Subgroups — C₂²×C₃²×#204 #13179116

Order 1728.

2092 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×#204	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₃²×SD₁₆	—	3	1	no	maximal
6	C₂²×C₃×#204	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²×D₃	—	2	1	no	—
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×Dic₃	—	2	1	no	—
10	C₂×C₃²×D₁₂	—	2	3	no	—
11	C₂×C₃²×D₁₂	C₂²	1	6	yes	—
12	C₂×C₃²×Dic₆	—	2	3	no	—
13	C₂×C₃²×Dic₆	C₂²	1	6	yes	complemented
14	C₂×C₃³×C₈	C₂²	1	6	yes	complemented
15	C₃²×#204	C₂²	1	16	yes	direct product factor
16	C₂²×C₃²×Q₈	—	3	1	no	—
17	C₂²×C₃²×D₄	—	3	1	no	—
18	C₂²×C₃×D₁₂	C₂×C₃	1	4	yes	—
19	C₂²×C₃×Dic₆	C₂×C₃	1	4	yes	complemented
20	C₂×C₃²×SD₁₆	—	3	12	no	—
21	C₂²×C₃²×C₈	—	2	4	no	—
22	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic, complemented
23	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	complemented
24	C₂×C₃×#204	C₂×C₃	1	48	yes	direct product factor
25	C₂³×C₃³	D₄	1	1	yes	characteristic
26	C₂²×C₃²×D₃	—	2	6	no	—
27	C₂²×C₃²×D₃	—	4	4	no	—
28	C₂×C₃³×C₄	D₄	1	3	yes	—
29	C₂×C₃³×C₄	C₂³	1	3	yes	—
30	C₂×C₃²×Dic₃	—	2	6	no	—
31	C₃²×D₁₂	—	2	6	no	—
32	C₃²×D₁₂	C₂³	1	4	yes	—
33	C₃²×Dic₆	—	2	6	no	—
34	C₃²×Dic₆	C₂³	1	4	yes	complemented
35	C₃³×C₈	C₂³	1	4	yes	complemented
36	C₂²×C₃×SD₁₆	—	3	4	no	—
37	C₂²×#204	C₃²	1	1	yes	characteristic, direct product factor
38	C₂⁴×C₃²	—	6	1	no	—
39	C₂³×C₃×D₃	—	2	4	no	—
40	C₂×C₃²×Q₈	—	3	6	no	—
41	C₂×C₃²×Q₈	—	6	3	no	—
42	C₂×C₃²×D₄	—	3	6	no	—
43	C₂×C₃²×D₄	—	6	3	no	—
44	C₂²×C₃²×C₄	—	2	4	no	—
45	C₂²×C₃²×C₄	—	6	1	no	—
46	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
47	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
48	C₂²×C₃×Dic₃	—	2	4	no	—
49	C₂×C₃×D₁₂	—	2	12	no	—
50	C₂×C₃×D₁₂	C₂²×C₃	1	24	yes	—
51	C₂×C₃×Dic₆	—	2	12	no	—
52	C₂×C₃×Dic₆	C₂²×C₃	1	24	yes	complemented
53	C₃²×SD₁₆	—	3	16	no	—
54	C₂×C₃²×C₈	—	2	24	no	—
55	C₂×C₃²×C₈	C₂×D₃	1	6	yes	complemented
56	C₂×C₃²×C₈	C₂²×C₃	1	24	yes	complemented
57	C₃×#204	C₂²×C₃	1	64	yes	direct product factor
58	C₂²×C₃³	SD₁₆	1	4	yes	complemented
59	C₂²×C₃³	C₂×D₄	1	3	yes	—
60	C₂×C₃²×D₃	—	2	4	no	—
61	C₂×C₃²×D₃	—	4	12	no	—
62	C₃³×C₄	C₂×D₄	1	3	yes	—
63	C₃³×C₄	C₂⁴	1	1	yes	characteristic
64	C₃²×Dic₃	—	2	4	no	—
65	C₂²×C₃×Q₈	—	3	4	no	—
66	C₂²×C₃×D₄	—	3	4	no	—
67	C₂²×D₁₂	C₂×C₃²	1	1	yes	characteristic
68	C₂²×Dic₆	C₂×C₃²	1	1	yes	characteristic, complemented
69	C₂×C₃×SD₁₆	—	3	48	no	—
70	C₂²×C₃×C₈	—	2	4	no	—
71	C₂²×C₃×C₈	C₃×D₃	1	4	yes	complemented
72	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
73	C₂×#204	C₂×C₃²	1	12	yes	direct product factor
74	C₂³×C₃²	—	2	4	no	—
75	C₂³×C₃²	—	6	6	no	—
76	C₂³×C₃²	—	12	4	no	—
77	C₂³×C₃²	D₁₂	1	1	yes	characteristic
78	C₂³×C₃²	C₃×D₄	1	4	yes	—
79	C₂²×C₃×D₃	—	2	24	no	—
80	C₂²×C₃×D₃	—	4	16	no	—
81	C₃²×Q₈	—	3	4	no	—
82	C₃²×Q₈	—	6	6	no	—
83	C₃²×D₄	—	3	4	no	—
84	C₃²×D₄	—	6	6	no	—
85	C₂×C₃²×C₄	—	2	12	no	—
86	C₂×C₃²×C₄	—	2	12	no	—
87	C₂×C₃²×C₄	—	6	6	no	—
88	C₂×C₃²×C₄	D₁₂	1	3	yes	—
89	C₂×C₃²×C₄	C₃×D₄	1	12	yes	—
90	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
91	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
92	C₂×C₃×Dic₃	—	2	24	no	—
93	C₃×D₁₂	—	2	24	no	—
94	C₃×D₁₂	C₂³×C₃	1	16	yes	—
95	C₃×Dic₆	—	2	24	no	—
96	C₃×Dic₆	C₂³×C₃	1	16	yes	complemented
97	C₃²×C₈	—	2	16	no	—
98	C₃²×C₈	C₂²×D₃	1	4	yes	complemented
99	C₃²×C₈	C₂³×C₃	1	16	yes	complemented
100	C₂²×SD₁₆	—	3	1	no	Sylow
101	C₂×C₃³	C₂×SD₁₆	1	6	yes	complemented
102	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
103	C₃²×D₃	—	4	4	no	—
104	C₂⁴×C₃	—	6	4	no	—
105	C₂³×D₃	—	2	1	no	—
106	C₂×C₃×Q₈	—	3	24	no	—
107	C₂×C₃×Q₈	—	6	12	no	—
108	C₂×C₃×D₄	—	3	24	no	—
109	C₂×C₃×D₄	—	6	12	no	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	—	6	4	no	—
112	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
113	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
114	C₂²×Dic₃	—	2	1	no	—
115	C₂×D₁₂	—	2	3	no	—
116	C₂×D₁₂	C₂²×C₃²	1	6	yes	—
117	C₂×Dic₆	—	2	3	no	—
118	C₂×Dic₆	C₂²×C₃²	1	6	yes	complemented
119	C₃×SD₁₆	—	3	64	no	—
120	C₂×C₃×C₈	—	2	24	no	—
121	C₂×C₃×C₈	C₂×C₃×D₃	1	24	yes	complemented
122	C₂×C₃×C₈	C₂²×C₃²	1	6	yes	complemented
123	#204	C₂²×C₃²	1	16	yes	direct product factor
124	C₂²×C₃²	—	2	16	no	—
125	C₂²×C₃²	—	2	12	no	—
126	C₂²×C₃²	—	6	4	no	—
127	C₂²×C₃²	—	12	12	no	—
128	C₂²×C₃²	#204	1	4	yes	direct product factor
129	C₂²×C₃²	C₃×SD₁₆	1	16	yes	complemented
130	C₂²×C₃²	C₂×D₁₂	1	3	yes	—
131	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
132	C₂×C₃×D₃	—	2	16	no	—
133	C₂×C₃×D₃	—	4	48	no	—
134	C₃²×C₄	—	2	4	no	—
135	C₃²×C₄	—	2	12	no	—
136	C₃²×C₄	—	6	4	no	—
137	C₃²×C₄	C₂×D₁₂	1	3	yes	—
138	C₃²×C₄	C₂×C₃×D₄	1	12	yes	—
139	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
140	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
141	C₃×Dic₃	—	2	16	no	—
142	C₂²×Q₈	—	3	1	no	—
143	C₂²×D₄	—	3	1	no	—
144	C₂×SD₁₆	—	3	12	no	—
145	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic, complemented
146	C₃³	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₃	—	2	4	no	—
148	C₂³×C₃	—	6	24	no	—
149	C₂³×C₃	—	12	16	no	—
150	C₂³×C₃	C₃×D₁₂	1	4	yes	—
151	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
152	C₂²×D₃	—	2	6	no	—
153	C₂²×D₃	—	4	4	no	—
154	C₃×Q₈	—	3	16	no	—
155	C₃×Q₈	—	6	24	no	—
156	C₃×D₄	—	3	16	no	—
157	C₃×D₄	—	6	24	no	—
158	C₂×C₃×C₄	—	2	12	no	—
159	C₂×C₃×C₄	—	2	12	no	—
160	C₂×C₃×C₄	—	6	24	no	—
161	C₂×C₃×C₄	C₃×D₁₂	1	12	yes	—
162	C₂×C₃×C₄	C₃²×D₄	1	3	yes	—
163	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
164	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
165	C₂×Dic₃	—	2	6	no	—
166	D₁₂	—	2	6	no	—
167	D₁₂	C₂³×C₃²	1	4	yes	—
168	Dic₆	—	2	6	no	—
169	Dic₆	C₂³×C₃²	1	4	yes	complemented
170	C₃×C₈	—	2	16	no	—
171	C₃×C₈	C₂²×C₃×D₃	1	16	yes	complemented
172	C₃×C₈	C₂³×C₃²	1	4	yes	complemented
173	C₂×C₃²	—	2	24	no	—
174	C₂×C₃²	—	2	4	no	—
175	C₂×C₃²	—	12	4	no	—
176	C₂×C₃²	C₂×#204	1	6	yes	direct product factor
177	C₂×C₃²	C₂×C₃×SD₁₆	1	24	yes	complemented
178	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
179	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
180	C₃×D₃	—	4	16	no	—
181	C₂⁴	—	6	1	no	—
182	C₂×Q₈	—	3	6	no	—
183	C₂×Q₈	—	6	3	no	—
184	C₂×D₄	—	3	6	no	—
185	C₂×D₄	—	6	3	no	—
186	C₂²×C₄	—	6	1	no	—
187	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
188	SD₁₆	—	3	16	no	—
189	C₂×C₈	C₂×C₃²×D₃	1	6	yes	complemented
190	C₂²×C₃	—	2	12	no	—
191	C₂²×C₃	—	2	16	no	—
192	C₂²×C₃	—	6	16	no	—
193	C₂²×C₃	—	12	48	no	—
194	C₂²×C₃	C₃×#204	1	16	yes	direct product factor
195	C₂²×C₃	C₃²×SD₁₆	1	4	yes	complemented
196	C₂²×C₃	C₂×C₃×D₁₂	1	12	yes	—
197	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
198	C₂×D₃	—	2	4	no	—
199	C₂×D₃	—	4	12	no	—
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	—	2	12	no	—
202	C₃×C₄	—	6	16	no	—
203	C₃×C₄	C₂×C₃×D₁₂	1	12	yes	—
204	C₃×C₄	C₂×C₃²×D₄	1	3	yes	—
205	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
206	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
207	Dic₃	—	2	4	no	—
208	C₃²	—	2	4	no	—
209	C₃²	C₂²×#204	1	1	yes	characteristic, direct product factor
210	C₃²	C₂²×C₃×SD₁₆	1	4	yes	complemented
211	C₂³	—	6	6	no	—
212	C₂³	—	12	4	no	—
213	C₂³	C₃²×D₁₂	1	1	yes	characteristic
214	Q₈	—	3	4	no	—
215	Q₈	—	6	6	no	—
216	D₄	—	3	4	no	—
217	D₄	—	6	6	no	—
218	C₂×C₄	—	6	6	no	—
219	C₂×C₄	C₃²×D₁₂	1	3	yes	—
220	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
221	C₈	C₂²×C₃²×D₃	1	4	yes	complemented
222	C₂×C₃	—	2	4	no	—
223	C₂×C₃	—	2	24	no	—
224	C₂×C₃	—	12	16	no	—
225	C₂×C₃	C₂×C₃×#204	1	24	yes	direct product factor
226	C₂×C₃	C₂×C₃²×SD₁₆	1	6	yes	complemented
227	C₂×C₃	C₂²×C₃×D₁₂	1	4	yes	—
228	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
229	D₃	—	4	4	no	—
230	C₂²	—	6	4	no	—
231	C₂²	—	12	12	no	—
232	C₂²	C₃²×#204	1	4	yes	direct product factor
233	C₂²	C₂×C₃²×D₁₂	1	3	yes	—
234	C₄	—	6	4	no	—
235	C₄	C₂×C₃²×D₁₂	1	3	yes	—
236	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
237	C₃	—	2	4	no	—
238	C₃	C₂²×C₃×#204	1	4	yes	direct product factor
239	C₃	C₂²×C₃²×SD₁₆	1	1	yes	characteristic, complemented
240	C₂	—	12	4	no	—
241	C₂	C₂×C₃²×#204	1	6	yes	direct product factor
242	C₂	C₂²×C₃²×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃²	D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₃²
Frattini	C₄	C₂³×C₃²×D₃
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂³×C₃³	D₄
prime radical	C₂²×C₃²×D₁₂	C₂
hypercenter	C₂²×C₃²×C₈	D₃
hypocenter	C₃	C₂²×C₃²×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×Dic₆	C₂	semidirect
3. C₂²×C₃³×C₈	C₂	semidirect
4. C₂×C₃²×#204	C₂	direct (N × H)
6. C₂²×C₃×#204	C₃	direct (N × H)
13. C₂×C₃²×Dic₆	C₂²	semidirect
14. C₂×C₃³×C₈	C₂²	semidirect
15. C₃²×#204	C₂²	direct (N × H)
19. C₂²×C₃×Dic₆	C₂×C₃	semidirect
22. C₂²×C₃²×C₈	D₃	semidirect
23. C₂²×C₃²×C₈	C₂×C₃	semidirect
24. C₂×C₃×#204	C₂×C₃	direct (N × H)
34. C₃²×Dic₆	C₂³	semidirect
35. C₃³×C₈	C₂³	semidirect
37. C₂²×#204	C₃²	direct (N × H)
52. C₂×C₃×Dic₆	C₂²×C₃	semidirect
55. C₂×C₃²×C₈	C₂×D₃	semidirect
56. C₂×C₃²×C₈	C₂²×C₃	semidirect
57. C₃×#204	C₂²×C₃	direct (N × H)
58. C₂²×C₃³	SD₁₆	semidirect
68. C₂²×Dic₆	C₂×C₃²	semidirect
71. C₂²×C₃×C₈	C₃×D₃	semidirect
72. C₂²×C₃×C₈	C₂×C₃²	semidirect
73. C₂×#204	C₂×C₃²	direct (N × H)
96. C₃×Dic₆	C₂³×C₃	semidirect
98. C₃²×C₈	C₂²×D₃	semidirect
99. C₃²×C₈	C₂³×C₃	semidirect
101. C₂×C₃³	C₂×SD₁₆	semidirect
118. C₂×Dic₆	C₂²×C₃²	semidirect
121. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
122. C₂×C₃×C₈	C₂²×C₃²	semidirect
123. #204	C₂²×C₃²	direct (N × H)
128. C₂²×C₃²	#204	direct (N × H)
129. C₂²×C₃²	C₃×SD₁₆	semidirect
145. C₂²×C₈	C₃²×D₃	semidirect
146. C₃³	C₂²×SD₁₆	semidirect
169. Dic₆	C₂³×C₃²	semidirect
171. C₃×C₈	C₂²×C₃×D₃	semidirect
172. C₃×C₈	C₂³×C₃²	semidirect
176. C₂×C₃²	C₂×#204	direct (N × H)
177. C₂×C₃²	C₂×C₃×SD₁₆	semidirect
189. C₂×C₈	C₂×C₃²×D₃	semidirect
194. C₂²×C₃	C₃×#204	direct (N × H)
195. C₂²×C₃	C₃²×SD₁₆	semidirect
209. C₃²	C₂²×#204	direct (N × H)
210. C₃²	C₂²×C₃×SD₁₆	semidirect
221. C₈	C₂²×C₃²×D₃	semidirect
225. C₂×C₃	C₂×C₃×#204	direct (N × H)
226. C₂×C₃	C₂×C₃²×SD₁₆	semidirect
232. C₂²	C₃²×#204	direct (N × H)
238. C₃	C₂²×C₃×#204	direct (N × H)
239. C₃	C₂²×C₃²×SD₁₆	semidirect
241. C₂	C₂×C₃²×#204	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×C₃²×D₁₂	2
C₂²×C₃²×Dic₆	2
C₂²×C₃³×C₈	2
C₂×C₃²×#204	2
C₂²×C₃²×SD₁₆	3
C₂²×C₃×#204	3

Subgroups — C22×C32×#204 #13179116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×#204 #13179116