Subgroups — C₃×#2358 #84222

Order 384.

928 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#120	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#388	C₂	1	2	yes	maximal, complemented
6	C₃×#386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#386	C₂	1	2	yes	maximal, complemented
8	C₃×#385	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
10	#2358	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂⁵×C₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	2	yes	—
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	2	yes	—
15	C₂³×C₃×C₄	C₂²	1	2	yes	—
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
17	C₃×#120	C₄	1	8	yes	complemented
18	C₃×C₄×D₄	—	2	8	no	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
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	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
29	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
31	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
32	C₃×#95	C₂²	1	2	yes	complemented
33	C₃×#95	C₂²	1	2	yes	complemented
34	C₃×#95	C₂²	1	2	yes	complemented
35	C₂×#120	C₂×C₃	1	1	yes	characteristic
36	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
37	C₂²×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
38	#390	C₂×C₃	1	1	yes	characteristic, complemented
39	#388	C₂×C₃	1	2	yes	complemented
40	#386	C₂×C₃	1	1	yes	characteristic, complemented
41	#386	C₂×C₃	1	2	yes	complemented
42	#385	C₂×C₃	1	2	yes	complemented
43	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
44	C₂⁴×C₃	—	2	4	no	—
45	C₂⁴×C₃	—	2	4	no	—
46	C₂⁴×C₃	—	4	1	no	—
47	C₂⁴×C₃	—	4	1	no	—
48	C₂⁴×C₃	C₂×C₄	1	4	yes	complemented
49	C₂⁴×C₃	C₂³	1	1	yes	characteristic
50	C₂⁴×C₃	C₂³	1	1	yes	characteristic
51	C₂⁴×C₃	C₂³	1	2	yes	—
52	C₂×C₃×D₄	—	2	4	no	—
53	C₂×C₃×D₄	—	2	4	no	—
54	C₂×C₃×D₄	—	4	1	no	—
55	C₂×C₃×D₄	—	4	1	no	—
56	C₂×C₃×D₄	C₂×C₄	1	8	yes	complemented
57	C₂×C₃×D₄	C₂×C₄	1	4	yes	complemented
58	C₂²×C₃×C₄	—	2	4	no	—
59	C₂²×C₃×C₄	—	2	4	no	—
60	C₂²×C₃×C₄	—	2	4	no	—
61	C₂²×C₃×C₄	—	2	4	no	—
62	C₂²×C₃×C₄	—	2	4	no	—
63	C₂²×C₃×C₄	—	4	2	no	—
64	C₂²×C₃×C₄	—	4	2	no	—
65	C₂²×C₃×C₄	—	4	1	no	—
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	2	yes	—
71	C₂²×C₃×C₄	C₂³	1	2	yes	—
72	C₃×C₄⋊C₄	—	2	4	no	—
73	C₃×C₄⋊C₄	—	4	1	no	—
74	C₃×C₂²⋊C₄	—	2	4	no	—
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	—	4	1	no	—
79	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	complemented
80	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
81	C₃×C₂²⋊C₄	D₄	1	8	yes	complemented
82	C₃×C₄²	—	2	4	no	—
83	C₃×C₄²	—	4	1	no	—
84	C₂⁵	C₂²×C₃	1	1	yes	characteristic
85	C₂²×D₄	C₂²×C₃	1	2	yes	—
86	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
87	C₂³×C₄	C₂²×C₃	1	2	yes	—
88	C₂³×C₄	C₂²×C₃	1	2	yes	—
89	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic, complemented
90	#120	C₃×C₄	1	8	yes	complemented
91	C₄×D₄	—	2	8	no	—
92	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
93	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
94	C₂×C₂²⋊C₄	—	2	8	no	—
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	2	yes	—
99	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
100	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
102	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
103	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
104	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
105	#95	C₂²×C₃	1	2	yes	complemented
106	#95	C₂²×C₃	1	2	yes	complemented
107	#95	C₂²×C₃	1	2	yes	complemented
108	C₂³×C₃	—	2	2	no	—
109	C₂³×C₃	—	2	2	no	—
110	C₂³×C₃	—	2	2	no	—
111	C₂³×C₃	—	2	2	no	—
112	C₂³×C₃	—	2	16	no	—
113	C₂³×C₃	—	2	2	no	—
114	C₂³×C₃	—	2	2	no	—
115	C₂³×C₃	—	2	2	no	—
116	C₂³×C₃	—	2	2	no	—
117	C₂³×C₃	—	2	1	no	—
118	C₂³×C₃	—	2	1	no	—
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	—	2	1	no	—
121	C₂³×C₃	—	4	4	no	—
122	C₂³×C₃	—	4	4	no	—
123	C₂³×C₃	—	4	8	no	—
124	C₂³×C₃	—	4	4	no	—
125	C₂³×C₃	C₂²×C₄	1	2	yes	—
126	C₂³×C₃	C₂²×C₄	1	2	yes	—
127	C₂³×C₃	C₂²×C₄	1	4	yes	—
128	C₂³×C₃	C₂×D₄	1	4	yes	—
129	C₂³×C₃	#41	1	4	yes	—
130	C₂³×C₃	C₂⁴	1	1	yes	characteristic
131	C₃×D₄	—	2	16	no	—
132	C₃×D₄	—	4	4	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	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	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	4	4	no	—
148	C₂×C₃×C₄	—	4	8	no	—
149	C₂×C₃×C₄	—	4	8	no	—
150	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
151	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
152	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
153	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
154	C₂⁴	—	2	4	no	—
155	C₂⁴	—	2	4	no	—
156	C₂⁴	—	4	1	no	—
157	C₂⁴	—	4	1	no	—
158	C₂⁴	C₂×C₃×C₄	1	4	yes	complemented
159	C₂⁴	C₂³×C₃	1	1	yes	characteristic
160	C₂⁴	C₂³×C₃	1	1	yes	characteristic
161	C₂⁴	C₂³×C₃	1	2	yes	—
162	C₂×D₄	—	2	4	no	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	—	4	1	no	—
165	C₂×D₄	—	4	1	no	—
166	C₂×D₄	C₂×C₃×C₄	1	8	yes	complemented
167	C₂×D₄	C₂×C₃×C₄	1	4	yes	complemented
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	C₂³×C₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₃	1	2	yes	—
178	C₂²×C₄	C₂³×C₃	1	2	yes	—
179	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₃	1	2	yes	—
181	C₂²×C₄	C₂³×C₃	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
190	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
191	C₂²⋊C₄	C₃×D₄	1	8	yes	complemented
192	C₄²	—	2	4	no	—
193	C₄²	—	4	1	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	—	2	4	no	—
196	C₂²×C₃	—	2	4	no	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	2	4	no	—
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	2	4	no	—
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	4	16	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	—	4	1	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	—	4	2	no	—
209	C₂²×C₃	—	4	1	no	—
210	C₂²×C₃	—	4	1	no	—
211	C₂²×C₃	C₄×D₄	1	8	yes	complemented
212	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
213	C₂²×C₃	C₂²×D₄	1	2	yes	—
214	C₂²×C₃	C₂×#41	1	2	yes	—
215	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	—	2	4	no	—
221	C₃×C₄	—	4	2	no	—
222	C₃×C₄	—	4	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	16	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	4	8	no	—
239	C₂³	—	4	4	no	—
240	C₂³	C₂²×C₃×C₄	1	2	yes	—
241	C₂³	C₂²×C₃×C₄	1	2	yes	—
242	C₂³	C₂²×C₃×C₄	1	4	yes	—
243	C₂³	C₂×C₃×D₄	1	4	yes	—
244	C₂³	C₃×#41	1	4	yes	—
245	C₂³	C₂⁴×C₃	1	1	yes	characteristic
246	D₄	—	2	16	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
266	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
267	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
268	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
269	C₂×C₃	—	2	8	no	—
270	C₂×C₃	—	4	2	no	—
271	C₂×C₃	—	4	2	no	—
272	C₂×C₃	C₂×C₄×D₄	1	2	yes	—
273	C₂×C₃	#518	1	1	yes	characteristic
274	C₂×C₃	D₄²	1	1	yes	characteristic
275	C₂×C₃	#546	1	2	yes	—
276	C₂×C₃	#551	1	1	yes	characteristic
277	C₂²	—	2	4	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	16	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	C₃×C₄×D₄	1	8	yes	complemented
295	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
296	C₂²	C₂²×C₃×D₄	1	2	yes	—
297	C₂²	C₂×C₃×#41	1	2	yes	—
298	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
299	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
300	C₄	—	2	4	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	4	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₃	#2358	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	2	8	no	—
308	C₂	—	4	2	no	—
309	C₂	—	4	2	no	—
310	C₂	C₂×C₃×C₄×D₄	1	2	yes	—
311	C₂	C₃×#518	1	1	yes	characteristic
312	C₂	C₃×D₄²	1	1	yes	characteristic
313	C₂	C₃×#546	1	2	yes	—
314	C₂	C₃×#551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂×C₃×#120	C₂
LCS	G ▷ C₂² ▷ 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. C₂²×C₃×C₂²⋊C₄	C₂	semidirect
4. C₃×#390	C₂	semidirect
5. C₃×#388	C₂	semidirect
6. C₃×#386	C₂	semidirect
7. C₃×#386	C₂	semidirect
8. C₃×#385	C₂	semidirect
9. C₃×C₄×C₂²⋊C₄	C₂	semidirect
10. #2358	C₃	direct (N × H)
16. C₂³×C₃×C₄	C₂²	semidirect
17. C₃×#120	C₄	semidirect
19. C₂×C₃×C₄⋊C₄	C₂²	semidirect
20. C₂×C₃×C₄⋊C₄	C₂²	semidirect
28. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
29. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
30. C₂×C₃×C₄²	C₂²	semidirect
31. C₂×C₃×C₄²	C₂²	semidirect
32. C₃×#95	C₂²	semidirect
33. C₃×#95	C₂²	semidirect
34. C₃×#95	C₂²	semidirect
36. C₂×C₄×D₄	C₂×C₃	semidirect
37. C₂²×C₂²⋊C₄	C₂×C₃	semidirect
38. #390	C₂×C₃	semidirect
39. #388	C₂×C₃	semidirect
40. #386	C₂×C₃	semidirect
41. #386	C₂×C₃	semidirect
42. #385	C₂×C₃	semidirect
43. C₄×C₂²⋊C₄	C₂×C₃	semidirect
48. C₂⁴×C₃	C₂×C₄	semidirect
56. C₂×C₃×D₄	C₂×C₄	semidirect
57. C₂×C₃×D₄	C₂×C₄	semidirect
79. C₃×C₂²⋊C₄	C₂×C₄	semidirect
80. C₃×C₂²⋊C₄	C₂×C₄	semidirect
81. C₃×C₂²⋊C₄	D₄	semidirect
89. C₂³×C₄	C₂²×C₃	semidirect
90. #120	C₃×C₄	semidirect
92. C₂×C₄⋊C₄	C₂²×C₃	semidirect
93. C₂×C₄⋊C₄	C₂²×C₃	semidirect
101. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
102. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
103. C₂×C₄²	C₂²×C₃	semidirect
104. C₂×C₄²	C₂²×C₃	semidirect
105. #95	C₂²×C₃	semidirect
106. #95	C₂²×C₃	semidirect
107. #95	C₂²×C₃	semidirect
158. C₂⁴	C₂×C₃×C₄	semidirect
166. C₂×D₄	C₂×C₃×C₄	semidirect
167. C₂×D₄	C₂×C₃×C₄	semidirect
189. C₂²⋊C₄	C₂×C₃×C₄	semidirect
190. C₂²⋊C₄	C₂×C₃×C₄	semidirect
191. C₂²⋊C₄	C₃×D₄	semidirect
211. C₂²×C₃	C₄×D₄	semidirect
294. C₂²	C₃×C₄×D₄	semidirect
306. C₃	#2358	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×#120	2
C₂×C₃×C₄×D₄	2
C₂²×C₃×C₂²⋊C₄	2
C₃×#390	2
C₃×#388	2
C₃×#386	2
C₃×#385	2
C₃×C₄×C₂²⋊C₄	2
#2358	3

Subgroups — C3×#2358 #84222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2358 #84222