Subgroups — C₃×#2442 #84306

Order 384.

520 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#400	C₂	1	2	yes	maximal, complemented
4	C₃×#398	C₂	1	2	yes	maximal, complemented
5	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#385	C₂	1	2	yes	maximal
8	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#380	C₂	1	2	yes	maximal
10	C₃×#380	C₂	1	1	yes	maximal, characteristic
11	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	#2442	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×#125	—	2	4	no	—
16	C₂×C₃×C₄⋊C₄	—	2	4	no	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₃×#95	C₂²	1	1	yes	characteristic
32	C₃×#95	C₂²	1	1	yes	characteristic
33	C₃×#95	C₂²	1	2	yes	—
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×#125	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
40	#400	C₂×C₃	1	2	yes	complemented
41	#398	C₂×C₃	1	2	yes	complemented
42	#389	C₂×C₃	1	1	yes	characteristic, complemented
43	#389	C₂×C₃	1	1	yes	characteristic, complemented
44	#385	C₂×C₃	1	2	yes	—
45	#382	C₂×C₃	1	1	yes	characteristic, complemented
46	#380	C₂×C₃	1	2	yes	—
47	#380	C₂×C₃	1	1	yes	characteristic
48	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
49	C₂⁴×C₃	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₄	—	2	2	no	—
51	C₂²×C₃×C₄	—	2	2	no	—
52	C₂²×C₃×C₄	—	2	4	no	—
53	C₂²×C₃×C₄	—	2	4	no	—
54	C₂²×C₃×C₄	—	4	2	no	—
55	C₂²×C₃×C₄	Q₈	1	4	yes	—
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄	C₂³	1	2	yes	—
61	C₂²×C₃×C₄	C₂³	1	2	yes	—
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₃×C₄⋊C₄	—	2	4	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	4	no	—
69	C₃×C₄⋊C₄	—	2	4	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	—	4	2	no	—
73	C₃×C₄⋊C₄	—	4	2	no	—
74	C₃×C₄⋊C₄	D₄	1	4	yes	—
75	C₃×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₂²⋊C₄	—	2	4	no	—
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₃×C₄²	—	2	2	no	—
79	C₃×C₄²	—	2	2	no	—
80	C₂³×C₄	C₂²×C₃	1	2	yes	—
81	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
82	#125	—	2	4	no	—
83	C₂×C₄⋊C₄	—	2	4	no	—
84	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
91	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
92	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
93	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
94	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
96	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
97	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
98	#95	C₂²×C₃	1	1	yes	characteristic
99	#95	C₂²×C₃	1	1	yes	characteristic
100	#95	C₂²×C₃	1	2	yes	—
101	#95	C₂²×C₃	1	2	yes	—
102	#95	C₂²×C₃	1	2	yes	—
103	#95	C₂²×C₃	1	2	yes	—
104	#95	C₂²×C₃	1	2	yes	—
105	C₂³×C₃	—	2	1	no	—
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	—	2	1	no	—
108	C₂³×C₃	—	2	1	no	—
109	C₂³×C₃	C₂×Q₈	1	2	yes	—
110	C₂³×C₃	#41	1	2	yes	—
111	C₂³×C₃	#41	1	2	yes	—
112	C₂³×C₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	8	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	1	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	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	4	8	no	—
140	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
141	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
142	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
143	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
144	C₂×C₃×C₄	#41	1	2	yes	—
145	C₂⁴	C₂³×C₃	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	C₃×Q₈	1	4	yes	—
152	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₃	1	2	yes	—
156	C₂²×C₄	C₂³×C₃	1	2	yes	—
157	C₂²×C₄	C₂³×C₃	1	2	yes	—
158	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	C₃×D₄	1	4	yes	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	#125	1	4	yes	complemented
183	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
184	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
185	C₂²×C₃	C₂×#41	1	1	yes	characteristic
186	C₂²×C₃	C₂×#41	1	1	yes	characteristic
187	C₂²×C₃	C₂×#41	1	1	yes	characteristic
188	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	4	no	—
192	C₃×C₄	—	4	1	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	—	4	1	no	—
195	C₃×C₄	—	4	2	no	—
196	C₃×C₄	—	4	1	no	—
197	C₃×C₄	—	4	1	no	—
198	C₃×C₄	—	4	2	no	—
199	C₃×C₄	C₂²⋊Q₈	1	4	yes	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	C₂×C₃×Q₈	1	2	yes	—
205	C₂³	C₃×#41	1	2	yes	—
206	C₂³	C₃×#41	1	2	yes	—
207	C₂³	C₂⁴×C₃	1	1	yes	characteristic
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
236	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
237	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
238	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
239	C₂×C₄	C₃×#41	1	2	yes	—
240	C₂×C₃	—	2	4	no	—
241	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
242	C₂×C₃	C₂×#125	1	1	yes	characteristic
243	C₂×C₃	#544	1	1	yes	characteristic
244	C₂×C₃	#546	1	1	yes	characteristic
245	C₂×C₃	#548	1	1	yes	characteristic
246	C₂×C₃	#552	1	1	yes	characteristic
247	C₂×C₃	#553	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	C₃×#125	1	4	yes	complemented
255	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
256	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
257	C₂²	C₂×C₃×#41	1	1	yes	characteristic
258	C₂²	C₂×C₃×#41	1	1	yes	characteristic
259	C₂²	C₂×C₃×#41	1	1	yes	characteristic
260	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
261	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	4	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	C₃×C₂²⋊Q₈	1	4	yes	—
272	C₃	#2442	1	1	yes	characteristic, direct product factor, Sylow
273	C₂	—	2	4	no	—
274	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂	C₂×C₃×#125	1	1	yes	characteristic
276	C₂	C₃×#544	1	1	yes	characteristic
277	C₂	C₃×#546	1	1	yes	characteristic
278	C₂	C₃×#548	1	1	yes	characteristic
279	C₂	C₃×#552	1	1	yes	characteristic
280	C₂	C₃×#553	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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
1. C₂×C₃×#125	C₂	semidirect
3. C₃×#400	C₂	semidirect
4. C₃×#398	C₂	semidirect
5. C₃×#389	C₂	semidirect
6. C₃×#389	C₂	semidirect
8. C₃×#382	C₂	semidirect
12. #2442	C₃	direct (N × H)
38. C₂×#125	C₂×C₃	semidirect
40. #400	C₂×C₃	semidirect
41. #398	C₂×C₃	semidirect
42. #389	C₂×C₃	semidirect
43. #389	C₂×C₃	semidirect
45. #382	C₂×C₃	semidirect
182. C₂²×C₃	#125	semidirect
254. C₂²	C₃×#125	semidirect
272. C₃	#2442	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×#125	2
C₂²×C₃×C₄⋊C₄	2
C₃×#400	2
C₃×#398	2
C₃×#389	2
C₃×#385	2
C₃×#382	2
C₃×#380	2
C₃×C₄×C₂²⋊C₄	2
#2442	3

Subgroups — C3×#2442 #84306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2442 #84306