Subgroups — C₂×C₃×#435 #84791

Order 384.

564 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#435	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#435	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₄	C₄	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×M₁₆	—	2	4	no	—
14	C₂×C₃×M₁₆	C₂²	1	4	yes	complemented
15	C₂²×C₃×C₈	—	2	4	no	—
16	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
20	C₂²×C₃×C₈	C₂²	1	4	yes	complemented
21	C₃×C₄×D₄	C₄	1	4	yes	—
22	C₃×C₄×D₄	C₂²	1	4	yes	—
23	C₂×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
24	C₂×C₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
25	C₂×C₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊C₈	C₂²	1	4	yes	complemented
28	C₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
29	C₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
30	C₃×#97	C₂²	1	4	yes	complemented
31	C₂²×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂³×C₈	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
34	#435	C₂×C₃	1	8	yes	direct product factor
35	C₂×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×#97	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
40	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×D₄	—	2	4	no	—
42	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
43	C₂²×C₃×C₄	—	2	2	no	—
44	C₂²×C₃×C₄	—	2	2	no	—
45	C₂²×C₃×C₄	—	2	2	no	—
46	C₂²×C₃×C₄	—	2	2	no	—
47	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
52	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₃×M₁₆	—	2	8	no	—
57	C₂×C₃×C₈	—	2	2	no	—
58	C₂×C₃×C₈	—	2	2	no	—
59	C₂×C₃×C₈	—	2	2	no	—
60	C₂×C₃×C₈	—	2	8	no	—
61	C₂×C₃×C₈	D₄	1	4	yes	complemented
62	C₂×C₃×C₈	C₂³	1	2	yes	—
63	C₂×C₃×C₈	C₂³	1	2	yes	—
64	C₂×C₃×C₈	C₂³	1	2	yes	—
65	C₂×C₃×C₈	C₂³	1	2	yes	—
66	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
67	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
68	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
69	C₃×C₄²	C₂×C₄	1	2	yes	—
70	C₃×C₄²	C₂³	1	2	yes	—
71	C₂²×D₄	C₃×C₄	1	1	yes	characteristic
72	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂×M₁₆	—	2	4	no	—
75	C₂×M₁₆	C₂²×C₃	1	4	yes	complemented
76	C₂²×C₈	—	2	4	no	—
77	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
78	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
80	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₃	1	4	yes	complemented
82	C₄×D₄	C₃×C₄	1	4	yes	—
83	C₄×D₄	C₂²×C₃	1	4	yes	—
84	C₂×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
87	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
88	C₄⋊C₈	C₂²×C₃	1	4	yes	complemented
89	C₂²⋊C₈	C₂²×C₃	1	4	yes	complemented
90	C₂²⋊C₈	C₂²×C₃	1	4	yes	complemented
91	#97	C₂²×C₃	1	4	yes	complemented
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	2	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	M₁₆	1	4	yes	complemented
98	C₂³×C₃	C₂²×C₄	1	2	yes	—
99	C₂³×C₃	C₂²×C₄	1	2	yes	—
100	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
101	C₃×D₄	—	2	8	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	2	4	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	2	no	—
112	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
113	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
114	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
115	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
116	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
118	C₂×C₃×C₄	#41	1	2	yes	—
119	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
120	C₃×C₈	—	2	2	no	—
121	C₃×C₈	—	2	2	no	—
122	C₃×C₈	—	2	2	no	—
123	C₃×C₈	C₂×D₄	1	4	yes	complemented
124	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
137	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
138	C₂²×C₄	C₂³×C₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
141	M₁₆	—	2	8	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	C₃×D₄	1	4	yes	complemented
147	C₂×C₈	C₂³×C₃	1	2	yes	—
148	C₂×C₈	C₂³×C₃	1	2	yes	—
149	C₂×C₈	C₂³×C₃	1	2	yes	—
150	C₂×C₈	C₂³×C₃	1	2	yes	—
151	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
152	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
153	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
154	C₄²	C₂×C₃×C₄	1	2	yes	—
155	C₄²	C₂³×C₃	1	2	yes	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	—	2	2	no	—
158	C₂²×C₃	—	2	8	no	—
159	C₂²×C₃	—	2	2	no	—
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	4	4	no	—
163	C₂²×C₃	C₄×D₄	1	2	yes	—
164	C₂²×C₃	C₂×M₁₆	1	2	yes	—
165	C₂²×C₃	C₂×M₁₆	1	4	yes	complemented
166	C₂²×C₃	#131	1	2	yes	—
167	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
168	C₃×C₄	—	2	2	no	—
169	C₃×C₄	—	2	2	no	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	C₄×D₄	1	2	yes	—
173	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
174	C₃×C₄	C₂×#41	1	1	yes	characteristic
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	C₃×M₁₆	1	4	yes	complemented
181	C₂³	C₂²×C₃×C₄	1	2	yes	—
182	C₂³	C₂²×C₃×C₄	1	2	yes	—
183	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
184	D₄	—	2	8	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
196	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
197	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
198	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
199	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
200	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
201	C₂×C₄	C₃×#41	1	2	yes	—
202	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	C₂×C₃×D₄	1	4	yes	complemented
207	C₂×C₃	—	2	4	no	—
208	C₂×C₃	—	4	2	no	—
209	C₂×C₃	#435	1	4	yes	direct product factor
210	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
212	C₂×C₃	C₂×#131	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	4	4	no	—
220	C₂²	C₃×C₄×D₄	1	2	yes	—
221	C₂²	C₂×C₃×M₁₆	1	2	yes	—
222	C₂²	C₂×C₃×M₁₆	1	4	yes	complemented
223	C₂²	C₃×#131	1	2	yes	—
224	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	C₃×C₄×D₄	1	2	yes	—
230	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
231	C₄	C₂×C₃×#41	1	1	yes	characteristic
232	C₃	C₂×#435	1	1	yes	characteristic, direct product factor, Sylow
233	C₂	—	2	4	no	—
234	C₂	—	4	2	no	—
235	C₂	C₃×#435	1	4	yes	direct product factor
236	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
237	C₂	C₂²×C₃×M₁₆	1	1	yes	characteristic
238	C₂	C₂×C₃×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂²×C₃×D₄	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×M₁₆	C₂	semidirect
2. C₂³×C₃×C₈	C₂	semidirect
4. C₃×#435	C₂	direct (N × H)
5. C₂×C₃×C₄⋊C₈	C₂	semidirect
6. C₂×C₃×C₂²⋊C₈	C₂	semidirect
7. C₂×C₃×C₂²⋊C₈	C₂	semidirect
8. C₂×C₃×#97	C₂	semidirect
9. C₂×#435	C₃	direct (N × H)
14. C₂×C₃×M₁₆	C₂²	semidirect
20. C₂²×C₃×C₈	C₂²	semidirect
27. C₃×C₄⋊C₈	C₂²	semidirect
28. C₃×C₂²⋊C₈	C₂²	semidirect
29. C₃×C₂²⋊C₈	C₂²	semidirect
30. C₃×#97	C₂²	semidirect
31. C₂²×M₁₆	C₂×C₃	semidirect
32. C₂³×C₈	C₂×C₃	semidirect
34. #435	C₂×C₃	direct (N × H)
35. C₂×C₄⋊C₈	C₂×C₃	semidirect
36. C₂×C₂²⋊C₈	C₂×C₃	semidirect
37. C₂×C₂²⋊C₈	C₂×C₃	semidirect
38. C₂×#97	C₂×C₃	semidirect
61. C₂×C₃×C₈	D₄	semidirect
75. C₂×M₁₆	C₂²×C₃	semidirect
81. C₂²×C₈	C₂²×C₃	semidirect
88. C₄⋊C₈	C₂²×C₃	semidirect
89. C₂²⋊C₈	C₂²×C₃	semidirect
90. C₂²⋊C₈	C₂²×C₃	semidirect
91. #97	C₂²×C₃	semidirect
97. C₂³×C₃	M₁₆	semidirect
123. C₃×C₈	C₂×D₄	semidirect
146. C₂×C₈	C₃×D₄	semidirect
165. C₂²×C₃	C₂×M₁₆	semidirect
180. C₂³	C₃×M₁₆	semidirect
206. C₈	C₂×C₃×D₄	semidirect
209. C₂×C₃	#435	direct (N × H)
222. C₂²	C₂×C₃×M₁₆	semidirect
232. C₃	C₂×#435	direct (N × H)
235. C₂	C₃×#435	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×M₁₆	2
C₂³×C₃×C₈	2
C₂×C₃×C₄×D₄	2
C₃×#435	2
C₂×C₃×C₄⋊C₈	2
C₂×C₃×C₂²⋊C₈	2
C₂×C₃×#97	2
C₂×#435	3

Subgroups — C2×C3×#435 #84791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#435 #84791