Subgroups — C₃×#2482 #84346

Order 384.

672 conjugacy classes of subgroups.

The subgroup lattice has 218 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, complemented
2	C₂²×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#399	C₂	1	2	yes	maximal, complemented
4	C₃×#397	C₂	1	2	yes	maximal, complemented
5	C₃×#388	C₂	1	4	yes	maximal, complemented
6	C₃×#385	C₂	1	2	yes	maximal, complemented
7	C₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#379	C₂	1	2	yes	maximal
9	#2482	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂⁵×C₃	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₃×#117	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
17	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
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	4	yes	—
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	4	yes	—
24	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
25	C₃×#95	C₂²	1	2	yes	complemented
26	C₃×#95	C₂²	1	2	yes	complemented
27	C₃×#95	C₂²	1	4	yes	complemented
28	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
30	#399	C₂×C₃	1	2	yes	complemented
31	#397	C₂×C₃	1	2	yes	complemented
32	#388	C₂×C₃	1	4	yes	complemented
33	#385	C₂×C₃	1	2	yes	complemented
34	#381	C₂×C₃	1	1	yes	characteristic, complemented
35	#379	C₂×C₃	1	2	yes	—
36	C₂⁴×C₃	—	2	2	no	—
37	C₂⁴×C₃	—	2	4	no	—
38	C₂⁴×C₃	—	2	4	no	—
39	C₂⁴×C₃	—	4	2	no	—
40	C₂⁴×C₃	C₂³	1	2	yes	—
41	C₂⁴×C₃	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₄	—	2	4	no	—
43	C₂²×C₃×C₄	—	2	2	no	—
44	C₂²×C₃×C₄	—	4	2	no	—
45	C₂²×C₃×C₄	D₄	1	4	yes	—
46	C₂²×C₃×C₄	C₂³	1	4	yes	—
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	C₂³	1	2	yes	—
50	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₃×C₄⋊C₄	—	2	2	no	—
53	C₃×C₄⋊C₄	—	2	2	no	—
54	C₃×C₄⋊C₄	—	4	4	no	—
55	C₃×C₂²⋊C₄	—	2	4	no	—
56	C₃×C₂²⋊C₄	—	2	8	no	—
57	C₃×C₂²⋊C₄	—	2	4	no	—
58	C₃×C₂²⋊C₄	—	2	4	no	—
59	C₃×C₂²⋊C₄	—	2	4	no	—
60	C₃×C₂²⋊C₄	—	4	4	no	—
61	C₃×C₄²	—	2	4	no	—
62	C₂⁵	C₂²×C₃	1	1	yes	characteristic
63	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
64	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
65	#117	—	2	4	no	—
66	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
67	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
69	C₂×C₂²⋊C₄	—	2	4	no	—
70	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
75	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
76	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
77	#95	C₂²×C₃	1	2	yes	complemented
78	#95	C₂²×C₃	1	2	yes	complemented
79	#95	C₂²×C₃	1	4	yes	complemented
80	C₂³×C₃	—	2	2	no	—
81	C₂³×C₃	—	2	2	no	—
82	C₂³×C₃	—	2	8	no	—
83	C₂³×C₃	—	2	2	no	—
84	C₂³×C₃	—	2	2	no	—
85	C₂³×C₃	—	2	4	no	—
86	C₂³×C₃	—	2	4	no	—
87	C₂³×C₃	—	4	16	no	—
88	C₂³×C₃	—	4	4	no	—
89	C₂³×C₃	—	4	4	no	—
90	C₂³×C₃	C₂×D₄	1	2	yes	—
91	C₂³×C₃	#41	1	4	yes	—
92	C₂³×C₃	#41	1	4	yes	—
93	C₂³×C₃	C₂⁴	1	1	yes	characteristic
94	C₂×C₃×C₄	—	2	8	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	—	2	4	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	4	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	4	4	no	—
107	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
108	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
109	C₂⁴	—	2	2	no	—
110	C₂⁴	—	2	4	no	—
111	C₂⁴	—	2	4	no	—
112	C₂⁴	—	4	2	no	—
113	C₂⁴	C₂³×C₃	1	2	yes	—
114	C₂⁴	C₂³×C₃	1	1	yes	characteristic
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	C₃×D₄	1	4	yes	—
119	C₂²×C₄	C₂³×C₃	1	4	yes	—
120	C₂²×C₄	C₂³×C₃	1	2	yes	—
121	C₂²×C₄	C₂³×C₃	1	2	yes	—
122	C₂²×C₄	C₂³×C₃	1	2	yes	—
123	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₂²⋊C₄	—	2	4	no	—
129	C₂²⋊C₄	—	2	8	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	4	4	no	—
134	C₄²	—	2	4	no	—
135	C₂²×C₃	—	2	2	no	—
136	C₂²×C₃	—	2	2	no	—
137	C₂²×C₃	—	2	8	no	—
138	C₂²×C₃	—	2	4	no	—
139	C₂²×C₃	—	2	8	no	—
140	C₂²×C₃	—	4	16	no	—
141	C₂²×C₃	—	4	2	no	—
142	C₂²×C₃	—	4	2	no	—
143	C₂²×C₃	—	4	4	no	—
144	C₂²×C₃	#123	1	4	yes	—
145	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
146	C₂²×C₃	C₂×#41	1	2	yes	—
147	C₂²×C₃	C₂×#41	1	2	yes	—
148	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
149	C₃×C₄	—	2	2	no	—
150	C₃×C₄	—	2	2	no	—
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	—	4	2	no	—
153	C₃×C₄	—	4	2	no	—
154	C₃×C₄	—	4	4	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	8	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	4	no	—
161	C₂³	—	2	4	no	—
162	C₂³	—	4	16	no	—
163	C₂³	—	4	4	no	—
164	C₂³	—	4	4	no	—
165	C₂³	C₂×C₃×D₄	1	2	yes	—
166	C₂³	C₃×#41	1	4	yes	—
167	C₂³	C₃×#41	1	4	yes	—
168	C₂³	C₂⁴×C₃	1	1	yes	characteristic
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	—	4	4	no	—
186	C₂×C₃	C₂×#123	1	1	yes	characteristic
187	C₂×C₃	#525	1	1	yes	characteristic
188	C₂×C₃	#535	1	1	yes	characteristic
189	C₂×C₃	#551	1	4	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	8	no	—
195	C₂²	—	4	16	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	4	no	—
199	C₂²	C₃×#123	1	4	yes	—
200	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
201	C₂²	C₂×C₃×#41	1	2	yes	—
202	C₂²	C₂×C₃×#41	1	2	yes	—
203	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	4	no	—
210	C₃	#2482	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	4	no	—
212	C₂	—	4	4	no	—
213	C₂	C₂×C₃×#123	1	1	yes	characteristic
214	C₂	C₃×#525	1	1	yes	characteristic
215	C₂	C₃×#535	1	1	yes	characteristic
216	C₂	C₃×#551	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×#117	C₂	semidirect
3. C₃×#399	C₂	semidirect
4. C₃×#397	C₂	semidirect
5. C₃×#388	C₂	semidirect
6. C₃×#385	C₂	semidirect
7. C₃×#381	C₂	semidirect
9. #2482	C₃	direct (N × H)
14. C₂×C₃×C₄⋊C₄	C₂²	semidirect
15. C₂×C₃×C₄⋊C₄	C₂²	semidirect
16. C₂×C₃×C₄⋊C₄	C₂²	semidirect
24. C₂×C₃×C₄²	C₂²	semidirect
25. C₃×#95	C₂²	semidirect
26. C₃×#95	C₂²	semidirect
27. C₃×#95	C₂²	semidirect
28. C₂×#117	C₂×C₃	semidirect
30. #399	C₂×C₃	semidirect
31. #397	C₂×C₃	semidirect
32. #388	C₂×C₃	semidirect
33. #385	C₂×C₃	semidirect
34. #381	C₂×C₃	semidirect
66. C₂×C₄⋊C₄	C₂²×C₃	semidirect
67. C₂×C₄⋊C₄	C₂²×C₃	semidirect
68. C₂×C₄⋊C₄	C₂²×C₃	semidirect
76. C₂×C₄²	C₂²×C₃	semidirect
77. #95	C₂²×C₃	semidirect
78. #95	C₂²×C₃	semidirect
79. #95	C₂²×C₃	semidirect
210. C₃	#2482	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
C₂²×C₃×C₂²⋊C₄	2
C₃×#399	2
C₃×#397	2
C₃×#388	2
C₃×#385	2
C₃×#381	2
C₃×#379	2
#2482	3

Subgroups — C3×#2482 #84346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2482 #84346