Subgroups — C₃×#2512 #84376

Order 384.

496 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#126	C₂	1	2	yes	maximal, complemented
2	C₃×#397	C₂	1	2	yes	maximal, complemented
3	C₃×#396	C₂	1	2	yes	maximal, complemented
4	C₃×#390	C₂	1	1	yes	maximal, characteristic
5	C₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#388	C₂	1	2	yes	maximal, complemented
9	C₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#2512	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₃×#126	—	2	8	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	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	1	yes	characteristic
30	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₄²	C₂²	1	2	yes	—
32	C₂×C₃×C₄²	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	1	yes	characteristic
36	C₃×#95	C₂²	1	2	yes	—
37	C₂×#126	C₂×C₃	1	2	yes	complemented
38	#397	C₂×C₃	1	2	yes	complemented
39	#396	C₂×C₃	1	2	yes	complemented
40	#390	C₂×C₃	1	1	yes	characteristic
41	#390	C₂×C₃	1	1	yes	characteristic
42	#390	C₂×C₃	1	1	yes	characteristic
43	#389	C₂×C₃	1	1	yes	characteristic, complemented
44	#388	C₂×C₃	1	2	yes	complemented
45	#384	C₂×C₃	1	1	yes	characteristic, complemented
46	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂⁴×C₃	C₂³	1	2	yes	—
49	C₂×C₃×D₄	—	2	2	no	—
50	C₂×C₃×D₄	—	2	2	no	—
51	C₂×C₃×D₄	—	2	2	no	—
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	2	yes	—
56	C₂²×C₃×C₄	C₂³	1	2	yes	—
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊C₄	—	2	4	no	—
63	C₃×C₄⋊C₄	—	2	4	no	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	4	2	no	—
69	C₃×C₄⋊C₄	D₄	1	4	yes	complemented
70	C₃×C₂²⋊C₄	—	2	4	no	—
71	C₃×C₂²⋊C₄	—	2	4	no	—
72	C₃×C₂²⋊C₄	—	2	4	no	—
73	C₃×C₂²⋊C₄	—	2	4	no	—
74	C₃×C₂²⋊C₄	—	4	2	no	—
75	C₃×C₂²⋊C₄	—	4	2	no	—
76	C₃×C₄²	—	2	4	no	—
77	C₃×C₄²	—	2	2	no	—
78	C₃×C₄²	—	2	2	no	—
79	C₃×C₄²	—	2	2	no	—
80	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
81	#126	—	2	8	no	—
82	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
91	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
92	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
93	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
94	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
95	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
96	C₂×C₄²	C₂²×C₃	1	2	yes	—
97	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
100	#95	C₂²×C₃	1	1	yes	characteristic
101	#95	C₂²×C₃	1	2	yes	—
102	#95	C₂²×C₃	1	1	yes	characteristic
103	#95	C₂²×C₃	1	2	yes	—
104	C₂³×C₃	—	2	2	no	—
105	C₂³×C₃	—	2	2	no	—
106	C₂³×C₃	—	2	2	no	—
107	C₂³×C₃	—	2	2	no	—
108	C₂³×C₃	—	2	2	no	—
109	C₂³×C₃	—	2	2	no	—
110	C₂³×C₃	—	2	2	no	—
111	C₂³×C₃	C₂⁴	1	1	yes	characteristic
112	C₃×D₄	—	4	4	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	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	2	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	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
137	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
138	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
139	C₂×C₃×C₄	#41	1	2	yes	—
140	C₂×C₃×C₄	#41	1	2	yes	—
141	C₂×C₃×C₄	#41	1	2	yes	—
142	C₂×C₃×C₄	#41	1	2	yes	—
143	C₂⁴	C₂³×C₃	1	2	yes	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₃	1	2	yes	—
151	C₂²×C₄	C₂³×C₃	1	2	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	1	yes	characteristic
156	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	C₃×D₄	1	4	yes	complemented
165	C₂²⋊C₄	—	2	4	no	—
166	C₂²⋊C₄	—	2	4	no	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₂²⋊C₄	—	4	2	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₄²	—	2	4	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₂²×C₃	—	4	2	no	—
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	—	4	2	no	—
179	C₂²×C₃	—	4	2	no	—
180	C₂²×C₃	—	4	2	no	—
181	C₂²×C₃	—	4	2	no	—
182	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
183	C₂²×C₃	C₂×#41	1	1	yes	characteristic
184	C₂²×C₃	C₂×#41	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	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	—	4	1	no	—
195	C₃×C₄	—	4	2	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	—	4	1	no	—
198	C₃×C₄	#126	1	4	yes	complemented
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₂⁴×C₃	1	1	yes	characteristic
207	D₄	—	4	4	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
232	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
233	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
234	C₂×C₄	C₃×#41	1	2	yes	—
235	C₂×C₄	C₃×#41	1	2	yes	—
236	C₂×C₄	C₃×#41	1	2	yes	—
237	C₂×C₄	C₃×#41	1	2	yes	—
238	C₂×C₃	—	8	2	no	—
239	C₂×C₃	C₂×#126	1	1	yes	characteristic
240	C₂×C₃	#532	1	1	yes	characteristic
241	C₂×C₃	#542	1	1	yes	characteristic
242	C₂×C₃	#547	1	1	yes	characteristic
243	C₂×C₃	#550	1	1	yes	characteristic
244	C₂×C₃	#553	1	1	yes	characteristic
245	C₂×C₃	#556	1	1	yes	characteristic
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
254	C₂²	C₂×C₃×#41	1	1	yes	characteristic
255	C₂²	C₂×C₃×#41	1	1	yes	characteristic
256	C₂²	C₂×C₃×#41	1	1	yes	characteristic
257	C₂²	C₂×C₃×#41	1	1	yes	characteristic
258	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	1	no	—
269	C₄	C₃×#126	1	4	yes	complemented
270	C₃	#2512	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	8	2	no	—
272	C₂	C₂×C₃×#126	1	1	yes	characteristic
273	C₂	C₃×#532	1	1	yes	characteristic
274	C₂	C₃×#542	1	1	yes	characteristic
275	C₂	C₃×#547	1	1	yes	characteristic
276	C₂	C₃×#550	1	1	yes	characteristic
277	C₂	C₃×#553	1	1	yes	characteristic
278	C₂	C₃×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²
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×#126	C₂	semidirect
2. C₃×#397	C₂	semidirect
3. C₃×#396	C₂	semidirect
7. C₃×#389	C₂	semidirect
8. C₃×#388	C₂	semidirect
9. C₃×#384	C₂	semidirect
10. C₃×C₄×C₄⋊C₄	C₂	semidirect
11. C₃×C₄×C₄⋊C₄	C₂	semidirect
12. #2512	C₃	direct (N × H)
37. C₂×#126	C₂×C₃	semidirect
38. #397	C₂×C₃	semidirect
39. #396	C₂×C₃	semidirect
43. #389	C₂×C₃	semidirect
44. #388	C₂×C₃	semidirect
45. #384	C₂×C₃	semidirect
46. C₄×C₄⋊C₄	C₂×C₃	semidirect
47. C₄×C₄⋊C₄	C₂×C₃	semidirect
69. C₃×C₄⋊C₄	D₄	semidirect
164. C₄⋊C₄	C₃×D₄	semidirect
198. C₃×C₄	#126	semidirect
269. C₄	C₃×#126	semidirect
270. C₃	#2512	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×#126	2
C₃×#397	2
C₃×#396	2
C₃×#390	2
C₃×#389	2
C₃×#388	2
C₃×#384	2
C₃×C₄×C₄⋊C₄	2
#2512	3

Subgroups — C3×#2512 #84376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2512 #84376