Subgroups — C₃×#2420 #84284

Order 384.

596 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×C₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#397	C₂	1	2	yes	maximal, complemented
4	C₃×#394	C₂	1	2	yes	maximal, complemented
5	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#386	C₂	1	2	yes	maximal, complemented
8	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#381	C₂	1	2	yes	maximal, complemented
10	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#2420	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	2	yes	—
17	C₃×#121	—	2	4	no	—
18	C₃×C₄×D₄	—	2	4	no	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
32	C₃×#95	C₂²	1	1	yes	characteristic
33	C₃×#95	C₂²	1	1	yes	characteristic
34	C₃×#95	C₂²	1	2	yes	—
35	C₃×#95	C₂²	1	2	yes	—
36	C₃×#95	C₂²	1	2	yes	—
37	C₃×#95	C₂²	1	2	yes	—
38	C₂×#121	C₂×C₃	1	1	yes	characteristic
39	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
40	#397	C₂×C₃	1	2	yes	complemented
41	#394	C₂×C₃	1	2	yes	complemented
42	#390	C₂×C₃	1	1	yes	characteristic, complemented
43	#389	C₂×C₃	1	1	yes	characteristic, complemented
44	#386	C₂×C₃	1	2	yes	complemented
45	#382	C₂×C₃	1	1	yes	characteristic, complemented
46	#381	C₂×C₃	1	2	yes	complemented
47	#380	C₂×C₃	1	1	yes	characteristic, complemented
48	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂⁴×C₃	C₂³	1	2	yes	—
50	C₂⁴×C₃	C₂³	1	1	yes	characteristic
51	C₂×C₃×D₄	—	2	4	no	—
52	C₂×C₃×D₄	—	2	4	no	—
53	C₂×C₃×D₄	—	2	2	no	—
54	C₂×C₃×D₄	—	2	2	no	—
55	C₂×C₃×D₄	—	4	2	no	—
56	C₂×C₃×D₄	D₄	1	4	yes	—
57	C₂²×C₃×C₄	—	2	4	no	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	—	2	4	no	—
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₂²×C₃×C₄	—	2	2	no	—
62	C₂²×C₃×C₄	—	4	2	no	—
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₂²×C₃×C₄	C₂³	1	2	yes	—
68	C₂²×C₃×C₄	C₂³	1	2	yes	—
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₄⋊C₄	—	4	2	no	—
76	C₃×C₂²⋊C₄	—	2	4	no	—
77	C₃×C₂²⋊C₄	—	2	4	no	—
78	C₃×C₂²⋊C₄	—	2	2	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	4	2	no	—
81	C₃×C₂²⋊C₄	—	4	2	no	—
82	C₃×C₂²⋊C₄	—	4	2	no	—
83	C₃×C₄²	—	2	2	no	—
84	C₃×C₄²	—	2	2	no	—
85	C₂²×D₄	C₂²×C₃	1	2	yes	—
86	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
87	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
88	C₂³×C₄	C₂²×C₃	1	2	yes	—
89	#121	—	2	4	no	—
90	C₄×D₄	—	2	4	no	—
91	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
92	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
93	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
95	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
96	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
102	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
104	#95	C₂²×C₃	1	1	yes	characteristic
105	#95	C₂²×C₃	1	1	yes	characteristic
106	#95	C₂²×C₃	1	2	yes	—
107	#95	C₂²×C₃	1	2	yes	—
108	#95	C₂²×C₃	1	2	yes	—
109	#95	C₂²×C₃	1	2	yes	—
110	C₂³×C₃	—	2	1	no	—
111	C₂³×C₃	—	2	1	no	—
112	C₂³×C₃	—	2	1	no	—
113	C₂³×C₃	—	2	2	no	—
114	C₂³×C₃	—	2	1	no	—
115	C₂³×C₃	—	2	2	no	—
116	C₂³×C₃	—	2	2	no	—
117	C₂³×C₃	—	2	2	no	—
118	C₂³×C₃	—	2	1	no	—
119	C₂³×C₃	—	2	2	no	—
120	C₂³×C₃	—	2	2	no	—
121	C₂³×C₃	—	2	1	no	—
122	C₂³×C₃	C₂×D₄	1	4	yes	—
123	C₂³×C₃	#41	1	2	yes	—
124	C₂³×C₃	C₂⁴	1	1	yes	characteristic
125	C₃×D₄	—	2	8	no	—
126	C₃×D₄	—	4	8	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	4	8	no	—
151	C₂×C₃×C₄	—	4	4	no	—
152	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
153	C₂×C₃×C₄	#41	1	2	yes	—
154	C₂×C₃×C₄	#41	1	2	yes	—
155	C₂×C₃×C₄	#41	1	2	yes	—
156	C₂⁴	C₂³×C₃	1	2	yes	—
157	C₂⁴	C₂³×C₃	1	1	yes	characteristic
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	C₃×D₄	1	4	yes	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₃	1	2	yes	—
174	C₂²×C₄	C₂³×C₃	1	2	yes	—
175	C₂²×C₄	C₂³×C₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₂²×C₃	—	2	4	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	2	4	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	—	4	1	no	—
201	C₂²×C₃	—	4	2	no	—
202	C₂²×C₃	—	4	1	no	—
203	C₂²×C₃	—	4	2	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
207	C₂²×C₃	C₂×#41	1	1	yes	characteristic
208	C₂²×C₃	C₂×#41	1	1	yes	characteristic
209	C₂²×C₃	C₂×#41	1	1	yes	characteristic
210	C₂²×C₃	C₂×#41	1	1	yes	characteristic
211	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	—	4	2	no	—
216	C₃×C₄	—	4	1	no	—
217	C₃×C₄	—	4	2	no	—
218	C₃×C₄	—	4	2	no	—
219	C₃×C₄	—	4	1	no	—
220	C₃×C₄	—	4	1	no	—
221	C₃×C₄	#123	1	4	yes	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	C₂×C₃×D₄	1	4	yes	—
235	C₂³	C₃×#41	1	2	yes	—
236	C₂³	C₂⁴×C₃	1	1	yes	characteristic
237	D₄	—	2	8	no	—
238	D₄	—	4	8	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
265	C₂×C₄	C₃×#41	1	2	yes	—
266	C₂×C₄	C₃×#41	1	2	yes	—
267	C₂×C₄	C₃×#41	1	2	yes	—
268	C₂×C₃	—	4	4	no	—
269	C₂×C₃	—	4	2	no	—
270	C₂×C₃	C₂×#123	1	1	yes	characteristic
271	C₂×C₃	#529	1	1	yes	characteristic
272	C₂×C₃	#540	1	1	yes	characteristic
273	C₂×C₃	#546	1	1	yes	characteristic
274	C₂×C₃	#546	1	1	yes	characteristic
275	C₂×C₃	#553	1	1	yes	characteristic
276	C₂×C₃	#555	1	1	yes	characteristic
277	C₂²	—	2	4	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
292	C₂²	C₂×C₃×#41	1	1	yes	characteristic
293	C₂²	C₂×C₃×#41	1	1	yes	characteristic
294	C₂²	C₂×C₃×#41	1	1	yes	characteristic
295	C₂²	C₂×C₃×#41	1	1	yes	characteristic
296	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
297	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	C₃×#123	1	4	yes	—
307	C₃	#2420	1	1	yes	characteristic, direct product factor, Sylow
308	C₂	—	4	4	no	—
309	C₂	—	4	2	no	—
310	C₂	C₂×C₃×#123	1	1	yes	characteristic
311	C₂	C₃×#529	1	1	yes	characteristic
312	C₂	C₃×#540	1	1	yes	characteristic
313	C₂	C₃×#546	1	1	yes	characteristic
314	C₂	C₃×#546	1	1	yes	characteristic
315	C₂	C₃×#553	1	1	yes	characteristic
316	C₂	C₃×#555	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×D₄	C₂	semidirect
3. C₃×#397	C₂	semidirect
4. C₃×#394	C₂	semidirect
5. C₃×#390	C₂	semidirect
6. C₃×#389	C₂	semidirect
7. C₃×#386	C₂	semidirect
8. C₃×#382	C₂	semidirect
9. C₃×#381	C₂	semidirect
10. C₃×#380	C₂	semidirect
11. C₃×C₄×C₂²⋊C₄	C₂	semidirect
12. #2420	C₃	direct (N × H)
39. C₂×C₄×D₄	C₂×C₃	semidirect
40. #397	C₂×C₃	semidirect
41. #394	C₂×C₃	semidirect
42. #390	C₂×C₃	semidirect
43. #389	C₂×C₃	semidirect
44. #386	C₂×C₃	semidirect
45. #382	C₂×C₃	semidirect
46. #381	C₂×C₃	semidirect
47. #380	C₂×C₃	semidirect
48. C₄×C₂²⋊C₄	C₂×C₃	semidirect
307. C₃	#2420	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×#121	2
C₂×C₃×C₄×D₄	2
C₃×#397	2
C₃×#394	2
C₃×#390	2
C₃×#389	2
C₃×#386	2
C₃×#382	2
C₃×#381	2
C₃×#380	2
C₃×C₄×C₂²⋊C₄	2
#2420	3

Subgroups — C3×#2420 #84284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2420 #84284