Subgroups — C₃×#2290 #84154

Order 384.

696 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#388	C₂	1	2	yes	maximal, complemented
5	C₃×#387	C₂	1	2	yes	maximal, complemented
6	C₃×#386	C₂	1	2	yes	maximal
7	C₃×#385	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
12	#2290	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	2	yes	—
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₃×C₄×D₄	C₄	1	8	yes	complemented
18	C₃×#117	—	2	4	no	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₄²	—	2	4	no	—
28	C₂×C₃×C₄²	C₂²	1	2	yes	—
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄²	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	C₃×#95	C₂²	1	2	yes	—
39	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
40	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic, complemented
42	#388	C₂×C₃	1	2	yes	complemented
43	#387	C₂×C₃	1	2	yes	complemented
44	#386	C₂×C₃	1	2	yes	—
45	#385	C₂×C₃	1	2	yes	complemented
46	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
48	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
49	#376	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂⁴×C₃	C₂³	1	1	yes	characteristic
51	C₂⁴×C₃	C₂³	1	1	yes	characteristic
52	C₂×C₃×D₄	—	2	4	no	—
53	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
54	C₂²×C₃×C₄	—	2	4	no	—
55	C₂²×C₃×C₄	—	2	4	no	—
56	C₂²×C₃×C₄	—	2	2	no	—
57	C₂²×C₃×C₄	—	2	2	no	—
58	C₂²×C₃×C₄	—	2	4	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
62	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	2	yes	—
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	2	yes	—
70	C₂²×C₃×C₄	C₂³	1	2	yes	—
71	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
72	C₃×C₄⋊C₄	—	2	4	no	—
73	C₃×C₄⋊C₄	—	2	4	no	—
74	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
75	C₃×C₂²⋊C₄	—	2	4	no	—
76	C₃×C₂²⋊C₄	—	2	4	no	—
77	C₃×C₂²⋊C₄	—	2	4	no	—
78	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
79	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	complemented
80	C₃×C₄²	—	2	4	no	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	2	2	no	—
84	C₃×C₄²	C₂×C₄	1	4	yes	complemented
85	C₃×C₄²	D₄	1	4	yes	complemented
86	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
87	C₂³×C₄	C₂²×C₃	1	2	yes	—
88	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
89	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
90	C₄×D₄	C₃×C₄	1	8	yes	complemented
91	#117	—	2	4	no	—
92	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
93	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
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	2	yes	—
98	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
100	C₂×C₄²	—	2	4	no	—
101	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₃	1	2	yes	—
105	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
107	#95	C₂²×C₃	1	2	yes	—
108	#95	C₂²×C₃	1	2	yes	—
109	#95	C₂²×C₃	1	2	yes	—
110	#95	C₂²×C₃	1	2	yes	—
111	#95	C₂²×C₃	1	2	yes	—
112	C₂³×C₃	—	2	1	no	—
113	C₂³×C₃	—	2	1	no	—
114	C₂³×C₃	—	2	2	no	—
115	C₂³×C₃	—	2	1	no	—
116	C₂³×C₃	—	2	2	no	—
117	C₂³×C₃	—	2	1	no	—
118	C₂³×C₃	—	2	2	no	—
119	C₂³×C₃	C₂²×C₄	1	2	yes	—
120	C₂³×C₃	C₂²×C₄	1	2	yes	—
121	C₂³×C₃	#41	1	4	yes	—
122	C₂³×C₃	C₂⁴	1	1	yes	characteristic
123	C₃×D₄	—	2	8	no	—
124	C₂×C₃×C₄	—	2	16	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	2	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	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	8	no	—
136	C₂×C₃×C₄	—	2	8	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
144	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
145	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
146	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
147	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
148	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
149	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
150	C₂×C₃×C₄	#41	1	4	yes	—
151	C₂×C₃×C₄	#41	1	2	yes	—
152	C₂⁴	C₂³×C₃	1	1	yes	characteristic
153	C₂⁴	C₂³×C₃	1	1	yes	characteristic
154	C₂×D₄	—	2	4	no	—
155	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
164	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
165	C₂²×C₄	C₂³×C₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃	1	2	yes	—
172	C₂²×C₄	C₂³×C₃	1	2	yes	—
173	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
177	C₂²⋊C₄	—	2	4	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
181	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
182	C₄²	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	C₂×C₃×C₄	1	4	yes	complemented
187	C₄²	C₃×D₄	1	4	yes	complemented
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	4	2	no	—
196	C₂²×C₃	—	4	2	no	—
197	C₂²×C₃	#117	1	4	yes	complemented
198	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
199	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
200	C₂²×C₃	C₂×#41	1	2	yes	—
201	C₂²×C₃	C₂×#41	1	1	yes	characteristic
202	C₂²×C₃	C₂×#41	1	2	yes	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	4	2	no	—
210	C₃×C₄	—	4	2	no	—
211	C₃×C₄	C₄×D₄	1	4	yes	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	2	no	—
219	C₂³	C₂²×C₃×C₄	1	2	yes	—
220	C₂³	C₂²×C₃×C₄	1	2	yes	—
221	C₂³	C₃×#41	1	4	yes	—
222	C₂³	C₂⁴×C₃	1	1	yes	characteristic
223	D₄	—	2	8	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
244	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
245	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
246	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
247	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
248	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
249	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
250	C₂×C₄	C₃×#41	1	4	yes	—
251	C₂×C₄	C₃×#41	1	2	yes	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	—	4	2	no	—
254	C₂×C₃	C₂×#117	1	1	yes	characteristic
255	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂×C₃	C₄×#41	1	1	yes	characteristic
257	C₂×C₃	#525	1	2	yes	—
258	C₂×C₃	#529	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	C₃×#117	1	4	yes	complemented
269	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
270	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
271	C₂²	C₂×C₃×#41	1	2	yes	—
272	C₂²	C₂×C₃×#41	1	1	yes	characteristic
273	C₂²	C₂×C₃×#41	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	4	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	2	no	—
282	C₄	C₃×C₄×D₄	1	4	yes	—
283	C₃	#2290	1	1	yes	characteristic, direct product factor, Sylow
284	C₂	—	2	4	no	—
285	C₂	—	4	2	no	—
286	C₂	C₂×C₃×#117	1	1	yes	characteristic
287	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
288	C₂	C₃×C₄×#41	1	1	yes	characteristic
289	C₂	C₃×#525	1	2	yes	—
290	C₂	C₃×#529	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#117	C₂	semidirect
3. C₂²×C₃×C₄²	C₂	semidirect
4. C₃×#388	C₂	semidirect
5. C₃×#387	C₂	semidirect
7. C₃×#385	C₂	semidirect
8. C₃×C₄×C₄⋊C₄	C₂	semidirect
9. C₃×C₄×C₂²⋊C₄	C₂	semidirect
10. C₃×C₄×C₂²⋊C₄	C₂	semidirect
11. C₃×#376	C₂	semidirect
12. #2290	C₃	direct (N × H)
17. C₃×C₄×D₄	C₄	semidirect
40. C₂×#117	C₂×C₃	semidirect
41. C₂²×C₄²	C₂×C₃	semidirect
42. #388	C₂×C₃	semidirect
43. #387	C₂×C₃	semidirect
45. #385	C₂×C₃	semidirect
46. C₄×C₄⋊C₄	C₂×C₃	semidirect
47. C₄×C₂²⋊C₄	C₂×C₃	semidirect
48. C₄×C₂²⋊C₄	C₂×C₃	semidirect
49. #376	C₂×C₃	semidirect
62. C₂²×C₃×C₄	C₂×C₄	semidirect
74. C₃×C₄⋊C₄	C₂×C₄	semidirect
79. C₃×C₂²⋊C₄	C₂×C₄	semidirect
84. C₃×C₄²	C₂×C₄	semidirect
85. C₃×C₄²	D₄	semidirect
90. C₄×D₄	C₃×C₄	semidirect
164. C₂²×C₄	C₂×C₃×C₄	semidirect
176. C₄⋊C₄	C₂×C₃×C₄	semidirect
181. C₂²⋊C₄	C₂×C₃×C₄	semidirect
186. C₄²	C₂×C₃×C₄	semidirect
187. C₄²	C₃×D₄	semidirect
197. C₂²×C₃	#117	semidirect
268. C₂²	C₃×#117	semidirect
283. C₃	#2290	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×C₄×D₄	2
C₂×C₃×#117	2
C₂²×C₃×C₄²	2
C₃×#388	2
C₃×#387	2
C₃×#386	2
C₃×#385	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
C₃×#376	2
#2290	3

Subgroups — C3×#2290 #84154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2290 #84154