Subgroups — C₂×C₃×#5068 #11873436

Order 1152.

937 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#875	C₂	1	1	yes	maximal, characteristic
3	C₃×#5068	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#810	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#809	C₂	1	2	yes	maximal
6	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#488	—	3	1	no	maximal
8	C₂×#5068	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₃×D₁₂	C₂²	1	1	yes	characteristic
10	C₂²×C₃×Dic₆	C₂²	1	1	yes	characteristic
11	C₃²×#125	—	2	2	no	—
12	C₃²×#125	C₂²	1	4	yes	complemented
13	C₂×C₃²×C₄⋊C₄	—	2	2	no	—
14	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
16	C₂×C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
17	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃×#875	—	2	2	no	—
19	C₃×#875	C₂²	1	4	yes	—
20	C₃×#810	C₂²	1	8	yes	complemented
21	C₃×#809	C₂²	1	8	yes	—
22	C₃×C₄×C₃⋊C₈	C₂²	1	4	yes	complemented
23	C₂×C₃×#125	—	2	1	no	—
24	C₂×C₃×#125	D₃	1	1	yes	characteristic, complemented
25	C₂×C₃×#125	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×C₃×#124	—	3	1	no	—
27	C₂×#875	C₂×C₃	1	1	yes	characteristic
28	C₃×#488	—	3	8	no	—
29	C₂×C₃×#103	—	3	2	no	—
30	C₂×C₃×#102	—	3	2	no	—
31	C₂×C₃×C₄×C₈	—	3	1	no	—
32	#5068	C₂×C₃	1	8	yes	direct product factor
33	C₂×#810	C₂×C₃	1	2	yes	complemented
34	C₂×#809	C₂×C₃	1	2	yes	—
35	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂³×C₃×D₃	—	2	1	no	—
37	C₂²×C₃²×C₄	—	2	2	no	—
38	C₂²×C₃²×C₄	D₄	1	2	yes	—
39	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₃×Dic₃	—	2	1	no	—
41	C₂×C₃×D₁₂	—	2	4	no	—
42	C₂×C₃×D₁₂	—	2	1	no	—
43	C₂×C₃×D₁₂	C₂³	1	2	yes	—
44	C₂×C₃×Dic₆	—	2	4	no	—
45	C₂×C₃×Dic₆	—	2	1	no	—
46	C₂×C₃×Dic₆	C₂³	1	2	yes	—
47	C₃²×C₄⋊C₄	—	2	8	no	—
48	C₃²×C₄⋊C₄	—	2	2	no	—
49	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
50	C₃²×C₄²	D₄	1	2	yes	—
51	C₃²×C₄²	C₂³	1	2	yes	—
52	C₃×(C₂×D₃)⋊C₄	—	2	8	no	—
53	C₂×C₃×C₃⋊C₈	—	2	4	no	—
54	C₂×C₃×C₃⋊C₈	C₂³	1	4	yes	—
55	C₂×#488	—	3	1	no	Sylow
56	C₂²×C₃×Q₈	—	3	1	no	—
57	C₂²×C₃×D₄	—	3	1	no	—
58	C₂²×D₁₂	C₂²×C₃	1	1	yes	characteristic
59	C₂²×Dic₆	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₃×C₈	—	3	2	no	—
61	C₃×#125	—	2	4	no	—
62	C₃×#125	—	2	2	no	—
63	C₃×#125	—	2	2	no	—
64	C₃×#125	—	2	4	no	—
65	C₃×#125	C₂×D₃	1	4	yes	complemented
66	C₃×#125	C₂²×C₃	1	4	yes	complemented
67	C₃×#124	—	3	4	no	—
68	C₃×#124	—	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×C₄⋊C₄	—	2	2	no	—
73	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
74	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₃×C₂²⋊C₄	—	6	2	no	—
76	C₂×C₃×C₄²	—	2	1	no	—
77	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
78	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
79	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
80	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
81	#875	—	2	2	no	—
82	#875	C₂²×C₃	1	4	yes	—
83	C₃×#103	—	3	8	no	—
84	C₃×#102	—	3	8	no	—
85	C₃×C₄×C₈	—	3	4	no	—
86	#810	C₂²×C₃	1	8	yes	complemented
87	#809	C₂²×C₃	1	8	yes	—
88	C₄×C₃⋊C₈	C₂²×C₃	1	4	yes	complemented
89	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₃×D₃	—	2	2	no	—
91	C₂²×C₃×D₃	—	4	2	no	—
92	C₂²×C₃×D₃	—	4	4	no	—
93	C₂×C₃²×C₄	—	2	4	no	—
94	C₂×C₃²×C₄	—	2	4	no	—
95	C₂×C₃²×C₄	—	4	4	no	—
96	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
97	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃²×C₄	#41	1	4	yes	—
99	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
100	C₂×C₃×Dic₃	—	2	2	no	—
101	C₂×C₃×Dic₃	—	4	2	no	—
102	C₃×D₁₂	—	2	4	no	—
103	C₃×D₁₂	—	4	2	no	—
104	C₃×Dic₆	—	2	4	no	—
105	C₃×Dic₆	—	4	2	no	—
106	C₃×C₃⋊C₈	—	2	4	no	—
107	C₂×#125	C₃×D₃	1	1	yes	characteristic, complemented
108	C₂×#124	—	3	1	no	—
109	#488	—	3	8	no	—
110	C₂×#103	—	3	2	no	—
111	C₂×#102	—	3	2	no	—
112	C₂×C₄×C₈	—	3	1	no	—
113	C₂⁴×C₃	—	6	1	no	—
114	C₂³×D₃	—	2	1	no	—
115	C₂×C₃×Q₈	—	3	2	no	—
116	C₂×C₃×Q₈	—	6	1	no	—
117	C₂×C₃×Q₈	—	6	4	no	—
118	C₂×C₃×D₄	—	3	2	no	—
119	C₂×C₃×D₄	—	6	4	no	—
120	C₂×C₃×D₄	—	6	1	no	—
121	C₂²×C₃×C₄	—	2	4	no	—
122	C₂²×C₃×C₄	—	2	2	no	—
123	C₂²×C₃×C₄	—	2	2	no	—
124	C₂²×C₃×C₄	—	2	2	no	—
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₂²×C₃×C₄	—	6	1	no	—
127	C₂²×C₃×C₄	#67	1	2	yes	—
128	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
129	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
131	C₂²×Dic₃	—	2	1	no	—
132	C₂×D₁₂	—	2	4	no	—
133	C₂×D₁₂	—	2	1	no	—
134	C₂×D₁₂	C₂³×C₃	1	2	yes	—
135	C₂×Dic₆	—	2	4	no	—
136	C₂×Dic₆	—	2	1	no	—
137	C₂×Dic₆	C₂³×C₃	1	2	yes	—
138	C₂×C₃×C₈	—	3	4	no	—
139	C₂×C₃×C₈	—	6	4	no	—
140	C₃×C₄⋊C₄	—	2	16	no	—
141	C₃×C₄⋊C₄	—	2	4	no	—
142	C₃×C₄⋊C₄	—	2	8	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	—	2	8	no	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄⋊C₄	—	2	4	no	—
147	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
148	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
149	C₃×C₂²⋊C₄	—	6	8	no	—
150	C₃×C₄²	—	2	2	no	—
151	C₃×C₄²	—	2	2	no	—
152	C₃×C₄²	#67	1	2	yes	—
153	C₃×C₄²	C₃×D₄	1	2	yes	—
154	C₃×C₄²	C₂²×D₃	1	2	yes	—
155	C₃×C₄²	C₂³×C₃	1	2	yes	—
156	(C₂×D₃)⋊C₄	—	2	8	no	—
157	C₂×C₃⋊C₈	—	2	4	no	—
158	C₂×C₃⋊C₈	C₂³×C₃	1	4	yes	—
159	C₂²×C₃²	#124	1	2	yes	—
160	C₂²×C₃²	#135	1	4	yes	—
161	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₃×D₃	—	4	2	no	—
163	C₂×C₃×D₃	—	4	4	no	—
164	C₂×C₃×D₃	—	8	4	no	—
165	C₃²×C₄	—	2	4	no	—
166	C₃²×C₄	—	4	4	no	—
167	C₃²×C₄	#124	1	2	yes	—
168	C₃²×C₄	C₂×#41	1	2	yes	—
169	C₃×Dic₃	—	4	2	no	—
170	C₂²×Q₈	—	3	1	no	—
171	C₂²×D₄	—	3	1	no	—
172	C₂²×C₈	—	3	2	no	—
173	#125	—	2	2	no	—
174	#125	C₂×C₃×D₃	1	4	yes	complemented
175	#124	—	3	4	no	—
176	#124	—	6	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
179	C₂×C₂²⋊C₄	—	6	2	no	—
180	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
181	#103	—	3	8	no	—
182	#102	—	3	8	no	—
183	C₄×C₈	—	3	4	no	—
184	C₂³×C₃	—	2	1	no	—
185	C₂³×C₃	—	6	2	no	—
186	C₂³×C₃	—	12	2	no	—
187	C₂³×C₃	—	12	4	no	—
188	C₂³×C₃	C₂×#67	1	1	yes	characteristic
189	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂²×D₃	—	2	2	no	—
191	C₂²×D₃	—	4	2	no	—
192	C₂²×D₃	—	4	4	no	—
193	C₃×Q₈	—	6	4	no	—
194	C₃×Q₈	—	12	2	no	—
195	C₃×D₄	—	6	4	no	—
196	C₃×D₄	—	12	2	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	—	2	8	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	4	4	no	—
207	C₂×C₃×C₄	—	4	8	no	—
208	C₂×C₃×C₄	—	4	4	no	—
209	C₂×C₃×C₄	—	4	4	no	—
210	C₂×C₃×C₄	—	6	2	no	—
211	C₂×C₃×C₄	—	12	2	no	—
212	C₂×C₃×C₄	#239	1	4	yes	—
213	C₂×C₃×C₄	C₂×#67	1	4	yes	—
214	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
216	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₃×#41	1	4	yes	—
218	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
220	C₂×Dic₃	—	2	2	no	—
221	C₂×Dic₃	—	4	2	no	—
222	D₁₂	—	2	4	no	—
223	D₁₂	—	4	2	no	—
224	Dic₆	—	2	4	no	—
225	Dic₆	—	4	2	no	—
226	C₃×C₈	—	6	4	no	—
227	C₃⋊C₈	—	2	4	no	—
228	C₂×C₃²	#488	1	4	yes	complemented
229	C₂×C₃²	C₂×#124	1	1	yes	characteristic
230	C₂×C₃²	C₂×#135	1	2	yes	—
231	C₃×D₃	—	8	2	no	—
232	C₂⁴	—	6	1	no	—
233	C₂×Q₈	—	3	2	no	—
234	C₂×Q₈	—	6	1	no	—
235	C₂×Q₈	—	6	4	no	—
236	C₂×D₄	—	3	2	no	—
237	C₂×D₄	—	6	4	no	—
238	C₂×D₄	—	6	1	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	6	1	no	—
241	C₂²×C₄	C₃×#67	1	2	yes	—
242	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
243	C₂×C₈	—	3	4	no	—
244	C₂×C₈	—	6	4	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	C₂²×C₃×D₃	1	4	yes	—
248	C₂²⋊C₄	—	6	8	no	—
249	C₄²	C₃×#67	1	2	yes	—
250	C₄²	C₂²×C₃×D₃	1	2	yes	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	4	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	12	2	no	—
255	C₂²×C₃	—	12	4	no	—
256	C₂²×C₃	—	24	4	no	—
257	C₂²×C₃	#947	1	2	yes	—
258	C₂²×C₃	#950	1	4	yes	—
259	C₂²×C₃	C₃×#124	1	2	yes	—
260	C₂²×C₃	C₃×#135	1	4	yes	—
261	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
262	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
263	C₂×D₃	—	4	4	no	—
264	C₂×D₃	—	4	2	no	—
265	C₂×D₃	—	8	4	no	—
266	C₃×C₄	—	2	4	no	—
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	4	4	no	—
271	C₃×C₄	—	4	8	no	—
272	C₃×C₄	—	4	4	no	—
273	C₃×C₄	—	4	4	no	—
274	C₃×C₄	—	12	2	no	—
275	C₃×C₄	#947	1	2	yes	—
276	C₃×C₄	C₃×#124	1	2	yes	—
277	C₃×C₄	C₂×#239	1	2	yes	—
278	C₃×C₄	C₂×C₃×#41	1	2	yes	—
279	Dic₃	—	4	2	no	—
280	C₃²	C₂×#488	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	6	2	no	—
282	C₂³	—	12	2	no	—
283	C₂³	—	12	4	no	—
284	C₂³	C₂×C₃×#67	1	1	yes	characteristic
285	Q₈	—	6	4	no	—
286	Q₈	—	12	2	no	—
287	D₄	—	6	4	no	—
288	D₄	—	12	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	6	2	no	—
293	C₂×C₄	—	12	2	no	—
294	C₂×C₄	C₃×#239	1	4	yes	—
295	C₂×C₄	C₂×C₃×#67	1	4	yes	—
296	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
297	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
298	C₈	—	6	4	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	—	2	4	no	—
302	C₂×C₃	—	24	2	no	—
303	C₂×C₃	#5068	1	4	yes	direct product factor
304	C₂×C₃	C₃×#488	1	4	yes	complemented
305	C₂×C₃	C₂×#947	1	1	yes	characteristic
306	C₂×C₃	C₂×#950	1	2	yes	—
307	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
308	C₂×C₃	C₂×C₃×#135	1	2	yes	—
309	D₃	—	8	2	no	—
310	C₂²	—	12	2	no	—
311	C₂²	—	12	4	no	—
312	C₂²	—	24	4	no	—
313	C₂²	C₃×#947	1	2	yes	—
314	C₂²	C₃×#950	1	4	yes	—
315	C₂²	C₂²×C₃×#67	1	1	yes	characteristic
316	C₄	—	2	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	12	2	no	—
319	C₄	C₃×#947	1	2	yes	—
320	C₄	C₂×C₃×#239	1	2	yes	—
321	C₃	—	2	1	no	—
322	C₃	C₂×#5068	1	1	yes	characteristic, direct product factor
323	C₃	C₂×C₃×#488	1	1	yes	characteristic, complemented
324	C₂	—	24	2	no	—
325	C₂	C₃×#5068	1	4	yes	direct product factor
326	C₂	C₂×C₃×#947	1	1	yes	characteristic
327	C₂	C₂×C₃×#950	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×#125	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂²×C₃×D₁₂	C₂²
hypercenter	C₂×C₃×#125	D₃
hypocenter	C₃	C₂×C₃×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#125	C₂	semidirect
3. C₃×#5068	C₂	direct (N × H)
4. C₂×C₃×#810	C₂	semidirect
6. C₂×C₃×C₄×C₃⋊C₈	C₂	semidirect
8. C₂×#5068	C₃	direct (N × H)
12. C₃²×#125	C₂²	semidirect
20. C₃×#810	C₂²	semidirect
22. C₃×C₄×C₃⋊C₈	C₂²	semidirect
24. C₂×C₃×#125	D₃	semidirect
25. C₂×C₃×#125	C₂×C₃	semidirect
32. #5068	C₂×C₃	direct (N × H)
33. C₂×#810	C₂×C₃	semidirect
35. C₂×C₄×C₃⋊C₈	C₂×C₃	semidirect
65. C₃×#125	C₂×D₃	semidirect
66. C₃×#125	C₂²×C₃	semidirect
86. #810	C₂²×C₃	semidirect
88. C₄×C₃⋊C₈	C₂²×C₃	semidirect
107. C₂×#125	C₃×D₃	semidirect
174. #125	C₂×C₃×D₃	semidirect
228. C₂×C₃²	#488	semidirect
280. C₃²	C₂×#488	semidirect
303. C₂×C₃	#5068	direct (N × H)
304. C₂×C₃	C₃×#488	semidirect
322. C₃	C₂×#5068	direct (N × H)
323. C₃	C₂×C₃×#488	semidirect
325. C₂	C₃×#5068	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×#125	2
C₂×C₃×#875	2
C₃×#5068	2
C₂×C₃×#810	2
C₂×C₃×#809	2
C₂×C₃×C₄×C₃⋊C₈	2
C₂×C₃×#488	3
C₂×#5068	3

Subgroups — C2×C3×#5068 #11873436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#5068 #11873436