Subgroups — C₃×#2367 #84231

Order 384.

632 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₃×#388	C₂	1	2	yes	maximal, complemented
7	C₃×#388	C₂	1	2	yes	maximal, complemented
8	C₃×#385	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	#2367	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	2	yes	—
16	C₃×#124	C₄	1	8	yes	complemented
17	C₃×C₄×Q₈	—	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×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₄²	C₂²	1	2	yes	—
30	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄²	C₂²	1	2	yes	—
33	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
34	C₃×#95	C₂²	1	1	yes	characteristic
35	C₃×#95	C₂²	1	2	yes	—
36	C₃×#95	C₂²	1	2	yes	—
37	C₃×#95	C₂²	1	1	yes	characteristic
38	C₃×#95	C₂²	1	2	yes	—
39	C₂×#124	C₂×C₃	1	1	yes	characteristic
40	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
42	#391	C₂×C₃	1	1	yes	characteristic, complemented
43	#390	C₂×C₃	1	1	yes	characteristic
44	#388	C₂×C₃	1	2	yes	complemented
45	#388	C₂×C₃	1	2	yes	complemented
46	#385	C₂×C₃	1	2	yes	complemented
47	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
48	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
49	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
50	C₂⁴×C₃	C₂³	1	2	yes	—
51	C₂×C₃×Q₈	—	2	2	no	—
52	C₂×C₃×Q₈	—	2	2	no	—
53	C₂×C₃×Q₈	C₂×C₄	1	4	yes	complemented
54	C₂×C₃×D₄	—	2	2	no	—
55	C₂×C₃×D₄	—	2	2	no	—
56	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
57	C₂²×C₃×C₄	—	2	4	no	—
58	C₂²×C₃×C₄	—	2	4	no	—
59	C₂²×C₃×C₄	—	4	2	no	—
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	2	yes	—
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³	1	2	yes	—
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₃×C₄⋊C₄	—	2	4	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	D₄	1	4	yes	—
75	C₃×C₂²⋊C₄	—	2	4	no	—
76	C₃×C₂²⋊C₄	—	2	4	no	—
77	C₃×C₂²⋊C₄	—	4	2	no	—
78	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
79	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
80	C₃×C₄²	—	2	2	no	—
81	C₃×C₄²	—	2	4	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	4	2	no	—
84	C₃×C₄²	C₂×C₄	1	4	yes	complemented
85	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
86	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
87	C₂³×C₄	C₂²×C₃	1	2	yes	—
88	#124	C₃×C₄	1	8	yes	complemented
89	C₄×Q₈	—	2	4	no	—
90	C₄×D₄	—	2	4	no	—
91	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
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	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
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	2	yes	—
100	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₃	1	2	yes	—
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	#95	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	1	yes	characteristic
110	#95	C₂²×C₃	1	2	yes	—
111	C₂³×C₃	—	2	2	no	—
112	C₂³×C₃	—	2	2	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	C₂²×C₄	1	4	yes	—
118	C₂³×C₃	C₂⁴	1	1	yes	characteristic
119	C₃×Q₈	—	2	8	no	—
120	C₃×D₄	—	2	8	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	2	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	2	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	4	8	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	4	yes	—
146	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
147	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
148	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
149	C₂×C₃×C₄	#41	1	2	yes	—
150	C₂×C₃×C₄	#41	1	2	yes	—
151	C₂×C₃×C₄	#41	1	2	yes	—
152	C₂⁴	C₂³×C₃	1	2	yes	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	C₂×C₃×C₄	1	4	yes	complemented
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₃	1	2	yes	—
164	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	C₃×D₄	1	4	yes	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
181	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
182	C₄²	—	2	2	no	—
183	C₄²	—	2	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	C₂×C₃×C₄	1	4	yes	complemented
187	C₂²×C₃	—	2	4	no	—
188	C₂²×C₃	—	2	4	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	4	2	no	—
191	C₂²×C₃	—	4	2	no	—
192	C₂²×C₃	—	4	2	no	—
193	C₂²×C₃	—	4	2	no	—
194	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
195	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
196	C₂²×C₃	C₂×#41	1	1	yes	characteristic
197	C₂²×C₃	C₂×#41	1	1	yes	characteristic
198	C₂²×C₃	C₂×#41	1	1	yes	characteristic
199	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	4	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	—	4	1	no	—
209	C₃×C₄	—	4	2	no	—
210	C₃×C₄	C₄×D₄	1	4	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	C₂²×C₃×C₄	1	4	yes	—
218	C₂³	C₂⁴×C₃	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	D₄	—	2	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	4	8	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	4	yes	—
246	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
247	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
248	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
249	C₂×C₄	C₃×#41	1	2	yes	—
250	C₂×C₄	C₃×#41	1	2	yes	—
251	C₂×C₄	C₃×#41	1	2	yes	—
252	C₂×C₃	—	4	4	no	—
253	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
254	C₂×C₃	C₄×#41	1	1	yes	characteristic
255	C₂×C₃	#520	1	1	yes	characteristic
256	C₂×C₃	#546	1	1	yes	characteristic
257	C₂×C₃	#548	1	1	yes	characteristic
258	C₂×C₃	#556	1	1	yes	characteristic
259	C₂×C₃	#559	1	1	yes	characteristic
260	C₂²	—	2	4	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
268	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
269	C₂²	C₂×C₃×#41	1	1	yes	characteristic
270	C₂²	C₂×C₃×#41	1	1	yes	characteristic
271	C₂²	C₂×C₃×#41	1	1	yes	characteristic
272	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
273	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	1	no	—
283	C₄	C₃×C₄×D₄	1	4	yes	—
284	C₃	#2367	1	1	yes	characteristic, direct product factor, Sylow
285	C₂	—	4	4	no	—
286	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
287	C₂	C₃×C₄×#41	1	1	yes	characteristic
288	C₂	C₃×#520	1	1	yes	characteristic
289	C₂	C₃×#546	1	1	yes	characteristic
290	C₂	C₃×#548	1	1	yes	characteristic
291	C₂	C₃×#556	1	1	yes	characteristic
292	C₂	C₃×#559	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²
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×Q₈	C₂	semidirect
4. C₃×#391	C₂	semidirect
6. C₃×#388	C₂	semidirect
7. C₃×#388	C₂	semidirect
8. C₃×#385	C₂	semidirect
9. C₃×C₄×C₄⋊C₄	C₂	semidirect
10. C₃×C₄×C₄⋊C₄	C₂	semidirect
11. C₃×C₄×C₂²⋊C₄	C₂	semidirect
12. #2367	C₃	direct (N × H)
16. C₃×#124	C₄	semidirect
40. C₂×C₄×Q₈	C₂×C₃	semidirect
42. #391	C₂×C₃	semidirect
44. #388	C₂×C₃	semidirect
45. #388	C₂×C₃	semidirect
46. #385	C₂×C₃	semidirect
47. C₄×C₄⋊C₄	C₂×C₃	semidirect
48. C₄×C₄⋊C₄	C₂×C₃	semidirect
49. C₄×C₂²⋊C₄	C₂×C₃	semidirect
53. C₂×C₃×Q₈	C₂×C₄	semidirect
78. C₃×C₂²⋊C₄	C₂×C₄	semidirect
79. C₃×C₂²⋊C₄	C₂×C₄	semidirect
84. C₃×C₄²	C₂×C₄	semidirect
88. #124	C₃×C₄	semidirect
155. C₂×Q₈	C₂×C₃×C₄	semidirect
180. C₂²⋊C₄	C₂×C₃×C₄	semidirect
181. C₂²⋊C₄	C₂×C₃×C₄	semidirect
186. C₄²	C₂×C₃×C₄	semidirect
284. C₃	#2367	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×#124	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×C₄×D₄	2
C₃×#391	2
C₃×#390	2
C₃×#388	2
C₃×#385	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2367	3

Subgroups — C3×#2367 #84231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2367 #84231