Subgroups — C₂²×C₃²×#215 #13179187

Order 1728.

2412 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×#215	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×#215	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²×D₃	—	2	1	no	—
8	C₂×C₃³×Q₈	—	2	3	no	—
9	C₂×C₃³×Q₈	C₂²	1	6	yes	complemented
10	C₂²×C₃³×C₄	—	2	1	no	—
11	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃²×D₁₂	—	2	3	no	—
13	C₂×C₃²×D₁₂	C₂²	1	6	yes	—
14	C₃²×#215	C₂²	1	16	yes	direct product factor
15	C₂×C₃²×C₃⋊C₈	C₂²	1	6	yes	complemented
16	C₂²×C₃²×Q₈	—	2	4	no	—
17	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic, complemented
18	C₂²×C₃²×Q₈	C₂×C₃	1	4	yes	complemented
19	C₂²×C₃²×D₄	—	3	1	no	—
20	C₂²×C₃×D₁₂	C₂×C₃	1	4	yes	—
21	C₂×C₃²×SD₁₆	—	3	12	no	—
22	C₂²×C₃²×C₈	—	3	1	no	—
23	C₂×C₃×#215	C₂×C₃	1	48	yes	direct product factor
24	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
25	C₂³×C₃³	D₄	1	1	yes	characteristic
26	C₂²×C₃²×D₃	—	2	6	no	—
27	C₂²×C₃²×D₃	—	4	4	no	—
28	C₃³×Q₈	—	2	6	no	—
29	C₃³×Q₈	C₂³	1	4	yes	complemented
30	C₂×C₃³×C₄	—	2	6	no	—
31	C₂×C₃³×C₄	D₄	1	3	yes	—
32	C₂×C₃³×C₄	C₂³	1	3	yes	—
33	C₃²×D₁₂	—	2	6	no	—
34	C₃²×D₁₂	C₂³	1	4	yes	—
35	C₃²×C₃⋊C₈	C₂³	1	4	yes	complemented
36	C₂²×C₃×SD₁₆	—	3	4	no	—
37	C₂²×#215	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	—	2	24	no	—
41	C₂×C₃²×Q₈	—	2	24	no	—
42	C₂×C₃²×Q₈	—	2	3	no	—
43	C₂×C₃²×Q₈	—	2	12	no	—
44	C₂×C₃²×Q₈	C₂×D₃	1	6	yes	complemented
45	C₂×C₃²×Q₈	C₂²×C₃	1	24	yes	complemented
46	C₂×C₃²×D₄	—	3	6	no	—
47	C₂×C₃²×D₄	—	6	3	no	—
48	C₂²×C₃²×C₄	—	2	8	no	—
49	C₂²×C₃²×C₄	—	2	4	no	—
50	C₂²×C₃²×C₄	—	2	1	no	—
51	C₂²×C₃²×C₄	—	2	4	no	—
52	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
54	C₂×C₃×D₁₂	—	2	12	no	—
55	C₂×C₃×D₁₂	C₂²×C₃	1	24	yes	—
56	C₃²×SD₁₆	—	3	16	no	—
57	C₂×C₃²×C₈	—	3	6	no	—
58	C₃×#215	C₂²×C₃	1	64	yes	direct product factor
59	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	24	yes	complemented
60	C₂²×C₃³	SD₁₆	1	4	yes	complemented
61	C₂²×C₃³	C₂×D₄	1	3	yes	—
62	C₂×C₃²×D₃	—	2	4	no	—
63	C₂×C₃²×D₃	—	4	12	no	—
64	C₃³×C₄	—	2	4	no	—
65	C₃³×C₄	C₂×D₄	1	3	yes	—
66	C₃³×C₄	C₂⁴	1	1	yes	characteristic
67	C₂²×C₃×Q₈	—	2	4	no	—
68	C₂²×C₃×Q₈	C₃×D₃	1	4	yes	complemented
69	C₂²×C₃×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
70	C₂²×C₃×D₄	—	3	4	no	—
71	C₂²×D₁₂	C₂×C₃²	1	1	yes	characteristic
72	C₂×C₃×SD₁₆	—	3	48	no	—
73	C₂²×C₃×C₈	—	3	4	no	—
74	C₂×#215	C₂×C₃²	1	12	yes	direct product factor
75	C₂²×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
76	C₂³×C₃²	—	2	4	no	—
77	C₂³×C₃²	—	6	6	no	—
78	C₂³×C₃²	—	12	4	no	—
79	C₂³×C₃²	#67	1	1	yes	characteristic
80	C₂³×C₃²	C₃×D₄	1	4	yes	—
81	C₂²×C₃×D₃	—	2	24	no	—
82	C₂²×C₃×D₃	—	4	16	no	—
83	C₃²×Q₈	—	2	48	no	—
84	C₃²×Q₈	—	2	6	no	—
85	C₃²×Q₈	—	2	24	no	—
86	C₃²×Q₈	—	2	16	no	—
87	C₃²×Q₈	C₂²×D₃	1	4	yes	complemented
88	C₃²×Q₈	C₂³×C₃	1	16	yes	complemented
89	C₃²×D₄	—	3	4	no	—
90	C₃²×D₄	—	6	6	no	—
91	C₂×C₃²×C₄	—	2	48	no	—
92	C₂×C₃²×C₄	—	2	6	no	—
93	C₂×C₃²×C₄	—	2	24	no	—
94	C₂×C₃²×C₄	—	2	12	no	—
95	C₂×C₃²×C₄	—	2	12	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	C₃×D₁₂	—	2	24	no	—
101	C₃×D₁₂	C₂³×C₃	1	16	yes	—
102	C₃²×C₈	—	3	4	no	—
103	C₃×C₃⋊C₈	C₂³×C₃	1	16	yes	complemented
104	C₂²×SD₁₆	—	3	1	no	Sylow
105	C₂×C₃³	C₂×SD₁₆	1	6	yes	complemented
106	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
107	C₃²×D₃	—	4	4	no	—
108	C₂⁴×C₃	—	6	4	no	—
109	C₂³×D₃	—	2	1	no	—
110	C₂×C₃×Q₈	—	2	24	no	—
111	C₂×C₃×Q₈	—	2	12	no	—
112	C₂×C₃×Q₈	—	2	3	no	—
113	C₂×C₃×Q₈	—	2	24	no	—
114	C₂×C₃×Q₈	C₂×C₃×D₃	1	24	yes	complemented
115	C₂×C₃×Q₈	C₂²×C₃²	1	6	yes	complemented
116	C₂×C₃×D₄	—	3	24	no	—
117	C₂×C₃×D₄	—	6	12	no	—
118	C₂²×C₃×C₄	—	2	8	no	—
119	C₂²×C₃×C₄	—	2	4	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	2	4	no	—
122	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
123	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
124	C₂×D₁₂	—	2	3	no	—
125	C₂×D₁₂	C₂²×C₃²	1	6	yes	—
126	C₃×SD₁₆	—	3	64	no	—
127	C₂×C₃×C₈	—	3	24	no	—
128	#215	C₂²×C₃²	1	16	yes	direct product factor
129	C₂×C₃⋊C₈	C₂²×C₃²	1	6	yes	complemented
130	C₂²×C₃²	—	2	16	no	—
131	C₂²×C₃²	—	2	12	no	—
132	C₂²×C₃²	—	6	4	no	—
133	C₂²×C₃²	—	12	12	no	—
134	C₂²×C₃²	#215	1	4	yes	direct product factor
135	C₂²×C₃²	C₃×SD₁₆	1	16	yes	complemented
136	C₂²×C₃²	C₂×#67	1	3	yes	—
137	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
138	C₂×C₃×D₃	—	2	16	no	—
139	C₂×C₃×D₃	—	4	48	no	—
140	C₃²×C₄	—	2	32	no	—
141	C₃²×C₄	—	2	4	no	—
142	C₃²×C₄	—	2	16	no	—
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	—	2	12	no	—
145	C₃²×C₄	C₂×#67	1	3	yes	—
146	C₃²×C₄	C₂×C₃×D₄	1	12	yes	—
147	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
148	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
149	C₂²×Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
150	C₂²×D₄	—	3	1	no	—
151	C₂×SD₁₆	—	3	12	no	—
152	C₂²×C₈	—	3	1	no	—
153	C₃³	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₃	—	2	4	no	—
155	C₂³×C₃	—	6	24	no	—
156	C₂³×C₃	—	12	16	no	—
157	C₂³×C₃	C₃×#67	1	4	yes	—
158	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
159	C₂²×D₃	—	2	6	no	—
160	C₂²×D₃	—	4	4	no	—
161	C₃×Q₈	—	2	48	no	—
162	C₃×Q₈	—	2	24	no	—
163	C₃×Q₈	—	2	6	no	—
164	C₃×Q₈	—	2	16	no	—
165	C₃×Q₈	C₂²×C₃×D₃	1	16	yes	complemented
166	C₃×Q₈	C₂³×C₃²	1	4	yes	complemented
167	C₃×D₄	—	3	16	no	—
168	C₃×D₄	—	6	24	no	—
169	C₂×C₃×C₄	—	2	48	no	—
170	C₂×C₃×C₄	—	2	24	no	—
171	C₂×C₃×C₄	—	2	6	no	—
172	C₂×C₃×C₄	—	2	12	no	—
173	C₂×C₃×C₄	—	2	12	no	—
174	C₂×C₃×C₄	C₃×#67	1	12	yes	—
175	C₂×C₃×C₄	C₃²×D₄	1	3	yes	—
176	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
177	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
178	D₁₂	—	2	6	no	—
179	D₁₂	C₂³×C₃²	1	4	yes	—
180	C₃×C₈	—	3	16	no	—
181	C₃⋊C₈	C₂³×C₃²	1	4	yes	complemented
182	C₂×C₃²	—	2	24	no	—
183	C₂×C₃²	—	2	4	no	—
184	C₂×C₃²	—	12	4	no	—
185	C₂×C₃²	C₂×#215	1	6	yes	direct product factor
186	C₂×C₃²	C₂×C₃×SD₁₆	1	24	yes	complemented
187	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
188	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
189	C₃×D₃	—	4	16	no	—
190	C₂⁴	—	6	1	no	—
191	C₂×Q₈	—	2	3	no	—
192	C₂×Q₈	C₂×C₃²×D₃	1	6	yes	complemented
193	C₂×D₄	—	3	6	no	—
194	C₂×D₄	—	6	3	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
197	SD₁₆	—	3	16	no	—
198	C₂×C₈	—	3	6	no	—
199	C₂²×C₃	—	2	12	no	—
200	C₂²×C₃	—	2	16	no	—
201	C₂²×C₃	—	6	16	no	—
202	C₂²×C₃	—	12	48	no	—
203	C₂²×C₃	C₃×#215	1	16	yes	direct product factor
204	C₂²×C₃	C₃²×SD₁₆	1	4	yes	complemented
205	C₂²×C₃	C₂×C₃×#67	1	12	yes	—
206	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
207	C₂×D₃	—	2	4	no	—
208	C₂×D₃	—	4	12	no	—
209	C₃×C₄	—	2	32	no	—
210	C₃×C₄	—	2	16	no	—
211	C₃×C₄	—	2	4	no	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	2	12	no	—
214	C₃×C₄	C₂×C₃×#67	1	12	yes	—
215	C₃×C₄	C₂×C₃²×D₄	1	3	yes	—
216	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
217	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
218	C₃²	—	2	4	no	—
219	C₃²	C₂²×#215	1	1	yes	characteristic, direct product factor
220	C₃²	C₂²×C₃×SD₁₆	1	4	yes	complemented
221	C₂³	—	6	6	no	—
222	C₂³	—	12	4	no	—
223	C₂³	C₃²×#67	1	1	yes	characteristic
224	Q₈	—	2	6	no	—
225	Q₈	C₂²×C₃²×D₃	1	4	yes	complemented
226	D₄	—	3	4	no	—
227	D₄	—	6	6	no	—
228	C₂×C₄	—	2	6	no	—
229	C₂×C₄	C₃²×#67	1	3	yes	—
230	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
231	C₈	—	3	4	no	—
232	C₂×C₃	—	2	24	no	—
233	C₂×C₃	—	2	4	no	—
234	C₂×C₃	—	12	16	no	—
235	C₂×C₃	C₂×C₃×#215	1	24	yes	direct product factor
236	C₂×C₃	C₂×C₃²×SD₁₆	1	6	yes	complemented
237	C₂×C₃	C₂²×C₃×#67	1	4	yes	—
238	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
239	D₃	—	4	4	no	—
240	C₂²	—	6	4	no	—
241	C₂²	—	12	12	no	—
242	C₂²	C₃²×#215	1	4	yes	direct product factor
243	C₂²	C₂×C₃²×#67	1	3	yes	—
244	C₄	—	2	4	no	—
245	C₄	C₂×C₃²×#67	1	3	yes	—
246	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
247	C₃	—	2	4	no	—
248	C₃	C₂²×C₃×#215	1	4	yes	direct product factor
249	C₃	C₂²×C₃²×SD₁₆	1	1	yes	characteristic, complemented
250	C₂	—	12	4	no	—
251	C₂	C₂×C₃²×#215	1	6	yes	direct product factor
252	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	C₂²×C₃³×Q₈	C₂
socle	C₂³×C₃³	D₄
prime radical	C₂²×C₃²×D₁₂	C₂
hypercenter	C₂²×C₃²×Q₈	D₃
hypocenter	C₃	C₂²×C₃²×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×Q₈	C₂	semidirect
3. C₂×C₃²×#215	C₂	direct (N × H)
4. C₂²×C₃²×C₃⋊C₈	C₂	semidirect
6. C₂²×C₃×#215	C₃	direct (N × H)
9. C₂×C₃³×Q₈	C₂²	semidirect
14. C₃²×#215	C₂²	direct (N × H)
15. C₂×C₃²×C₃⋊C₈	C₂²	semidirect
17. C₂²×C₃²×Q₈	D₃	semidirect
18. C₂²×C₃²×Q₈	C₂×C₃	semidirect
23. C₂×C₃×#215	C₂×C₃	direct (N × H)
24. C₂²×C₃×C₃⋊C₈	C₂×C₃	semidirect
29. C₃³×Q₈	C₂³	semidirect
35. C₃²×C₃⋊C₈	C₂³	semidirect
37. C₂²×#215	C₃²	direct (N × H)
44. C₂×C₃²×Q₈	C₂×D₃	semidirect
45. C₂×C₃²×Q₈	C₂²×C₃	semidirect
58. C₃×#215	C₂²×C₃	direct (N × H)
59. C₂×C₃×C₃⋊C₈	C₂²×C₃	semidirect
60. C₂²×C₃³	SD₁₆	semidirect
68. C₂²×C₃×Q₈	C₃×D₃	semidirect
69. C₂²×C₃×Q₈	C₂×C₃²	semidirect
74. C₂×#215	C₂×C₃²	direct (N × H)
75. C₂²×C₃⋊C₈	C₂×C₃²	semidirect
87. C₃²×Q₈	C₂²×D₃	semidirect
88. C₃²×Q₈	C₂³×C₃	semidirect
103. C₃×C₃⋊C₈	C₂³×C₃	semidirect
105. C₂×C₃³	C₂×SD₁₆	semidirect
114. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
115. C₂×C₃×Q₈	C₂²×C₃²	semidirect
128. #215	C₂²×C₃²	direct (N × H)
129. C₂×C₃⋊C₈	C₂²×C₃²	semidirect
134. C₂²×C₃²	#215	direct (N × H)
135. C₂²×C₃²	C₃×SD₁₆	semidirect
149. C₂²×Q₈	C₃²×D₃	semidirect
153. C₃³	C₂²×SD₁₆	semidirect
165. C₃×Q₈	C₂²×C₃×D₃	semidirect
166. C₃×Q₈	C₂³×C₃²	semidirect
181. C₃⋊C₈	C₂³×C₃²	semidirect
185. C₂×C₃²	C₂×#215	direct (N × H)
186. C₂×C₃²	C₂×C₃×SD₁₆	semidirect
192. C₂×Q₈	C₂×C₃²×D₃	semidirect
203. C₂²×C₃	C₃×#215	direct (N × H)
204. C₂²×C₃	C₃²×SD₁₆	semidirect
219. C₃²	C₂²×#215	direct (N × H)
220. C₃²	C₂²×C₃×SD₁₆	semidirect
225. Q₈	C₂²×C₃²×D₃	semidirect
235. C₂×C₃	C₂×C₃×#215	direct (N × H)
236. C₂×C₃	C₂×C₃²×SD₁₆	semidirect
242. C₂²	C₃²×#215	direct (N × H)
248. C₃	C₂²×C₃×#215	direct (N × H)
249. C₃	C₂²×C₃²×SD₁₆	semidirect
251. C₂	C₂×C₃²×#215	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×Q₈	2
C₂²×C₃²×D₁₂	2
C₂×C₃²×#215	2
C₂²×C₃²×C₃⋊C₈	2
C₂²×C₃²×SD₁₆	3
C₂²×C₃×#215	3

Subgroups — C22×C32×#215 #13179187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×#215 #13179187