Subgroups — #11785966

Order 1152.

589 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×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589155	C₂	1	4	yes	maximal
5	#70443	—	3	1	no	maximal
6	#69392	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₃	C₄	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
13	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
14	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
15	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
16	C₂×C₃×C₄⋊C₈	—	3	1	no	—
17	C₃×C₄³	—	3	1	no	—
18	C₄²×Dic₃	D₃	1	1	yes	characteristic
19	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
20	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
22	C₂×C₄⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
23	#4550	—	3	4	no	—
24	#4524	—	3	4	no	—
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
29	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
30	C₃²×C₄²	D₄	1	2	yes	—
31	C₃²×C₄²	Q₈	1	2	yes	—
32	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
33	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
34	C₃×C₄×Dic₃	—	2	2	no	—
35	C₃×C₄×Dic₃	—	2	2	no	—
36	C₃×C₄×Dic₃	—	2	2	no	—
37	C₃×C₄×Dic₃	—	2	4	no	—
38	C₃×C₄×Dic₃	C₈	1	4	yes	complemented
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	—	2	4	no	—
41	#1831	—	9	1	no	Sylow
42	C₂²×C₃×C₈	—	3	2	no	—
43	C₂×C₃×C₄²	—	2	1	no	—
44	C₂×C₃×C₄²	—	3	1	no	—
45	C₂×C₃×C₄²	—	3	2	no	—
46	C₂×C₃×C₄²	—	3	1	no	—
47	C₂×C₃×C₄²	—	3	2	no	—
48	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
51	C₂×C₄×Dic₃	Dic₃	1	2	yes	—
52	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
53	C₂×C₄×Dic₃	C₂×D₃	1	2	yes	—
54	C₂²×C₃⋊C₈	—	3	2	no	—
55	C₂²×C₃⋊C₈	—	3	2	no	—
56	C₂²×C₃⋊C₈	—	6	2	no	—
57	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
58	C₃×C₄⋊C₈	—	6	2	no	—
59	C₄⋊(C₃⋊C₈)	—	2	2	no	—
60	C₄⋊(C₃⋊C₈)	—	6	4	no	—
61	C₄⋊(C₃⋊C₈)	—	6	2	no	—
62	C₄⋊(C₃⋊C₈)	—	6	2	no	—
63	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
66	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
67	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
68	C₂×C₃²×C₄	M₁₆	1	2	yes	—
69	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃²×C₄	#41	1	2	yes	—
72	C₂×C₃²×C₄	#41	1	2	yes	—
73	C₂×C₃×Dic₃	—	2	2	no	—
74	C₂×C₃×Dic₃	—	2	2	no	—
75	C₂×C₃×Dic₃	—	2	2	no	—
76	C₂×C₃×Dic₃	—	2	2	no	—
77	C₂×C₃×Dic₃	C₂×C₈	1	4	yes	—
78	C₂×C₃×Dic₃	M₁₆	1	4	yes	—
79	C₃⋊(C₃⋊C₈)	—	4	2	no	—
80	C₃×C₃⋊C₈	—	4	2	no	—
81	C₂×C₄⋊C₈	—	3	1	no	—
82	C₂×C₄⋊C₈	—	9	1	no	—
83	C₄³	—	3	1	no	—
84	#336	—	9	4	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	—	3	2	no	—
89	C₂²×C₃×C₄	—	3	2	no	—
90	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
97	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
98	C₂×C₃×C₈	—	6	2	no	—
99	C₂×C₃×C₈	—	6	4	no	—
100	C₃×C₄²	—	2	2	no	—
101	C₃×C₄²	—	2	2	no	—
102	C₃×C₄²	—	3	4	no	—
103	C₃×C₄²	—	6	4	no	—
104	C₃×C₄²	—	6	2	no	—
105	C₃×C₄²	—	6	2	no	—
106	C₃×C₄²	—	6	2	no	—
107	C₃×C₄²	Dic₆	1	2	yes	—
108	C₃×C₄²	Dic₆	1	2	yes	—
109	C₃×C₄²	D₁₂	1	2	yes	—
110	C₃×C₄²	#67	1	2	yes	—
111	C₄×Dic₃	—	2	2	no	—
112	C₄×Dic₃	—	2	2	no	—
113	C₄×Dic₃	—	2	2	no	—
114	C₄×Dic₃	—	2	4	no	—
115	C₄×Dic₃	C₃⋊C₈	1	4	yes	complemented
116	C₂×C₃⋊C₈	—	2	4	no	—
117	C₂×C₃⋊C₈	—	2	2	no	—
118	C₂×C₃⋊C₈	—	6	4	no	—
119	C₂×C₃⋊C₈	—	6	2	no	—
120	C₂×C₃⋊C₈	—	6	4	no	—
121	C₂×C₃⋊C₈	—	6	4	no	—
122	C₂×C₃⋊C₈	—	6	2	no	—
123	C₂×C₃⋊C₈	—	12	4	no	—
124	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₃²	#117	1	2	yes	—
126	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
127	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
128	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
129	C₃²×C₄	—	2	2	no	—
130	C₃²×C₄	C₄⋊C₈	1	4	yes	—
131	C₃²×C₄	#124	1	1	yes	characteristic
132	C₃²×C₄	#124	1	1	yes	characteristic
133	C₃²×C₄	#125	1	2	yes	—
134	C₃×Dic₃	—	2	4	no	—
135	C₃×Dic₃	—	2	4	no	—
136	C₂²×C₈	—	3	2	no	—
137	C₂²×C₈	—	9	2	no	—
138	C₂×C₄²	—	3	1	no	—
139	C₂×C₄²	—	3	2	no	—
140	C₂×C₄²	—	3	1	no	—
141	C₂×C₄²	—	3	2	no	—
142	C₂×C₄²	D₃²	1	1	yes	characteristic
143	C₄⋊C₈	—	6	2	no	—
144	C₄⋊C₈	—	18	2	no	—
145	C₂³×C₃	—	2	1	no	—
146	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	3	4	no	—
159	C₂×C₃×C₄	—	3	4	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	6	2	no	—
162	C₂×C₃×C₄	—	6	2	no	—
163	C₂×C₃×C₄	—	6	2	no	—
164	C₂×C₃×C₄	—	6	2	no	—
165	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
166	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
167	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
168	C₂×C₃×C₄	#208	1	2	yes	—
169	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
170	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
171	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
172	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
173	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
176	C₂×C₃×C₄	#235	1	2	yes	—
177	C₂×C₃×C₄	#235	1	2	yes	—
178	C₂×C₃×C₄	#237	1	2	yes	—
179	C₂×C₃×C₄	#239	1	2	yes	—
180	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
181	C₂×Dic₃	—	2	2	no	—
182	C₂×Dic₃	—	2	2	no	—
183	C₂×Dic₃	—	2	2	no	—
184	C₂×Dic₃	—	2	2	no	—
185	C₂×Dic₃	C₂×C₃⋊C₈	1	4	yes	—
186	C₂×Dic₃	#208	1	4	yes	—
187	C₃×C₈	—	12	2	no	—
188	C₃⋊C₈	—	4	2	no	—
189	C₃⋊C₈	—	12	4	no	—
190	C₃⋊C₈	—	12	2	no	—
191	C₃⋊C₈	—	12	2	no	—
192	C₂×C₃²	#382	1	1	yes	characteristic
193	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
194	C₂×C₃²	#423	1	1	yes	characteristic
195	C₂×C₃²	#431	1	2	yes	—
196	C₂×C₃²	#432	1	2	yes	—
197	C₂²×C₄	—	3	2	no	—
198	C₂²×C₄	—	3	2	no	—
199	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
200	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
202	C₂×C₈	—	6	4	no	—
203	C₂×C₈	—	6	2	no	—
204	C₂×C₈	—	18	4	no	—
205	C₂×C₈	—	18	2	no	—
206	C₄²	—	3	4	no	—
207	C₄²	—	6	4	no	—
208	C₄²	—	6	2	no	—
209	C₄²	—	6	2	no	—
210	C₄²	—	6	2	no	—
211	C₄²	#626	1	2	yes	—
212	C₄²	#627	1	2	yes	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	#892	1	2	yes	—
219	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
220	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
221	C₂²×C₃	D₃×M₁₆	1	2	yes	—
222	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
223	C₂²×C₃	C₂×#208	1	2	yes	—
224	C₂²×C₃	C₂×#208	1	1	yes	characteristic
225	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
226	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
227	C₂²×C₃	#926	1	2	yes	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	4	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	4	2	no	—
235	C₃×C₄	—	6	4	no	—
236	C₃×C₄	—	6	4	no	—
237	C₃×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
238	C₃×C₄	Dic₃⋊C₈	1	4	yes	—
239	C₃×C₄	#870	1	2	yes	—
240	C₃×C₄	#875	1	1	yes	characteristic
241	C₃×C₄	#875	1	1	yes	characteristic
242	C₃×C₄	#890	1	2	yes	—
243	C₃×C₄	#936	1	1	yes	characteristic
244	C₃×C₄	#947	1	1	yes	characteristic
245	Dic₃	—	2	4	no	—
246	Dic₃	—	2	4	no	—
247	C₃²	#1831	1	1	yes	characteristic, complemented, Sylow
248	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	3	4	no	—
251	C₂×C₄	—	3	4	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	—	6	2	no	—
254	C₂×C₄	—	6	2	no	—
255	C₂×C₄	—	6	2	no	—
256	C₂×C₄	D₃×C₃⋊C₈	1	2	yes	—
257	C₂×C₄	#3711	1	2	yes	—
258	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
259	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
260	C₂×C₄	#3795	1	2	yes	—
261	C₂×C₄	#3799	1	2	yes	—
262	C₂×C₄	C₂×#626	1	1	yes	characteristic
263	C₂×C₄	C₂×#627	1	1	yes	characteristic
264	C₈	—	12	2	no	—
265	C₈	—	36	2	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	#4833	1	2	yes	—
272	C₂×C₃	#4835	1	2	yes	—
273	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
274	C₂×C₃	#4924	1	1	yes	characteristic
275	C₂×C₃	#4925	1	2	yes	—
276	C₂×C₃	#4926	1	2	yes	—
277	C₂×C₃	#4936	1	1	yes	characteristic
278	C₂×C₃	#4977	1	1	yes	characteristic
279	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
280	C₂×C₃	#5118	1	1	yes	characteristic
281	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
282	C₂²	D₃×#208	1	2	yes	—
283	C₂²	C₂×#3711	1	1	yes	characteristic
284	C₂²	#63841	1	2	yes	—
285	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	6	4	no	—
288	C₄	—	6	4	no	—
289	C₄	Dic₃⋊(C₃⋊C₈)	1	4	yes	—
290	C₄	#63860	1	1	yes	characteristic
291	C₄	#63864	1	1	yes	characteristic
292	C₄	#63869	1	2	yes	—
293	C₃	—	2	1	no	—
294	C₃	#69392	1	1	yes	characteristic, complemented
295	C₃	#70443	1	1	yes	characteristic, complemented
296	C₂	#10590248	1	2	yes	—
297	C₂	#10590263	1	2	yes	—
298	C₂	C₂×Dic₃⋊(C₃⋊C₈)	1	1	yes	characteristic
299	C₂	#10591115	1	1	yes	characteristic
300	C₂	#10591224	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃²
commutator	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1831
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
38. C₃×C₄×Dic₃	C₈	semidirect
115. C₄×Dic₃	C₃⋊C₈	semidirect
247. C₃²	#1831	semidirect
294. C₃	#69392	semidirect
295. C₃	#70443	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⋊C₈))	2
C₃×C₄²×Dic₃	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
#10589155	2
#70443	3
#69392	3