Subgroups — C₃×#2664 #84528

Order 384.

434 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×#126	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#399	C₂	1	1	yes	maximal, characteristic
5	C₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#398	C₂	1	2	yes	maximal, complemented
7	C₃×#397	C₂	1	1	yes	maximal, characteristic
8	C₃×#396	C₂	1	2	yes	maximal
9	C₃×#389	C₂	1	2	yes	maximal, complemented
10	C₃×#388	C₂	1	2	yes	maximal
11	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#2664	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃×#126	—	2	4	no	—
14	C₃×#125	—	2	4	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₃×C₄²	C₂²	1	2	yes	—
31	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
32	C₃×#95	C₂²	1	1	yes	characteristic
33	C₃×#95	C₂²	1	1	yes	characteristic
34	C₃×#95	C₂²	1	2	yes	—
35	C₃×#95	C₂²	1	2	yes	—
36	C₃×#95	C₂²	1	2	yes	—
37	C₃×#95	C₂²	1	2	yes	—
38	C₂×#126	C₂×C₃	1	1	yes	characteristic
39	C₂×#125	C₂×C₃	1	1	yes	characteristic, complemented
40	#400	C₂×C₃	1	1	yes	characteristic, complemented
41	#399	C₂×C₃	1	1	yes	characteristic
42	#398	C₂×C₃	1	1	yes	characteristic, complemented
43	#398	C₂×C₃	1	2	yes	complemented
44	#397	C₂×C₃	1	1	yes	characteristic
45	#396	C₂×C₃	1	2	yes	—
46	#389	C₂×C₃	1	2	yes	complemented
47	#388	C₂×C₃	1	2	yes	—
48	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂⁴×C₃	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₃×C₄⋊C₄	—	2	4	no	—
61	C₃×C₄⋊C₄	—	2	4	no	—
62	C₃×C₄⋊C₄	—	2	2	no	—
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	4	no	—
68	C₃×C₄⋊C₄	—	2	4	no	—
69	C₃×C₄⋊C₄	—	4	2	no	—
70	C₃×C₂²⋊C₄	—	2	2	no	—
71	C₃×C₂²⋊C₄	—	2	2	no	—
72	C₃×C₂²⋊C₄	—	2	2	no	—
73	C₃×C₂²⋊C₄	—	2	4	no	—
74	C₃×C₂²⋊C₄	—	4	2	no	—
75	C₃×C₄²	—	4	2	no	—
76	C₃×C₄²	D₄	1	4	yes	—
77	#126	—	2	4	no	—
78	#125	—	2	4	no	—
79	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
92	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
93	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
94	C₂×C₄²	C₂²×C₃	1	2	yes	—
95	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
96	#95	C₂²×C₃	1	1	yes	characteristic
97	#95	C₂²×C₃	1	1	yes	characteristic
98	#95	C₂²×C₃	1	2	yes	—
99	#95	C₂²×C₃	1	2	yes	—
100	#95	C₂²×C₃	1	2	yes	—
101	#95	C₂²×C₃	1	2	yes	—
102	C₂³×C₃	—	2	1	no	—
103	C₂³×C₃	—	2	2	no	—
104	C₂³×C₃	—	2	1	no	—
105	C₂³×C₃	—	2	2	no	—
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	C₂⁴	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	1	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	1	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	—	2	1	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
131	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
133	C₂×C₃×C₄	#41	1	4	yes	—
134	C₂⁴	C₂³×C₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₃	1	2	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	C₃×D₄	1	4	yes	—
162	C₂²×C₃	—	4	1	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	C₂²×D₄	1	1	yes	characteristic
168	C₂²×C₃	C₂×#41	1	2	yes	—
169	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
171	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	4	no	—
176	C₃×C₄	—	4	1	no	—
177	C₃×C₄	—	4	2	no	—
178	C₃×C₄	—	4	1	no	—
179	C₃×C₄	—	4	1	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	—	4	2	no	—
182	C₂³	—	2	1	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	1	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	1	no	—
187	C₂³	C₂⁴×C₃	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	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	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
211	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
212	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₄	C₃×#41	1	4	yes	—
214	C₂×C₃	—	8	1	no	—
215	C₂×C₃	#532	1	1	yes	characteristic
216	C₂×C₃	#536	1	1	yes	characteristic
217	C₂×C₃	#537	1	1	yes	characteristic
218	C₂×C₃	#539	1	2	yes	—
219	C₂×C₃	#541	1	2	yes	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
226	C₂²	C₂×C₃×#41	1	2	yes	—
227	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
228	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
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	4	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₃	#2664	1	1	yes	characteristic, direct product factor, Sylow
241	C₂	—	8	1	no	—
242	C₂	C₃×#532	1	1	yes	characteristic
243	C₂	C₃×#536	1	1	yes	characteristic
244	C₂	C₃×#537	1	1	yes	characteristic
245	C₂	C₃×#539	1	2	yes	—
246	C₂	C₃×#541	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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
2. C₂×C₃×#125	C₂	semidirect
3. C₃×#400	C₂	semidirect
5. C₃×#398	C₂	semidirect
6. C₃×#398	C₂	semidirect
9. C₃×#389	C₂	semidirect
11. C₃×C₄×C₄⋊C₄	C₂	semidirect
12. #2664	C₃	direct (N × H)
39. C₂×#125	C₂×C₃	semidirect
40. #400	C₂×C₃	semidirect
42. #398	C₂×C₃	semidirect
43. #398	C₂×C₃	semidirect
46. #389	C₂×C₃	semidirect
48. C₄×C₄⋊C₄	C₂×C₃	semidirect
240. C₃	#2664	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×#126	2
C₂×C₃×#125	2
C₃×#400	2
C₃×#399	2
C₃×#398	2
C₃×#397	2
C₃×#396	2
C₃×#389	2
C₃×#388	2
C₃×C₄×C₄⋊C₄	2
#2664	3

Subgroups — C3×#2664 #84528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2664 #84528