Subgroups — C₃×#2343 #84207

Order 384.

532 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₃×#387	C₂	1	2	yes	maximal, complemented
3	C₃×#387	C₂	1	2	yes	maximal, complemented
4	C₃×#385	C₂	1	2	yes	maximal
5	C₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#381	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal
10	C₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#95	C₂	1	1	yes	maximal, characteristic
12	#2343	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×#117	C₄	1	8	yes	complemented
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
23	C₂×C₃×C₄²	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	2	yes	—
25	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
27	C₃×#95	C₂²	1	1	yes	characteristic
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	C₃×#95	C₂²	1	1	yes	characteristic
35	C₃×#95	C₂²	1	2	yes	—
36	C₃×#95	C₂²	1	2	yes	—
37	C₂×#117	C₂×C₃	1	1	yes	characteristic
38	#387	C₂×C₃	1	2	yes	complemented
39	#387	C₂×C₃	1	2	yes	complemented
40	#385	C₂×C₃	1	2	yes	—
41	#383	C₂×C₃	1	1	yes	characteristic, complemented
42	#382	C₂×C₃	1	1	yes	characteristic, complemented
43	#381	C₂×C₃	1	1	yes	characteristic
44	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
45	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	—
46	#376	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×#95	C₂×C₃	1	1	yes	characteristic
48	C₂⁴×C₃	C₂³	1	1	yes	characteristic
49	C₂²×C₃×C₄	—	2	4	no	—
50	C₂²×C₃×C₄	—	2	4	no	—
51	C₂²×C₃×C₄	—	2	4	no	—
52	C₂²×C₃×C₄	—	4	2	no	—
53	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
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	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₃×C₄⋊C₄	—	2	4	no	—
65	C₃×C₄⋊C₄	—	4	2	no	—
66	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
67	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
68	C₃×C₂²⋊C₄	—	2	4	no	—
69	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
70	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
71	C₃×C₄²	—	2	4	no	—
72	C₃×C₄²	—	4	2	no	—
73	C₃×C₄²	C₂×C₄	1	4	yes	complemented
74	C₃×C₄²	C₂×C₄	1	4	yes	complemented
75	C₂³×C₄	C₂²×C₃	1	2	yes	—
76	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
77	#117	C₃×C₄	1	8	yes	complemented
78	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
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	1	yes	characteristic
88	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
89	#95	C₂²×C₃	1	1	yes	characteristic
90	#95	C₂²×C₃	1	1	yes	characteristic
91	#95	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	2	yes	—
95	#95	C₂²×C₃	1	2	yes	—
96	#95	C₂²×C₃	1	1	yes	characteristic
97	#95	C₂²×C₃	1	2	yes	—
98	#95	C₂²×C₃	1	2	yes	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	1	no	—
101	C₂³×C₃	—	2	1	no	—
102	C₂³×C₃	C₂²×C₄	1	2	yes	—
103	C₂³×C₃	#41	1	4	yes	—
104	C₂³×C₃	C₂⁴	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	8	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	1	no	—
112	C₂×C₃×C₄	—	2	2	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	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	—	4	8	no	—
126	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
127	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
128	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
129	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
130	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
131	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
132	C₂×C₃×C₄	#41	1	4	yes	—
133	C₂⁴	C₂³×C₃	1	1	yes	characteristic
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
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	1	yes	characteristic
143	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₃	1	2	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
152	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
153	C₂²⋊C₄	—	2	4	no	—
154	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
155	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
156	C₄²	—	2	4	no	—
157	C₄²	—	4	2	no	—
158	C₄²	C₂×C₃×C₄	1	4	yes	complemented
159	C₄²	C₂×C₃×C₄	1	4	yes	complemented
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	2	4	no	—
163	C₂²×C₃	—	2	4	no	—
164	C₂²×C₃	#117	1	4	yes	—
165	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
166	C₂²×C₃	C₂×#41	1	2	yes	—
167	C₂²×C₃	C₂×#41	1	2	yes	—
168	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	4	no	—
177	C₃×C₄	—	4	2	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	—	4	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	1	no	—
182	C₂³	—	2	1	no	—
183	C₂³	C₂²×C₃×C₄	1	2	yes	—
184	C₂³	C₃×#41	1	4	yes	—
185	C₂³	C₂⁴×C₃	1	1	yes	characteristic
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	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	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
208	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
209	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
210	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
211	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
212	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
213	C₂×C₄	C₃×#41	1	4	yes	—
214	C₂×C₃	—	2	4	no	—
215	C₂×C₃	C₂×#117	1	1	yes	characteristic
216	C₂×C₃	C₄×#41	1	1	yes	characteristic
217	C₂×C₃	#520	1	1	yes	characteristic
218	C₂×C₃	#551	1	2	yes	—
219	C₂×C₃	#552	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	4	no	—
224	C₂²	C₃×#117	1	4	yes	—
225	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
226	C₂²	C₂×C₃×#41	1	2	yes	—
227	C₂²	C₂×C₃×#41	1	2	yes	—
228	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₃	#2343	1	1	yes	characteristic, direct product factor, Sylow
241	C₂	—	2	4	no	—
242	C₂	C₂×C₃×#117	1	1	yes	characteristic
243	C₂	C₃×C₄×#41	1	1	yes	characteristic
244	C₂	C₃×#520	1	1	yes	characteristic
245	C₂	C₃×#551	1	2	yes	—
246	C₂	C₃×#552	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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. C₃×#387	C₂	semidirect
3. C₃×#387	C₂	semidirect
5. C₃×#383	C₂	semidirect
6. C₃×#382	C₂	semidirect
8. C₃×C₄×C₄⋊C₄	C₂	semidirect
10. C₃×#376	C₂	semidirect
12. #2343	C₃	direct (N × H)
15. C₃×#117	C₄	semidirect
38. #387	C₂×C₃	semidirect
39. #387	C₂×C₃	semidirect
41. #383	C₂×C₃	semidirect
42. #382	C₂×C₃	semidirect
44. C₄×C₄⋊C₄	C₂×C₃	semidirect
46. #376	C₂×C₃	semidirect
66. C₃×C₄⋊C₄	C₂×C₄	semidirect
67. C₃×C₄⋊C₄	C₂×C₄	semidirect
73. C₃×C₄²	C₂×C₄	semidirect
74. C₃×C₄²	C₂×C₄	semidirect
77. #117	C₃×C₄	semidirect
151. C₄⋊C₄	C₂×C₃×C₄	semidirect
152. C₄⋊C₄	C₂×C₃×C₄	semidirect
158. C₄²	C₂×C₃×C₄	semidirect
159. C₄²	C₂×C₃×C₄	semidirect
240. C₃	#2343	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×#117	2
C₃×#387	2
C₃×#385	2
C₃×#383	2
C₃×#382	2
C₃×#381	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
C₃×#376	2
C₂×C₃×#95	2
#2343	3

Subgroups — C3×#2343 #84207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2343 #84207