Subgroups — C₃×#3056 #84920

Order 384.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#525	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#470	C₂	1	2	yes	maximal, complemented
6	C₃×#468	C₂	1	2	yes	maximal, complemented
7	C₃×#467	C₂	1	2	yes	maximal, complemented
8	C₃×#465	C₂	1	2	yes	maximal, complemented
9	C₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#419	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
12	#3056	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₃×#137	—	2	4	no	—
17	C₂×C₃×Q₁₆	C₂²	1	2	yes	complemented
18	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
19	C₂×C₃×M₁₆	—	2	4	no	—
20	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×M₁₆	C₂²	1	2	yes	—
22	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂²	1	2	yes	—
24	C₃×#123	—	2	1	no	—
25	C₃×C₂²⋊Q₈	—	2	1	no	—
26	C₃×C₂²⋊Q₈	—	2	1	no	—
27	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
30	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
31	C₃×#121	C₂²	1	1	yes	characteristic
32	C₃×#121	C₂²	1	1	yes	characteristic
33	C₃×#120	—	2	1	no	—
34	C₃×C₄×D₄	—	2	1	no	—
35	C₃×C₄×D₄	—	2	1	no	—
36	C₃×#117	C₂²	1	1	yes	characteristic
37	C₂×C₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
40	C₃×#107	C₂²	1	2	yes	complemented
41	C₃×#106	C₂²	1	2	yes	complemented
42	C₃×#103	C₂²	1	2	yes	complemented
43	C₃×#103	C₂²	1	2	yes	complemented
44	C₃×#103	C₂²	1	2	yes	complemented
45	C₃×#102	C₂²	1	2	yes	complemented
46	C₂×#137	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂²×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
48	#525	C₂×C₃	1	1	yes	characteristic
49	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
50	#470	C₂×C₃	1	2	yes	complemented
51	#468	C₂×C₃	1	2	yes	complemented
52	#467	C₂×C₃	1	2	yes	complemented
53	#465	C₂×C₃	1	2	yes	complemented
54	#428	C₂×C₃	1	1	yes	characteristic, complemented
55	#419	C₂×C₃	1	1	yes	characteristic, complemented
56	#417	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂⁴×C₃	D₄	1	1	yes	characteristic
58	C₃×#41	—	2	2	no	—
59	C₃×#41	—	2	2	no	—
60	C₂×C₃×Q₈	—	2	1	no	—
61	C₂×C₃×Q₈	—	2	2	no	—
62	C₂×C₃×Q₈	—	2	2	no	—
63	C₂×C₃×Q₈	C₂³	1	2	yes	—
64	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×D₄	—	2	1	no	—
66	C₂×C₃×D₄	—	2	1	no	—
67	C₂×C₃×D₄	—	2	1	no	—
68	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	—	2	2	no	—
70	C₂²×C₃×C₄	—	2	2	no	—
71	C₂²×C₃×C₄	—	2	1	no	—
72	C₂²×C₃×C₄	—	2	1	no	—
73	C₂²×C₃×C₄	—	2	1	no	—
74	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
80	C₃×Q₁₆	—	2	4	no	—
81	C₃×SD₁₆	—	2	4	no	—
82	C₃×M₁₆	—	2	2	no	—
83	C₃×M₁₆	—	2	4	no	—
84	C₃×M₁₆	D₄	1	4	yes	complemented
85	C₂×C₃×C₈	—	2	4	no	—
86	C₂×C₃×C₈	C₂³	1	2	yes	—
87	C₂×C₃×C₈	C₂³	1	2	yes	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	—	2	1	no	—
91	C₃×C₄⋊C₄	—	2	1	no	—
92	C₃×C₄⋊C₄	—	2	1	no	—
93	C₃×C₄⋊C₄	C₂³	1	2	yes	—
94	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
95	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
96	C₃×C₂²⋊C₄	—	2	2	no	—
97	C₃×C₂²⋊C₄	—	2	2	no	—
98	C₃×C₂²⋊C₄	—	2	1	no	—
99	C₃×C₂²⋊C₄	—	2	1	no	—
100	C₃×C₂²⋊C₄	—	2	1	no	—
101	C₃×C₂²⋊C₄	—	2	1	no	—
102	C₃×C₂²⋊C₄	—	2	1	no	—
103	C₃×C₄²	—	2	1	no	—
104	C₂×#41	C₂²×C₃	1	1	yes	characteristic
105	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
106	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
107	#137	—	2	4	no	—
108	C₂×Q₁₆	C₂²×C₃	1	2	yes	complemented
109	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
110	C₂×M₁₆	—	2	4	no	—
111	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
112	C₂×M₁₆	C₂²×C₃	1	2	yes	—
113	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²×C₃	1	2	yes	—
115	#123	—	2	1	no	—
116	C₂²⋊Q₈	—	2	1	no	—
117	C₂²⋊Q₈	—	2	1	no	—
118	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
119	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
120	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
121	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
122	#121	C₂²×C₃	1	1	yes	characteristic
123	#121	C₂²×C₃	1	1	yes	characteristic
124	#120	—	2	1	no	—
125	C₄×D₄	—	2	1	no	—
126	C₄×D₄	—	2	1	no	—
127	#117	C₂²×C₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	—	2	1	no	—
131	#107	C₂²×C₃	1	2	yes	complemented
132	#106	C₂²×C₃	1	2	yes	complemented
133	#103	C₂²×C₃	1	2	yes	complemented
134	#103	C₂²×C₃	1	2	yes	complemented
135	#103	C₂²×C₃	1	2	yes	complemented
136	#102	C₂²×C₃	1	2	yes	complemented
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	—	2	2	no	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	—	4	1	no	—
141	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
142	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
143	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
144	C₃×Q₈	—	2	2	no	—
145	C₃×Q₈	—	2	1	no	—
146	C₃×Q₈	—	2	2	no	—
147	C₃×Q₈	—	2	1	no	—
148	C₃×Q₈	—	4	2	no	—
149	C₃×D₄	—	2	1	no	—
150	C₃×D₄	—	2	1	no	—
151	C₃×D₄	—	4	2	no	—
152	C₃×D₄	—	4	2	no	—
153	C₃×D₄	—	4	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	4	2	no	—
167	C₂×C₃×C₄	—	4	2	no	—
168	C₂×C₃×C₄	—	4	2	no	—
169	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
173	C₂×C₃×C₄	#41	1	2	yes	—
174	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	C₂×D₄	1	4	yes	—
177	C₂⁴	C₃×D₄	1	1	yes	characteristic
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	C₂³×C₃	1	2	yes	—
184	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
196	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	SD₁₆	—	2	4	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	C₃×D₄	1	4	yes	complemented
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	C₂³×C₃	1	2	yes	—
207	C₂×C₈	C₂³×C₃	1	2	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	C₂³×C₃	1	2	yes	—
214	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	4	4	no	—
232	C₂²×C₃	—	4	1	no	—
233	C₂²×C₃	—	4	1	no	—
234	C₂²×C₃	—	4	1	no	—
235	C₂²×C₃	#121	1	2	yes	—
236	C₂²×C₃	#137	1	2	yes	complemented
237	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	—	4	2	no	—
242	C₃×C₄	—	4	1	no	—
243	C₃×C₄	—	4	2	no	—
244	C₃×C₄	—	4	1	no	—
245	C₃×C₄	#121	1	2	yes	—
246	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
247	C₃×C₄	C₂×#41	1	1	yes	characteristic
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
253	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
254	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	2	1	no	—
257	Q₈	—	2	2	no	—
258	Q₈	—	2	1	no	—
259	Q₈	—	4	2	no	—
260	D₄	—	2	1	no	—
261	D₄	—	2	1	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	4	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
282	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
283	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
284	C₂×C₄	C₃×#41	1	2	yes	—
285	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	C₂×C₃×D₄	1	4	yes	—
288	C₂×C₃	—	2	2	no	—
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	4	1	no	—
291	C₂×C₃	—	8	1	no	—
292	C₂×C₃	C₂×#121	1	1	yes	characteristic
293	C₂×C₃	C₂×#137	1	1	yes	characteristic
294	C₂×C₃	#575	1	1	yes	characteristic
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	C₃×#121	1	2	yes	—
307	C₂²	C₃×#137	1	2	yes	complemented
308	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	2	no	—
316	C₄	C₃×#121	1	2	yes	—
317	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
318	C₄	C₂×C₃×#41	1	1	yes	characteristic
319	C₃	#3056	1	1	yes	characteristic, direct product factor, Sylow
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	8	1	no	—
324	C₂	C₂×C₃×#121	1	1	yes	characteristic
325	C₂	C₂×C₃×#137	1	1	yes	characteristic
326	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#137	C₂	semidirect
2. C₂²×C₃×M₁₆	C₂	semidirect
4. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
5. C₃×#470	C₂	semidirect
6. C₃×#468	C₂	semidirect
7. C₃×#467	C₂	semidirect
8. C₃×#465	C₂	semidirect
9. C₃×#428	C₂	semidirect
10. C₃×#419	C₂	semidirect
11. C₃×#417	C₂	semidirect
12. #3056	C₃	direct (N × H)
17. C₂×C₃×Q₁₆	C₂²	semidirect
18. C₂×C₃×SD₁₆	C₂²	semidirect
40. C₃×#107	C₂²	semidirect
41. C₃×#106	C₂²	semidirect
42. C₃×#103	C₂²	semidirect
43. C₃×#103	C₂²	semidirect
44. C₃×#103	C₂²	semidirect
45. C₃×#102	C₂²	semidirect
46. C₂×#137	C₂×C₃	semidirect
47. C₂²×M₁₆	C₂×C₃	semidirect
49. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
50. #470	C₂×C₃	semidirect
51. #468	C₂×C₃	semidirect
52. #467	C₂×C₃	semidirect
53. #465	C₂×C₃	semidirect
54. #428	C₂×C₃	semidirect
55. #419	C₂×C₃	semidirect
56. #417	C₂×C₃	semidirect
84. C₃×M₁₆	D₄	semidirect
108. C₂×Q₁₆	C₂²×C₃	semidirect
109. C₂×SD₁₆	C₂²×C₃	semidirect
131. #107	C₂²×C₃	semidirect
132. #106	C₂²×C₃	semidirect
133. #103	C₂²×C₃	semidirect
134. #103	C₂²×C₃	semidirect
135. #103	C₂²×C₃	semidirect
136. #102	C₂²×C₃	semidirect
204. M₁₆	C₃×D₄	semidirect
236. C₂²×C₃	#137	semidirect
307. C₂²	C₃×#137	semidirect
319. C₃	#3056	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×#137	2
C₂²×C₃×M₁₆	2
C₃×#525	2
C₂×C₃×C₂²⋊Q₈	2
C₃×#470	2
C₃×#468	2
C₃×#467	2
C₃×#465	2
C₃×#428	2
C₃×#419	2
C₃×#417	2
#3056	3

Subgroups — C3×#3056 #84920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3056 #84920