Subgroups — C₂×C₃×#4697 #11873430

Order 1152.

905 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#875	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#821	C₂	1	2	yes	maximal
5	C₂×C₃×#816	C₂	1	2	yes	maximal, complemented
6	C₃×#4697	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₃×#488	—	3	1	no	maximal
8	C₂×#4697	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	C₂²×C₃²×C₈	C₂²	1	2	yes	—
12	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
14	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₂×C₃×Dic₃⋊C₄	—	2	2	no	—
16	C₃×#875	—	2	2	no	—
17	C₃×#875	C₂²	1	4	yes	—
18	C₃×#870	—	2	2	no	—
19	C₃×#870	C₂²	1	4	yes	complemented
20	C₃²×C₄×C₈	C₂²	1	4	yes	complemented
21	C₃×#821	C₂²	1	8	yes	—
22	C₃×#816	C₂²	1	8	yes	complemented
23	C₂×C₃×#125	—	3	1	no	—
24	C₂×C₃×#124	—	3	1	no	—
25	C₂×#875	C₂×C₃	1	1	yes	characteristic
26	C₂×#870	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×#488	—	3	8	no	—
28	C₂×C₃×#103	—	3	2	no	—
29	C₂×C₃×#102	—	3	2	no	—
30	C₂×C₃×C₄×C₈	—	2	1	no	—
31	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×#821	C₂×C₃	1	2	yes	—
34	C₂×#816	C₂×C₃	1	2	yes	complemented
35	#4697	C₂×C₃	1	8	yes	direct product factor
36	C₂³×C₃×D₃	—	2	1	no	—
37	C₂²×C₃²×C₄	D₄	1	2	yes	—
38	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₃×Dic₃	—	2	2	no	—
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	C₂×C₃²×C₈	—	2	4	no	—
48	C₂×C₃²×C₈	C₂³	1	4	yes	—
49	C₃²×C₄²	D₄	1	2	yes	—
50	C₃²×C₄²	C₂³	1	2	yes	—
51	C₃×(C₂×D₃)⋊C₄	—	2	8	no	—
52	C₃×C₄⋊Dic₃	—	2	2	no	—
53	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
54	C₃×Dic₃⋊C₄	—	2	8	no	—
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	—	2	2	no	—
61	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
62	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
63	C₃×#125	—	3	4	no	—
64	C₃×#125	—	6	2	no	—
65	C₃×#124	—	3	4	no	—
66	C₃×#124	—	6	2	no	—
67	C₂×C₃×C₄⋊C₄	—	3	2	no	—
68	C₂×C₃×C₄⋊C₄	—	6	2	no	—
69	C₂×C₃×C₂²⋊C₄	—	6	2	no	—
70	C₂×C₃×C₄²	—	2	1	no	—
71	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
72	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
73	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
74	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
75	C₂×Dic₃⋊C₄	—	2	2	no	—
76	#875	—	2	2	no	—
77	#875	C₂²×C₃	1	4	yes	—
78	#870	—	2	2	no	—
79	#870	C₂²×C₃	1	4	yes	complemented
80	C₃×#103	—	3	8	no	—
81	C₃×#102	—	3	8	no	—
82	C₃×C₄×C₈	—	2	4	no	—
83	C₃×C₄×C₈	C₂×D₃	1	4	yes	complemented
84	C₃×C₄×C₈	C₂²×C₃	1	4	yes	complemented
85	#821	C₂²×C₃	1	8	yes	—
86	#816	C₂²×C₃	1	8	yes	complemented
87	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₃×D₃	—	2	2	no	—
89	C₂²×C₃×D₃	—	4	2	no	—
90	C₂²×C₃×D₃	—	4	4	no	—
91	C₂×C₃²×C₄	—	2	4	no	—
92	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
93	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	#41	1	4	yes	—
95	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
96	C₂×C₃×Dic₃	—	2	4	no	—
97	C₂×C₃×Dic₃	—	2	2	no	—
98	C₂×C₃×Dic₃	—	4	2	no	—
99	C₂×C₃×Dic₃	—	4	4	no	—
100	C₃×D₁₂	—	2	4	no	—
101	C₃×D₁₂	—	4	2	no	—
102	C₃×Dic₆	—	2	4	no	—
103	C₃×Dic₆	—	4	2	no	—
104	C₃²×C₈	—	2	4	no	—
105	C₂×#125	—	3	1	no	—
106	C₂×#124	—	3	1	no	—
107	#488	—	3	8	no	—
108	C₂×#103	—	3	2	no	—
109	C₂×#102	—	3	2	no	—
110	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic, complemented
111	C₂⁴×C₃	—	6	1	no	—
112	C₂³×D₃	—	2	1	no	—
113	C₂×C₃×Q₈	—	3	2	no	—
114	C₂×C₃×Q₈	—	6	1	no	—
115	C₂×C₃×Q₈	—	6	4	no	—
116	C₂×C₃×D₄	—	3	2	no	—
117	C₂×C₃×D₄	—	6	1	no	—
118	C₂×C₃×D₄	—	6	4	no	—
119	C₂²×C₃×C₄	—	2	2	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	6	1	no	—
122	C₂²×C₃×C₄	—	6	2	no	—
123	C₂²×C₃×C₄	D₁₂	1	2	yes	—
124	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
125	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
127	C₂²×Dic₃	—	2	2	no	—
128	C₂²×Dic₃	—	2	1	no	—
129	C₂×D₁₂	—	2	4	no	—
130	C₂×D₁₂	—	2	1	no	—
131	C₂×D₁₂	C₂³×C₃	1	2	yes	—
132	C₂×Dic₆	—	2	4	no	—
133	C₂×Dic₆	—	2	1	no	—
134	C₂×Dic₆	C₂³×C₃	1	2	yes	—
135	C₂×C₃×C₈	—	2	8	no	—
136	C₂×C₃×C₈	—	2	4	no	—
137	C₂×C₃×C₈	—	2	4	no	—
138	C₂×C₃×C₈	—	2	4	no	—
139	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
140	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
141	C₃×C₄⋊C₄	—	3	4	no	—
142	C₃×C₄⋊C₄	—	6	8	no	—
143	C₃×C₄⋊C₄	—	6	2	no	—
144	C₃×C₂²⋊C₄	—	6	8	no	—
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	—	2	2	no	—
147	C₃×C₄²	D₁₂	1	2	yes	—
148	C₃×C₄²	C₃×D₄	1	2	yes	—
149	C₃×C₄²	C₂²×D₃	1	2	yes	—
150	C₃×C₄²	C₂³×C₃	1	2	yes	—
151	(C₂×D₃)⋊C₄	—	2	8	no	—
152	C₄⋊Dic₃	—	2	2	no	—
153	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
154	Dic₃⋊C₄	—	2	8	no	—
155	C₂²×C₃²	#124	1	2	yes	—
156	C₂²×C₃²	#135	1	4	yes	—
157	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₃×D₃	—	4	2	no	—
159	C₂×C₃×D₃	—	4	4	no	—
160	C₂×C₃×D₃	—	8	4	no	—
161	C₃²×C₄	—	2	4	no	—
162	C₃²×C₄	#124	1	2	yes	—
163	C₃²×C₄	C₂×#41	1	2	yes	—
164	C₃×Dic₃	—	4	2	no	—
165	C₃×Dic₃	—	4	4	no	—
166	C₂²×Q₈	—	3	1	no	—
167	C₂²×D₄	—	3	1	no	—
168	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
169	#125	—	3	4	no	—
170	#125	—	6	2	no	—
171	#124	—	3	4	no	—
172	#124	—	6	2	no	—
173	C₂×C₄⋊C₄	—	3	2	no	—
174	C₂×C₄⋊C₄	—	6	2	no	—
175	C₂×C₂²⋊C₄	—	6	2	no	—
176	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
177	#103	—	3	8	no	—
178	#102	—	3	8	no	—
179	C₄×C₈	C₂×C₃×D₃	1	4	yes	complemented
180	C₂³×C₃	—	2	1	no	—
181	C₂³×C₃	—	6	2	no	—
182	C₂³×C₃	—	12	2	no	—
183	C₂³×C₃	—	12	4	no	—
184	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
185	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂²×D₃	—	2	2	no	—
187	C₂²×D₃	—	4	2	no	—
188	C₂²×D₃	—	4	4	no	—
189	C₃×Q₈	—	6	4	no	—
190	C₃×Q₈	—	12	2	no	—
191	C₃×D₄	—	6	4	no	—
192	C₃×D₄	—	12	2	no	—
193	C₂×C₃×C₄	—	2	8	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	6	2	no	—
201	C₂×C₃×C₄	—	6	4	no	—
202	C₂×C₃×C₄	—	12	2	no	—
203	C₂×C₃×C₄	—	12	4	no	—
204	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
205	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
206	C₂×C₃×C₄	#235	1	4	yes	—
207	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
208	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₃×#41	1	4	yes	—
210	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
212	C₂×Dic₃	—	2	4	no	—
213	C₂×Dic₃	—	2	2	no	—
214	C₂×Dic₃	—	4	2	no	—
215	C₂×Dic₃	—	4	4	no	—
216	D₁₂	—	2	4	no	—
217	D₁₂	—	4	2	no	—
218	Dic₆	—	2	4	no	—
219	Dic₆	—	4	2	no	—
220	C₃×C₈	—	2	8	no	—
221	C₃×C₈	—	2	4	no	—
222	C₃×C₈	—	2	4	no	—
223	C₂×C₃²	#488	1	4	yes	complemented
224	C₂×C₃²	C₂×#124	1	1	yes	characteristic
225	C₂×C₃²	C₂×#135	1	2	yes	—
226	C₃×D₃	—	8	2	no	—
227	C₂⁴	—	6	1	no	—
228	C₂×Q₈	—	3	2	no	—
229	C₂×Q₈	—	6	1	no	—
230	C₂×Q₈	—	6	4	no	—
231	C₂×D₄	—	3	2	no	—
232	C₂×D₄	—	6	1	no	—
233	C₂×D₄	—	6	4	no	—
234	C₂²×C₄	—	6	1	no	—
235	C₂²×C₄	—	6	2	no	—
236	C₂²×C₄	C₃×D₁₂	1	2	yes	—
237	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
240	C₄⋊C₄	—	3	4	no	—
241	C₄⋊C₄	—	6	2	no	—
242	C₄⋊C₄	—	6	8	no	—
243	C₂²⋊C₄	—	6	8	no	—
244	C₄²	C₃×D₁₂	1	2	yes	—
245	C₄²	C₂²×C₃×D₃	1	2	yes	—
246	C₂²×C₃	—	2	4	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	12	2	no	—
250	C₂²×C₃	—	12	4	no	—
251	C₂²×C₃	—	24	4	no	—
252	C₂²×C₃	#875	1	2	yes	—
253	C₂²×C₃	#904	1	4	yes	—
254	C₂²×C₃	C₃×#124	1	2	yes	—
255	C₂²×C₃	C₃×#135	1	4	yes	—
256	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
257	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
258	C₂×D₃	—	4	4	no	—
259	C₂×D₃	—	4	2	no	—
260	C₂×D₃	—	8	4	no	—
261	C₃×C₄	—	2	8	no	—
262	C₃×C₄	—	2	4	no	—
263	C₃×C₄	—	2	4	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	12	2	no	—
267	C₃×C₄	—	12	4	no	—
268	C₃×C₄	#875	1	2	yes	—
269	C₃×C₄	C₃×#124	1	2	yes	—
270	C₃×C₄	C₂×#235	1	2	yes	—
271	C₃×C₄	C₂×C₃×#41	1	2	yes	—
272	Dic₃	—	4	4	no	—
273	Dic₃	—	4	2	no	—
274	C₃²	C₂×#488	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	6	2	no	—
276	C₂³	—	12	4	no	—
277	C₂³	—	12	2	no	—
278	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
279	Q₈	—	6	4	no	—
280	Q₈	—	12	2	no	—
281	D₄	—	6	4	no	—
282	D₄	—	12	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	6	2	no	—
285	C₂×C₄	—	6	4	no	—
286	C₂×C₄	—	12	4	no	—
287	C₂×C₄	—	12	2	no	—
288	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
289	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
290	C₂×C₄	C₃×#235	1	4	yes	—
291	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₂×C₃	—	2	4	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	—	24	2	no	—
297	C₂×C₃	#4697	1	4	yes	direct product factor
298	C₂×C₃	C₃×#488	1	4	yes	complemented
299	C₂×C₃	C₂×#875	1	1	yes	characteristic
300	C₂×C₃	C₂×#904	1	2	yes	—
301	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
302	C₂×C₃	C₂×C₃×#135	1	2	yes	—
303	D₃	—	8	2	no	—
304	C₂²	—	12	2	no	—
305	C₂²	—	12	4	no	—
306	C₂²	—	24	4	no	—
307	C₂²	C₃×#875	1	2	yes	—
308	C₂²	C₃×#904	1	4	yes	—
309	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	12	2	no	—
312	C₄	—	12	4	no	—
313	C₄	C₃×#875	1	2	yes	—
314	C₄	C₂×C₃×#235	1	2	yes	—
315	C₃	—	2	1	no	—
316	C₃	C₂×#4697	1	1	yes	characteristic, direct product factor
317	C₃	C₂×C₃×#488	1	1	yes	characteristic, complemented
318	C₂	—	24	2	no	—
319	C₂	C₃×#4697	1	4	yes	direct product factor
320	C₂	C₂×C₃×#875	1	1	yes	characteristic
321	C₂	C₂×C₃×#904	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂²×C₃×D₁₂	C₂²
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₂×C₃×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#870	C₂	semidirect
3. C₂×C₃²×C₄×C₈	C₂	semidirect
5. C₂×C₃×#816	C₂	semidirect
6. C₃×#4697	C₂	direct (N × H)
8. C₂×#4697	C₃	direct (N × H)
19. C₃×#870	C₂²	semidirect
20. C₃²×C₄×C₈	C₂²	semidirect
22. C₃×#816	C₂²	semidirect
26. C₂×#870	C₂×C₃	semidirect
31. C₂×C₃×C₄×C₈	D₃	semidirect
32. C₂×C₃×C₄×C₈	C₂×C₃	semidirect
34. C₂×#816	C₂×C₃	semidirect
35. #4697	C₂×C₃	direct (N × H)
79. #870	C₂²×C₃	semidirect
83. C₃×C₄×C₈	C₂×D₃	semidirect
84. C₃×C₄×C₈	C₂²×C₃	semidirect
86. #816	C₂²×C₃	semidirect
110. C₂×C₄×C₈	C₃×D₃	semidirect
179. C₄×C₈	C₂×C₃×D₃	semidirect
223. C₂×C₃²	#488	semidirect
274. C₃²	C₂×#488	semidirect
297. C₂×C₃	#4697	direct (N × H)
298. C₂×C₃	C₃×#488	semidirect
316. C₃	C₂×#4697	direct (N × H)
317. C₃	C₂×C₃×#488	semidirect
319. C₂	C₃×#4697	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×C₃×#875	2
C₂×C₃×#870	2
C₂×C₃²×C₄×C₈	2
C₂×C₃×#821	2
C₂×C₃×#816	2
C₃×#4697	2
C₂×C₃×#488	3
C₂×#4697	3

Subgroups — C2×C3×#4697 #11873430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4697 #11873430