Subgroups — C₃×#2377 #84241

Order 384.

576 conjugacy classes of subgroups.

The subgroup lattice has 260 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	C₃×#390	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×#381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	#2377	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	2	yes	—
13	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×#121	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×C₂²⋊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	2	yes	—
26	C₂×C₃×C₄²	C₂²	1	2	yes	—
27	C₃×#95	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	C₂×#121	C₂×C₃	1	1	yes	characteristic
34	#390	C₂×C₃	1	2	yes	complemented
35	#388	C₂×C₃	1	2	yes	complemented
36	#387	C₂×C₃	1	2	yes	complemented
37	#386	C₂×C₃	1	2	yes	complemented
38	#386	C₂×C₃	1	2	yes	complemented
39	#385	C₂×C₃	1	2	yes	complemented
40	#381	C₂×C₃	1	1	yes	characteristic, complemented
41	#380	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂⁴×C₃	C₂³	1	1	yes	characteristic
43	C₂⁴×C₃	C₂³	1	2	yes	—
44	C₂×C₃×D₄	—	4	2	no	—
45	C₂×C₃×D₄	—	4	2	no	—
46	C₂×C₃×D₄	—	4	2	no	—
47	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
48	C₂×C₃×D₄	C₂×C₄	1	8	yes	complemented
49	C₂²×C₃×C₄	—	4	2	no	—
50	C₂²×C₃×C₄	—	4	2	no	—
51	C₂²×C₃×C₄	—	4	2	no	—
52	C₂²×C₃×C₄	—	4	2	no	—
53	C₂²×C₃×C₄	C₂×C₄	1	4	yes	complemented
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	2	yes	—
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₃×C₄⋊C₄	—	4	2	no	—
62	C₃×C₄⋊C₄	—	4	2	no	—
63	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
64	C₃×C₂²⋊C₄	—	4	2	no	—
65	C₃×C₂²⋊C₄	—	4	1	no	—
66	C₃×C₂²⋊C₄	—	4	2	no	—
67	C₃×C₂²⋊C₄	—	4	2	no	—
68	C₃×C₂²⋊C₄	—	4	2	no	—
69	C₃×C₂²⋊C₄	—	4	1	no	—
70	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
71	C₃×C₄²	—	4	2	no	—
72	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
73	C₂²×D₄	C₂²×C₃	1	2	yes	—
74	C₂³×C₄	C₂²×C₃	1	2	yes	—
75	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
76	#121	C₃×C₄	1	8	yes	complemented
77	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
86	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₄²	C₂²×C₃	1	2	yes	—
88	#95	C₂²×C₃	1	2	yes	—
89	#95	C₂²×C₃	1	1	yes	characteristic
90	#95	C₂²×C₃	1	2	yes	—
91	#95	C₂²×C₃	1	1	yes	characteristic
92	#95	C₂²×C₃	1	2	yes	—
93	#95	C₂²×C₃	1	2	yes	—
94	C₂³×C₃	—	2	1	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	—	2	2	no	—
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	1	no	—
103	C₂³×C₃	—	2	2	no	—
104	C₂³×C₃	C₂²×C₄	1	4	yes	—
105	C₂³×C₃	C₂²×C₄	1	2	yes	—
106	C₂³×C₃	C₂⁴	1	1	yes	characteristic
107	C₃×D₄	—	4	4	no	—
108	C₃×D₄	—	4	8	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	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	—	4	4	no	—
126	C₂×C₃×C₄	—	4	8	no	—
127	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
128	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
129	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
130	C₂⁴	C₂³×C₃	1	1	yes	characteristic
131	C₂⁴	C₂³×C₃	1	2	yes	—
132	C₂×D₄	—	4	2	no	—
133	C₂×D₄	—	4	2	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
136	C₂×D₄	C₂×C₃×C₄	1	8	yes	complemented
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂×C₃×C₄	1	4	yes	complemented
142	C₂²×C₄	C₂³×C₃	1	2	yes	—
143	C₂²×C₄	C₂³×C₃	1	2	yes	—
144	C₂²×C₄	C₂³×C₃	1	2	yes	—
145	C₂²×C₄	C₂³×C₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
152	C₂²⋊C₄	—	4	2	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₂²⋊C₄	—	4	2	no	—
156	C₂²⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	4	1	no	—
158	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
159	C₄²	—	4	2	no	—
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	—	2	2	no	—
165	C₂²×C₃	—	2	4	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	2	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	—	4	2	no	—
171	C₂²×C₃	—	4	2	no	—
172	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
173	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
174	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
175	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	4	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	—	4	2	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	—	4	2	no	—
184	C₂³	—	2	1	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	2	no	—
194	C₂³	C₂²×C₃×C₄	1	4	yes	—
195	C₂³	C₂²×C₃×C₄	1	2	yes	—
196	C₂³	C₂⁴×C₃	1	1	yes	characteristic
197	D₄	—	4	4	no	—
198	D₄	—	4	8	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	—	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	—	4	4	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
218	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
219	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
220	C₂×C₃	—	4	2	no	—
221	C₂×C₃	—	4	4	no	—
222	C₂×C₃	#518	1	1	yes	characteristic
223	C₂×C₃	#518	1	1	yes	characteristic
224	C₂×C₃	#520	1	1	yes	characteristic
225	C₂×C₃	#560	1	2	yes	—
226	C₂×C₃	#562	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
240	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
242	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
243	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₃	#2377	1	1	yes	characteristic, direct product factor, Sylow
252	C₂	—	4	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₃×#518	1	1	yes	characteristic
255	C₂	C₃×#518	1	1	yes	characteristic
256	C₂	C₃×#520	1	1	yes	characteristic
257	C₂	C₃×#560	1	2	yes	—
258	C₂	C₃×#562	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. C₃×#390	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×#381	C₂	semidirect
9. C₃×#380	C₂	semidirect
10. #2377	C₃	direct (N × H)
15. C₃×#121	C₄	semidirect
34. #390	C₂×C₃	semidirect
35. #388	C₂×C₃	semidirect
36. #387	C₂×C₃	semidirect
37. #386	C₂×C₃	semidirect
38. #386	C₂×C₃	semidirect
39. #385	C₂×C₃	semidirect
40. #381	C₂×C₃	semidirect
41. #380	C₂×C₃	semidirect
48. C₂×C₃×D₄	C₂×C₄	semidirect
53. C₂²×C₃×C₄	C₂×C₄	semidirect
63. C₃×C₄⋊C₄	C₂×C₄	semidirect
70. C₃×C₂²⋊C₄	C₂×C₄	semidirect
76. #121	C₃×C₄	semidirect
136. C₂×D₄	C₂×C₃×C₄	semidirect
141. C₂²×C₄	C₂×C₃×C₄	semidirect
151. C₄⋊C₄	C₂×C₃×C₄	semidirect
158. C₂²⋊C₄	C₂×C₃×C₄	semidirect
251. C₃	#2377	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
C₃×#390	2
C₃×#388	2
C₃×#387	2
C₃×#386	2
C₃×#385	2
C₃×#381	2
C₃×#380	2
#2377	3

Subgroups — C3×#2377 #84241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2377 #84241