Subgroups — C₃×#68081 #11777622

Order 1152.

397 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#4688	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4459	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4459	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4457	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1586	—	3	1	no	maximal
9	#68081	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×D₂₄	C₄	1	1	yes	characteristic
11	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
12	C₂×C₃×D₃×C₈	—	2	1	no	—
13	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
14	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₃×#821	C₄	1	1	yes	characteristic
19	C₃×#821	C₄	1	1	yes	characteristic
20	C₃×(C₂×D₃)⋊C₈	—	2	1	no	—
21	C₃×(C₂×D₃)⋊C₈	—	2	1	no	—
22	C₃×#818	C₄	1	1	yes	characteristic
23	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₃×C₄×D₈	—	3	1	no	—
26	C₃×#436	—	3	1	no	—
27	C₃×C₈×D₄	—	3	1	no	—
28	C₄×D₂₄	C₂×C₃	1	1	yes	characteristic
29	#4688	C₂×C₃	1	1	yes	characteristic, complemented
30	C₈×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×#334	—	3	1	no	—
32	C₃×D₄⋊C₈	—	3	1	no	—
33	C₃×D₄⋊C₈	—	3	1	no	—
34	C₃×#322	—	2	1	no	—
35	C₃×#322	D₃	1	1	yes	characteristic, complemented
36	C₃×#322	C₂×C₃	1	1	yes	characteristic, complemented
37	#4459	C₂×C₃	1	1	yes	characteristic, complemented
38	#4459	C₂×C₃	1	1	yes	characteristic, complemented
39	#4457	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃×D₁₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×D₁₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	—	2	1	no	—
43	C₂×C₃×C₄×D₃	—	2	1	no	—
44	C₂×C₃²×C₈	—	2	1	no	—
45	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
48	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
49	C₃²×C₄²	C₂³	1	1	yes	characteristic
50	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
51	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
52	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃×C₃⋊C₈	—	2	1	no	—
56	C₃×D₂₄	—	2	2	no	—
57	C₃×ΓL(1,𝔽₂₅)	—	4	2	no	—
58	C₃×D₃×C₈	—	4	2	no	—
59	#1586	—	3	1	no	Sylow
60	C₂×C₃×D₈	—	3	1	no	—
61	C₂×C₃×M₁₆	—	6	1	no	—
62	C₂²×C₃×C₈	—	6	1	no	—
63	C₃×C₄×D₄	—	3	1	no	—
64	C₃×C₄×D₄	—	3	1	no	—
65	C₂×D₂₄	C₃×C₄	1	1	yes	characteristic
66	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
67	C₂×D₃×C₈	—	2	1	no	—
68	C₄×D₁₂	C₂²×C₃	1	1	yes	characteristic
69	C₄×D₁₂	C₂²×C₃	1	1	yes	characteristic
70	C₃×#107	—	3	1	no	—
71	C₃×C₄⋊C₈	—	3	1	no	—
72	C₃×C₄⋊C₈	—	3	1	no	—
73	C₃×#102	—	3	1	no	—
74	C₃×#102	—	3	1	no	—
75	C₃×C₂²⋊C₈	—	6	1	no	—
76	C₃×C₂²⋊C₈	—	6	1	no	—
77	C₃×C₄×C₈	—	2	1	no	—
78	C₃×C₄×C₈	—	2	1	no	—
79	C₃×C₄×C₈	—	2	1	no	—
80	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
82	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
83	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
84	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
86	#821	C₃×C₄	1	1	yes	characteristic
87	#821	C₃×C₄	1	1	yes	characteristic
88	(C₂×D₃)⋊C₈	—	2	1	no	—
89	(C₂×D₃)⋊C₈	—	2	1	no	—
90	#818	C₃×C₄	1	1	yes	characteristic
91	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
92	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
93	C₂²×C₃×D₃	—	2	1	no	—
94	C₂²×C₃×D₃	—	2	1	no	—
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	#41	1	1	yes	characteristic
98	C₂×C₃×Dic₃	—	2	1	no	—
99	C₂×C₃×Dic₃	—	2	1	no	—
100	C₃×D₁₂	—	2	1	no	—
101	C₃×D₁₂	—	2	1	no	—
102	C₃×D₁₂	—	2	1	no	—
103	C₃×D₁₂	M₁₆	1	2	yes	complemented
104	C₃×C₄×D₃	—	4	1	no	—
105	C₃×C₄×D₃	—	4	1	no	—
106	C₃×C₄×D₃	—	4	1	no	—
107	C₃×C₄×D₃	—	4	1	no	—
108	C₃²×C₈	—	2	2	no	—
109	C₃²×C₈	—	2	1	no	—
110	C₃²×C₈	—	2	1	no	—
111	C₃²×C₈	—	2	1	no	—
112	C₃²×C₈	—	2	1	no	—
113	C₃×C₃⋊C₈	—	4	1	no	—
114	C₃×C₃⋊C₈	—	4	1	no	—
115	C₄×D₈	—	3	1	no	—
116	#436	—	3	1	no	—
117	C₈×D₄	—	3	1	no	—
118	#334	—	3	1	no	—
119	D₄⋊C₈	—	3	1	no	—
120	D₄⋊C₈	—	3	1	no	—
121	#322	C₃×D₃	1	1	yes	characteristic, complemented
122	C₂×C₃×D₄	—	3	1	no	—
123	C₂×C₃×D₄	—	3	1	no	—
124	C₂²×C₃×C₄	—	6	1	no	—
125	C₂²×C₃×C₄	—	6	1	no	—
126	C₂×D₁₂	C₂×C₃×C₄	1	1	yes	characteristic
127	C₂×D₁₂	C₂×C₃×C₄	1	1	yes	characteristic
128	C₂×C₄×D₃	—	2	1	no	—
129	C₂×C₄×D₃	—	2	1	no	—
130	C₃×D₈	—	6	2	no	—
131	C₃×M₁₆	—	12	2	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	—	2	1	no	—
138	C₂×C₃×C₈	—	2	1	no	—
139	C₂×C₃×C₈	—	6	1	no	—
140	C₂×C₃×C₈	—	6	1	no	—
141	C₂×C₃×C₈	—	12	2	no	—
142	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
143	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
144	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
145	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
146	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
148	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
149	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	3	1	no	—
151	C₃×C₄⋊C₄	—	3	1	no	—
152	C₃×C₂²⋊C₄	—	6	1	no	—
153	C₃×C₂²⋊C₄	—	6	1	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
156	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
157	(C₂×D₃)⋊C₄	—	2	1	no	—
158	(C₂×D₃)⋊C₄	—	2	1	no	—
159	C₄⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
160	C₄⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
161	C₂×C₃⋊C₈	—	2	1	no	—
162	C₂×C₃⋊C₈	—	2	1	no	—
163	D₂₄	—	2	2	no	—
164	ΓL(1,𝔽₂₅)	—	4	2	no	—
165	D₃×C₈	—	4	2	no	—
166	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
167	C₂×C₃×D₃	—	2	2	no	—
168	C₂×C₃×D₃	—	4	1	no	—
169	C₂×C₃×D₃	—	4	1	no	—
170	C₂×C₃×D₃	—	4	1	no	—
171	C₂×C₃×D₃	—	4	1	no	—
172	C₂×C₃×D₃	—	4	1	no	—
173	C₃²×C₄	—	2	1	no	—
174	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
175	C₃²×C₄	#131	1	1	yes	characteristic
176	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₃×Dic₃	—	4	1	no	—
179	C₃×Dic₃	—	4	1	no	—
180	C₂×D₈	—	3	1	no	—
181	C₂×M₁₆	—	6	1	no	—
182	C₂²×C₈	—	6	1	no	—
183	C₄×D₄	—	3	1	no	—
184	C₄×D₄	—	3	1	no	—
185	#107	—	3	1	no	—
186	C₄⋊C₈	—	3	1	no	—
187	C₄⋊C₈	—	3	1	no	—
188	#102	—	3	1	no	—
189	#102	—	3	1	no	—
190	C₂²⋊C₈	—	6	1	no	—
191	C₂²⋊C₈	—	6	1	no	—
192	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
193	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
194	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
195	C₂³×C₃	—	6	1	no	—
196	C₂³×C₃	—	6	1	no	—
197	C₂²×D₃	—	2	1	no	—
198	C₂²×D₃	—	2	1	no	—
199	C₃×D₄	—	3	2	no	—
200	C₃×D₄	—	6	1	no	—
201	C₃×D₄	—	6	1	no	—
202	C₃×D₄	—	6	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	6	1	no	—
207	C₂×C₃×C₄	—	6	1	no	—
208	C₂×C₃×C₄	—	12	1	no	—
209	C₂×C₃×C₄	—	12	1	no	—
210	C₂×C₃×C₄	—	12	1	no	—
211	C₂×C₃×C₄	—	12	1	no	—
212	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
214	C₂×C₃×C₄	#235	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
218	C₂×Dic₃	—	2	1	no	—
219	C₂×Dic₃	—	2	1	no	—
220	D₁₂	—	2	1	no	—
221	D₁₂	—	2	1	no	—
222	D₁₂	—	2	1	no	—
223	D₁₂	C₃×M₁₆	1	2	yes	complemented
224	C₄×D₃	—	4	1	no	—
225	C₄×D₃	—	4	1	no	—
226	C₄×D₃	—	4	1	no	—
227	C₄×D₃	—	4	1	no	—
228	C₃×C₈	—	2	4	no	—
229	C₃×C₈	—	2	2	no	—
230	C₃×C₈	—	2	2	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	2	1	no	—
234	C₃×C₈	—	2	1	no	—
235	C₃×C₈	—	2	1	no	—
236	C₃×C₈	—	2	1	no	—
237	C₃×C₈	—	2	1	no	—
238	C₃×C₈	—	2	1	no	—
239	C₃×C₈	—	4	1	no	—
240	C₃×C₈	—	4	1	no	—
241	C₃×C₈	—	4	1	no	—
242	C₃×C₈	—	4	1	no	—
243	C₃×C₈	—	12	1	no	—
244	C₃×C₈	—	12	1	no	—
245	C₃⋊C₈	—	4	1	no	—
246	C₃⋊C₈	—	4	1	no	—
247	C₂×C₃²	#435	1	1	yes	characteristic
248	C₂×C₃²	#442	1	1	yes	characteristic
249	C₂×C₃²	#443	1	1	yes	characteristic
250	C₃×D₃	—	4	2	no	—
251	C₃×D₃	—	8	1	no	—
252	C₂×D₄	—	3	1	no	—
253	C₂×D₄	—	3	1	no	—
254	C₂²×C₄	—	6	1	no	—
255	C₂²×C₄	—	6	1	no	—
256	D₈	—	6	2	no	—
257	M₁₆	—	12	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	6	1	no	—
260	C₂×C₈	—	6	1	no	—
261	C₂×C₈	—	12	2	no	—
262	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
263	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
264	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
265	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
266	C₄⋊C₄	—	3	1	no	—
267	C₄⋊C₄	—	3	1	no	—
268	C₂²⋊C₄	—	6	1	no	—
269	C₂²⋊C₄	—	6	1	no	—
270	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	6	2	no	—
273	C₂²×C₃	—	12	1	no	—
274	C₂²×C₃	—	12	1	no	—
275	C₂²×C₃	—	12	1	no	—
276	C₂²×C₃	—	12	1	no	—
277	C₂²×C₃	—	12	1	no	—
278	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
279	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
280	C₂×D₃	—	2	2	no	—
281	C₂×D₃	—	4	1	no	—
282	C₂×D₃	—	4	1	no	—
283	C₂×D₃	—	4	1	no	—
284	C₂×D₃	—	4	1	no	—
285	C₂×D₃	—	4	1	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	12	1	no	—
294	C₃×C₄	—	12	1	no	—
295	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
296	C₃×C₄	#907	1	1	yes	characteristic
297	C₃×C₄	#908	1	1	yes	characteristic
298	C₃×C₄	#908	1	1	yes	characteristic
299	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
300	C₃×C₄	C₃×#131	1	1	yes	characteristic
301	C₃×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₃×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
303	Dic₃	—	4	1	no	—
304	Dic₃	—	4	1	no	—
305	C₃²	#1586	1	1	yes	characteristic, complemented, Sylow
306	C₂³	—	6	1	no	—
307	C₂³	—	6	1	no	—
308	D₄	—	3	2	no	—
309	D₄	—	6	1	no	—
310	D₄	—	6	1	no	—
311	D₄	—	6	1	no	—
312	C₂×C₄	—	6	1	no	—
313	C₂×C₄	—	6	1	no	—
314	C₂×C₄	—	12	1	no	—
315	C₂×C₄	—	12	1	no	—
316	C₂×C₄	—	12	1	no	—
317	C₂×C₄	—	12	1	no	—
318	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
319	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
320	C₂×C₄	C₃×#235	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	12	1	no	—
327	C₈	—	12	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	12	2	no	—
332	C₂×C₃	—	24	1	no	—
333	C₂×C₃	#4700	1	1	yes	characteristic
334	C₂×C₃	#4706	1	1	yes	characteristic
335	C₂×C₃	#4711	1	1	yes	characteristic
336	C₂×C₃	C₃×#435	1	1	yes	characteristic
337	C₂×C₃	C₃×#442	1	1	yes	characteristic
338	C₂×C₃	C₃×#443	1	1	yes	characteristic
339	D₃	—	4	2	no	—
340	D₃	—	8	1	no	—
341	C₂²	—	6	2	no	—
342	C₂²	—	12	1	no	—
343	C₂²	—	12	1	no	—
344	C₂²	—	12	1	no	—
345	C₂²	—	12	1	no	—
346	C₂²	—	12	1	no	—
347	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	12	1	no	—
350	C₄	—	12	1	no	—
351	C₄	C₃×D₃×M₁₆	1	1	yes	characteristic
352	C₄	C₃×#907	1	1	yes	characteristic
353	C₄	C₃×#908	1	1	yes	characteristic
354	C₄	C₃×#908	1	1	yes	characteristic
355	C₃	—	2	1	no	—
356	C₃	#68081	1	1	yes	characteristic, direct product factor
357	C₃	C₃×#1586	1	1	yes	characteristic, complemented
358	C₂	—	12	2	no	—
359	C₂	—	24	1	no	—
360	C₂	C₃×#4700	1	1	yes	characteristic
361	C₂	C₃×#4706	1	1	yes	characteristic
362	C₂	C₃×#4711	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#322	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×D₂₄	C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#322	D₃
hypocenter	C₃	C₃×#1586
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ C₃×#322

Complements

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

N	H ≅ G/N	type
2. C₃×#4688	C₂	semidirect
3. C₃×C₈×D₁₂	C₂	semidirect
4. C₃²×#322	C₂	semidirect
5. C₃×#4459	C₂	semidirect
6. C₃×#4459	C₂	semidirect
7. C₃×#4457	C₂	semidirect
9. #68081	C₃	direct (N × H)
29. #4688	C₂×C₃	semidirect
30. C₈×D₁₂	C₂×C₃	semidirect
35. C₃×#322	D₃	semidirect
36. C₃×#322	C₂×C₃	semidirect
37. #4459	C₂×C₃	semidirect
38. #4459	C₂×C₃	semidirect
39. #4457	C₂×C₃	semidirect
103. C₃×D₁₂	M₁₆	semidirect
121. #322	C₃×D₃	semidirect
223. D₁₂	C₃×M₁₆	semidirect
305. C₃²	#1586	semidirect
356. C₃	#68081	direct (N × H)
357. C₃	C₃×#1586	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₂₄	2
C₃×#4688	2
C₃×C₈×D₁₂	2
C₃²×#322	2
C₃×#4459	2
C₃×#4457	2
C₃×#1586	3
#68081	3

Subgroups — C3×#68081 #11777622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68081 #11777622