Subgroups — #13145657

Order 1728.

860 conjugacy classes of subgroups.

The subgroup lattice has 302 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	C₂×C₃×#3748	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10589396	—	3	1	no	maximal
5	#10589167	—	3	4	no	maximal
6	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃²×Dic₃	C₄	1	2	yes	—
8	#88855	—	2	1	no	—
9	#88855	C₄	1	2	yes	complemented
10	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
11	C₃×#3748	—	2	1	no	—
12	C₃×#3748	C₄	1	2	yes	complemented
13	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
14	C₃²×C₄⋊Dic₃	—	2	1	no	—
15	C₃²×C₄⋊Dic₃	—	2	1	no	—
16	C₂×C₃²×C₄⋊C₄	—	3	1	no	—
17	C₂×#3748	—	3	1	no	—
18	C₂×#3748	—	3	4	no	—
19	C₂×#3748	—	6	4	no	—
20	C₂×#3748	D₃	1	1	yes	characteristic
21	C₂×C₃×C₄⋊Dic₃	D₃	1	4	yes	—
22	C₂×C₃×#208	—	3	4	no	—
23	C₂³×C₃³	C₂×C₄	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	D₄	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	Q₈	1	1	yes	characteristic
30	C₂×C₃²×Dic₃	—	2	2	no	—
31	C₂×C₃²×Dic₃	—	4	2	no	—
32	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
33	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
34	#4554	—	9	1	no	—
35	#4530	—	9	4	no	—
36	C₂²×C₃²×C₄	—	2	4	no	—
37	C₂²×C₃²×C₄	—	3	2	no	—
38	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃×Dic₃	Dic₃	1	8	yes	—
41	C₃²×C₄⋊C₄	—	6	1	no	—
42	C₃²×C₄⋊C₄	—	6	1	no	—
43	#3748	—	2	1	no	—
44	#3748	—	3	2	no	—
45	#3748	—	3	8	no	—
46	#3748	—	6	8	no	—
47	#3748	—	6	8	no	—
48	#3748	—	6	4	no	—
49	#3748	—	6	1	no	—
50	#3748	Dic₃	1	2	yes	complemented
51	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
52	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
53	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
54	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
55	C₃×C₄⋊Dic₃	—	2	4	no	—
56	C₃×C₄⋊Dic₃	—	2	4	no	—
57	C₃×#208	—	3	8	no	—
58	C₃×#208	—	6	4	no	—
59	C₂×C₃×C₃⋊C₈	—	6	4	no	—
60	C₂²×C₃³	—	2	1	no	—
61	C₂²×C₃³	—	2	1	no	—
62	C₂²×C₃³	C₄²	1	1	yes	characteristic
63	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
65	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
69	C₃²×Dic₃	—	4	2	no	—
70	C₂×C₃×M₁₆	—	9	1	no	—
71	C₂×C₃×C₄⋊C₄	—	3	4	no	—
72	C₂×C₄⋊Dic₃	C₃⋊D₃	1	1	yes	characteristic
73	C₂×#208	—	3	4	no	—
74	C₂×#208	—	9	4	no	—
75	C₂×#208	—	9	1	no	—
76	C₂×#208	—	18	4	no	—
77	C₂³×C₃²	—	2	4	no	—
78	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
79	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
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	—	2	4	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	6	2	no	—
87	C₂×C₃²×C₄	—	12	2	no	—
88	C₂×C₃²×C₄	Dic₆	1	4	yes	—
89	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
91	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
92	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃²×C₄	D₁₂	1	4	yes	—
94	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₃²×C₄	#67	1	4	yes	—
97	C₂×C₃²×C₄	#67	1	4	yes	—
98	C₂×C₃²×C₄	#67	1	1	yes	characteristic
99	C₂×C₃²×C₄	#67	1	1	yes	characteristic
100	C₂×C₃×Dic₃	—	2	8	no	—
101	C₂×C₃×Dic₃	—	4	8	no	—
102	C₃⋊(C₃⋊C₈)	—	2	2	no	—
103	C₃⋊(C₃⋊C₈)	—	6	16	no	—
104	C₃⋊(C₃⋊C₈)	—	6	8	no	—
105	C₃⋊(C₃⋊C₈)	—	6	2	no	—
106	C₃×C₃⋊C₈	—	6	8	no	—
107	#343	—	27	1	no	Sylow
108	C₂×C₃³	—	2	1	no	—
109	C₂×C₃³	—	2	1	no	—
110	C₂×C₃³	#95	1	1	yes	characteristic
111	C₂×C₃³	#100	1	1	yes	characteristic
112	C₂×C₃³	#101	1	1	yes	characteristic
113	C₂²×C₃×C₄	—	2	4	no	—
114	C₂²×C₃×C₄	—	3	8	no	—
115	C₂²×C₃×C₄	D₃²	1	4	yes	—
116	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
117	C₂²×Dic₃	C₃⋊Dic₃	1	2	yes	—
118	C₃×M₁₆	—	9	2	no	—
119	C₃×M₁₆	—	18	1	no	—
120	C₂×C₃×C₈	—	18	1	no	—
121	C₃×C₄⋊C₄	—	6	4	no	—
122	C₃×C₄⋊C₄	—	6	4	no	—
123	C₄⋊Dic₃	—	2	1	no	—
124	C₄⋊Dic₃	—	2	1	no	—
125	#208	—	3	8	no	—
126	#208	—	6	4	no	—
127	#208	—	9	8	no	—
128	#208	—	9	2	no	—
129	#208	—	18	8	no	—
130	#208	—	18	8	no	—
131	#208	—	18	4	no	—
132	#208	—	18	1	no	—
133	C₂×C₃⋊C₈	—	6	4	no	—
134	C₂×C₃⋊C₈	—	18	8	no	—
135	C₂×C₃⋊C₈	—	18	4	no	—
136	C₂×C₃⋊C₈	—	18	1	no	—
137	C₂²×C₃²	—	2	1	no	—
138	C₂²×C₃²	—	2	1	no	—
139	C₂²×C₃²	—	2	4	no	—
140	C₂²×C₃²	—	2	4	no	—
141	C₂²×C₃²	—	2	4	no	—
142	C₂²×C₃²	—	2	4	no	—
143	C₂²×C₃²	—	2	4	no	—
144	C₂²×C₃²	—	4	4	no	—
145	C₂²×C₃²	—	4	4	no	—
146	C₂²×C₃²	C₄×Dic₃	1	4	yes	—
147	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
148	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
149	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
150	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
151	C₂²×C₃²	#217	1	4	yes	—
152	C₃²×C₄	—	2	4	no	—
153	C₃²×C₄	—	2	4	no	—
154	C₃²×C₄	—	2	4	no	—
155	C₃²×C₄	—	2	4	no	—
156	C₃²×C₄	—	12	2	no	—
157	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
158	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
159	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
160	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
161	C₃²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
162	C₃²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
163	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
164	C₃²×C₄	#217	1	1	yes	characteristic
165	C₃×Dic₃	—	4	8	no	—
166	C₂×M₁₆	—	9	1	no	—
167	C₂×M₁₆	—	27	1	no	—
168	C₂×C₄⋊C₄	—	3	1	no	—
169	C₃³	#343	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	2	4	no	—
171	C₂³×C₃	D₃×Dic₃	1	4	yes	—
172	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	6	8	no	—
180	C₂×C₃×C₄	—	12	8	no	—
181	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
182	C₂×C₃×C₄	#624	1	4	yes	—
183	C₂×C₃×C₄	#625	1	4	yes	—
184	C₂×C₃×C₄	#626	1	4	yes	—
185	C₂×C₃×C₄	#626	1	4	yes	—
186	C₂×C₃×C₄	#627	1	4	yes	—
187	C₂×C₃×C₄	#634	1	1	yes	characteristic
188	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
189	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
190	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
191	C₂×C₃×C₄	#638	1	1	yes	characteristic
192	C₂×C₃×C₄	#638	1	1	yes	characteristic
193	C₂×Dic₃	—	2	2	no	—
194	C₂×Dic₃	—	4	2	no	—
195	C₃×C₈	—	18	2	no	—
196	C₃⋊C₈	—	6	8	no	—
197	C₃⋊C₈	—	18	16	no	—
198	C₃⋊C₈	—	18	8	no	—
199	C₃⋊C₈	—	18	2	no	—
200	C₂×C₃²	—	2	1	no	—
201	C₂×C₃²	—	2	1	no	—
202	C₂×C₃²	—	2	4	no	—
203	C₂×C₃²	—	2	4	no	—
204	C₂×C₃²	—	2	4	no	—
205	C₂×C₃²	—	2	4	no	—
206	C₂×C₃²	—	2	4	no	—
207	C₂×C₃²	—	4	4	no	—
208	C₂×C₃²	—	4	4	no	—
209	C₂×C₃²	#823	1	1	yes	characteristic
210	C₂×C₃²	#824	1	1	yes	characteristic
211	C₂×C₃²	C₃⋊#95	1	4	yes	—
212	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
213	C₂×C₃²	#833	1	4	yes	—
214	C₂×C₃²	#836	1	4	yes	—
215	C₂²×C₄	—	3	2	no	—
216	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
217	M₁₆	—	9	2	no	—
218	M₁₆	—	18	1	no	—
219	M₁₆	—	27	2	no	—
220	M₁₆	—	54	1	no	—
221	C₂×C₈	—	18	1	no	—
222	C₂×C₈	—	54	1	no	—
223	C₄⋊C₄	—	6	1	no	—
224	C₄⋊C₄	—	6	1	no	—
225	C₂²×C₃	—	2	4	no	—
226	C₂²×C₃	—	2	4	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	4	no	—
230	C₂²×C₃	—	2	4	no	—
231	C₂²×C₃	—	2	4	no	—
232	C₂²×C₃	—	4	4	no	—
233	C₂²×C₃	—	4	4	no	—
234	C₂²×C₃	Dic₃²	1	4	yes	—
235	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	4	yes	—
236	C₂²×C₃	Dic₃⋊Dic₃	1	4	yes	—
237	C₂²×C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic
238	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
239	C₂²×C₃	#3757	1	1	yes	characteristic
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	2	4	no	—
242	C₃×C₄	—	2	4	no	—
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	12	8	no	—
245	C₃×C₄	(C₂×D₃)⋊Dic₃	1	4	yes	—
246	C₃×C₄	#3722	1	4	yes	—
247	C₃×C₄	Dic₃⋊Dic₃	1	4	yes	—
248	C₃×C₄	#3724	1	4	yes	—
249	C₃×C₄	#3750	1	1	yes	characteristic
250	C₃×C₄	#3750	1	1	yes	characteristic
251	C₃×C₄	#3752	1	1	yes	characteristic
252	C₃×C₄	#3752	1	1	yes	characteristic
253	Dic₃	—	4	2	no	—
254	C₃²	—	2	4	no	—
255	C₃²	#4530	1	4	yes	complemented
256	C₃²	#4554	1	1	yes	characteristic, complemented
257	C₂³	D₃×C₃⋊Dic₃	1	1	yes	characteristic
258	C₂×C₄	—	6	2	no	—
259	C₂×C₄	—	12	2	no	—
260	C₂×C₄	Dic₃×C₃⋊D₃	1	1	yes	characteristic
261	C₂×C₄	#6289	1	1	yes	characteristic
262	C₂×C₄	#6290	1	1	yes	characteristic
263	C₂×C₄	#6291	1	1	yes	characteristic
264	C₂×C₄	#6292	1	1	yes	characteristic
265	C₂×C₄	#6293	1	1	yes	characteristic
266	C₈	—	18	2	no	—
267	C₈	—	54	2	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	4	no	—
271	C₂×C₃	—	2	4	no	—
272	C₂×C₃	—	2	4	no	—
273	C₂×C₃	—	2	4	no	—
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	4	4	no	—
276	C₂×C₃	—	4	4	no	—
277	C₂×C₃	#63558	1	4	yes	—
278	C₂×C₃	#63560	1	4	yes	—
279	C₂×C₃	#63579	1	4	yes	—
280	C₂×C₃	#63658	1	1	yes	characteristic
281	C₂×C₃	#63660	1	1	yes	characteristic
282	C₂×C₃	#63663	1	1	yes	characteristic
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	Dic₃×C₃⋊Dic₃	1	1	yes	characteristic
286	C₂²	(C₂×D₃)⋊(C₃⋊Dic₃)	1	1	yes	characteristic
287	C₂²	Dic₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
288	C₄	—	12	2	no	—
289	C₄	#88803	1	1	yes	characteristic
290	C₄	#88804	1	1	yes	characteristic
291	C₄	#88806	1	1	yes	characteristic
292	C₄	#88807	1	1	yes	characteristic
293	C₃	—	2	4	no	—
294	C₃	#10589167	1	4	yes	complemented
295	C₃	#10589396	1	1	yes	characteristic, complemented
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	#11722025	1	1	yes	characteristic
299	C₂	#11722028	1	1	yes	characteristic
300	C₂	#11722061	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₃)⋊(C₃⋊Dic₃)
commutator	C₂²×C₃³	C₄²
Frattini, hypercenter	C₂²×C₄	D₃×C₃⋊D₃
Fitting, cosocle	C₂²×C₃³×C₄	C₂²
socle	C₂²×C₃³	C₂²⋊C₄
prime radical	C₂³×C₃³	C₂×C₄
hypocenter	C₃³	#343
LCS	G ▷ C₂²×C₃³ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
9. #88855	C₄	semidirect
12. C₃×#3748	C₄	semidirect
50. #3748	Dic₃	semidirect
169. C₃³	#343	semidirect
255. C₃²	#4530	semidirect
256. C₃²	#4554	semidirect
294. C₃	#10589167	semidirect
295. C₃	#10589396	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
C₂×C₃×#3748	2
C₂×C₃²×C₄⋊Dic₃	2
#10589396	3
#10589167	3