Subgroups — C₃×#21184 #10639791

Order 768.

844 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#430	C₂	1	2	yes	maximal
2	C₂×C₃×#430	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#430	C₂	1	1	yes	maximal, characteristic
4	C₃×#2873	C₂	1	1	yes	maximal, characteristic
5	C₃×#1953	C₂	1	8	yes	maximal, complemented
6	C₂×C₃×#344	C₂	1	2	yes	maximal
7	#21184	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×#117	C₂²	1	1	yes	characteristic
13	C₃×#430	—	2	4	no	—
14	C₃×#430	—	2	2	no	—
15	C₃×#430	—	2	2	no	—
16	C₃×#430	C₄	1	8	yes	complemented
17	C₃×#430	C₂²	1	8	yes	—
18	C₂×C₃×#108	—	2	4	no	—
19	C₂×C₃×#108	C₂²	1	4	yes	—
20	C₂×C₃×#108	C₂²	1	4	yes	—
21	C₃×#405	—	2	2	no	—
22	C₃×#405	C₂²	1	4	yes	—
23	C₃×C₄×M₁₆	—	2	2	no	—
24	C₃×#344	C₂²	1	8	yes	—
25	C₂×#430	C₂×C₃	1	2	yes	—
26	C₂×#430	C₂×C₃	1	1	yes	characteristic
27	C₂×#430	C₂×C₃	1	1	yes	characteristic
28	#2873	C₂×C₃	1	1	yes	characteristic
29	#1953	C₂×C₃	1	8	yes	complemented
30	C₂×#344	C₂×C₃	1	2	yes	—
31	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃×M₁₆	—	2	1	no	—
33	C₂×C₃×M₁₆	—	2	2	no	—
34	C₂×C₃×M₁₆	—	2	2	no	—
35	C₂×C₃×M₁₆	—	2	2	no	—
36	C₂×C₃×M₁₆	—	2	2	no	—
37	C₂×C₃×M₁₆	—	2	2	no	—
38	C₂×C₃×M₁₆	—	2	2	no	—
39	C₂×C₃×M₁₆	—	2	2	no	—
40	C₂×C₃×M₁₆	—	2	2	no	—
41	C₂×C₃×M₁₆	—	2	2	no	—
42	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
43	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
44	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
45	C₂×C₃×M₁₆	D₄	1	4	yes	—
46	C₂×C₃×M₁₆	D₄	1	2	yes	—
47	C₂×C₃×M₁₆	D₄	1	2	yes	—
48	C₂×C₃×M₁₆	Q₈	1	4	yes	—
49	C₂×C₃×M₁₆	C₂³	1	4	yes	—
50	C₂×C₃×M₁₆	C₂³	1	4	yes	—
51	C₂×C₃×M₁₆	C₂³	1	2	yes	—
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₂²×C₃×C₈	—	2	2	no	—
54	C₂²×C₃×C₈	—	2	2	no	—
55	C₂²×C₃×C₈	C₂³	1	2	yes	—
56	C₃×#117	—	2	1	no	—
57	C₃×#117	—	2	1	no	—
58	C₃×#117	Q₈	1	2	yes	—
59	C₃×#117	C₂³	1	2	yes	—
60	C₂×C₃×C₄⋊C₄	—	2	1	no	—
61	C₂×C₃×C₂²⋊C₄	Q₈	1	2	yes	—
62	C₂×C₃×C₄²	—	2	1	no	—
63	C₃×#108	—	2	16	no	—
64	C₃×#108	—	2	8	no	—
65	C₃×#108	C₂×C₄	1	16	yes	complemented
66	C₃×#97	—	2	4	no	—
67	C₃×C₄×C₈	—	2	4	no	—
68	C₂²×M₁₆	C₂²×C₃	1	2	yes	—
69	C₂²×M₁₆	C₂²×C₃	1	2	yes	—
70	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
71	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
72	C₂×#117	C₂²×C₃	1	1	yes	characteristic
73	#430	—	2	4	no	—
74	#430	—	2	2	no	—
75	#430	—	2	2	no	—
76	#430	C₃×C₄	1	8	yes	complemented
77	#430	C₂²×C₃	1	8	yes	—
78	C₂×#108	—	2	4	no	—
79	C₂×#108	C₂²×C₃	1	4	yes	—
80	C₂×#108	C₂²×C₃	1	4	yes	—
81	#405	—	2	2	no	—
82	#405	C₂²×C₃	1	4	yes	—
83	C₄×M₁₆	—	2	2	no	—
84	#344	C₂²×C₃	1	8	yes	—
85	C₂⁴×C₃	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
88	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
91	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
92	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₃×C₄	#41	1	1	yes	characteristic
94	C₂²×C₃×C₄	#41	1	1	yes	characteristic
95	C₂²×C₃×C₄	#41	1	2	yes	—
96	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
97	C₃×M₁₆	—	2	4	no	—
98	C₃×M₁₆	—	2	4	no	—
99	C₃×M₁₆	—	2	4	no	—
100	C₃×M₁₆	—	2	4	no	—
101	C₃×M₁₆	—	2	2	no	—
102	C₃×M₁₆	—	2	4	no	—
103	C₃×M₁₆	—	2	4	no	—
104	C₃×M₁₆	—	4	2	no	—
105	C₃×M₁₆	—	4	2	no	—
106	C₃×M₁₆	C₄⋊C₄	1	8	yes	—
107	C₃×M₁₆	C₄⋊C₄	1	8	yes	—
108	C₃×M₁₆	C₂²×C₄	1	8	yes	—
109	C₃×M₁₆	C₂×D₄	1	4	yes	—
110	C₂×C₃×C₈	—	2	4	no	—
111	C₂×C₃×C₈	—	2	4	no	—
112	C₂×C₃×C₈	—	2	4	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	4	no	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	—	4	2	no	—
117	C₂×C₃×C₈	—	4	2	no	—
118	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
119	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
120	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₂²⋊C₄	—	2	2	no	—
124	C₃×C₂²⋊C₄	C₂×Q₈	1	4	yes	—
125	C₃×C₄²	—	2	2	no	—
126	C₃×C₄²	—	2	2	no	—
127	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	C₂×C₃×C₄	1	4	yes	—
139	C₂×M₁₆	C₂×C₃×C₄	1	4	yes	—
140	C₂×M₁₆	C₂×C₃×C₄	1	4	yes	—
141	C₂×M₁₆	C₃×D₄	1	4	yes	—
142	C₂×M₁₆	C₃×D₄	1	2	yes	—
143	C₂×M₁₆	C₃×D₄	1	2	yes	—
144	C₂×M₁₆	C₃×Q₈	1	4	yes	—
145	C₂×M₁₆	C₂³×C₃	1	4	yes	—
146	C₂×M₁₆	C₂³×C₃	1	4	yes	—
147	C₂×M₁₆	C₂³×C₃	1	2	yes	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	C₂³×C₃	1	2	yes	—
152	#117	—	2	1	no	—
153	#117	—	2	1	no	—
154	#117	C₃×Q₈	1	2	yes	—
155	#117	C₂³×C₃	1	2	yes	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₂²⋊C₄	C₃×Q₈	1	2	yes	—
158	C₂×C₄²	—	2	1	no	—
159	#108	—	2	16	no	—
160	#108	—	2	8	no	—
161	#108	C₂×C₃×C₄	1	16	yes	complemented
162	#97	—	2	4	no	—
163	C₄×C₈	—	2	4	no	—
164	C₂³×C₃	—	8	1	no	—
165	C₂³×C₃	C₄×Q₈	1	2	yes	—
166	C₂³×C₃	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂³×C₃	C₂²⋊Q₈	1	1	yes	characteristic
168	C₂³×C₃	C₄⋊Q₈	1	2	yes	—
169	C₂³×C₃	C₂²×Q₈	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	8	2	no	—
172	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
174	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
175	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
176	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
177	C₂×C₃×C₄	#121	1	2	yes	—
178	C₂×C₃×C₄	#121	1	2	yes	—
179	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
181	C₂×C₃×C₄	#125	1	2	yes	—
182	C₂×C₃×C₄	#125	1	2	yes	—
183	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
184	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
190	C₃×C₈	—	2	8	no	—
191	C₃×C₈	—	2	4	no	—
192	C₃×C₈	—	4	4	no	—
193	C₃×C₈	C₂×C₄⋊C₄	1	8	yes	—
194	C₂⁴	C₂×C₃×Q₈	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
197	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
200	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
201	C₂²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
202	C₂²×C₄	C₃×#41	1	1	yes	characteristic
203	C₂²×C₄	C₃×#41	1	1	yes	characteristic
204	C₂²×C₄	C₃×#41	1	2	yes	—
205	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	C₃×C₄⋊C₄	1	8	yes	—
216	M₁₆	C₃×C₄⋊C₄	1	8	yes	—
217	M₁₆	C₂²×C₃×C₄	1	8	yes	—
218	M₁₆	C₂×C₃×D₄	1	4	yes	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
228	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
229	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	C₂×C₃×Q₈	1	4	yes	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₂²×C₃	—	4	1	no	—
237	C₂²×C₃	—	4	2	no	—
238	C₂²×C₃	—	4	1	no	—
239	C₂²×C₃	—	4	2	no	—
240	C₂²×C₃	—	4	1	no	—
241	C₂²×C₃	#389	1	2	yes	—
242	C₂²×C₃	#389	1	2	yes	—
243	C₂²×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
244	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂²×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
246	C₃×C₄	—	4	4	no	—
247	C₃×C₄	#389	1	2	yes	—
248	C₃×C₄	#389	1	2	yes	—
249	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
250	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
251	C₃×C₄	C₂×#121	1	1	yes	characteristic
252	C₃×C₄	C₂×#125	1	1	yes	characteristic
253	C₂³	—	8	1	no	—
254	C₂³	C₃×C₄×Q₈	1	2	yes	—
255	C₂³	C₃×C₂²⋊Q₈	1	1	yes	characteristic
256	C₂³	C₃×C₂²⋊Q₈	1	1	yes	characteristic
257	C₂³	C₃×C₄⋊Q₈	1	2	yes	—
258	C₂³	C₂²×C₃×Q₈	1	1	yes	characteristic
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	8	2	no	—
261	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
262	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
263	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
264	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
265	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
266	C₂×C₄	C₃×#121	1	2	yes	—
267	C₂×C₄	C₃×#121	1	2	yes	—
268	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
269	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
270	C₂×C₄	C₃×#125	1	2	yes	—
271	C₂×C₄	C₃×#125	1	2	yes	—
272	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
273	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
277	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
278	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
279	C₈	—	2	8	no	—
280	C₈	—	2	4	no	—
281	C₈	—	4	4	no	—
282	C₈	C₂×C₃×C₄⋊C₄	1	8	yes	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	2	no	—
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	C₂×#389	1	1	yes	characteristic
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	C₃×#389	1	2	yes	—
295	C₂²	C₃×#389	1	2	yes	—
296	C₂²	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
297	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
298	C₂²	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
299	C₄	—	4	4	no	—
300	C₄	C₃×#389	1	2	yes	—
301	C₄	C₃×#389	1	2	yes	—
302	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
303	C₄	C₂×C₃×C₄×D₄	1	1	yes	characteristic
304	C₄	C₂×C₃×#121	1	1	yes	characteristic
305	C₄	C₂×C₃×#125	1	1	yes	characteristic
306	C₃	#21184	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	C₂×C₃×#389	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×#121
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	C₂×#389
prime radical	C₂⁴×C₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃×#1953	C₂	semidirect
7. #21184	C₃	direct (N × H)
16. C₃×#430	C₄	semidirect
29. #1953	C₂×C₃	semidirect
65. C₃×#108	C₂×C₄	semidirect
76. #430	C₃×C₄	semidirect
161. #108	C₂×C₃×C₄	semidirect
306. C₃	#21184	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×#430	2
C₃×#2873	2
C₃×#1953	2
C₂×C₃×#344	2
#21184	3

Subgroups — C3×#21184 #10639791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21184 #10639791