Subgroups — C₃×#82028 #11899754

Order 1152.

741 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5766	C₂	1	2	yes	maximal, complemented
2	C₂×C₃²×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5179	C₂	1	2	yes	maximal, complemented
4	C₃×#4919	C₂	1	2	yes	maximal, complemented
5	C₃×#4918	C₂	1	4	yes	maximal, complemented
6	C₃×D₃×Q₃₂	C₂	1	2	yes	maximal, complemented
7	C₃×#4904	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4901	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#3417	—	3	1	no	maximal
10	#82028	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₃×#1007	—	2	2	no	—
12	C₂×C₃²×Q₁₆	C₂²	1	2	yes	—
13	C₃×#942	—	2	2	no	—
14	C₃×#919	C₂²	1	4	yes	—
15	C₃×#918	—	2	4	no	—
16	C₃×D₃×Q₁₆	C₂²	1	4	yes	—
17	C₃×#904	C₂²	1	2	yes	—
18	C₃×#901	C₂²	1	1	yes	characteristic
19	C₃²×Q₃₂	C₂²	1	2	yes	—
20	C₃²×Q₃₂	C₂²	1	2	yes	—
21	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
22	C₃×#829	C₂²	1	4	yes	—
23	C₃×#828	C₂²	1	4	yes	—
24	C₃×#812	C₂²	1	1	yes	characteristic
25	C₃×Dic₂₄	C₂²	1	1	yes	characteristic
26	C₃×#800	C₂²	1	2	yes	—
27	C₃×D₄₈	C₂²	1	1	yes	characteristic
28	C₃×#798	C₂²	1	2	yes	—
29	C₃×D₃×C₁₆	C₂²	1	2	yes	—
30	C₃×#578	—	3	2	no	—
31	#5766	C₂×C₃	1	2	yes	complemented
32	C₃×#510	—	3	2	no	—
33	C₃×#510	—	3	4	no	—
34	C₃×#508	—	3	2	no	—
35	C₃×#508	—	3	1	no	—
36	C₂×C₃×Q₃₂	—	2	1	no	—
37	C₂×C₃×Q₃₂	—	3	2	no	—
38	C₂×C₃×Q₃₂	D₃	1	1	yes	characteristic, complemented
39	C₂×C₃×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
40	C₃×#504	—	3	1	no	—
41	#5179	C₂×C₃	1	2	yes	complemented
42	#4919	C₂×C₃	1	2	yes	complemented
43	#4918	C₂×C₃	1	4	yes	complemented
44	D₃×Q₃₂	C₂×C₃	1	2	yes	complemented
45	#4904	C₂×C₃	1	1	yes	characteristic, complemented
46	#4901	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×C₃²×Q₈	—	2	2	no	—
48	C₃×#239	—	2	4	no	—
49	C₃×#239	—	4	2	no	—
50	C₃×D₃×Q₈	—	2	4	no	—
51	C₃×D₃×Q₈	—	4	2	no	—
52	C₃×#235	—	2	2	no	—
53	C₃×#235	—	4	4	no	—
54	C₃×#235	D₄	1	1	yes	characteristic
55	C₃²×Q₁₆	—	2	2	no	—
56	C₃²×Q₁₆	C₂³	1	4	yes	—
57	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
58	C₃×#216	—	2	4	no	—
59	C₃×#215	—	2	4	no	—
60	C₃×#208	D₄	1	1	yes	characteristic
61	C₃×Dic₁₂	C₂³	1	2	yes	—
62	C₃×D₂₄	C₂³	1	2	yes	—
63	C₃×#204	—	2	2	no	—
64	C₃×ΓL(1,𝔽₂₅)	D₄	1	2	yes	—
65	C₃×D₃×C₈	C₂³	1	2	yes	—
66	C₃²×C₁₆	C₂³	1	1	yes	characteristic
67	C₃²×C₁₆	C₂³	1	1	yes	characteristic
68	C₃×C₃⋊C₁₆	C₂³	1	2	yes	—
69	#3417	—	3	1	no	Sylow
70	C₃×2₋¹⁺⁴	—	6	2	no	—
71	#1007	—	2	2	no	—
72	C₃×#137	—	6	4	no	—
73	C₃×#137	—	6	2	no	—
74	C₃×#135	—	3	4	no	—
75	C₃×#135	—	3	2	no	—
76	C₂×C₃×Q₁₆	—	2	2	no	—
77	C₂×C₃×Q₁₆	—	3	4	no	—
78	C₂×C₃×Q₁₆	C₂×D₃	1	2	yes	—
79	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
80	C₃×#131	—	3	1	no	—
81	#942	—	2	2	no	—
82	#919	C₂²×C₃	1	4	yes	—
83	#918	—	2	4	no	—
84	D₃×Q₁₆	C₂²×C₃	1	4	yes	—
85	#904	C₂²×C₃	1	2	yes	—
86	#901	C₂²×C₃	1	1	yes	characteristic
87	C₃×Q₃₂	—	2	2	no	—
88	C₃×Q₃₂	—	2	2	no	—
89	C₃×Q₃₂	—	3	4	no	—
90	C₃×Q₃₂	—	3	1	no	—
91	C₃×Q₃₂	C₂×D₃	1	2	yes	—
92	C₃×Q₃₂	C₂×D₃	1	2	yes	—
93	C₃×Q₃₂	C₂²×C₃	1	2	yes	—
94	C₃×Q₃₂	C₂²×C₃	1	2	yes	—
95	C₃×SD₃₂	—	3	4	no	—
96	C₃×SD₃₂	—	3	2	no	—
97	C₃×D₁₆	—	3	1	no	—
98	C₃×M₃₂	—	3	1	no	—
99	C₃×M₃₂	—	3	2	no	—
100	C₂×C₃×C₁₆	—	2	1	no	—
101	C₂×C₃×C₁₆	—	3	2	no	—
102	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
103	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
104	#829	C₂²×C₃	1	4	yes	—
105	#828	C₂²×C₃	1	4	yes	—
106	#812	C₂²×C₃	1	1	yes	characteristic
107	Dic₂₄	C₂²×C₃	1	1	yes	characteristic
108	#800	C₂²×C₃	1	2	yes	—
109	D₄₈	C₂²×C₃	1	1	yes	characteristic
110	#798	C₂²×C₃	1	2	yes	—
111	D₃×C₁₆	C₂²×C₃	1	2	yes	—
112	C₃²×Q₈	—	2	4	no	—
113	C₃²×Q₈	—	4	2	no	—
114	C₂×C₃²×C₄	—	4	2	no	—
115	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
116	C₃×#67	—	4	2	no	—
117	C₃×#67	D₈	1	2	yes	—
118	C₃×D₁₂	—	2	2	no	—
119	C₃×D₁₂	—	4	4	no	—
120	C₃×D₁₂	D₈	1	1	yes	characteristic
121	C₃×C₄×D₃	—	4	4	no	—
122	C₃×C₄×D₃	—	4	2	no	—
123	C₃×C₄×D₃	—	4	4	no	—
124	C₃×C₄×D₃	C₂×D₄	1	2	yes	—
125	C₃×Dic₆	—	2	2	no	—
126	C₃×Dic₆	—	4	4	no	—
127	C₃×Dic₆	D₈	1	1	yes	characteristic
128	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
129	C₃²×C₈	C₂⁴	1	1	yes	characteristic
130	C₃×C₃⋊C₈	C₂×D₄	1	2	yes	—
131	#578	—	3	2	no	—
132	#510	—	3	2	no	—
133	#510	—	3	4	no	—
134	#508	—	3	2	no	—
135	#508	—	3	1	no	—
136	C₂×Q₃₂	—	3	2	no	—
137	C₂×Q₃₂	C₃×D₃	1	1	yes	characteristic, complemented
138	#504	—	3	1	no	—
139	C₃×#41	—	3	1	no	—
140	C₃×#41	—	6	4	no	—
141	C₃×#41	—	6	2	no	—
142	C₃×#41	—	12	4	no	—
143	C₃×#41	—	12	2	no	—
144	C₂×C₃×Q₈	—	2	2	no	—
145	C₂×C₃×Q₈	—	2	2	no	—
146	C₂×C₃×Q₈	—	4	2	no	—
147	C₂×C₃×Q₈	—	6	4	no	—
148	C₂×C₃×Q₈	—	12	2	no	—
149	#239	—	2	4	no	—
150	#239	—	4	2	no	—
151	D₃×Q₈	—	2	4	no	—
152	D₃×Q₈	—	4	2	no	—
153	#235	—	2	2	no	—
154	#235	—	4	4	no	—
155	#235	C₃×D₄	1	1	yes	characteristic
156	C₃×Q₁₆	—	2	2	no	—
157	C₃×Q₁₆	—	2	2	no	—
158	C₃×Q₁₆	—	2	4	no	—
159	C₃×Q₁₆	—	3	2	no	—
160	C₃×Q₁₆	—	4	2	no	—
161	C₃×Q₁₆	—	6	4	no	—
162	C₃×Q₁₆	C₂²×D₃	1	4	yes	—
163	C₃×Q₁₆	C₂³×C₃	1	4	yes	—
164	C₃×SD₁₆	—	6	4	no	—
165	C₃×SD₁₆	—	6	2	no	—
166	C₃×D₈	—	3	2	no	—
167	C₃×M₁₆	—	3	1	no	—
168	C₃×M₁₆	—	3	2	no	—
169	C₂×C₃×C₈	—	2	1	no	—
170	C₂×C₃×C₈	—	3	2	no	—
171	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
172	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
173	#216	—	2	4	no	—
174	#215	—	2	4	no	—
175	#208	C₃×D₄	1	1	yes	characteristic
176	Dic₁₂	C₂³×C₃	1	2	yes	—
177	D₂₄	C₂³×C₃	1	2	yes	—
178	#204	—	2	2	no	—
179	ΓL(1,𝔽₂₅)	C₃×D₄	1	2	yes	—
180	D₃×C₈	C₂³×C₃	1	2	yes	—
181	C₃×C₁₆	—	2	1	no	—
182	C₃×C₁₆	—	2	1	no	—
183	C₃×C₁₆	—	3	2	no	—
184	C₃×C₁₆	C₂²×D₃	1	1	yes	characteristic
185	C₃×C₁₆	C₂²×D₃	1	1	yes	characteristic
186	C₃×C₁₆	C₂³×C₃	1	1	yes	characteristic
187	C₃×C₁₆	C₂³×C₃	1	1	yes	characteristic
188	C₃⋊C₁₆	C₂³×C₃	1	2	yes	—
189	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
190	C₂×C₃×D₃	—	4	2	no	—
191	C₂×C₃×D₃	C₂×D₈	1	2	yes	—
192	C₃²×C₄	—	4	4	no	—
193	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
194	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
195	C₃×Dic₃	—	4	2	no	—
196	C₃×Dic₃	C₂×D₈	1	2	yes	—
197	2₋¹⁺⁴	—	6	2	no	—
198	#137	—	6	4	no	—
199	#137	—	6	2	no	—
200	#135	—	3	4	no	—
201	#135	—	3	2	no	—
202	C₂×Q₁₆	—	3	4	no	—
203	C₂×Q₁₆	C₂×C₃×D₃	1	2	yes	—
204	#131	—	3	1	no	—
205	Q₃₂	—	3	4	no	—
206	Q₃₂	—	3	1	no	—
207	Q₃₂	C₂×C₃×D₃	1	2	yes	—
208	Q₃₂	C₂×C₃×D₃	1	2	yes	—
209	SD₃₂	—	3	4	no	—
210	SD₃₂	—	3	2	no	—
211	D₁₆	—	3	1	no	—
212	M₃₂	—	3	1	no	—
213	M₃₂	—	3	2	no	—
214	C₂×C₁₆	—	3	2	no	—
215	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
216	C₃×Q₈	—	2	4	no	—
217	C₃×Q₈	—	2	4	no	—
218	C₃×Q₈	—	3	1	no	—
219	C₃×Q₈	—	4	4	no	—
220	C₃×Q₈	—	4	2	no	—
221	C₃×Q₈	—	4	2	no	—
222	C₃×Q₈	—	6	2	no	—
223	C₃×Q₈	—	8	2	no	—
224	C₃×Q₈	—	12	4	no	—
225	C₃×D₄	—	3	2	no	—
226	C₃×D₄	—	3	1	no	—
227	C₃×D₄	—	6	2	no	—
228	C₃×D₄	—	12	4	no	—
229	C₃×D₄	—	12	2	no	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	3	2	no	—
232	C₂×C₃×C₄	—	4	2	no	—
233	C₂×C₃×C₄	—	4	2	no	—
234	C₂×C₃×C₄	—	8	2	no	—
235	C₂×C₃×C₄	—	12	4	no	—
236	C₂×C₃×C₄	—	12	4	no	—
237	C₂×C₃×C₄	—	12	2	no	—
238	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	#67	—	4	2	no	—
241	#67	C₃×D₈	1	2	yes	—
242	D₁₂	—	2	2	no	—
243	D₁₂	—	4	4	no	—
244	D₁₂	C₃×D₈	1	1	yes	characteristic
245	C₄×D₃	—	4	4	no	—
246	C₄×D₃	—	4	2	no	—
247	C₄×D₃	—	4	4	no	—
248	C₄×D₃	C₂×C₃×D₄	1	2	yes	—
249	Dic₆	—	2	2	no	—
250	Dic₆	—	4	4	no	—
251	Dic₆	C₃×D₈	1	1	yes	characteristic
252	C₃×C₈	—	2	1	no	—
253	C₃×C₈	—	2	1	no	—
254	C₃×C₈	—	3	2	no	—
255	C₃×C₈	D₃×D₄	1	1	yes	characteristic
256	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
257	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
258	C₃×C₈	C₂⁴×C₃	1	1	yes	characteristic
259	C₃⋊C₈	C₂×C₃×D₄	1	2	yes	—
260	C₂×C₃²	—	2	1	no	—
261	C₂×C₃²	C₂²×D₈	1	1	yes	characteristic
262	C₃×D₃	—	2	2	no	—
263	C₃×D₃	—	8	2	no	—
264	#41	—	3	1	no	—
265	#41	—	6	4	no	—
266	#41	—	6	2	no	—
267	#41	—	12	4	no	—
268	#41	—	12	2	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	6	4	no	—
271	C₂×Q₈	—	12	2	no	—
272	Q₁₆	—	2	2	no	—
273	Q₁₆	—	3	2	no	—
274	Q₁₆	—	6	4	no	—
275	Q₁₆	C₂²×C₃×D₃	1	4	yes	—
276	SD₁₆	—	6	4	no	—
277	SD₁₆	—	6	2	no	—
278	D₈	—	3	2	no	—
279	M₁₆	—	3	1	no	—
280	M₁₆	—	3	2	no	—
281	C₂×C₈	—	3	2	no	—
282	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
283	C₁₆	—	3	2	no	—
284	C₁₆	C₂²×C₃×D₃	1	1	yes	characteristic
285	C₁₆	C₂²×C₃×D₃	1	1	yes	characteristic
286	C₂²×C₃	—	2	1	no	—
287	C₂²×C₃	—	3	2	no	—
288	C₂²×C₃	—	12	2	no	—
289	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
290	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
291	C₂×D₃	—	4	2	no	—
292	C₂×D₃	C₂×C₃×D₈	1	2	yes	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	3	2	no	—
296	C₃×C₄	—	4	4	no	—
297	C₃×C₄	—	4	4	no	—
298	C₃×C₄	—	8	4	no	—
299	C₃×C₄	—	12	2	no	—
300	C₃×C₄	D₃×D₈	1	1	yes	characteristic
301	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
302	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
303	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
304	Dic₃	—	4	2	no	—
305	Dic₃	C₂×C₃×D₈	1	2	yes	—
306	C₃²	#3417	1	1	yes	characteristic, complemented, Sylow
307	Q₈	—	2	4	no	—
308	Q₈	—	3	1	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	6	2	no	—
311	Q₈	—	12	4	no	—
312	D₄	—	3	2	no	—
313	D₄	—	3	1	no	—
314	D₄	—	6	2	no	—
315	D₄	—	12	4	no	—
316	D₄	—	12	2	no	—
317	C₂×C₄	—	3	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	12	4	no	—
320	C₂×C₄	—	12	4	no	—
321	C₂×C₄	—	12	2	no	—
322	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
323	C₈	—	3	2	no	—
324	C₈	C₃×D₃×D₄	1	1	yes	characteristic
325	C₈	C₂³×C₃×D₃	1	1	yes	characteristic
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	6	2	no	—
331	C₂×C₃	—	24	2	no	—
332	C₂×C₃	C₂×D₃×D₈	1	1	yes	characteristic
333	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
334	D₃	—	2	2	no	—
335	D₃	—	8	2	no	—
336	C₂²	—	3	2	no	—
337	C₂²	—	12	2	no	—
338	C₂²	C₃×D₃×D₈	1	1	yes	characteristic
339	C₄	—	3	2	no	—
340	C₄	—	4	4	no	—
341	C₄	—	12	2	no	—
342	C₄	C₃×D₃×D₈	1	1	yes	characteristic
343	C₄	C₂×C₃×D₃×D₄	1	1	yes	characteristic
344	C₃	—	2	1	no	—
345	C₃	#82028	1	1	yes	characteristic, direct product factor
346	C₃	C₃×#3417	1	1	yes	characteristic, complemented
347	C₂	—	2	1	no	—
348	C₂	—	6	2	no	—
349	C₂	—	24	2	no	—
350	C₂	C₂×C₃×D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₃
Fitting	C₂×C₃²×Q₃₂	C₂
socle	C₂×C₃²	C₂²×D₈
hypercenter	C₂×C₃×Q₃₂	D₃
hypocenter	C₃	C₃×#3417
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×Q₃₂

Complements

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

N	H ≅ G/N	type
1. C₃×#5766	C₂	semidirect
2. C₂×C₃²×Q₃₂	C₂	semidirect
3. C₃×#5179	C₂	semidirect
4. C₃×#4919	C₂	semidirect
5. C₃×#4918	C₂	semidirect
6. C₃×D₃×Q₃₂	C₂	semidirect
7. C₃×#4904	C₂	semidirect
8. C₃×#4901	C₂	semidirect
10. #82028	C₃	direct (N × H)
31. #5766	C₂×C₃	semidirect
38. C₂×C₃×Q₃₂	D₃	semidirect
39. C₂×C₃×Q₃₂	C₂×C₃	semidirect
41. #5179	C₂×C₃	semidirect
42. #4919	C₂×C₃	semidirect
43. #4918	C₂×C₃	semidirect
44. D₃×Q₃₂	C₂×C₃	semidirect
45. #4904	C₂×C₃	semidirect
46. #4901	C₂×C₃	semidirect
137. C₂×Q₃₂	C₃×D₃	semidirect
306. C₃²	#3417	semidirect
345. C₃	#82028	direct (N × H)
346. C₃	C₃×#3417	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5766	2
C₂×C₃²×Q₃₂	2
C₃×#5179	2
C₃×#4919	2
C₃×#4918	2
C₃×D₃×Q₃₂	2
C₃×#4904	2
C₃×#4901	2
C₃×#3417	3
#82028	3

Subgroups — C3×#82028 #11899754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#82028 #11899754