Subgroups — C₃×#2364 #84228

Order 384.

716 conjugacy classes of subgroups.

The subgroup lattice has 268 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×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
10	#2364	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	2	yes	—
14	C₂³×C₃×C₄	C₂²	1	2	yes	—
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₃×#121	C₄	1	8	yes	complemented
17	C₃×C₄×D₄	—	2	8	no	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
26	C₂×C₃×C₄²	C₂²	1	2	yes	—
27	C₂×C₃×C₄²	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	1	yes	characteristic
29	C₃×#95	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	1	yes	characteristic
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	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
41	C₄×C₂²⋊C₄	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	2	no	—
48	C₂×C₃×D₄	C₂×C₄	1	8	yes	complemented
49	C₂×C₃×D₄	C₂×C₄	1	4	yes	complemented
50	C₂²×C₃×C₄	—	2	4	no	—
51	C₂²×C₃×C₄	—	2	4	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	—	4	2	no	—
56	C₂²×C₃×C₄	—	4	2	no	—
57	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂³	1	2	yes	—
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	2	yes	—
65	C₃×C₄⋊C₄	—	2	4	no	—
66	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
67	C₃×C₂²⋊C₄	—	2	2	no	—
68	C₃×C₂²⋊C₄	—	2	4	no	—
69	C₃×C₂²⋊C₄	—	2	4	no	—
70	C₃×C₂²⋊C₄	—	4	2	no	—
71	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
72	C₃×C₂²⋊C₄	D₄	1	4	yes	complemented
73	C₃×C₄²	—	2	4	no	—
74	C₃×C₄²	—	4	2	no	—
75	C₂²×D₄	C₂²×C₃	1	2	yes	—
76	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
77	C₂³×C₄	C₂²×C₃	1	2	yes	—
78	C₂³×C₄	C₂²×C₃	1	2	yes	—
79	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
80	#121	C₃×C₄	1	8	yes	complemented
81	C₄×D₄	—	2	8	no	—
82	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₄²	C₂²×C₃	1	2	yes	—
91	C₂×C₄²	C₂²×C₃	1	2	yes	—
92	#95	C₂²×C₃	1	1	yes	characteristic
93	#95	C₂²×C₃	1	2	yes	—
94	#95	C₂²×C₃	1	1	yes	characteristic
95	#95	C₂²×C₃	1	2	yes	—
96	#95	C₂²×C₃	1	2	yes	—
97	#95	C₂²×C₃	1	2	yes	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	2	2	no	—
102	C₂³×C₃	—	2	2	no	—
103	C₂³×C₃	—	2	2	no	—
104	C₂³×C₃	—	2	2	no	—
105	C₂³×C₃	—	2	2	no	—
106	C₂³×C₃	C₂²×C₄	1	4	yes	—
107	C₂³×C₃	C₂²×C₄	1	2	yes	—
108	C₂³×C₃	C₂×D₄	1	2	yes	—
109	C₂³×C₃	#41	1	2	yes	—
110	C₂³×C₃	C₂⁴	1	1	yes	characteristic
111	C₃×D₄	—	2	16	no	—
112	C₃×D₄	—	4	4	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	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	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	4	4	no	—
128	C₂×C₃×C₄	—	4	8	no	—
129	C₂×C₃×C₄	—	4	8	no	—
130	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
131	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
132	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
133	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
134	C₂×C₃×C₄	#41	1	4	yes	—
135	C₂⁴	C₂³×C₃	1	2	yes	—
136	C₂⁴	C₂³×C₃	1	1	yes	characteristic
137	C₂×D₄	—	2	4	no	—
138	C₂×D₄	—	2	4	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂×D₄	C₂×C₃×C₄	1	8	yes	complemented
141	C₂×D₄	C₂×C₃×C₄	1	4	yes	complemented
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
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	2	yes	—
155	C₂²×C₄	C₂³×C₃	1	2	yes	—
156	C₂²×C₄	C₂³×C₃	1	2	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
164	C₂²⋊C₄	C₃×D₄	1	4	yes	complemented
165	C₄²	—	2	4	no	—
166	C₄²	—	4	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	2	4	no	—
170	C₂²×C₃	—	2	4	no	—
171	C₂²×C₃	—	2	4	no	—
172	C₂²×C₃	—	2	4	no	—
173	C₂²×C₃	—	2	4	no	—
174	C₂²×C₃	—	4	2	no	—
175	C₂²×C₃	—	4	2	no	—
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	C₄×D₄	1	4	yes	—
179	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
180	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
181	C₂²×C₃	C₂×#41	1	1	yes	characteristic
182	C₂²×C₃	C₂×#41	1	2	yes	—
183	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
184	C₃×C₄	—	2	4	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	2	4	no	—
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	4	2	no	—
190	C₃×C₄	—	4	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	C₂²×C₃×C₄	1	4	yes	—
200	C₂³	C₂²×C₃×C₄	1	2	yes	—
201	C₂³	C₂×C₃×D₄	1	2	yes	—
202	C₂³	C₃×#41	1	2	yes	—
203	C₂³	C₂⁴×C₃	1	1	yes	characteristic
204	D₄	—	2	16	no	—
205	D₄	—	4	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
224	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
225	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
226	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
227	C₂×C₄	C₃×#41	1	4	yes	—
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	—	4	4	no	—
230	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
231	C₂×C₃	C₄×#41	1	1	yes	characteristic
232	C₂×C₃	#518	1	1	yes	characteristic
233	C₂×C₃	#546	1	2	yes	—
234	C₂×C₃	#553	1	2	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	C₃×C₄×D₄	1	4	yes	—
247	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
248	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
249	C₂²	C₂×C₃×#41	1	1	yes	characteristic
250	C₂²	C₂×C₃×#41	1	2	yes	—
251	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	2	no	—
259	C₃	#2364	1	1	yes	characteristic, direct product factor, Sylow
260	C₂	—	2	4	no	—
261	C₂	—	4	4	no	—
262	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
263	C₂	C₃×C₄×#41	1	1	yes	characteristic
264	C₂	C₃×#518	1	1	yes	characteristic
265	C₂	C₃×#546	1	2	yes	—
266	C₂	C₃×#553	1	2	yes	—

	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×C₄×C₂²⋊C₄	C₂	semidirect
8. C₃×C₄×C₂²⋊C₄	C₂	semidirect
9. C₂×C₃×#95	C₂	semidirect
10. #2364	C₃	direct (N × H)
16. 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. C₄×C₂²⋊C₄	C₂×C₃	semidirect
41. C₄×C₂²⋊C₄	C₂×C₃	semidirect
42. C₂×#95	C₂×C₃	semidirect
48. C₂×C₃×D₄	C₂×C₄	semidirect
49. C₂×C₃×D₄	C₂×C₄	semidirect
57. C₂²×C₃×C₄	C₂×C₄	semidirect
66. C₃×C₄⋊C₄	C₂×C₄	semidirect
71. C₃×C₂²⋊C₄	C₂×C₄	semidirect
72. C₃×C₂²⋊C₄	D₄	semidirect
80. #121	C₃×C₄	semidirect
140. C₂×D₄	C₂×C₃×C₄	semidirect
141. C₂×D₄	C₂×C₃×C₄	semidirect
149. C₂²×C₄	C₂×C₃×C₄	semidirect
158. C₄⋊C₄	C₂×C₃×C₄	semidirect
163. C₂²⋊C₄	C₂×C₃×C₄	semidirect
164. C₂²⋊C₄	C₃×D₄	semidirect
259. C₃	#2364	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×C₄×C₂²⋊C₄	2
C₂×C₃×#95	2
#2364	3

Subgroups — C3×#2364 #84228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2364 #84228