Subgroups — C₃×#68531 #11776937

Order 1152.

359 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×#4837	C₂	1	1	yes	maximal, characteristic
2	C₃×#4835	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
4	C₃²×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4480	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4458	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1569	—	3	1	no	maximal
9	#68531	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
11	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
12	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
13	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
14	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×(C₂×D₃)⋊C₈	—	2	1	no	—
18	C₃×#815	—	2	1	no	—
19	C₃×#815	C₄	1	1	yes	characteristic
20	C₃×#815	C₄	1	1	yes	characteristic
21	C₃×Dic₃⋊C₈	—	2	1	no	—
22	C₃×C₄⋊(C₃⋊C₈)	C₂²	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×#432	—	3	1	no	—
26	C₃×#423	—	3	1	no	—
27	C₃×C₄×M₁₆	—	3	1	no	—
28	#4837	C₂×C₃	1	1	yes	characteristic
29	#4835	C₂×C₃	1	1	yes	characteristic
30	C₄×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
31	C₃×#335	—	3	1	no	—
32	C₃×#335	—	3	1	no	—
33	C₃×#334	—	2	1	no	—
34	C₃×#334	—	3	1	no	—
35	C₃×#334	D₃	1	1	yes	characteristic, complemented
36	C₃×#334	C₂×C₃	1	1	yes	characteristic, complemented
37	#4480	C₂×C₃	1	1	yes	characteristic, complemented
38	#4479	C₂×C₃	1	1	yes	characteristic, complemented
39	#4458	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	Q₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	—	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	C₃²×C₄²	C₂³	1	1	yes	characteristic
48	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×Dic₃	D₄	1	1	yes	characteristic
50	C₃×C₄×Dic₃	Q₈	1	1	yes	characteristic
51	C₂×C₃×C₃⋊C₈	—	2	1	no	—
52	C₂×C₃×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
56	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
57	C₃×ΓL(1,𝔽₂₅)	—	4	2	no	—
58	#1569	—	3	1	no	Sylow
59	C₂×C₃×M₁₆	—	3	1	no	—
60	C₂×C₃×M₁₆	—	3	1	no	—
61	C₂×C₃×M₁₆	—	6	1	no	—
62	C₂×C₃×C₄²	—	3	1	no	—
63	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
64	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	1	yes	characteristic
65	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	1	yes	characteristic
66	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
67	C₃×C₄⋊C₈	—	2	1	no	—
68	C₃×C₄⋊C₈	—	2	1	no	—
69	C₃×C₄⋊C₈	—	3	1	no	—
70	C₃×C₄⋊C₈	—	3	1	no	—
71	C₃×C₄⋊C₈	—	6	1	no	—
72	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
73	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
74	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
76	C₃×C₂²⋊C₈	—	6	1	no	—
77	C₃×#97	—	3	1	no	—
78	C₃×#97	—	3	1	no	—
79	C₃×#97	—	6	1	no	—
80	C₃×C₄×C₈	—	2	1	no	—
81	C₃×C₄×C₈	—	3	1	no	—
82	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
83	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
84	(C₂×D₃)⋊C₈	—	2	1	no	—
85	#815	—	2	1	no	—
86	#815	C₃×C₄	1	1	yes	characteristic
87	#815	C₃×C₄	1	1	yes	characteristic
88	Dic₃⋊C₈	—	2	1	no	—
89	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
91	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
92	C₂²×C₃×D₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
99	C₃×C₄×D₃	—	2	1	no	—
100	C₃×C₄×D₃	—	2	1	no	—
101	C₃×C₄×D₃	—	2	1	no	—
102	C₃×C₄×D₃	—	2	1	no	—
103	C₃×C₄×D₃	—	4	1	no	—
104	C₃²×C₈	—	2	1	no	—
105	C₃²×C₈	—	4	1	no	—
106	C₃²×C₈	—	4	1	no	—
107	C₃²×C₈	M₁₆	1	2	yes	complemented
108	C₃×C₃⋊C₈	—	2	1	no	—
109	C₃×C₃⋊C₈	—	4	1	no	—
110	C₃×C₃⋊C₈	—	4	1	no	—
111	C₃×C₃⋊C₈	M₁₆	1	2	yes	complemented
112	#432	—	3	1	no	—
113	#423	—	3	1	no	—
114	C₄×M₁₆	—	3	1	no	—
115	#335	—	3	1	no	—
116	#335	—	3	1	no	—
117	#334	—	3	1	no	—
118	#334	C₃×D₃	1	1	yes	characteristic, complemented
119	C₂²×C₃×C₄	—	3	1	no	—
120	C₂²×C₃×C₄	—	3	1	no	—
121	C₂²×C₃×C₄	—	3	1	no	—
122	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂×C₄×D₃	C₃×D₄	1	1	yes	characteristic
124	C₂×C₄×D₃	C₃×Q₈	1	1	yes	characteristic
125	C₃×M₁₆	—	6	2	no	—
126	C₃×M₁₆	—	6	2	no	—
127	C₃×M₁₆	—	12	2	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	3	1	no	—
135	C₂×C₃×C₈	—	3	1	no	—
136	C₂×C₃×C₈	—	4	1	no	—
137	C₂×C₃×C₈	—	4	1	no	—
138	C₂×C₃×C₈	—	6	1	no	—
139	C₂×C₃×C₈	—	6	1	no	—
140	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
141	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	—	3	1	no	—
146	C₃×C₄²	—	3	1	no	—
147	C₃×C₄²	—	3	1	no	—
148	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
149	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
150	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
151	C₄×Dic₃	C₃×D₄	1	1	yes	characteristic
152	C₄×Dic₃	C₃×Q₈	1	1	yes	characteristic
153	C₂×C₃⋊C₈	—	2	1	no	—
154	C₂×C₃⋊C₈	—	2	1	no	—
155	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
156	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
157	ΓL(1,𝔽₂₅)	—	2	2	no	—
158	ΓL(1,𝔽₂₅)	—	2	2	no	—
159	ΓL(1,𝔽₂₅)	—	4	2	no	—
160	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₃×D₃	—	2	1	no	—
162	C₂×C₃×D₃	—	2	1	no	—
163	C₂×C₃×D₃	—	2	1	no	—
164	C₃²×C₄	—	2	1	no	—
165	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
166	C₃²×C₄	C₂×M₁₆	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	1	no	—
170	C₃×Dic₃	—	2	1	no	—
171	C₃×Dic₃	—	2	1	no	—
172	C₂×M₁₆	—	3	1	no	—
173	C₂×M₁₆	—	3	1	no	—
174	C₂×M₁₆	—	6	1	no	—
175	C₂×C₄²	—	3	1	no	—
176	C₄⋊C₈	—	3	1	no	—
177	C₄⋊C₈	—	3	1	no	—
178	C₄⋊C₈	—	6	1	no	—
179	C₄⋊C₈	C₂×C₃×D₃	1	1	yes	characteristic
180	C₄⋊C₈	C₂×C₃×D₃	1	1	yes	characteristic
181	C₂²⋊C₈	—	6	1	no	—
182	#97	—	3	1	no	—
183	#97	—	3	1	no	—
184	#97	—	6	1	no	—
185	C₄×C₈	—	3	1	no	—
186	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
187	C₂³×C₃	—	3	1	no	—
188	C₂²×D₃	C₃×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	3	1	no	—
193	C₂×C₃×C₄	—	3	1	no	—
194	C₂×C₃×C₄	—	3	1	no	—
195	C₂×C₃×C₄	—	6	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	C₂×C₄×D₃	1	1	yes	characteristic
201	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
202	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
206	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
207	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
208	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
209	C₄×D₃	—	2	1	no	—
210	C₄×D₃	—	2	1	no	—
211	C₄×D₃	—	2	1	no	—
212	C₄×D₃	—	2	1	no	—
213	C₄×D₃	—	4	1	no	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	2	2	no	—
217	C₃×C₈	—	3	2	no	—
218	C₃×C₈	—	4	1	no	—
219	C₃×C₈	—	4	1	no	—
220	C₃×C₈	—	4	1	no	—
221	C₃×C₈	—	4	1	no	—
222	C₃×C₈	—	4	1	no	—
223	C₃×C₈	—	6	1	no	—
224	C₃×C₈	—	8	1	no	—
225	C₃×C₈	—	8	1	no	—
226	C₃×C₈	—	12	1	no	—
227	C₃×C₈	—	12	1	no	—
228	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
229	C₃×C₈	C₃×M₁₆	1	2	yes	complemented
230	C₃⋊C₈	—	2	1	no	—
231	C₃⋊C₈	—	4	1	no	—
232	C₃⋊C₈	—	4	1	no	—
233	C₃⋊C₈	C₃×M₁₆	1	2	yes	complemented
234	C₂×C₃²	#423	1	1	yes	characteristic
235	C₂×C₃²	#428	1	1	yes	characteristic
236	C₂×C₃²	#430	1	1	yes	characteristic
237	C₃×D₃	—	4	1	no	—
238	C₂²×C₄	—	3	1	no	—
239	C₂²×C₄	—	3	1	no	—
240	C₂²×C₄	—	3	1	no	—
241	M₁₆	—	6	2	no	—
242	M₁₆	—	6	2	no	—
243	M₁₆	—	12	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	3	1	no	—
247	C₂×C₈	—	3	1	no	—
248	C₂×C₈	—	6	1	no	—
249	C₂×C₈	—	6	1	no	—
250	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
251	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
252	C₄²	—	3	1	no	—
253	C₄²	—	3	1	no	—
254	C₄²	—	3	1	no	—
255	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	6	1	no	—
258	C₂²×C₃	—	6	1	no	—
259	C₂²×C₃	—	6	1	no	—
260	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
261	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
262	C₂×D₃	—	2	1	no	—
263	C₂×D₃	—	2	1	no	—
264	C₂×D₃	—	2	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	1	no	—
273	C₃×C₄	—	6	1	no	—
274	C₃×C₄	—	6	1	no	—
275	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
276	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
277	C₃×C₄	#914	1	1	yes	characteristic
278	C₃×C₄	#915	1	1	yes	characteristic
279	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
280	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
281	C₃×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₃×C₄	C₃×#137	1	1	yes	characteristic
283	Dic₃	—	2	1	no	—
284	Dic₃	—	2	1	no	—
285	Dic₃	—	2	1	no	—
286	C₃²	#1569	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	3	1	no	—
288	C₂×C₄	—	3	1	no	—
289	C₂×C₄	—	3	1	no	—
290	C₂×C₄	—	3	1	no	—
291	C₂×C₄	—	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	C₂×C₃×C₄×D₃	1	1	yes	characteristic
297	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
298	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	3	2	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	1	no	—
303	C₈	—	6	1	no	—
304	C₈	—	12	1	no	—
305	C₈	—	12	1	no	—
306	C₈	C₃×ΓL(1,𝔽₂₅)	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	1	no	—
311	C₂×C₃	#4837	1	1	yes	characteristic
312	C₂×C₃	#4861	1	1	yes	characteristic
313	C₂×C₃	#4893	1	1	yes	characteristic
314	C₂×C₃	C₃×#423	1	1	yes	characteristic
315	C₂×C₃	C₃×#428	1	1	yes	characteristic
316	C₂×C₃	C₃×#430	1	1	yes	characteristic
317	D₃	—	4	1	no	—
318	C₂²	—	6	1	no	—
319	C₂²	—	6	1	no	—
320	C₂²	—	6	1	no	—
321	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	6	1	no	—
324	C₄	—	6	1	no	—
325	C₄	—	6	1	no	—
326	C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
327	C₄	C₃×D₃×M₁₆	1	1	yes	characteristic
328	C₄	C₃×#914	1	1	yes	characteristic
329	C₄	C₃×#915	1	1	yes	characteristic
330	C₃	—	2	1	no	—
331	C₃	#68531	1	1	yes	characteristic, direct product factor
332	C₃	C₃×#1569	1	1	yes	characteristic, complemented
333	C₂	—	12	1	no	—
334	C₂	C₃×#4837	1	1	yes	characteristic
335	C₂	C₃×#4861	1	1	yes	characteristic
336	C₂	C₃×#4893	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#334	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#334	D₃
hypocenter	C₃	C₃×#1569
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#334

Complements

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

N	H ≅ G/N	type
4. C₃²×#334	C₂	semidirect
5. C₃×#4480	C₂	semidirect
6. C₃×#4479	C₂	semidirect
7. C₃×#4458	C₂	semidirect
9. #68531	C₃	direct (N × H)
35. C₃×#334	D₃	semidirect
36. C₃×#334	C₂×C₃	semidirect
37. #4480	C₂×C₃	semidirect
38. #4479	C₂×C₃	semidirect
39. #4458	C₂×C₃	semidirect
107. C₃²×C₈	M₁₆	semidirect
111. C₃×C₃⋊C₈	M₁₆	semidirect
118. #334	C₃×D₃	semidirect
228. C₃×C₈	ΓL(1,𝔽₂₅)	semidirect
229. C₃×C₈	C₃×M₁₆	semidirect
233. C₃⋊C₈	C₃×M₁₆	semidirect
286. C₃²	#1569	semidirect
306. C₈	C₃×ΓL(1,𝔽₂₅)	semidirect
331. C₃	#68531	direct (N × H)
332. C₃	C₃×#1569	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4837	2
C₃×#4835	2
C₃×C₄×ΓL(1,𝔽₂₅)	2
C₃²×#334	2
C₃×#4480	2
C₃×#4479	2
C₃×#4458	2
C₃×#1569	3
#68531	3

Subgroups — C3×#68531 #11776937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68531 #11776937