Subgroups — #13145656

Order 1728.

812 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×#88855	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×#208	C₂	1	1	yes	maximal, characteristic
4	#10589424	—	3	1	no	maximal
5	#10589166	—	3	4	no	maximal
6	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
7	#88855	—	2	1	no	—
8	#88855	C₄	1	2	yes	—
9	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
10	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
11	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
12	C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
13	C₃²×#208	—	2	1	no	—
14	C₃²×#208	C₄	1	2	yes	—
15	C₂×C₃²×C₃⋊C₈	—	2	1	no	—
16	C₂×C₃²×M₁₆	—	3	1	no	—
17	C₂×#3748	—	3	1	no	—
18	C₂×#3748	—	3	4	no	—
19	C₂×#3748	—	6	4	no	—
20	C₂²×C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
21	C₂×C₃×#208	D₃	1	4	yes	—
22	C₂²×C₃×C₃⋊C₈	—	3	4	no	—
23	C₂³×C₃³	Q₈	1	1	yes	characteristic
24	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
30	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
31	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
32	C₃²×C₃⋊C₈	—	2	2	no	—
33	#4558	—	9	4	no	—
34	#4552	—	9	1	no	—
35	C₂²×C₃²×C₄	—	2	4	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₃²×M₁₆	—	3	2	no	—
39	C₃²×M₁₆	—	6	1	no	—
40	C₂×C₃²×C₈	—	6	1	no	—
41	#3748	—	3	2	no	—
42	#3748	—	3	8	no	—
43	#3748	—	6	8	no	—
44	#3748	—	6	8	no	—
45	#3748	—	6	4	no	—
46	#3748	—	6	1	no	—
47	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
51	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
52	C₂×C₃⋊(C₃⋊C₈)	Dic₃	1	2	yes	—
53	C₃×#208	—	2	4	no	—
54	C₃×#208	Dic₃	1	8	yes	—
55	C₂×C₃×C₃⋊C₈	—	2	4	no	—
56	C₂×C₃×C₃⋊C₈	—	3	8	no	—
57	C₂×C₃×C₃⋊C₈	—	6	4	no	—
58	C₂×C₃×C₃⋊C₈	—	6	4	no	—
59	C₂²×C₃³	—	2	2	no	—
60	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
63	C₃³×C₄	C₄²	1	1	yes	characteristic
64	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃×M₁₆	—	3	4	no	—
68	C₂²×C₃×C₈	—	9	1	no	—
69	C₂×#208	—	9	4	no	—
70	C₂×#208	—	9	1	no	—
71	C₂×#208	—	18	4	no	—
72	C₂×#208	C₃⋊D₃	1	1	yes	characteristic
73	C₂²×C₃⋊C₈	—	3	4	no	—
74	C₂³×C₃²	—	2	4	no	—
75	C₂³×C₃²	Dic₆	1	4	yes	—
76	C₂³×C₃²	Dic₆	1	1	yes	characteristic
77	C₂×C₃²×C₄	—	2	4	no	—
78	C₂×C₃²×C₄	—	2	4	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
84	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
85	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	D₁₂	1	4	yes	—
88	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
90	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
91	C₂×C₃²×C₄	#67	1	4	yes	—
92	C₂×C₃²×C₄	#67	1	4	yes	—
93	C₂×C₃²×C₄	#67	1	1	yes	characteristic
94	C₂×C₃²×C₄	#67	1	1	yes	characteristic
95	C₃²×C₈	—	6	2	no	—
96	C₃⋊(C₃⋊C₈)	—	2	2	no	—
97	C₃⋊(C₃⋊C₈)	—	6	16	no	—
98	C₃⋊(C₃⋊C₈)	—	6	8	no	—
99	C₃⋊(C₃⋊C₈)	—	6	2	no	—
100	C₃×C₃⋊C₈	—	2	8	no	—
101	C₃×C₃⋊C₈	—	6	8	no	—
102	#341	—	27	1	no	Sylow
103	C₂×C₃³	—	2	1	no	—
104	C₂×C₃³	—	2	1	no	—
105	C₂×C₃³	#95	1	1	yes	characteristic
106	C₂×C₃³	#108	1	2	yes	—
107	C₂²×C₃×C₄	—	2	4	no	—
108	C₂²×C₃×C₄	D₃²	1	4	yes	—
109	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
110	C₃×M₁₆	—	3	8	no	—
111	C₃×M₁₆	—	6	4	no	—
112	C₂×C₃×C₈	—	6	4	no	—
113	C₂×C₃×C₈	—	9	2	no	—
114	C₂×C₃×C₈	—	18	1	no	—
115	C₂×C₃×C₈	—	18	1	no	—
116	#208	—	2	1	no	—
117	#208	—	9	8	no	—
118	#208	—	9	2	no	—
119	#208	—	18	8	no	—
120	#208	—	18	8	no	—
121	#208	—	18	4	no	—
122	#208	—	18	1	no	—
123	#208	C₃⋊Dic₃	1	2	yes	—
124	C₂×C₃⋊C₈	—	2	1	no	—
125	C₂×C₃⋊C₈	—	3	8	no	—
126	C₂×C₃⋊C₈	—	6	4	no	—
127	C₂×C₃⋊C₈	—	6	4	no	—
128	C₂×C₃⋊C₈	—	18	8	no	—
129	C₂×C₃⋊C₈	—	18	4	no	—
130	C₂×C₃⋊C₈	—	18	1	no	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	—	2	4	no	—
133	C₂²×C₃²	—	2	4	no	—
134	C₂²×C₃²	—	2	4	no	—
135	C₂²×C₃²	—	2	8	no	—
136	C₂²×C₃²	—	4	8	no	—
137	C₂²×C₃²	Dic₃⋊C₄	1	4	yes	—
138	C₂²×C₃²	Dic₃⋊C₄	1	4	yes	—
139	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
140	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
141	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
142	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	—	2	4	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	C₄×Dic₃	1	4	yes	—
148	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
149	C₃²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
150	C₃²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
151	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
152	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
153	C₃²×C₄	#217	1	4	yes	—
154	C₃²×C₄	#217	1	1	yes	characteristic
155	C₂×M₁₆	—	3	1	no	—
156	C₂×M₁₆	—	27	1	no	—
157	C₂²×C₈	—	9	1	no	—
158	C₃³	#341	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₃	—	2	4	no	—
160	C₂³×C₃	#627	1	4	yes	—
161	C₂³×C₃	#634	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
169	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
170	C₂×C₃×C₄	#624	1	4	yes	—
171	C₂×C₃×C₄	#625	1	4	yes	—
172	C₂×C₃×C₄	#626	1	4	yes	—
173	C₂×C₃×C₄	#626	1	4	yes	—
174	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
178	C₂×C₃×C₄	#638	1	1	yes	characteristic
179	C₂×C₃×C₄	#638	1	1	yes	characteristic
180	C₃×C₈	—	6	8	no	—
181	C₃×C₈	—	18	2	no	—
182	C₃⋊C₈	—	2	2	no	—
183	C₃⋊C₈	—	6	8	no	—
184	C₃⋊C₈	—	18	16	no	—
185	C₃⋊C₈	—	18	8	no	—
186	C₃⋊C₈	—	18	2	no	—
187	C₂×C₃²	—	2	1	no	—
188	C₂×C₃²	—	2	1	no	—
189	C₂×C₃²	—	2	8	no	—
190	C₂×C₃²	—	2	4	no	—
191	C₂×C₃²	—	2	4	no	—
192	C₂×C₃²	—	2	4	no	—
193	C₂×C₃²	—	4	4	no	—
194	C₂×C₃²	—	4	4	no	—
195	C₂×C₃²	#819	1	2	yes	—
196	C₂×C₃²	#822	1	8	yes	—
197	C₂×C₃²	C₃⋊#95	1	4	yes	—
198	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
199	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
200	M₁₆	—	3	2	no	—
201	M₁₆	—	6	1	no	—
202	M₁₆	—	27	2	no	—
203	M₁₆	—	54	1	no	—
204	C₂×C₈	—	6	1	no	—
205	C₂×C₈	—	9	2	no	—
206	C₂×C₈	—	18	1	no	—
207	C₂×C₈	—	18	1	no	—
208	C₂×C₈	—	54	1	no	—
209	C₂²×C₃	—	2	8	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	4	no	—
212	C₂²×C₃	—	2	4	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	—	4	8	no	—
215	C₂²×C₃	Dic₃⋊Dic₃	1	4	yes	—
216	C₂²×C₃	Dic₃⋊Dic₃	1	4	yes	—
217	C₂²×C₃	#3724	1	4	yes	—
218	C₂²×C₃	#3750	1	1	yes	characteristic
219	C₂²×C₃	#3750	1	1	yes	characteristic
220	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
221	C₃×C₄	—	2	4	no	—
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	Dic₃²	1	4	yes	—
226	C₃×C₄	(C₂×D₃)⋊Dic₃	1	4	yes	—
227	C₃×C₄	(C₂×D₃)⋊Dic₃	1	4	yes	—
228	C₃×C₄	#3722	1	4	yes	—
229	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
230	C₃×C₄	#3752	1	1	yes	characteristic
231	C₃×C₄	#3752	1	1	yes	characteristic
232	C₃×C₄	#3757	1	1	yes	characteristic
233	C₃²	—	2	4	no	—
234	C₃²	#4552	1	1	yes	characteristic, complemented
235	C₃²	#4558	1	4	yes	complemented
236	C₂³	#6293	1	1	yes	characteristic
237	C₂×C₄	D₃×C₃⋊Dic₃	1	1	yes	characteristic
238	C₂×C₄	Dic₃×C₃⋊D₃	1	1	yes	characteristic
239	C₂×C₄	#6289	1	1	yes	characteristic
240	C₂×C₄	#6290	1	1	yes	characteristic
241	C₂×C₄	#6291	1	1	yes	characteristic
242	C₂×C₄	#6292	1	1	yes	characteristic
243	C₈	—	6	2	no	—
244	C₈	—	18	2	no	—
245	C₈	—	54	2	no	—
246	C₂×C₃	—	2	4	no	—
247	C₂×C₃	—	2	4	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	8	no	—
251	C₂×C₃	—	2	4	no	—
252	C₂×C₃	—	4	4	no	—
253	C₂×C₃	—	4	4	no	—
254	C₂×C₃	#63577	1	8	yes	—
255	C₂×C₃	#63579	1	4	yes	—
256	C₂×C₃	#63649	1	2	yes	—
257	C₂×C₃	#63658	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	Dic₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
260	C₂²	#88806	1	1	yes	characteristic
261	C₂²	#88807	1	1	yes	characteristic
262	C₄	Dic₃×C₃⋊Dic₃	1	1	yes	characteristic
263	C₄	(C₂×D₃)⋊(C₃⋊Dic₃)	1	1	yes	characteristic
264	C₄	#88803	1	1	yes	characteristic
265	C₄	#88804	1	1	yes	characteristic
266	C₃	—	2	4	no	—
267	C₃	#10589166	1	4	yes	complemented
268	C₃	#10589424	1	1	yes	characteristic, complemented
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	#11722060	1	2	yes	—
272	C₂	#11722061	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6291
commutator	C₃³×C₄	C₄²
Frattini, hypercenter	C₂²×C₄	D₃×C₃⋊D₃
Fitting, cosocle	C₂²×C₃³×C₄	C₂²
socle	C₂²×C₃³	C₄⋊C₄
prime radical	C₂³×C₃³	Q₈
hypocenter	C₃³	#341
LCS	G ▷ C₃³×C₄ ▷ 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
158. C₃³	#341	semidirect
234. C₃²	#4552	semidirect
235. C₃²	#4558	semidirect
267. C₃	#10589166	semidirect
268. C₃	#10589424	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88855	2
C₂²×C₃×C₃⋊(C₃⋊C₈)	2
C₂×C₃²×#208	2
#10589424	3
#10589166	3