Subgroups — C₂²×C₃²×#214 #13179177

Order 1728.

2732 conjugacy classes of subgroups.

The subgroup lattice has 260 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
2	C₂²×C₃²×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#214	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃²×SD₁₆	—	3	1	no	maximal
6	C₂²×C₃×#214	C₃	1	4	yes	maximal, direct product factor
7	C₂⁴×C₃³	—	2	1	no	—
8	C₂×C₃³×D₄	—	2	3	no	—
9	C₂×C₃³×D₄	C₂²	1	6	yes	—
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×Dic₃	—	2	1	no	—
12	C₂×C₃²×Dic₆	—	2	3	no	—
13	C₂×C₃²×Dic₆	C₂²	1	6	yes	complemented
14	C₃²×#214	C₂²	1	16	yes	direct product factor
15	C₂×C₃²×C₃⋊C₈	C₂²	1	6	yes	complemented
16	C₂²×C₃²×Q₈	—	3	1	no	—
17	C₂²×C₃²×D₄	—	2	4	no	—
18	C₂²×C₃²×D₄	D₃	1	1	yes	characteristic
19	C₂²×C₃²×D₄	C₂×C₃	1	4	yes	—
20	C₂²×C₃×Dic₆	C₂×C₃	1	4	yes	complemented
21	C₂×C₃²×SD₁₆	—	3	12	no	—
22	C₂²×C₃²×C₈	—	3	1	no	—
23	C₂×C₃×#214	C₂×C₃	1	48	yes	direct product factor
24	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
25	C₂³×C₃³	—	2	6	no	—
26	C₂³×C₃³	—	4	4	no	—
27	C₂³×C₃³	D₄	1	1	yes	characteristic
28	C₃³×D₄	—	2	6	no	—
29	C₃³×D₄	C₂³	1	4	yes	—
30	C₂×C₃³×C₄	D₄	1	3	yes	—
31	C₂×C₃³×C₄	C₂³	1	3	yes	—
32	C₂×C₃²×Dic₃	—	2	6	no	—
33	C₃²×Dic₆	—	2	6	no	—
34	C₃²×Dic₆	C₂³	1	4	yes	complemented
35	C₃²×C₃⋊C₈	C₂³	1	4	yes	complemented
36	C₂²×C₃×SD₁₆	—	3	4	no	—
37	C₂²×#214	C₃²	1	1	yes	characteristic, direct product factor
38	C₂⁴×C₃²	—	2	8	no	—
39	C₂⁴×C₃²	—	2	1	no	—
40	C₂⁴×C₃²	—	2	4	no	—
41	C₂×C₃²×Q₈	—	3	6	no	—
42	C₂×C₃²×Q₈	—	6	3	no	—
43	C₂×C₃²×D₄	—	2	24	no	—
44	C₂×C₃²×D₄	—	2	24	no	—
45	C₂×C₃²×D₄	—	2	3	no	—
46	C₂×C₃²×D₄	—	2	12	no	—
47	C₂×C₃²×D₄	C₂×D₃	1	6	yes	—
48	C₂×C₃²×D₄	C₂²×C₃	1	24	yes	—
49	C₂²×C₃²×C₄	—	2	4	no	—
50	C₂²×C₃²×C₄	—	6	1	no	—
51	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
53	C₂²×C₃×Dic₃	—	2	4	no	—
54	C₂×C₃×Dic₆	—	2	12	no	—
55	C₂×C₃×Dic₆	C₂²×C₃	1	24	yes	complemented
56	C₃²×SD₁₆	—	3	16	no	—
57	C₂×C₃²×C₈	—	3	6	no	—
58	C₃×#214	C₂²×C₃	1	64	yes	direct product factor
59	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	24	yes	complemented
60	C₂²×C₃³	—	2	4	no	—
61	C₂²×C₃³	—	4	12	no	—
62	C₂²×C₃³	SD₁₆	1	4	yes	complemented
63	C₂²×C₃³	C₂×D₄	1	3	yes	—
64	C₃³×C₄	C₂×D₄	1	3	yes	—
65	C₃³×C₄	C₂⁴	1	1	yes	characteristic
66	C₃²×Dic₃	—	2	4	no	—
67	C₂²×C₃×Q₈	—	3	4	no	—
68	C₂²×C₃×D₄	—	2	4	no	—
69	C₂²×C₃×D₄	C₃×D₃	1	4	yes	—
70	C₂²×C₃×D₄	C₂×C₃²	1	1	yes	characteristic
71	C₂²×Dic₆	C₂×C₃²	1	1	yes	characteristic, complemented
72	C₂×C₃×SD₁₆	—	3	48	no	—
73	C₂²×C₃×C₈	—	3	4	no	—
74	C₂×#214	C₂×C₃²	1	12	yes	direct product factor
75	C₂²×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
76	C₂³×C₃²	—	2	48	no	—
77	C₂³×C₃²	—	2	6	no	—
78	C₂³×C₃²	—	2	24	no	—
79	C₂³×C₃²	—	2	4	no	—
80	C₂³×C₃²	—	4	32	no	—
81	C₂³×C₃²	—	4	4	no	—
82	C₂³×C₃²	—	4	16	no	—
83	C₂³×C₃²	#67	1	1	yes	characteristic
84	C₂³×C₃²	C₃×D₄	1	4	yes	—
85	C₃²×Q₈	—	3	4	no	—
86	C₃²×Q₈	—	6	6	no	—
87	C₃²×D₄	—	2	48	no	—
88	C₃²×D₄	—	2	6	no	—
89	C₃²×D₄	—	2	24	no	—
90	C₃²×D₄	—	2	16	no	—
91	C₃²×D₄	C₂²×D₃	1	4	yes	—
92	C₃²×D₄	C₂³×C₃	1	16	yes	—
93	C₂×C₃²×C₄	—	2	12	no	—
94	C₂×C₃²×C₄	—	2	12	no	—
95	C₂×C₃²×C₄	—	6	6	no	—
96	C₂×C₃²×C₄	#67	1	3	yes	—
97	C₂×C₃²×C₄	C₃×D₄	1	12	yes	—
98	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
99	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
100	C₂×C₃×Dic₃	—	2	24	no	—
101	C₃×Dic₆	—	2	24	no	—
102	C₃×Dic₆	C₂³×C₃	1	16	yes	complemented
103	C₃²×C₈	—	3	4	no	—
104	C₃×C₃⋊C₈	C₂³×C₃	1	16	yes	complemented
105	C₂²×SD₁₆	—	3	1	no	Sylow
106	C₂×C₃³	—	4	4	no	—
107	C₂×C₃³	C₂×SD₁₆	1	6	yes	complemented
108	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
109	C₂⁴×C₃	—	2	8	no	—
110	C₂⁴×C₃	—	2	4	no	—
111	C₂⁴×C₃	—	2	1	no	—
112	C₂×C₃×Q₈	—	3	24	no	—
113	C₂×C₃×Q₈	—	6	12	no	—
114	C₂×C₃×D₄	—	2	24	no	—
115	C₂×C₃×D₄	—	2	12	no	—
116	C₂×C₃×D₄	—	2	3	no	—
117	C₂×C₃×D₄	—	2	24	no	—
118	C₂×C₃×D₄	C₂×C₃×D₃	1	24	yes	—
119	C₂×C₃×D₄	C₂²×C₃²	1	6	yes	—
120	C₂²×C₃×C₄	—	2	4	no	—
121	C₂²×C₃×C₄	—	6	4	no	—
122	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
123	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
124	C₂²×Dic₃	—	2	1	no	—
125	C₂×Dic₆	—	2	3	no	—
126	C₂×Dic₆	C₂²×C₃²	1	6	yes	complemented
127	C₃×SD₁₆	—	3	64	no	—
128	C₂×C₃×C₈	—	3	24	no	—
129	#214	C₂²×C₃²	1	16	yes	direct product factor
130	C₂×C₃⋊C₈	C₂²×C₃²	1	6	yes	complemented
131	C₂²×C₃²	—	2	32	no	—
132	C₂²×C₃²	—	2	4	no	—
133	C₂²×C₃²	—	2	16	no	—
134	C₂²×C₃²	—	2	16	no	—
135	C₂²×C₃²	—	2	12	no	—
136	C₂²×C₃²	—	4	96	no	—
137	C₂²×C₃²	—	4	12	no	—
138	C₂²×C₃²	—	4	48	no	—
139	C₂²×C₃²	#214	1	4	yes	direct product factor
140	C₂²×C₃²	C₃×SD₁₆	1	16	yes	complemented
141	C₂²×C₃²	C₂×#67	1	3	yes	—
142	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	—	2	12	no	—
145	C₃²×C₄	—	6	4	no	—
146	C₃²×C₄	C₂×#67	1	3	yes	—
147	C₃²×C₄	C₂×C₃×D₄	1	12	yes	—
148	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
149	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
150	C₃×Dic₃	—	2	16	no	—
151	C₂²×Q₈	—	3	1	no	—
152	C₂²×D₄	C₃²×D₃	1	1	yes	characteristic
153	C₂×SD₁₆	—	3	12	no	—
154	C₂²×C₈	—	3	1	no	—
155	C₃³	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	2	48	no	—
157	C₂³×C₃	—	2	24	no	—
158	C₂³×C₃	—	2	6	no	—
159	C₂³×C₃	—	2	4	no	—
160	C₂³×C₃	—	4	32	no	—
161	C₂³×C₃	—	4	16	no	—
162	C₂³×C₃	—	4	4	no	—
163	C₂³×C₃	C₃×#67	1	4	yes	—
164	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
165	C₃×Q₈	—	3	16	no	—
166	C₃×Q₈	—	6	24	no	—
167	C₃×D₄	—	2	48	no	—
168	C₃×D₄	—	2	24	no	—
169	C₃×D₄	—	2	6	no	—
170	C₃×D₄	—	2	16	no	—
171	C₃×D₄	C₂²×C₃×D₃	1	16	yes	—
172	C₃×D₄	C₂³×C₃²	1	4	yes	—
173	C₂×C₃×C₄	—	2	12	no	—
174	C₂×C₃×C₄	—	2	12	no	—
175	C₂×C₃×C₄	—	6	24	no	—
176	C₂×C₃×C₄	C₃×#67	1	12	yes	—
177	C₂×C₃×C₄	C₃²×D₄	1	3	yes	—
178	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
179	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
180	C₂×Dic₃	—	2	6	no	—
181	Dic₆	—	2	6	no	—
182	Dic₆	C₂³×C₃²	1	4	yes	complemented
183	C₃×C₈	—	3	16	no	—
184	C₃⋊C₈	C₂³×C₃²	1	4	yes	complemented
185	C₂×C₃²	—	2	24	no	—
186	C₂×C₃²	—	2	4	no	—
187	C₂×C₃²	—	4	32	no	—
188	C₂×C₃²	—	4	16	no	—
189	C₂×C₃²	—	4	4	no	—
190	C₂×C₃²	C₂×#214	1	6	yes	direct product factor
191	C₂×C₃²	C₂×C₃×SD₁₆	1	24	yes	complemented
192	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
193	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
194	C₂⁴	—	2	1	no	—
195	C₂×Q₈	—	3	6	no	—
196	C₂×Q₈	—	6	3	no	—
197	C₂×D₄	—	2	3	no	—
198	C₂×D₄	C₂×C₃²×D₃	1	6	yes	—
199	C₂²×C₄	—	6	1	no	—
200	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
201	SD₁₆	—	3	16	no	—
202	C₂×C₈	—	3	6	no	—
203	C₂²×C₃	—	2	32	no	—
204	C₂²×C₃	—	2	16	no	—
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	2	12	no	—
207	C₂²×C₃	—	2	16	no	—
208	C₂²×C₃	—	4	96	no	—
209	C₂²×C₃	—	4	48	no	—
210	C₂²×C₃	—	4	12	no	—
211	C₂²×C₃	C₃×#214	1	16	yes	direct product factor
212	C₂²×C₃	C₃²×SD₁₆	1	4	yes	complemented
213	C₂²×C₃	C₂×C₃×#67	1	12	yes	—
214	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	2	12	no	—
217	C₃×C₄	—	6	16	no	—
218	C₃×C₄	C₂×C₃×#67	1	12	yes	—
219	C₃×C₄	C₂×C₃²×D₄	1	3	yes	—
220	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
221	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
222	Dic₃	—	2	4	no	—
223	C₃²	—	2	4	no	—
224	C₃²	C₂²×#214	1	1	yes	characteristic, direct product factor
225	C₃²	C₂²×C₃×SD₁₆	1	4	yes	complemented
226	C₂³	—	2	6	no	—
227	C₂³	—	4	4	no	—
228	C₂³	C₃²×#67	1	1	yes	characteristic
229	Q₈	—	3	4	no	—
230	Q₈	—	6	6	no	—
231	D₄	—	2	6	no	—
232	D₄	C₂²×C₃²×D₃	1	4	yes	—
233	C₂×C₄	—	6	6	no	—
234	C₂×C₄	C₃²×#67	1	3	yes	—
235	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
236	C₈	—	3	4	no	—
237	C₂×C₃	—	2	24	no	—
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	—	4	32	no	—
240	C₂×C₃	—	4	4	no	—
241	C₂×C₃	—	4	16	no	—
242	C₂×C₃	C₂×C₃×#214	1	24	yes	direct product factor
243	C₂×C₃	C₂×C₃²×SD₁₆	1	6	yes	complemented
244	C₂×C₃	C₂²×C₃×#67	1	4	yes	—
245	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
246	C₂²	—	2	4	no	—
247	C₂²	—	4	12	no	—
248	C₂²	C₃²×#214	1	4	yes	direct product factor
249	C₂²	C₂×C₃²×#67	1	3	yes	—
250	C₄	—	6	4	no	—
251	C₄	C₂×C₃²×#67	1	3	yes	—
252	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
253	C₃	—	2	4	no	—
254	C₃	C₂²×C₃×#214	1	4	yes	direct product factor
255	C₃	C₂²×C₃²×SD₁₆	1	1	yes	characteristic, complemented
256	C₂	—	4	4	no	—
257	C₂	C₂×C₃²×#214	1	6	yes	direct product factor
258	C₂	C₂²×C₃²×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃²	#67
commutator, cosocle	C₃×C₄	C₂⁴×C₃²
Frattini	C₄	C₂³×C₃²×D₃
Fitting, prime radical	C₂²×C₃³×D₄	C₂
socle	C₂³×C₃³	D₄
hypercenter	C₂²×C₃²×D₄	D₃
hypocenter	C₃	C₂²×C₃²×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×Dic₆	C₂	semidirect
3. C₂×C₃²×#214	C₂	direct (N × H)
4. C₂²×C₃²×C₃⋊C₈	C₂	semidirect
6. C₂²×C₃×#214	C₃	direct (N × H)
13. C₂×C₃²×Dic₆	C₂²	semidirect
14. C₃²×#214	C₂²	direct (N × H)
15. C₂×C₃²×C₃⋊C₈	C₂²	semidirect
20. C₂²×C₃×Dic₆	C₂×C₃	semidirect
23. C₂×C₃×#214	C₂×C₃	direct (N × H)
24. C₂²×C₃×C₃⋊C₈	C₂×C₃	semidirect
34. C₃²×Dic₆	C₂³	semidirect
35. C₃²×C₃⋊C₈	C₂³	semidirect
37. C₂²×#214	C₃²	direct (N × H)
55. C₂×C₃×Dic₆	C₂²×C₃	semidirect
58. C₃×#214	C₂²×C₃	direct (N × H)
59. C₂×C₃×C₃⋊C₈	C₂²×C₃	semidirect
62. C₂²×C₃³	SD₁₆	semidirect
71. C₂²×Dic₆	C₂×C₃²	semidirect
74. C₂×#214	C₂×C₃²	direct (N × H)
75. C₂²×C₃⋊C₈	C₂×C₃²	semidirect
102. C₃×Dic₆	C₂³×C₃	semidirect
104. C₃×C₃⋊C₈	C₂³×C₃	semidirect
107. C₂×C₃³	C₂×SD₁₆	semidirect
126. C₂×Dic₆	C₂²×C₃²	semidirect
129. #214	C₂²×C₃²	direct (N × H)
130. C₂×C₃⋊C₈	C₂²×C₃²	semidirect
139. C₂²×C₃²	#214	direct (N × H)
140. C₂²×C₃²	C₃×SD₁₆	semidirect
155. C₃³	C₂²×SD₁₆	semidirect
182. Dic₆	C₂³×C₃²	semidirect
184. C₃⋊C₈	C₂³×C₃²	semidirect
190. C₂×C₃²	C₂×#214	direct (N × H)
191. C₂×C₃²	C₂×C₃×SD₁₆	semidirect
211. C₂²×C₃	C₃×#214	direct (N × H)
212. C₂²×C₃	C₃²×SD₁₆	semidirect
224. C₃²	C₂²×#214	direct (N × H)
225. C₃²	C₂²×C₃×SD₁₆	semidirect
242. C₂×C₃	C₂×C₃×#214	direct (N × H)
243. C₂×C₃	C₂×C₃²×SD₁₆	semidirect
248. C₂²	C₃²×#214	direct (N × H)
254. C₃	C₂²×C₃×#214	direct (N × H)
255. C₃	C₂²×C₃²×SD₁₆	semidirect
257. C₂	C₂×C₃²×#214	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
C₂²×C₃²×Dic₆	2
C₂×C₃²×#214	2
C₂²×C₃²×C₃⋊C₈	2
C₂²×C₃²×SD₁₆	3
C₂²×C₃×#214	3

Subgroups — C22×C32×#214 #13179177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×#214 #13179177