Subgroups — #11777293

Order 1152.

379 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#204	C₂	1	1	yes	maximal, characteristic
2	C₃⋊C₈×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃⋊C₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10589324	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4455	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589084	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589081	C₂	1	1	yes	maximal, characteristic
8	SD₁₆⋊(C₃⋊C₈)	—	3	1	no	maximal
9	#68076	—	3	1	no	maximal
10	C₂×C₃×#204	C₄	1	1	yes	characteristic
11	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
12	C₃×C₄×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×D₃×C₃⋊C₈	—	2	1	no	—
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
17	C₃×#821	C₄	1	1	yes	characteristic
18	C₃×#817	C₄	1	1	yes	characteristic
19	C₃×#816	C₄	1	1	yes	characteristic
20	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	Dic₃⋊(C₃⋊C₈)	—	2	1	no	—
23	(C₂×D₃)⋊(C₃⋊C₈)	—	2	1	no	—
24	Dic₃×C₃⋊C₈	—	2	1	no	—
25	C₃×C₄×SD₁₆	—	3	1	no	—
26	Q₈×C₃⋊C₈	—	3	1	no	—
27	D₄×C₃⋊C₈	—	3	1	no	—
28	C₄×#204	D₃	1	1	yes	characteristic
29	C₈×D₁₂	—	3	1	no	—
30	C₈×Dic₆	—	3	1	no	—
31	C₃×#322	—	3	1	no	—
32	#4535	—	3	1	no	—
33	D₄⋊(C₃⋊C₈)	—	3	1	no	—
34	#4459	—	3	1	no	—
35	#4458	—	3	1	no	—
36	#4458	—	3	1	no	—
37	#4458	—	6	1	no	—
38	#4456	—	3	1	no	—
39	#4455	D₃	1	1	yes	characteristic, complemented
40	C₂×C₃×D₁₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	—	2	1	no	—
42	C₂×C₃×Dic₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
45	C₃²×C₄²	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
47	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
48	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
49	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
50	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₃×Dic₃⋊C₄	—	2	1	no	—
52	C₃×C₄×Dic₃	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
55	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
56	C₃×#204	C₈	1	4	yes	complemented
57	D₃×C₃⋊C₈	—	4	2	no	—
58	SD₁₆⋊C₈	—	9	1	no	Sylow
59	C₂×C₃×SD₁₆	—	3	1	no	—
60	C₃×C₄×Q₈	—	3	1	no	—
61	C₃×C₄×D₄	—	3	1	no	—
62	C₂²×C₃⋊C₈	—	6	1	no	—
63	C₂×#204	Dic₃	1	1	yes	characteristic
64	C₂×D₃×C₈	—	6	1	no	—
65	C₄×D₁₂	C₂×D₃	1	1	yes	characteristic
66	C₄×Dic₆	C₂×D₃	1	1	yes	characteristic
67	C₃×#106	—	3	1	no	—
68	C₃×#103	—	3	1	no	—
69	C₃×#102	—	3	1	no	—
70	C₃×C₄×C₈	—	2	1	no	—
71	C₃×C₄×C₈	—	3	1	no	—
72	C₃×C₄×C₈	—	3	1	no	—
73	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
75	#830	—	6	1	no	—
76	#821	Dic₃	1	1	yes	characteristic
77	(C₂×D₃)⋊C₈	—	6	1	no	—
78	#817	Dic₃	1	1	yes	characteristic
79	#816	Dic₃	1	1	yes	characteristic
80	Dic₃⋊C₈	—	6	1	no	—
81	C₈×Dic₃	—	6	1	no	—
82	C₄⋊(C₃⋊C₈)	—	3	1	no	—
83	C₄⋊(C₃⋊C₈)	—	3	1	no	—
84	C₄⋊(C₃⋊C₈)	—	3	1	no	—
85	C₄⋊(C₃⋊C₈)	—	3	1	no	—
86	C₄⋊(C₃⋊C₈)	—	6	1	no	—
87	C₄⋊(C₃⋊C₈)	—	6	1	no	—
88	C₄⋊(C₃⋊C₈)	—	6	1	no	—
89	C₄×C₃⋊C₈	—	6	1	no	—
90	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
91	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
92	C₂²×C₃×D₃	—	2	1	no	—
93	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	#41	1	1	yes	characteristic
96	C₂×C₃×Dic₃	—	2	1	no	—
97	C₂×C₃×Dic₃	—	2	1	no	—
98	C₂×C₃×Dic₃	—	2	1	no	—
99	C₃×D₁₂	—	2	1	no	—
100	C₃×D₁₂	C₂×C₈	1	2	yes	—
101	C₃×C₄×D₃	—	4	1	no	—
102	C₃×C₄×D₃	—	4	1	no	—
103	C₃×Dic₆	—	2	1	no	—
104	C₃×Dic₆	C₂×C₈	1	2	yes	complemented
105	C₃²×C₈	—	2	1	no	—
106	C₃²×C₈	C₂×C₈	1	2	yes	complemented
107	C₃⋊(C₃⋊C₈)	—	4	1	no	—
108	C₃⋊(C₃⋊C₈)	—	4	1	no	—
109	C₃×C₃⋊C₈	—	2	1	no	—
110	C₃×C₃⋊C₈	—	2	1	no	—
111	C₃×C₃⋊C₈	—	4	1	no	—
112	C₈×Q₈	—	9	1	no	—
113	C₄×SD₁₆	—	3	1	no	—
114	C₈×D₄	—	9	1	no	—
115	#335	—	9	1	no	—
116	Q₈⋊C₈	—	9	1	no	—
117	D₄⋊C₈	—	9	1	no	—
118	#322	—	3	1	no	—
119	C₂×C₃×Q₈	—	3	1	no	—
120	C₂×C₃×D₄	—	3	1	no	—
121	C₂²×C₃×C₄	—	6	1	no	—
122	C₂×D₁₂	C₂×Dic₃	1	1	yes	characteristic
123	C₂×C₄×D₃	—	2	1	no	—
124	C₂×Dic₆	C₂×Dic₃	1	1	yes	characteristic
125	C₃×SD₁₆	—	3	4	no	—
126	C₂×C₃×C₈	—	2	1	no	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	3	1	no	—
129	C₂×C₃×C₈	—	3	1	no	—
130	C₂×C₃×C₈	—	6	1	no	—
131	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
132	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
135	C₃×C₄⋊C₄	—	3	1	no	—
136	C₃×C₄⋊C₄	—	3	1	no	—
137	C₃×C₄⋊C₄	—	6	1	no	—
138	C₃×C₂²⋊C₄	—	6	1	no	—
139	C₃×C₄²	—	2	1	no	—
140	C₃×C₄²	—	6	1	no	—
141	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
142	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
143	(C₂×D₃)⋊C₄	—	2	1	no	—
144	C₄⋊Dic₃	C₂×Dic₃	1	1	yes	characteristic
145	C₄⋊Dic₃	C₂×Dic₃	1	1	yes	characteristic
146	Dic₃⋊C₄	—	2	1	no	—
147	C₄×Dic₃	—	2	1	no	—
148	C₂×C₃⋊C₈	—	2	1	no	—
149	C₂×C₃⋊C₈	—	6	1	no	—
150	C₂×C₃⋊C₈	—	6	1	no	—
151	C₂×C₃⋊C₈	—	6	1	no	—
152	C₂×C₃⋊C₈	—	6	1	no	—
153	C₂×C₃⋊C₈	—	12	1	no	—
154	C₂×C₃⋊C₈	—	12	1	no	—
155	C₂×C₃⋊C₈	—	12	2	no	—
156	C₂×C₃⋊C₈	D₁₂	1	1	yes	characteristic
157	C₂×C₃⋊C₈	D₁₂	1	1	yes	characteristic
158	#204	C₃⋊C₈	1	4	yes	complemented
159	D₃×C₈	—	12	2	no	—
160	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
161	C₂×C₃×D₃	—	2	2	no	—
162	C₂×C₃×D₃	—	4	1	no	—
163	C₂×C₃×D₃	—	4	1	no	—
164	C₃²×C₄	—	2	1	no	—
165	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
166	C₃²×C₄	#131	1	1	yes	characteristic
167	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₃²×C₄	#137	1	1	yes	characteristic
169	C₃×Dic₃	—	2	2	no	—
170	C₃×Dic₃	—	4	1	no	—
171	C₃×Dic₃	—	4	1	no	—
172	C₂×SD₁₆	—	3	1	no	—
173	C₂²×C₈	—	18	1	no	—
174	C₄×Q₈	—	3	1	no	—
175	C₄×D₄	—	3	1	no	—
176	#106	—	3	1	no	—
177	C₄⋊C₈	—	9	1	no	—
178	C₄⋊C₈	—	9	1	no	—
179	C₄⋊C₈	—	18	1	no	—
180	#103	—	3	1	no	—
181	#102	—	3	1	no	—
182	C₂²⋊C₈	—	18	1	no	—
183	C₄×C₈	—	3	1	no	—
184	C₄×C₈	—	3	1	no	—
185	C₄×C₈	—	18	1	no	—
186	C₄×C₈	D₃²	1	1	yes	characteristic
187	C₂³×C₃	—	6	1	no	—
188	C₂²×D₃	—	2	1	no	—
189	C₃×Q₈	—	3	2	no	—
190	C₃×Q₈	—	6	1	no	—
191	C₃×D₄	—	3	2	no	—
192	C₃×D₄	—	6	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	6	1	no	—
197	C₂×C₃×C₄	—	6	1	no	—
198	C₂×C₃×C₄	—	6	1	no	—
199	C₂×C₃×C₄	—	12	1	no	—
200	C₂×C₃×C₄	—	12	1	no	—
201	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
203	C₂×C₃×C₄	#235	1	1	yes	characteristic
204	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	#237	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
207	C₂×Dic₃	—	2	1	no	—
208	C₂×Dic₃	—	2	1	no	—
209	C₂×Dic₃	—	2	1	no	—
210	D₁₂	—	2	1	no	—
211	D₁₂	C₂×C₃⋊C₈	1	2	yes	—
212	C₄×D₃	—	4	1	no	—
213	C₄×D₃	—	4	1	no	—
214	Dic₆	—	2	1	no	—
215	Dic₆	C₂×C₃⋊C₈	1	2	yes	complemented
216	C₃×C₈	—	2	1	no	—
217	C₃×C₈	—	2	1	no	—
218	C₃×C₈	—	2	2	no	—
219	C₃×C₈	—	4	1	no	—
220	C₃×C₈	—	6	1	no	—
221	C₃×C₈	—	6	1	no	—
222	C₃×C₈	—	12	1	no	—
223	C₃×C₈	D₃×C₈	1	2	yes	complemented
224	C₃×C₈	C₂×C₃⋊C₈	1	2	yes	complemented
225	C₃⋊C₈	—	2	1	no	—
226	C₃⋊C₈	—	2	1	no	—
227	C₃⋊C₈	—	4	1	no	—
228	C₃⋊C₈	—	12	1	no	—
229	C₃⋊C₈	—	12	1	no	—
230	C₃⋊C₈	—	12	1	no	—
231	C₃⋊C₈	—	12	1	no	—
232	C₃⋊C₈	—	24	1	no	—
233	C₃⋊C₈	—	24	1	no	—
234	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
235	C₂×C₃²	#440	1	1	yes	characteristic
236	C₂×C₃²	#444	1	1	yes	characteristic
237	C₃×D₃	—	4	2	no	—
238	C₂×Q₈	—	3	1	no	—
239	C₂×D₄	—	3	1	no	—
240	C₂²×C₄	—	6	1	no	—
241	SD₁₆	—	3	4	no	—
242	C₂×C₈	—	3	1	no	—
243	C₂×C₈	—	3	1	no	—
244	C₂×C₈	—	6	1	no	—
245	C₂×C₈	—	18	1	no	—
246	C₂×C₈	—	18	1	no	—
247	C₂×C₈	—	36	2	no	—
248	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
249	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
250	C₄⋊C₄	—	3	1	no	—
251	C₄⋊C₄	—	3	1	no	—
252	C₄⋊C₄	—	6	1	no	—
253	C₂²⋊C₄	—	6	1	no	—
254	C₄²	—	6	1	no	—
255	C₄²	C₂×D₃²	1	1	yes	characteristic
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	6	2	no	—
258	C₂²×C₃	—	12	1	no	—
259	C₂²×C₃	—	12	1	no	—
260	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
261	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
262	C₂×D₃	—	2	2	no	—
263	C₂×D₃	—	4	1	no	—
264	C₂×D₃	—	4	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	4	1	no	—
272	C₃×C₄	—	6	2	no	—
273	C₃×C₄	—	12	1	no	—
274	C₃×C₄	—	12	1	no	—
275	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
276	C₃×C₄	#901	1	1	yes	characteristic
277	C₃×C₄	#908	1	1	yes	characteristic
278	C₃×C₄	#909	1	1	yes	characteristic
279	C₃×C₄	#914	1	1	yes	characteristic
280	C₃×C₄	#915	1	1	yes	characteristic
281	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
282	C₃×C₄	#948	1	1	yes	characteristic
283	Dic₃	—	2	2	no	—
284	Dic₃	—	4	1	no	—
285	Dic₃	—	4	1	no	—
286	C₃²	SD₁₆⋊C₈	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	6	1	no	—
288	Q₈	—	3	2	no	—
289	Q₈	—	6	1	no	—
290	D₄	—	3	2	no	—
291	D₄	—	6	1	no	—
292	C₂×C₄	—	6	1	no	—
293	C₂×C₄	—	6	1	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	12	1	no	—
296	C₂×C₄	—	12	1	no	—
297	C₂×C₄	#3795	1	1	yes	characteristic
298	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
299	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	6	1	no	—
302	C₈	—	6	1	no	—
303	C₈	—	12	1	no	—
304	C₈	—	36	1	no	—
305	C₈	—	36	1	no	—
306	C₈	D₃×C₃⋊C₈	1	2	yes	complemented
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	12	2	no	—
311	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
312	C₂×C₃	#4710	1	1	yes	characteristic
313	C₂×C₃	#4717	1	1	yes	characteristic
314	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
315	C₂×C₃	#5164	1	1	yes	characteristic
316	C₂×C₃	#5197	1	1	yes	characteristic
317	D₃	—	4	2	no	—
318	C₂²	—	6	2	no	—
319	C₂²	—	12	1	no	—
320	C₂²	—	12	1	no	—
321	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	6	2	no	—
324	C₄	—	12	1	no	—
325	C₄	—	12	1	no	—
326	C₄	#63794	1	1	yes	characteristic
327	C₄	#63800	1	1	yes	characteristic
328	C₄	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
329	C₄	#63814	1	1	yes	characteristic
330	C₃	—	2	1	no	—
331	C₃	#68076	1	1	yes	characteristic, complemented
332	C₃	SD₁₆⋊(C₃⋊C₈)	1	1	yes	characteristic, complemented
333	C₂	—	12	2	no	—
334	C₂	C₃⋊C₈×D₁₂	1	1	yes	characteristic
335	C₂	#10590766	1	1	yes	characteristic
336	C₂	#10590916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₁₂
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×D₁₂	C₂×C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	SD₁₆⋊C₈
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₃⋊C₈×Dic₆	C₂	semidirect
4. #10589324	C₂	semidirect
5. C₃×#4455	C₂	semidirect
6. #10589084	C₂	semidirect
39. #4455	D₃	semidirect
56. C₃×#204	C₈	semidirect
104. C₃×Dic₆	C₂×C₈	semidirect
106. C₃²×C₈	C₂×C₈	semidirect
158. #204	C₃⋊C₈	semidirect
215. Dic₆	C₂×C₃⋊C₈	semidirect
223. C₃×C₈	D₃×C₈	semidirect
224. C₃×C₈	C₂×C₃⋊C₈	semidirect
286. C₃²	SD₁₆⋊C₈	semidirect
306. C₈	D₃×C₃⋊C₈	semidirect
331. C₃	#68076	semidirect
332. C₃	SD₁₆⋊(C₃⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#204	2
C₃⋊C₈×D₁₂	2
C₃⋊C₈×Dic₆	2
#10589324	2
C₃×#4455	2
#10589084	2
#10589081	2
SD₁₆⋊(C₃⋊C₈)	3
#68076	3