Subgroups — #13160283

Order 1728.

1284 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3748	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×#208	C₂	1	1	yes	maximal, characteristic
4	#11722021	C₂	1	2	yes	maximal, complemented
5	#11722018	C₂	1	2	yes	maximal, complemented
6	M₁₆×C₃⋊Dic₃	—	3	1	no	maximal
7	#10591060	—	3	4	no	maximal
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
10	C₄×C₃⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
11	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
13	C₃×#3748	C₄	1	4	yes	complemented
14	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₃²×#208	C₄	1	4	yes	complemented
16	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃²×M₁₆	—	3	1	no	—
18	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
19	C₂×C₄×C₃⋊Dic₃	—	3	4	no	—
20	C₂×C₄×C₃⋊Dic₃	—	6	4	no	—
21	C₂×#3748	D₃	1	1	yes	characteristic
22	C₂×C₃×#208	—	3	4	no	—
23	C₂×C₃×#208	D₃	1	4	yes	—
24	#63642	—	3	2	no	—
25	C₈×C₃⋊Dic₃	—	3	2	no	—
26	#63555	—	3	8	no	—
27	#63553	—	3	8	no	—
28	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
31	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
34	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
35	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
36	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	complemented
37	C₃²×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
38	Dic₃×M₁₆	—	9	4	no	—
39	Dic₃×M₁₆	—	9	1	no	—
40	C₂²×C₃²×C₄	—	2	4	no	—
41	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃⋊Dic₃	—	3	2	no	—
44	C₂²×C₃⋊Dic₃	—	3	8	no	—
45	C₂²×C₃⋊Dic₃	—	6	8	no	—
46	C₃²×M₁₆	—	3	4	no	—
47	C₂×C₃²×C₈	—	3	2	no	—
48	C₄×C₃⋊Dic₃	—	3	2	no	—
49	C₄×C₃⋊Dic₃	—	3	8	no	—
50	C₄×C₃⋊Dic₃	—	3	1	no	—
51	C₄×C₃⋊Dic₃	—	3	1	no	—
52	C₄×C₃⋊Dic₃	—	3	4	no	—
53	C₄×C₃⋊Dic₃	—	3	4	no	—
54	C₄×C₃⋊Dic₃	—	6	8	no	—
55	C₄×C₃⋊Dic₃	—	6	4	no	—
56	C₄×C₃⋊Dic₃	—	6	4	no	—
57	#3748	Dic₃	1	4	yes	complemented
58	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
59	C₃×#208	—	3	16	no	—
60	C₃×#208	Dic₃	1	16	yes	complemented
61	C₂×C₃×C₃⋊C₈	—	3	8	no	—
62	C₂×C₃×C₃⋊C₈	C₂×D₃	1	8	yes	—
63	C₂²×C₃³	—	2	2	no	—
64	C₂²×C₃³	C₄²	1	2	yes	—
65	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
66	C₃³×C₄	C₄²	1	2	yes	—
67	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
70	C₃⋊(C₃⋊Dic₃)	M₁₆	1	4	yes	complemented
71	C₂×C₃×M₁₆	—	3	4	no	—
72	C₂×C₃×M₁₆	—	9	1	no	—
73	C₂×C₄×Dic₃	—	9	4	no	—
74	C₂×C₄×Dic₃	—	9	1	no	—
75	C₂×C₄×Dic₃	—	18	4	no	—
76	C₂×#208	—	3	4	no	—
77	C₂×#208	C₃⋊D₃	1	1	yes	characteristic
78	#815	—	9	8	no	—
79	#815	—	9	2	no	—
80	C₈×Dic₃	—	9	8	no	—
81	C₈×Dic₃	—	9	2	no	—
82	C₂³×C₃²	—	2	4	no	—
83	C₂³×C₃²	C₄×D₃	1	4	yes	—
84	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	—	2	8	no	—
86	C₂×C₃²×C₄	—	2	8	no	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	—	2	4	no	—
89	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
90	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
91	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
92	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
94	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
95	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
96	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃⋊Dic₃	—	3	2	no	—
98	C₂×C₃⋊Dic₃	—	3	2	no	—
99	C₂×C₃⋊Dic₃	—	3	8	no	—
100	C₂×C₃⋊Dic₃	—	3	8	no	—
101	C₂×C₃⋊Dic₃	—	6	8	no	—
102	C₂×C₃⋊Dic₃	—	6	8	no	—
103	C₂×C₃⋊Dic₃	—	6	4	no	—
104	C₂×C₃⋊Dic₃	—	6	16	no	—
105	C₂×C₃⋊Dic₃	—	6	32	no	—
106	C₃²×C₈	—	3	4	no	—
107	C₃⋊(C₃⋊C₈)	C₂×Dic₃	1	4	yes	complemented
108	C₃×C₃⋊C₈	—	3	16	no	—
109	C₃×C₃⋊C₈	C₂×Dic₃	1	16	yes	complemented
110	C₄×M₁₆	—	27	1	no	Sylow
111	C₂×C₃³	—	2	2	no	—
112	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₃³	C₂×M₁₆	1	2	yes	—
114	C₂²×C₃×C₄	—	2	4	no	—
115	C₂²×C₃×C₄	D₃²	1	4	yes	—
116	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
117	C₂²×Dic₃	—	9	8	no	—
118	C₂²×Dic₃	—	9	2	no	—
119	C₂²×Dic₃	—	18	8	no	—
120	C₃×M₁₆	—	3	16	no	—
121	C₃×M₁₆	—	9	4	no	—
122	C₂×C₃×C₈	—	3	8	no	—
123	C₂×C₃×C₈	—	9	2	no	—
124	C₄×Dic₃	—	9	8	no	—
125	C₄×Dic₃	—	9	2	no	—
126	C₄×Dic₃	—	9	4	no	—
127	C₄×Dic₃	—	9	4	no	—
128	C₄×Dic₃	—	9	1	no	—
129	C₄×Dic₃	—	9	1	no	—
130	C₄×Dic₃	—	18	8	no	—
131	C₄×Dic₃	—	18	4	no	—
132	C₄×Dic₃	—	18	4	no	—
133	#208	—	3	16	no	—
134	#208	C₃⋊Dic₃	1	4	yes	complemented
135	C₂×C₃⋊C₈	—	3	8	no	—
136	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
137	C₂²×C₃²	—	2	16	no	—
138	C₂²×C₃²	—	2	8	no	—
139	C₂²×C₃²	—	2	2	no	—
140	C₂²×C₃²	—	2	8	no	—
141	C₂²×C₃²	—	2	4	no	—
142	C₂²×C₃²	C₄×Dic₃	1	8	yes	—
143	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
144	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
145	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
146	C₃²×C₄	—	2	8	no	—
147	C₃²×C₄	—	2	4	no	—
148	C₃²×C₄	—	2	4	no	—
149	C₃²×C₄	C₄×Dic₃	1	8	yes	—
150	C₃²×C₄	C₄×Dic₃	1	2	yes	—
151	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
152	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
153	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
154	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
155	C₃⋊Dic₃	—	3	4	no	—
156	C₃⋊Dic₃	—	3	16	no	—
157	C₃⋊Dic₃	—	6	16	no	—
158	C₃⋊Dic₃	—	6	16	no	—
159	C₃⋊Dic₃	—	6	8	no	—
160	C₃⋊Dic₃	—	6	2	no	—
161	C₂×M₁₆	—	3	1	no	—
162	C₂×M₁₆	—	9	1	no	—
163	C₂×C₄²	—	27	1	no	—
164	#97	—	27	2	no	—
165	C₄×C₈	—	27	2	no	—
166	C₃³	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₃	—	2	4	no	—
168	C₂³×C₃	#624	1	4	yes	—
169	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	8	no	—
171	C₂×C₃×C₄	—	2	8	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
175	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
176	C₂×C₃×C₄	#624	1	4	yes	—
177	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
178	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
180	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
181	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
182	C₂×Dic₃	—	9	8	no	—
183	C₂×Dic₃	—	9	2	no	—
184	C₂×Dic₃	—	9	8	no	—
185	C₂×Dic₃	—	9	2	no	—
186	C₂×Dic₃	—	18	8	no	—
187	C₂×Dic₃	—	18	8	no	—
188	C₂×Dic₃	—	18	16	no	—
189	C₂×Dic₃	—	18	4	no	—
190	C₂×Dic₃	—	18	32	no	—
191	C₃×C₈	—	3	16	no	—
192	C₃×C₈	—	9	4	no	—
193	C₃⋊C₈	—	3	16	no	—
194	C₃⋊C₈	C₂×C₃⋊Dic₃	1	4	yes	complemented
195	C₂×C₃²	—	2	16	no	—
196	C₂×C₃²	—	2	8	no	—
197	C₂×C₃²	—	2	2	no	—
198	C₂×C₃²	—	2	8	no	—
199	C₂×C₃²	—	2	4	no	—
200	C₂×C₃²	D₃×M₁₆	1	8	yes	—
201	C₂×C₃²	D₃×M₁₆	1	2	yes	—
202	C₂×C₃²	C₂×C₄×Dic₃	1	4	yes	—
203	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
204	C₂²×C₄	—	27	2	no	—
205	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
206	M₁₆	—	3	4	no	—
207	M₁₆	—	9	4	no	—
208	C₂×C₈	—	3	2	no	—
209	C₂×C₈	—	9	2	no	—
210	C₄²	—	27	2	no	—
211	C₄²	—	27	1	no	—
212	C₄²	—	27	1	no	—
213	C₂²×C₃	—	2	16	no	—
214	C₂²×C₃	—	2	8	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	4	no	—
217	C₂²×C₃	—	2	8	no	—
218	C₂²×C₃	Dic₃²	1	8	yes	—
219	C₂²×C₃	C₄×C₃⋊Dic₃	1	2	yes	—
220	C₂²×C₃	C₂×#624	1	4	yes	—
221	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	2	8	no	—
225	C₃×C₄	Dic₃²	1	8	yes	—
226	C₃×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
227	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
228	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
229	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
230	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
231	Dic₃	—	9	16	no	—
232	Dic₃	—	9	4	no	—
233	Dic₃	—	18	16	no	—
234	Dic₃	—	18	16	no	—
235	Dic₃	—	18	8	no	—
236	Dic₃	—	18	2	no	—
237	C₃²	—	2	4	no	—
238	C₃²	Dic₃×M₁₆	1	4	yes	complemented
239	C₃²	Dic₃×M₁₆	1	1	yes	characteristic, complemented
240	C₂³	#6289	1	1	yes	characteristic
241	C₂×C₄	—	27	2	no	—
242	C₂×C₄	—	27	2	no	—
243	C₂×C₄	—	54	4	no	—
244	C₂×C₄	D₃×C₃⋊Dic₃	1	2	yes	—
245	C₂×C₄	Dic₃×C₃⋊D₃	1	2	yes	—
246	C₂×C₄	#6289	1	1	yes	characteristic
247	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
248	C₈	—	3	4	no	—
249	C₈	—	9	4	no	—
250	C₂×C₃	—	2	16	no	—
251	C₂×C₃	—	2	2	no	—
252	C₂×C₃	—	2	8	no	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	—	2	8	no	—
255	C₂×C₃	#63818	1	8	yes	—
256	C₂×C₃	C₂×Dic₃²	1	4	yes	—
257	C₂×C₃	M₁₆×C₃⋊D₃	1	2	yes	—
258	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	Dic₃×C₃⋊Dic₃	1	2	yes	—
261	C₂²	C₂×#6289	1	1	yes	characteristic
262	C₄	—	27	4	no	—
263	C₄	—	54	2	no	—
264	C₄	Dic₃×C₃⋊Dic₃	1	2	yes	—
265	C₄	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
266	C₄	C₂×Dic₃×C₃⋊D₃	1	1	yes	characteristic
267	C₃	—	2	4	no	—
268	C₃	#10591060	1	4	yes	complemented
269	C₃	M₁₆×C₃⋊Dic₃	1	1	yes	characteristic, complemented
270	C₂	—	2	2	no	—
271	C₂	#11723358	1	2	yes	—
272	C₂	C₂×Dic₃×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
commutator	C₂×C₃³	C₂×C₄²
Fitting	C₂²×C₃³×C₄	C₂²
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂²×C₄	D₃×C₃⋊D₃
hypocenter	C₃³	C₄×M₁₆
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11722021	C₂	semidirect
5. #11722018	C₂	semidirect
13. C₃×#3748	C₄	semidirect
15. C₃²×#208	C₄	semidirect
36. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
37. C₃²×C₃⋊C₈	C₂×C₄	semidirect
57. #3748	Dic₃	semidirect
60. C₃×#208	Dic₃	semidirect
70. C₃⋊(C₃⋊Dic₃)	M₁₆	semidirect
107. C₃⋊(C₃⋊C₈)	C₂×Dic₃	semidirect
109. C₃×C₃⋊C₈	C₂×Dic₃	semidirect
134. #208	C₃⋊Dic₃	semidirect
166. C₃³	C₄×M₁₆	semidirect
194. C₃⋊C₈	C₂×C₃⋊Dic₃	semidirect
238. C₃²	Dic₃×M₁₆	semidirect
239. C₃²	Dic₃×M₁₆	semidirect
268. C₃	#10591060	semidirect
269. C₃	M₁₆×C₃⋊Dic₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊(C₃⋊Dic₃)	2
C₂×C₃×#3748	2
C₂×C₃²×#208	2
#11722021	2
#11722018	2
M₁₆×C₃⋊Dic₃	3
#10591060	3