Subgroups — #11784468

Order 1152.

617 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
2	Dic₃×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₂×#63575	C₂	1	2	yes	maximal
4	#10589417	C₂	1	1	yes	maximal, characteristic
5	C₃×#4551	C₂	1	1	yes	maximal, characteristic
6	C₄×#818	—	3	1	no	maximal
7	#68785	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×Dic₃²	—	2	2	no	—
14	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
15	#63579	—	2	2	no	—
16	#63575	C₄	1	8	yes	complemented
17	C₂×C₃×C₄×C₈	—	3	1	no	—
18	C₂×#818	—	3	2	no	—
19	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
20	Dic₃×C₄⋊C₄	—	3	2	no	—
21	C₂×#807	—	3	2	no	—
22	C₄×C₄⋊Dic₃	—	3	2	no	—
23	C₃×#340	—	3	1	no	—
24	#4551	—	3	1	no	—
25	#4551	—	3	1	no	—
26	#4551	—	6	1	no	—
27	#4551	D₃	1	1	yes	characteristic
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃⋊Dic₃	—	2	2	no	—
30	C₂²×C₃×Dic₃	—	2	2	no	—
31	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
36	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
37	C₃×C₄⋊Dic₃	—	2	2	no	—
38	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
39	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
40	C₃×C₄×Dic₃	Q₈	1	2	yes	—
41	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	Dic₃²	—	4	4	no	—
44	C₄×#107	—	9	1	no	Sylow
45	C₂²×C₃×C₈	—	2	1	no	—
46	C₂²×C₃×C₈	—	3	1	no	—
47	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄⋊C₄	—	3	2	no	—
50	C₂×C₃×C₄²	—	3	1	no	—
51	C₂×C₄⋊Dic₃	—	3	2	no	—
52	C₂×C₄⋊Dic₃	—	3	2	no	—
53	C₂×C₄⋊Dic₃	—	6	2	no	—
54	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
55	C₂×C₄×Dic₃	—	6	2	no	—
56	C₂×C₄×Dic₃	—	6	2	no	—
57	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
59	C₃×C₄×C₈	—	3	4	no	—
60	C₃⋊#95	—	6	2	no	—
61	C₃⋊#95	—	6	2	no	—
62	#818	—	3	8	no	—
63	C₈×Dic₃	Dic₃	1	4	yes	complemented
64	#807	—	3	8	no	—
65	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
67	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	#41	1	2	yes	—
70	C₂×C₃⋊Dic₃	—	2	4	no	—
71	C₂×C₃⋊Dic₃	—	4	2	no	—
72	C₂×C₃⋊Dic₃	—	4	2	no	—
73	C₂×C₃×Dic₃	—	2	2	no	—
74	C₂×C₃×Dic₃	—	2	4	no	—
75	C₂×C₃×Dic₃	—	4	2	no	—
76	C₂×C₃×Dic₃	—	4	2	no	—
77	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₃×Dic₃	D₈	1	2	yes	—
80	C₂×C₃×Dic₃	Q₁₆	1	2	yes	—
81	C₃²×C₈	—	2	2	no	—
82	C₃×C₃⋊C₈	C₄²	1	4	yes	complemented
83	C₂×#107	—	9	2	no	—
84	C₂×C₄×C₈	—	3	1	no	—
85	C₄×C₄⋊C₄	—	9	2	no	—
86	#340	—	3	1	no	—
87	#340	—	9	1	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	3	1	no	—
90	C₂²×C₃×C₄	—	3	1	no	—
91	C₂²×C₃×C₄	—	6	2	no	—
92	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
94	C₂²×Dic₃	—	2	2	no	—
95	C₂²×Dic₃	—	6	2	no	—
96	C₂²×Dic₃	—	6	2	no	—
97	C₂²×Dic₃	—	12	2	no	—
98	C₂²×Dic₃	D₁₂	1	1	yes	characteristic
99	C₂²×Dic₃	D₁₂	1	1	yes	characteristic
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	3	4	no	—
104	C₂×C₃×C₈	—	3	2	no	—
105	C₂×C₃×C₈	—	4	2	no	—
106	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
107	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
108	C₃×C₄⋊C₄	—	3	4	no	—
109	C₃×C₄⋊C₄	—	6	2	no	—
110	C₃×C₄²	—	3	2	no	—
111	C₃×C₄²	—	3	2	no	—
112	C₄⋊Dic₃	—	2	2	no	—
113	C₄⋊Dic₃	—	3	4	no	—
114	C₄⋊Dic₃	—	3	4	no	—
115	C₄⋊Dic₃	—	6	4	no	—
116	C₄⋊Dic₃	—	6	2	no	—
117	C₄⋊Dic₃	—	6	2	no	—
118	C₄⋊Dic₃	—	12	2	no	—
119	C₄⋊Dic₃	C₄×D₃	1	4	yes	—
120	C₄×Dic₃	—	12	4	no	—
121	C₄×Dic₃	—	12	4	no	—
122	C₄×Dic₃	Dic₆	1	2	yes	—
123	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
124	C₂×C₃⋊C₈	C₄×D₃	1	4	yes	—
125	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
126	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₃²	C₄×D₄	1	2	yes	—
128	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
129	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
130	C₂²×C₃²	#135	1	2	yes	—
131	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
132	C₃²×C₄	#117	1	1	yes	characteristic
133	C₃²×C₄	C₄×Q₈	1	2	yes	—
134	C₃⋊Dic₃	—	4	4	no	—
135	C₃×Dic₃	—	2	2	no	—
136	C₃×Dic₃	—	4	4	no	—
137	C₃×Dic₃	#107	1	4	yes	complemented
138	C₂²×C₈	—	3	1	no	—
139	C₂²×C₈	D₃²	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	3	2	no	—
141	C₂×C₄⋊C₄	—	9	2	no	—
142	C₂×C₄²	—	3	1	no	—
143	C₂×C₄²	—	18	2	no	—
144	#107	—	9	8	no	—
145	C₄×C₈	—	3	4	no	—
146	#95	—	18	2	no	—
147	C₂³×C₃	—	2	1	no	—
148	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
149	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	3	2	no	—
155	C₂×C₃×C₄	—	3	2	no	—
156	C₂×C₃×C₄	—	3	2	no	—
157	C₂×C₃×C₄	—	3	2	no	—
158	C₂×C₃×C₄	—	6	2	no	—
159	C₂×C₃×C₄	—	6	4	no	—
160	C₂×C₃×C₄	—	12	2	no	—
161	C₂×C₃×C₄	—	12	2	no	—
162	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
164	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
165	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
166	C₂×C₃×C₄	#235	1	2	yes	—
167	C₂×C₃×C₄	#237	1	2	yes	—
168	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
170	C₂×Dic₃	—	2	2	no	—
171	C₂×Dic₃	—	2	4	no	—
172	C₂×Dic₃	—	4	2	no	—
173	C₂×Dic₃	—	4	2	no	—
174	C₂×Dic₃	—	6	4	no	—
175	C₂×Dic₃	—	6	4	no	—
176	C₂×Dic₃	—	12	4	no	—
177	C₂×Dic₃	—	12	2	no	—
178	C₂×Dic₃	—	12	2	no	—
179	C₂×Dic₃	—	12	2	no	—
180	C₂×Dic₃	—	12	2	no	—
181	C₂×Dic₃	—	24	2	no	—
182	C₂×Dic₃	—	24	2	no	—
183	C₂×Dic₃	D₂₄	1	2	yes	—
184	C₂×Dic₃	Dic₁₂	1	2	yes	—
185	C₂×Dic₃	C₄⋊Dic₃	1	2	yes	—
186	C₂×Dic₃	C₄⋊Dic₃	1	2	yes	—
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	3	4	no	—
190	C₃×C₈	—	4	2	no	—
191	C₃⋊C₈	C₄×Dic₃	1	4	yes	complemented
192	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₃²	C₂×#107	1	1	yes	characteristic
194	C₂×C₃²	#427	1	1	yes	characteristic
195	C₂×C₃²	C₄×D₈	1	2	yes	—
196	C₂×C₃²	C₄×Q₁₆	1	2	yes	—
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	—	3	1	no	—
199	C₂²×C₄	—	6	2	no	—
200	C₂²×C₄	—	18	2	no	—
201	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	3	4	no	—
204	C₂×C₈	—	3	2	no	—
205	C₂×C₈	D₃×Dic₃	1	2	yes	—
206	C₄⋊C₄	—	3	4	no	—
207	C₄⋊C₄	—	6	2	no	—
208	C₄⋊C₄	—	9	4	no	—
209	C₄⋊C₄	—	18	2	no	—
210	C₄²	—	3	2	no	—
211	C₄²	—	3	2	no	—
212	C₄²	—	36	4	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	C₄×D₁₂	1	2	yes	—
219	C₂²×C₃	#67⋊C₄	1	2	yes	—
220	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
221	C₂²×C₃	C₂×D₂₄	1	1	yes	characteristic
222	C₂²×C₃	#904	1	2	yes	—
223	C₂²×C₃	C₂×Dic₁₂	1	1	yes	characteristic
224	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
225	C₂²×C₃	#916	1	2	yes	—
226	C₂²×C₃	D₃×Q₁₆	1	1	yes	characteristic
227	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	3	4	no	—
232	C₃×C₄	—	6	2	no	—
233	C₃×C₄	—	12	4	no	—
234	C₃×C₄	C₄×Dic₆	1	2	yes	—
235	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
236	C₃×C₄	#877	1	1	yes	characteristic
237	C₃×C₄	#887	1	2	yes	—
238	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
239	C₃×C₄	#926	1	1	yes	characteristic
240	Dic₃	—	2	2	no	—
241	Dic₃	—	4	4	no	—
242	Dic₃	—	12	4	no	—
243	Dic₃	—	12	4	no	—
244	Dic₃	—	24	4	no	—
245	Dic₃	#818	1	4	yes	complemented
246	C₃²	C₄×#107	1	1	yes	characteristic, complemented, Sylow
247	C₂³	D₃×D₁₂	1	1	yes	characteristic
248	C₂×C₄	—	3	2	no	—
249	C₂×C₄	—	3	2	no	—
250	C₂×C₄	—	3	2	no	—
251	C₂×C₄	—	3	2	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	—	6	4	no	—
254	C₂×C₄	—	12	2	no	—
255	C₂×C₄	—	12	2	no	—
256	C₂×C₄	—	18	4	no	—
257	C₂×C₄	—	36	2	no	—
258	C₂×C₄	—	36	2	no	—
259	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
260	C₂×C₄	C₄×D₃²	1	2	yes	—
261	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
262	C₂×C₄	#3804	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	—	3	4	no	—
265	C₂×C₃	—	2	1	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	#4649	1	1	yes	characteristic
271	C₂×C₃	C₄×D₂₄	1	2	yes	—
272	C₂×C₃	C₄×Dic₁₂	1	2	yes	—
273	C₂×C₃	#4756	1	2	yes	—
274	C₂×C₃	#4790	1	2	yes	—
275	C₂×C₃	#4860	1	1	yes	characteristic
276	C₂×C₃	D₃×#107	1	1	yes	characteristic
277	C₂×C₃	C₄×C₄⋊Dic₃	1	1	yes	characteristic
278	C₂×C₃	C₂×#818	1	1	yes	characteristic
279	C₂×C₃	#5106	1	1	yes	characteristic
280	C₂²	D₃×D₂₄	1	1	yes	characteristic
281	C₂²	D₃×Dic₁₂	1	1	yes	characteristic
282	C₂²	#63806	1	2	yes	—
283	C₂²	#63880	1	2	yes	—
284	C₂²	D₃×C₄⋊Dic₃	1	1	yes	characteristic
285	C₄	—	3	4	no	—
286	C₄	—	6	2	no	—
287	C₄	—	12	4	no	—
288	C₄	—	36	4	no	—
289	C₄	#63837	1	2	yes	—
290	C₄	C₄×D₃×Dic₃	1	1	yes	characteristic
291	C₄	#63955	1	1	yes	characteristic
292	C₃	—	2	1	no	—
293	C₃	#68785	1	1	yes	characteristic, complemented
294	C₃	C₄×#818	1	1	yes	characteristic, complemented
295	C₂	D₃×#818	1	1	yes	characteristic
296	C₂	#10590706	1	1	yes	characteristic
297	C₂	#10590745	1	2	yes	—
298	C₂	#10590793	1	2	yes	—
299	C₂	#10591242	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₁₂
commutator	C₃²×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	C₄×#107
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
14. C₃×C₈×Dic₃	C₄	semidirect
16. #63575	C₄	semidirect
63. C₈×Dic₃	Dic₃	semidirect
82. C₃×C₃⋊C₈	C₄²	semidirect
137. C₃×Dic₃	#107	semidirect
191. C₃⋊C₈	C₄×Dic₃	semidirect
245. Dic₃	#818	semidirect
246. C₃²	C₄×#107	semidirect
293. C₃	#68785	semidirect
294. C₃	C₄×#818	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Dic₃	2
Dic₃×C₄⋊Dic₃	2
C₂×#63575	2
#10589417	2
C₃×#4551	2
C₄×#818	3
#68785	3