Subgroups — C₃×#2374 #84238

Order 384.

552 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#123	C₂	1	1	yes	maximal, characteristic
2	C₃×#390	C₂	1	1	yes	maximal, characteristic
3	C₃×#388	C₂	1	2	yes	maximal, complemented
4	C₃×#388	C₂	1	2	yes	maximal, complemented
5	C₃×#387	C₂	1	2	yes	maximal, complemented
6	C₃×#386	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
10	#2374	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₃×#123	C₄	1	8	yes	complemented
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄²	C₂²	1	2	yes	—
25	C₂×C₃×C₄²	C₂²	1	2	yes	—
26	C₂×C₃×C₄²	C₂²	1	2	yes	—
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₃×#95	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	C₂×#123	C₂×C₃	1	1	yes	characteristic
34	#390	C₂×C₃	1	1	yes	characteristic
35	#388	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	1	yes	characteristic
39	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
40	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
41	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
42	C₂⁴×C₃	C₂³	1	1	yes	characteristic
43	C₂⁴×C₃	C₂³	1	1	yes	characteristic
44	C₂×C₃×D₄	—	4	1	no	—
45	C₂×C₃×D₄	—	4	1	no	—
46	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
47	C₂²×C₃×C₄	—	2	4	no	—
48	C₂²×C₃×C₄	—	4	1	no	—
49	C₂²×C₃×C₄	—	4	1	no	—
50	C₂²×C₃×C₄	—	4	1	no	—
51	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	2	yes	—
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₃×C₄⋊C₄	—	2	4	no	—
60	C₃×C₄⋊C₄	—	4	1	no	—
61	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
62	C₃×C₂²⋊C₄	—	2	4	no	—
63	C₃×C₂²⋊C₄	—	2	4	no	—
64	C₃×C₂²⋊C₄	—	2	4	no	—
65	C₃×C₂²⋊C₄	—	4	1	no	—
66	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
67	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	complemented
68	C₃×C₄²	—	2	4	no	—
69	C₃×C₄²	—	4	1	no	—
70	C₃×C₄²	—	4	2	no	—
71	C₃×C₄²	—	4	2	no	—
72	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
73	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
75	#123	C₃×C₄	1	8	yes	complemented
76	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄²	C₂²×C₃	1	2	yes	—
86	C₂×C₄²	C₂²×C₃	1	2	yes	—
87	C₂×C₄²	C₂²×C₃	1	2	yes	—
88	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
89	#95	C₂²×C₃	1	2	yes	—
90	#95	C₂²×C₃	1	2	yes	—
91	#95	C₂²×C₃	1	2	yes	—
92	#95	C₂²×C₃	1	2	yes	—
93	#95	C₂²×C₃	1	2	yes	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	—	2	2	no	—
98	C₂³×C₃	—	2	1	no	—
99	C₂³×C₃	—	2	1	no	—
100	C₂³×C₃	—	2	1	no	—
101	C₂³×C₃	—	2	1	no	—
102	C₂³×C₃	C₂²×C₄	1	2	yes	—
103	C₂³×C₃	C₂²×C₄	1	2	yes	—
104	C₂³×C₃	C₂⁴	1	1	yes	characteristic
105	C₃×D₄	—	4	4	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	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	—	4	4	no	—
123	C₂×C₃×C₄	—	4	4	no	—
124	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
125	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
126	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
127	C₂×C₃×C₄	#41	1	4	yes	—
128	C₂×C₃×C₄	#41	1	4	yes	—
129	C₂⁴	C₂³×C₃	1	1	yes	characteristic
130	C₂⁴	C₂³×C₃	1	1	yes	characteristic
131	C₂×D₄	—	4	1	no	—
132	C₂×D₄	—	4	1	no	—
133	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
139	C₂²×C₄	C₂³×C₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₃	1	2	yes	—
143	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₃	1	2	yes	—
145	C₂²×C₄	C₂³×C₃	1	2	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
149	C₂²⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₂²⋊C₄	—	4	1	no	—
153	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
154	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
155	C₄²	—	2	4	no	—
156	C₄²	—	4	1	no	—
157	C₄²	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₂²×C₃	—	2	4	no	—
160	C₂²×C₃	—	2	4	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	—	4	1	no	—
164	C₂²×C₃	—	4	1	no	—
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	1	no	—
168	C₂²×C₃	—	4	1	no	—
169	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
170	C₂²×C₃	C₂×#41	1	2	yes	—
171	C₂²×C₃	C₂×#41	1	2	yes	—
172	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
174	C₃×C₄	—	2	4	no	—
175	C₃×C₄	—	2	4	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	2	4	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	—	4	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	1	no	—
186	C₂³	—	2	1	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	C₂²×C₃×C₄	1	2	yes	—
190	C₂³	C₂²×C₃×C₄	1	2	yes	—
191	C₂³	C₂⁴×C₃	1	1	yes	characteristic
192	D₄	—	4	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
212	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
213	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
214	C₂×C₄	C₃×#41	1	4	yes	—
215	C₂×C₄	C₃×#41	1	4	yes	—
216	C₂×C₃	—	4	2	no	—
217	C₂×C₃	—	4	2	no	—
218	C₂×C₃	C₄×#41	1	2	yes	—
219	C₂×C₃	#518	1	1	yes	characteristic
220	C₂×C₃	#551	1	1	yes	characteristic
221	C₂×C₃	#553	1	2	yes	—
222	C₂×C₃	#559	1	1	yes	characteristic
223	C₂²	—	2	4	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
234	C₂²	C₂×C₃×#41	1	2	yes	—
235	C₂²	C₂×C₃×#41	1	2	yes	—
236	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
237	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	2	4	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	2	4	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₃	#2374	1	1	yes	characteristic, direct product factor, Sylow
246	C₂	—	4	2	no	—
247	C₂	—	4	2	no	—
248	C₂	C₃×C₄×#41	1	2	yes	—
249	C₂	C₃×#518	1	1	yes	characteristic
250	C₂	C₃×#551	1	1	yes	characteristic
251	C₂	C₃×#553	1	2	yes	—
252	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
3. C₃×#388	C₂	semidirect
4. C₃×#388	C₂	semidirect
5. C₃×#387	C₂	semidirect
7. C₃×C₄×C₄⋊C₄	C₂	semidirect
8. C₃×C₄×C₂²⋊C₄	C₂	semidirect
9. C₃×C₄×C₂²⋊C₄	C₂	semidirect
10. #2374	C₃	direct (N × H)
14. C₃×#123	C₄	semidirect
35. #388	C₂×C₃	semidirect
36. #388	C₂×C₃	semidirect
37. #387	C₂×C₃	semidirect
39. C₄×C₄⋊C₄	C₂×C₃	semidirect
40. C₄×C₂²⋊C₄	C₂×C₃	semidirect
41. C₄×C₂²⋊C₄	C₂×C₃	semidirect
51. C₂²×C₃×C₄	C₂×C₄	semidirect
61. C₃×C₄⋊C₄	C₂×C₄	semidirect
67. C₃×C₂²⋊C₄	C₂×C₄	semidirect
75. #123	C₃×C₄	semidirect
138. C₂²×C₄	C₂×C₃×C₄	semidirect
148. C₄⋊C₄	C₂×C₃×C₄	semidirect
154. C₂²⋊C₄	C₂×C₃×C₄	semidirect
245. C₃	#2374	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×#123	2
C₃×#390	2
C₃×#388	2
C₃×#387	2
C₃×#386	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2374	3

Subgroups — C3×#2374 #84238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2374 #84238