Subgroups — C₂×C₃³×#121 #13179977

Order 1728.

5964 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×C₃³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃³×D₄	C₂	1	2	yes	maximal, complemented
3	C₂³×C₃³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×#121	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃³×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃³×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂×C₃²×#121	C₃	1	13	yes	maximal, direct product factor
8	C₂⁴×C₃³	C₂²	1	2	yes	—
9	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
10	C₂×C₃³×D₄	—	2	8	no	—
11	C₂×C₃³×D₄	—	2	4	no	—
12	C₂×C₃³×D₄	C₂²	1	8	yes	complemented
13	C₂×C₃³×D₄	C₂²	1	4	yes	complemented
14	C₂²×C₃³×C₄	—	2	4	no	—
15	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃³×C₄	C₂²	1	2	yes	—
17	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
18	C₂²×C₃³×C₄	C₂²	1	4	yes	complemented
19	C₃³×C₄⋊C₄	C₂²	1	4	yes	complemented
20	C₃³×C₂²⋊C₄	C₂²	1	8	yes	complemented
21	C₂²×C₃²×D₄	C₂×C₃	1	13	yes	complemented
22	C₂²×C₃²×D₄	C₂×C₃	1	26	yes	complemented
23	C₂³×C₃²×C₄	C₂×C₃	1	13	yes	complemented
24	C₃²×#121	C₂×C₃	1	104	yes	direct product factor
25	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	13	yes	complemented
26	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	26	yes	complemented
27	C₂³×C₃³	—	2	2	no	—
28	C₂³×C₃³	—	2	2	no	—
29	C₂³×C₃³	—	2	4	no	—
30	C₂³×C₃³	—	2	4	no	—
31	C₂³×C₃³	—	2	4	no	—
32	C₂³×C₃³	D₄	1	4	yes	complemented
33	C₂³×C₃³	C₂³	1	4	yes	—
34	C₂³×C₃³	C₂³	1	2	yes	—
35	C₂³×C₃³	C₂³	1	1	yes	characteristic
36	C₃³×D₄	—	2	16	no	—
37	C₃³×D₄	—	2	8	no	—
38	C₂×C₃³×C₄	—	2	2	no	—
39	C₂×C₃³×C₄	—	2	4	no	—
40	C₂×C₃³×C₄	—	2	8	no	—
41	C₂×C₃³×C₄	D₄	1	4	yes	complemented
42	C₂×C₃³×C₄	C₂³	1	2	yes	—
43	C₂×C₃³×C₄	C₂³	1	4	yes	—
44	C₂×C₃³×C₄	C₂³	1	2	yes	—
45	C₂×C₃×#121	C₃²	1	13	yes	direct product factor
46	C₂⁴×C₃²	C₂²×C₃	1	26	yes	—
47	C₂⁴×C₃²	C₂²×C₃	1	13	yes	—
48	C₂×C₃²×D₄	—	2	104	no	—
49	C₂×C₃²×D₄	—	2	52	no	—
50	C₂×C₃²×D₄	C₂²×C₃	1	104	yes	complemented
51	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	complemented
52	C₂²×C₃²×C₄	—	2	52	no	—
53	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
54	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
55	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
56	C₂²×C₃²×C₄	C₂²×C₃	1	52	yes	complemented
57	C₃²×C₄⋊C₄	C₂²×C₃	1	52	yes	complemented
58	C₃²×C₂²⋊C₄	C₂²×C₃	1	104	yes	complemented
59	C₂²×C₃³	—	2	8	no	—
60	C₂²×C₃³	—	2	2	no	—
61	C₂²×C₃³	—	2	2	no	—
62	C₂²×C₃³	—	2	4	no	—
63	C₂²×C₃³	—	2	4	no	—
64	C₂²×C₃³	—	2	4	no	—
65	C₂²×C₃³	—	4	8	no	—
66	C₂²×C₃³	C₂×D₄	1	2	yes	—
67	C₂²×C₃³	C₂×D₄	1	2	yes	—
68	C₂²×C₃³	C₂×D₄	1	4	yes	complemented
69	C₂²×C₃³	#41	1	2	yes	—
70	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
71	C₃³×C₄	—	2	2	no	—
72	C₃³×C₄	—	2	4	no	—
73	C₃³×C₄	C₂×D₄	1	4	yes	complemented
74	C₂²×C₃×D₄	C₂×C₃²	1	26	yes	complemented
75	C₂²×C₃×D₄	C₂×C₃²	1	13	yes	complemented
76	C₂³×C₃×C₄	C₂×C₃²	1	13	yes	complemented
77	C₃×#121	C₂×C₃²	1	104	yes	direct product factor
78	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	13	yes	complemented
79	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	26	yes	complemented
80	C₂³×C₃²	—	2	26	no	—
81	C₂³×C₃²	—	2	26	no	—
82	C₂³×C₃²	—	2	52	no	—
83	C₂³×C₃²	—	2	52	no	—
84	C₂³×C₃²	—	2	52	no	—
85	C₂³×C₃²	C₃×D₄	1	52	yes	complemented
86	C₂³×C₃²	C₂³×C₃	1	52	yes	—
87	C₂³×C₃²	C₂³×C₃	1	26	yes	—
88	C₂³×C₃²	C₂³×C₃	1	13	yes	—
89	C₃²×D₄	—	2	208	no	—
90	C₃²×D₄	—	2	104	no	—
91	C₂×C₃²×C₄	—	2	26	no	—
92	C₂×C₃²×C₄	—	2	52	no	—
93	C₂×C₃²×C₄	—	2	104	no	—
94	C₂×C₃²×C₄	C₃×D₄	1	52	yes	complemented
95	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
96	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
97	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
98	C₂×#121	C₃³	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×C₃³	—	2	4	no	—
100	C₂×C₃³	—	4	4	no	—
101	C₂×C₃³	#121	1	4	yes	direct product factor
102	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₃³	C₂×#41	1	1	yes	characteristic
105	C₂⁴×C₃	C₂²×C₃²	1	26	yes	—
106	C₂⁴×C₃	C₂²×C₃²	1	13	yes	—
107	C₂×C₃×D₄	—	2	104	no	—
108	C₂×C₃×D₄	—	2	52	no	—
109	C₂×C₃×D₄	C₂²×C₃²	1	104	yes	complemented
110	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	complemented
111	C₂²×C₃×C₄	—	2	52	no	—
112	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
113	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
114	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
115	C₂²×C₃×C₄	C₂²×C₃²	1	52	yes	complemented
116	C₃×C₄⋊C₄	C₂²×C₃²	1	52	yes	complemented
117	C₃×C₂²⋊C₄	C₂²×C₃²	1	104	yes	complemented
118	C₂²×C₃²	—	2	104	no	—
119	C₂²×C₃²	—	2	26	no	—
120	C₂²×C₃²	—	2	26	no	—
121	C₂²×C₃²	—	2	52	no	—
122	C₂²×C₃²	—	2	52	no	—
123	C₂²×C₃²	—	2	52	no	—
124	C₂²×C₃²	—	4	104	no	—
125	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
126	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
127	C₂²×C₃²	C₂×C₃×D₄	1	52	yes	complemented
128	C₂²×C₃²	C₃×#41	1	26	yes	—
129	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
130	C₃²×C₄	—	2	26	no	—
131	C₃²×C₄	—	2	52	no	—
132	C₃²×C₄	C₂×C₃×D₄	1	52	yes	complemented
133	C₂²×D₄	C₂×C₃³	1	2	yes	complemented
134	C₂²×D₄	C₂×C₃³	1	1	yes	characteristic, complemented
135	C₂³×C₄	C₂×C₃³	1	1	yes	characteristic, complemented
136	#121	C₂×C₃³	1	8	yes	direct product factor
137	C₂×C₄⋊C₄	C₂×C₃³	1	1	yes	characteristic, complemented
138	C₂×C₂²⋊C₄	C₂×C₃³	1	2	yes	complemented
139	C₃³	C₂×#121	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³×C₃	—	2	26	no	—
141	C₂³×C₃	—	2	26	no	—
142	C₂³×C₃	—	2	52	no	—
143	C₂³×C₃	—	2	52	no	—
144	C₂³×C₃	—	2	52	no	—
145	C₂³×C₃	C₃²×D₄	1	52	yes	complemented
146	C₂³×C₃	C₂³×C₃²	1	52	yes	—
147	C₂³×C₃	C₂³×C₃²	1	26	yes	—
148	C₂³×C₃	C₂³×C₃²	1	13	yes	—
149	C₃×D₄	—	2	208	no	—
150	C₃×D₄	—	2	104	no	—
151	C₂×C₃×C₄	—	2	26	no	—
152	C₂×C₃×C₄	—	2	52	no	—
153	C₂×C₃×C₄	—	2	104	no	—
154	C₂×C₃×C₄	C₃²×D₄	1	52	yes	complemented
155	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
156	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
157	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
158	C₂×C₃²	—	2	52	no	—
159	C₂×C₃²	—	4	52	no	—
160	C₂×C₃²	C₃×#121	1	52	yes	direct product factor
161	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
162	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
163	C₂×C₃²	C₂×C₃×#41	1	13	yes	—
164	C₂⁴	C₂²×C₃³	1	2	yes	—
165	C₂⁴	C₂²×C₃³	1	1	yes	characteristic
166	C₂×D₄	—	2	8	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×C₃³	1	8	yes	complemented
169	C₂×D₄	C₂²×C₃³	1	4	yes	complemented
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₃³	1	2	yes	—
173	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₃³	1	4	yes	complemented
175	C₄⋊C₄	C₂²×C₃³	1	4	yes	complemented
176	C₂²⋊C₄	C₂²×C₃³	1	8	yes	complemented
177	C₂²×C₃	—	2	26	no	—
178	C₂²×C₃	—	2	26	no	—
179	C₂²×C₃	—	2	104	no	—
180	C₂²×C₃	—	2	52	no	—
181	C₂²×C₃	—	2	52	no	—
182	C₂²×C₃	—	2	52	no	—
183	C₂²×C₃	—	4	104	no	—
184	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
185	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
186	C₂²×C₃	C₂×C₃²×D₄	1	52	yes	complemented
187	C₂²×C₃	C₃²×#41	1	26	yes	—
188	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
189	C₃×C₄	—	2	26	no	—
190	C₃×C₄	—	2	52	no	—
191	C₃×C₄	C₂×C₃²×D₄	1	52	yes	complemented
192	C₃²	C₂×C₃×#121	1	13	yes	direct product factor
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	4	no	—
198	C₂³	C₃³×D₄	1	4	yes	complemented
199	C₂³	C₂³×C₃³	1	4	yes	—
200	C₂³	C₂³×C₃³	1	2	yes	—
201	C₂³	C₂³×C₃³	1	1	yes	characteristic
202	D₄	—	2	16	no	—
203	D₄	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	C₃³×D₄	1	4	yes	complemented
208	C₂×C₄	C₂³×C₃³	1	2	yes	—
209	C₂×C₄	C₂³×C₃³	1	4	yes	—
210	C₂×C₄	C₂³×C₃³	1	2	yes	—
211	C₂×C₃	—	2	52	no	—
212	C₂×C₃	—	4	52	no	—
213	C₂×C₃	C₃²×#121	1	52	yes	direct product factor
214	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
215	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
216	C₂×C₃	C₂×C₃²×#41	1	13	yes	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	8	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	4	8	no	—
224	C₂²	C₂×C₃³×D₄	1	2	yes	—
225	C₂²	C₂×C₃³×D₄	1	2	yes	—
226	C₂²	C₂×C₃³×D₄	1	4	yes	complemented
227	C₂²	C₃³×#41	1	2	yes	—
228	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	C₂×C₃³×D₄	1	4	yes	complemented
232	C₃	C₂×C₃²×#121	1	13	yes	direct product factor
233	C₂	—	2	4	no	—
234	C₂	—	4	4	no	—
235	C₂	C₃³×#121	1	4	yes	direct product factor
236	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic
237	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic
238	C₂	C₂×C₃³×#41	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃³ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×D₄	C₂	semidirect
2. C₂²×C₃³×D₄	C₂	semidirect
3. C₂³×C₃³×C₄	C₂	semidirect
4. C₃³×#121	C₂	direct (N × H)
5. C₂×C₃³×C₄⋊C₄	C₂	semidirect
6. C₂×C₃³×C₂²⋊C₄	C₂	semidirect
7. C₂×C₃²×#121	C₃	direct (N × H)
12. C₂×C₃³×D₄	C₂²	semidirect
13. C₂×C₃³×D₄	C₂²	semidirect
18. C₂²×C₃³×C₄	C₂²	semidirect
19. C₃³×C₄⋊C₄	C₂²	semidirect
20. C₃³×C₂²⋊C₄	C₂²	semidirect
21. C₂²×C₃²×D₄	C₂×C₃	semidirect
22. C₂²×C₃²×D₄	C₂×C₃	semidirect
23. C₂³×C₃²×C₄	C₂×C₃	semidirect
24. C₃²×#121	C₂×C₃	direct (N × H)
25. C₂×C₃²×C₄⋊C₄	C₂×C₃	semidirect
26. C₂×C₃²×C₂²⋊C₄	C₂×C₃	semidirect
32. C₂³×C₃³	D₄	semidirect
41. C₂×C₃³×C₄	D₄	semidirect
45. C₂×C₃×#121	C₃²	direct (N × H)
50. C₂×C₃²×D₄	C₂²×C₃	semidirect
51. C₂×C₃²×D₄	C₂²×C₃	semidirect
56. C₂²×C₃²×C₄	C₂²×C₃	semidirect
57. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
58. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
68. C₂²×C₃³	C₂×D₄	semidirect
73. C₃³×C₄	C₂×D₄	semidirect
74. C₂²×C₃×D₄	C₂×C₃²	semidirect
75. C₂²×C₃×D₄	C₂×C₃²	semidirect
76. C₂³×C₃×C₄	C₂×C₃²	semidirect
77. C₃×#121	C₂×C₃²	direct (N × H)
78. C₂×C₃×C₄⋊C₄	C₂×C₃²	semidirect
79. C₂×C₃×C₂²⋊C₄	C₂×C₃²	semidirect
85. C₂³×C₃²	C₃×D₄	semidirect
94. C₂×C₃²×C₄	C₃×D₄	semidirect
98. C₂×#121	C₃³	direct (N × H)
101. C₂×C₃³	#121	direct (N × H)
109. C₂×C₃×D₄	C₂²×C₃²	semidirect
110. C₂×C₃×D₄	C₂²×C₃²	semidirect
115. C₂²×C₃×C₄	C₂²×C₃²	semidirect
116. C₃×C₄⋊C₄	C₂²×C₃²	semidirect
117. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
127. C₂²×C₃²	C₂×C₃×D₄	semidirect
132. C₃²×C₄	C₂×C₃×D₄	semidirect
133. C₂²×D₄	C₂×C₃³	semidirect
134. C₂²×D₄	C₂×C₃³	semidirect
135. C₂³×C₄	C₂×C₃³	semidirect
136. #121	C₂×C₃³	direct (N × H)
137. C₂×C₄⋊C₄	C₂×C₃³	semidirect
138. C₂×C₂²⋊C₄	C₂×C₃³	semidirect
139. C₃³	C₂×#121	direct (N × H)
145. C₂³×C₃	C₃²×D₄	semidirect
154. C₂×C₃×C₄	C₃²×D₄	semidirect
160. C₂×C₃²	C₃×#121	direct (N × H)
168. C₂×D₄	C₂²×C₃³	semidirect
169. C₂×D₄	C₂²×C₃³	semidirect
174. C₂²×C₄	C₂²×C₃³	semidirect
175. C₄⋊C₄	C₂²×C₃³	semidirect
176. C₂²⋊C₄	C₂²×C₃³	semidirect
186. C₂²×C₃	C₂×C₃²×D₄	semidirect
191. C₃×C₄	C₂×C₃²×D₄	semidirect
192. C₃²	C₂×C₃×#121	direct (N × H)
198. C₂³	C₃³×D₄	semidirect
207. C₂×C₄	C₃³×D₄	semidirect
213. C₂×C₃	C₃²×#121	direct (N × H)
226. C₂²	C₂×C₃³×D₄	semidirect
231. C₄	C₂×C₃³×D₄	semidirect
232. C₃	C₂×C₃²×#121	direct (N × H)
235. C₂	C₃³×#121	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×C₃³×D₄	2
C₂³×C₃³×C₄	2
C₃³×#121	2
C₂×C₃³×C₄⋊C₄	2
C₂×C₃³×C₂²⋊C₄	2
C₂×C₃²×#121	3

Subgroups — C2×C33×#121 #13179977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×#121 #13179977