Subgroups — C₃×#2562 #84426

Order 384.

532 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#400	C₂	1	2	yes	maximal, complemented
4	C₃×#394	C₂	1	2	yes	maximal
5	C₃×#388	C₂	1	4	yes	maximal, complemented
6	C₃×#380	C₂	1	2	yes	maximal, complemented
7	C₃×#380	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#2562	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₃×#125	—	2	4	no	—
13	C₃×C₄×D₄	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
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	4	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₃×C₄²	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#95	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	4	yes	—
27	C₃×#95	C₂²	1	4	yes	—
28	C₂×#125	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
30	#400	C₂×C₃	1	2	yes	complemented
31	#394	C₂×C₃	1	2	yes	—
32	#388	C₂×C₃	1	4	yes	complemented
33	#380	C₂×C₃	1	2	yes	complemented
34	#380	C₂×C₃	1	2	yes	complemented
35	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂⁴×C₃	C₂³	1	2	yes	—
37	C₂×C₃×D₄	—	2	4	no	—
38	C₂×C₃×D₄	—	2	2	no	—
39	C₂²×C₃×C₄	—	2	4	no	—
40	C₂²×C₃×C₄	—	2	4	no	—
41	C₂²×C₃×C₄	—	2	4	no	—
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₂²×C₃×C₄	C₂³	1	4	yes	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	2	yes	—
46	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₃×C₄⋊C₄	—	2	2	no	—
50	C₃×C₄⋊C₄	—	2	4	no	—
51	C₃×C₄⋊C₄	—	2	4	no	—
52	C₃×C₄⋊C₄	—	2	2	no	—
53	C₃×C₄⋊C₄	—	2	2	no	—
54	C₃×C₄⋊C₄	—	2	4	no	—
55	C₃×C₂²⋊C₄	—	2	8	no	—
56	C₃×C₂²⋊C₄	—	2	4	no	—
57	C₃×C₂²⋊C₄	—	4	4	no	—
58	C₃×C₄²	—	4	2	no	—
59	C₃×C₄²	D₄	1	4	yes	—
60	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
61	C₂³×C₄	C₂²×C₃	1	2	yes	—
62	#125	—	2	4	no	—
63	C₄×D₄	—	2	4	no	—
64	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
69	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
72	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
73	C₂×C₄²	C₂²×C₃	1	2	yes	—
74	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
75	#95	C₂²×C₃	1	2	yes	—
76	#95	C₂²×C₃	1	4	yes	—
77	#95	C₂²×C₃	1	4	yes	—
78	C₂³×C₃	—	2	2	no	—
79	C₂³×C₃	—	2	2	no	—
80	C₂³×C₃	—	2	2	no	—
81	C₂³×C₃	—	2	2	no	—
82	C₂³×C₃	—	2	2	no	—
83	C₂³×C₃	—	2	2	no	—
84	C₂³×C₃	#41	1	4	yes	—
85	C₂³×C₃	C₂⁴	1	1	yes	characteristic
86	C₃×D₄	—	2	8	no	—
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	—	2	1	no	—
89	C₂×C₃×C₄	—	2	1	no	—
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	4	no	—
94	C₂×C₃×C₄	—	2	4	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	4	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	4	8	no	—
102	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
103	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
104	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
105	C₂×C₃×C₄	#41	1	4	yes	—
106	C₂⁴	C₂³×C₃	1	2	yes	—
107	C₂×D₄	—	2	4	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	C₂³×C₃	1	2	yes	—
113	C₂²×C₄	C₂³×C₃	1	4	yes	—
114	C₂²×C₄	C₂³×C₃	1	2	yes	—
115	C₂²×C₄	C₂³×C₃	1	2	yes	—
116	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	—	2	8	no	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	4	4	no	—
128	C₄²	—	4	2	no	—
129	C₄²	C₃×D₄	1	4	yes	—
130	C₂²×C₃	—	2	4	no	—
131	C₂²×C₃	—	2	4	no	—
132	C₂²×C₃	—	2	4	no	—
133	C₂²×C₃	—	4	2	no	—
134	C₂²×C₃	—	4	2	no	—
135	C₂²×C₃	—	4	2	no	—
136	C₂²×C₃	—	4	2	no	—
137	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₃	C₂×#41	1	2	yes	—
139	C₂²×C₃	C₂×#41	1	2	yes	—
140	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	2	2	no	—
144	C₃×C₄	—	4	2	no	—
145	C₃×C₄	—	4	2	no	—
146	C₃×C₄	—	4	2	no	—
147	C₃×C₄	—	4	4	no	—
148	C₃×C₄	#124	1	4	yes	—
149	C₂³	—	2	2	no	—
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	2	no	—
154	C₂³	—	2	2	no	—
155	C₂³	C₃×#41	1	4	yes	—
156	C₂³	C₂⁴×C₃	1	1	yes	characteristic
157	D₄	—	2	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
174	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₄	C₃×#41	1	4	yes	—
177	C₂×C₃	—	4	4	no	—
178	C₂×C₃	#525	1	1	yes	characteristic
179	C₂×C₃	C₂×#124	1	1	yes	characteristic
180	C₂×C₃	#537	1	1	yes	characteristic
181	C₂×C₃	#552	1	2	yes	—
182	C₂×C₃	#553	1	2	yes	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	4	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
191	C₂²	C₂×C₃×#41	1	2	yes	—
192	C₂²	C₂×C₃×#41	1	2	yes	—
193	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
194	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	4	no	—
201	C₄	C₃×#124	1	4	yes	—
202	C₃	#2562	1	1	yes	characteristic, direct product factor, Sylow
203	C₂	—	4	4	no	—
204	C₂	C₃×#525	1	1	yes	characteristic
205	C₂	C₂×C₃×#124	1	1	yes	characteristic
206	C₂	C₃×#537	1	1	yes	characteristic
207	C₂	C₃×#552	1	2	yes	—
208	C₂	C₃×#553	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	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×#125	C₂	semidirect
3. C₃×#400	C₂	semidirect
5. C₃×#388	C₂	semidirect
6. C₃×#380	C₂	semidirect
7. C₃×#380	C₂	semidirect
8. C₃×C₄×C₄⋊C₄	C₂	semidirect
9. #2562	C₃	direct (N × H)
28. C₂×#125	C₂×C₃	semidirect
30. #400	C₂×C₃	semidirect
32. #388	C₂×C₃	semidirect
33. #380	C₂×C₃	semidirect
34. #380	C₂×C₃	semidirect
35. C₄×C₄⋊C₄	C₂×C₃	semidirect
202. C₃	#2562	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×#125	2
C₂×C₃×C₄×D₄	2
C₃×#400	2
C₃×#394	2
C₃×#388	2
C₃×#380	2
C₃×C₄×C₄⋊C₄	2
#2562	3

Subgroups — C3×#2562 #84426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2562 #84426