×#5076	C₃	direct (N × H)
15. C₂×C₃×#214	C₂²	semidirect
19. C₃×#869	C₂²	semidirect
20. C₃×C₄×C₃⋊C₈	C₂²	semidirect
26. C₂²×#214	C₂×C₃	semidirect
27. C₂×#869	C₂×C₃	semidirect
30. #5076	C₂×C₃	direct (N × H)
31. C₂×C₄×C₃⋊C₈	C₂×C₃	semidirect
48. C₂×C₃×C₃⋊C₈	D₄	semidirect
75. C₂×#214	C₂²×C₃	semidirect
79. #869	C₂²×C₃	semidirect
81. C₄×C₃⋊C₈	C₂²×C₃	semidirect
89. C₂×C₃²×C₄	SD₁₆	semidirect
98. C₃×C₃⋊C₈	C₂×D₄	semidirect
148. C₂×C₃⋊C₈	C₃×D₄	semidirect
157. C₃²×C₄	C₂×SD₁₆	semidirect
203. C₂×C₃×C₄	#214	semidirect
204. C₂×C₃×C₄	C₃×SD₁₆	semidirect
218. C₃⋊C₈	C₂×C₃×D₄	semidirect
220. C₂×C₃²	#492	semidirect
263. C₃×C₄	C₂×#214	semidirect
266. C₃×C₄	C₂×C₃×SD₁₆	semidirect
270. C₃²	C₂×#492	semidirect
282. C₂×C₄	C₃×#214	semidirect
294. C₂×C₃	#5076	direct (N × H)
295. C₂×C₃	C₃×#492	semidirect
307. C₄	C₂×C₃×#214	semidirect
311. C₃	C₂×#5076	direct (N × H)
312. C₃	C₂×C₃×#492	semidirect
314. C₂	C₃×#5076	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×#127	2
C₂²×C₃×#214	2
C₂×C₃×#869	2
C₃×#5076	2
C₂×C₃×C₄×C₃⋊C₈	2
C₂×C₃×#492	3
C₂×#5076	3

Subgroups — C2×C3×#5076 #11874144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#5076 #11874144