Subgroups — C₃×#2325 #84189

Order 384.

696 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
3	C₃×#388	C₂	1	2	yes	maximal, complemented
4	C₃×#387	C₂	1	2	yes	maximal, complemented
5	C₃×#386	C₂	1	2	yes	maximal, complemented
6	C₃×#386	C₂	1	2	yes	maximal, complemented
7	C₃×#385	C₂	1	2	yes	maximal, complemented
8	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
10	#2325	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×D₄	C₂²	1	2	yes	—
12	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	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×#121	C₄	1	8	yes	complemented
18	C₃×C₄×D₄	—	2	8	no	—
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	1	yes	characteristic
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×C₄²	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	2	yes	—
32	C₃×#95	C₂²	1	2	yes	—
33	C₃×#95	C₂²	1	2	yes	—
34	C₂×#121	C₂×C₃	1	1	yes	characteristic
35	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
36	#388	C₂×C₃	1	2	yes	complemented
37	#387	C₂×C₃	1	2	yes	complemented
38	#386	C₂×C₃	1	2	yes	complemented
39	#386	C₂×C₃	1	2	yes	complemented
40	#385	C₂×C₃	1	2	yes	complemented
41	#380	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×#95	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂⁴×C₃	C₂³	1	2	yes	—
44	C₂⁴×C₃	C₂³	1	1	yes	characteristic
45	C₂×C₃×D₄	—	2	4	no	—
46	C₂×C₃×D₄	—	2	4	no	—
47	C₂×C₃×D₄	—	4	1	no	—
48	C₂×C₃×D₄	—	4	1	no	—
49	C₂×C₃×D₄	C₂×C₄	1	8	yes	complemented
50	C₂×C₃×D₄	C₂×C₄	1	4	yes	complemented
51	C₂²×C₃×C₄	—	2	2	no	—
52	C₂²×C₃×C₄	—	2	4	no	—
53	C₂²×C₃×C₄	—	4	1	no	—
54	C₂²×C₃×C₄	—	4	1	no	—
55	C₂²×C₃×C₄	—	4	1	no	—
56	C₂²×C₃×C₄	—	4	1	no	—
57	C₂²×C₃×C₄	—	4	2	no	—
58	C₂²×C₃×C₄	—	4	2	no	—
59	C₂²×C₃×C₄	—	4	1	no	—
60	C₂²×C₃×C₄	—	4	1	no	—
61	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
62	C₂²×C₃×C₄	D₄	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	2	yes	—
66	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂³	1	2	yes	—
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	2	yes	—
70	C₃×C₄⋊C₄	—	2	4	no	—
71	C₃×C₄⋊C₄	—	4	1	no	—
72	C₃×C₄⋊C₄	—	4	1	no	—
73	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
74	C₃×C₂²⋊C₄	—	2	4	no	—
75	C₃×C₂²⋊C₄	—	2	4	no	—
76	C₃×C₂²⋊C₄	—	4	2	no	—
77	C₃×C₂²⋊C₄	—	4	2	no	—
78	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
79	C₃×C₄²	—	2	4	no	—
80	C₂²×D₄	C₂²×C₃	1	2	yes	—
81	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
82	C₂³×C₄	C₂²×C₃	1	2	yes	—
83	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
86	#121	C₃×C₄	1	8	yes	complemented
87	C₄×D₄	—	2	8	no	—
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	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
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×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂×C₄²	C₂²×C₃	1	2	yes	—
98	#95	C₂²×C₃	1	2	yes	—
99	#95	C₂²×C₃	1	2	yes	—
100	#95	C₂²×C₃	1	2	yes	—
101	#95	C₂²×C₃	1	2	yes	—
102	#95	C₂²×C₃	1	2	yes	—
103	C₂³×C₃	—	2	1	no	—
104	C₂³×C₃	—	2	1	no	—
105	C₂³×C₃	—	2	1	no	—
106	C₂³×C₃	—	2	1	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	—	2	2	no	—
112	C₂³×C₃	—	2	2	no	—
113	C₂³×C₃	C₂²×C₄	1	4	yes	—
114	C₂³×C₃	C₂²×C₄	1	2	yes	—
115	C₂³×C₃	C₂×D₄	1	2	yes	—
116	C₂³×C₃	#41	1	2	yes	—
117	C₂³×C₃	C₂⁴	1	1	yes	characteristic
118	C₃×D₄	—	2	16	no	—
119	C₃×D₄	—	4	4	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	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	2	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	2	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	—	4	4	no	—
138	C₂×C₃×C₄	—	4	4	no	—
139	C₂×C₃×C₄	—	4	8	no	—
140	C₂×C₃×C₄	—	4	4	no	—
141	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
142	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
143	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
144	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
145	C₂⁴	C₂³×C₃	1	2	yes	—
146	C₂⁴	C₂³×C₃	1	1	yes	characteristic
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	—	2	4	no	—
149	C₂×D₄	—	4	1	no	—
150	C₂×D₄	—	4	1	no	—
151	C₂×D₄	C₂×C₃×C₄	1	8	yes	complemented
152	C₂×D₄	C₂×C₃×C₄	1	4	yes	complemented
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	1	no	—
156	C₂²×C₄	—	4	1	no	—
157	C₂²×C₄	—	4	1	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
164	C₂²×C₄	C₃×D₄	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	2	yes	—
168	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
176	C₂²⋊C₄	—	2	4	no	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
181	C₄²	—	2	4	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	4	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	—	4	2	no	—
192	C₂²×C₃	—	4	2	no	—
193	C₂²×C₃	—	4	2	no	—
194	C₂²×C₃	—	4	2	no	—
195	C₂²×C₃	C₄×D₄	1	4	yes	—
196	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
197	C₂²×C₃	C₂²×D₄	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	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
202	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	2	4	no	—
207	C₃×C₄	—	4	2	no	—
208	C₃×C₄	—	4	2	no	—
209	C₃×C₄	—	4	2	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	C₂²×C₃×C₄	1	4	yes	—
221	C₂³	C₂²×C₃×C₄	1	2	yes	—
222	C₂³	C₂×C₃×D₄	1	2	yes	—
223	C₂³	C₃×#41	1	2	yes	—
224	C₂³	C₂⁴×C₃	1	1	yes	characteristic
225	D₄	—	2	16	no	—
226	D₄	—	4	4	no	—
227	C₂×C₄	—	2	2	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	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	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	—	2	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
249	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
250	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
251	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	—	4	4	no	—
254	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂×C₃	#518	1	1	yes	characteristic
256	C₂×C₃	#520	1	1	yes	characteristic
257	C₂×C₃	#534	1	1	yes	characteristic
258	C₂×C₃	#537	1	1	yes	characteristic
259	C₂×C₃	#538	1	1	yes	characteristic
260	C₂×C₃	#539	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	C₃×C₄×D₄	1	4	yes	—
275	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
276	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
277	C₂²	C₂×C₃×#41	1	1	yes	characteristic
278	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
279	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
280	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
281	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
282	C₄	—	2	4	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₃	#2325	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	—	2	4	no	—
291	C₂	—	4	4	no	—
292	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
293	C₂	C₃×#518	1	1	yes	characteristic
294	C₂	C₃×#520	1	1	yes	characteristic
295	C₂	C₃×#534	1	1	yes	characteristic
296	C₂	C₃×#537	1	1	yes	characteristic
297	C₂	C₃×#538	1	1	yes	characteristic
298	C₂	C₃×#539	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×#121	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. C₃×#388	C₂	semidirect
4. C₃×#387	C₂	semidirect
5. C₃×#386	C₂	semidirect
6. C₃×#386	C₂	semidirect
7. C₃×#385	C₂	semidirect
8. C₃×#380	C₂	semidirect
9. C₂×C₃×#95	C₂	semidirect
10. #2325	C₃	direct (N × H)
17. C₃×#121	C₄	semidirect
35. C₂×C₄×D₄	C₂×C₃	semidirect
36. #388	C₂×C₃	semidirect
37. #387	C₂×C₃	semidirect
38. #386	C₂×C₃	semidirect
39. #386	C₂×C₃	semidirect
40. #385	C₂×C₃	semidirect
41. #380	C₂×C₃	semidirect
42. C₂×#95	C₂×C₃	semidirect
49. C₂×C₃×D₄	C₂×C₄	semidirect
50. C₂×C₃×D₄	C₂×C₄	semidirect
61. C₂²×C₃×C₄	C₂×C₄	semidirect
62. C₂²×C₃×C₄	D₄	semidirect
73. C₃×C₄⋊C₄	C₂×C₄	semidirect
78. C₃×C₂²⋊C₄	C₂×C₄	semidirect
86. #121	C₃×C₄	semidirect
151. C₂×D₄	C₂×C₃×C₄	semidirect
152. C₂×D₄	C₂×C₃×C₄	semidirect
163. C₂²×C₄	C₂×C₃×C₄	semidirect
164. C₂²×C₄	C₃×D₄	semidirect
175. C₄⋊C₄	C₂×C₃×C₄	semidirect
180. C₂²⋊C₄	C₂×C₃×C₄	semidirect
289. C₃	#2325	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×#121	2
C₂×C₃×C₄×D₄	2
C₃×#388	2
C₃×#387	2
C₃×#386	2
C₃×#385	2
C₃×#380	2
C₂×C₃×#95	2
#2325	3

Subgroups — C3×#2325 #84189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2325 #84189