Subgroups — C₃×#33751 #10652358

Order 768.

548 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3169	C₂	1	1	yes	maximal, characteristic
3	C₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2220	C₂	1	4	yes	maximal, complemented
8	C₃×#2175	C₂	1	4	yes	maximal, complemented
9	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#33751	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×#533	—	2	1	no	—
12	C₃×#532	—	2	1	no	—
13	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
14	C₂×C₃×M₃₂	C₂²	1	2	yes	—
15	C₃×#500	C₂²	1	1	yes	characteristic
16	C₃×#500	C₂²	1	1	yes	characteristic
17	C₃×#494	C₂²	1	1	yes	characteristic
18	C₃×#493	C₂²	1	1	yes	characteristic
19	C₃×#487	—	2	1	no	—
20	C₃×#486	—	2	1	no	—
21	C₃×#467	—	2	1	no	—
22	C₃×#467	C₂²	1	2	yes	—
23	C₃×#466	—	2	1	no	—
24	C₃×#466	C₂²	1	2	yes	—
25	C₃×C₄×Q₁₆	—	2	1	no	—
26	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
27	C₃×C₄×D₈	—	2	1	no	—
28	C₃×C₄×D₈	C₂²	1	2	yes	—
29	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₃×#366	C₂²	1	4	yes	—
31	C₃×#358	C₂²	1	4	yes	—
32	C₃×#357	C₂²	1	4	yes	—
33	C₃×#346	C₂²	1	2	yes	—
34	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
35	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
36	#3170	C₂×C₃	1	1	yes	characteristic, complemented
37	#3169	C₂×C₃	1	1	yes	characteristic
38	#2252	C₂×C₃	1	1	yes	characteristic, complemented
39	#2248	C₂×C₃	1	1	yes	characteristic, complemented
40	#2241	C₂×C₃	1	1	yes	characteristic, complemented
41	#2240	C₂×C₃	1	1	yes	characteristic, complemented
42	#2220	C₂×C₃	1	4	yes	complemented
43	#2175	C₂×C₃	1	4	yes	complemented
44	C₄×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×#41	—	4	1	no	—
46	C₂×C₃×Q₁₆	—	2	1	no	—
47	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
48	C₂×C₃×D₈	—	2	1	no	—
49	C₂×C₃×D₈	C₂³	1	2	yes	—
50	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
52	C₃×C₄⋊Q₈	—	2	1	no	—
53	C₃×C₄⋊Q₈	—	2	1	no	—
54	C₃×C₄⋊Q₈	—	2	1	no	—
55	C₃×#127	—	2	1	no	—
56	C₃×#125	—	4	1	no	—
57	C₃×#124	—	4	1	no	—
58	C₃×C₂²⋊Q₈	—	2	2	no	—
59	C₃×C₂²⋊Q₈	—	4	1	no	—
60	C₃×#121	—	2	2	no	—
61	C₃×#121	—	4	1	no	—
62	C₃×C₄×Q₈	—	2	2	no	—
63	C₃×C₄×Q₈	—	4	1	no	—
64	C₃×C₄×D₄	—	2	2	no	—
65	C₃×C₄×D₄	—	4	1	no	—
66	C₃×#117	—	4	1	no	—
67	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
68	C₃×SD₃₂	—	2	8	no	—
69	C₃×M₃₂	—	2	4	no	—
70	C₂×C₃×C₁₆	C₂³	1	2	yes	—
71	C₂×C₃×C₁₆	C₂³	1	2	yes	—
72	C₃×#107	—	2	1	no	—
73	C₃×#107	—	2	1	no	—
74	C₃×#107	C₂³	1	2	yes	—
75	C₃×#107	C₂³	1	2	yes	—
76	C₃×#103	—	2	2	no	—
77	C₃×#103	—	2	2	no	—
78	C₃×#102	—	2	2	no	—
79	C₃×#102	—	2	2	no	—
80	C₃×C₄×C₈	D₄	1	1	yes	characteristic
81	C₃×C₄×C₈	D₄	1	1	yes	characteristic
82	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
83	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
84	#533	—	2	1	no	—
85	#532	—	2	1	no	—
86	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
87	C₂×M₃₂	C₂²×C₃	1	2	yes	—
88	#500	C₂²×C₃	1	1	yes	characteristic
89	#500	C₂²×C₃	1	1	yes	characteristic
90	#494	C₂²×C₃	1	1	yes	characteristic
91	#493	C₂²×C₃	1	1	yes	characteristic
92	#487	—	2	1	no	—
93	#486	—	2	1	no	—
94	#467	—	2	1	no	—
95	#467	C₂²×C₃	1	2	yes	—
96	#466	—	2	1	no	—
97	#466	C₂²×C₃	1	2	yes	—
98	C₄×Q₁₆	—	2	1	no	—
99	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
100	C₄×D₈	—	2	1	no	—
101	C₄×D₈	C₂²×C₃	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
103	#366	C₂²×C₃	1	4	yes	—
104	#358	C₂²×C₃	1	4	yes	—
105	#357	C₂²×C₃	1	4	yes	—
106	#346	C₂²×C₃	1	2	yes	—
107	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
108	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
109	C₃×#41	—	8	4	no	—
110	C₂×C₃×Q₈	—	2	2	no	—
111	C₂×C₃×Q₈	—	4	1	no	—
112	C₂×C₃×Q₈	—	4	1	no	—
113	C₂×C₃×D₄	—	2	2	no	—
114	C₂×C₃×D₄	—	4	2	no	—
115	C₂×C₃×D₄	—	4	1	no	—
116	C₂×C₃×D₄	—	4	1	no	—
117	C₂²×C₃×C₄	—	4	2	no	—
118	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
119	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₃×Q₁₆	—	2	4	no	—
122	C₃×Q₁₆	—	4	2	no	—
123	C₃×D₈	—	2	4	no	—
124	C₃×D₈	—	4	2	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
128	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
129	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
131	C₃×C₄⋊C₄	—	2	2	no	—
132	C₃×C₄⋊C₄	—	2	2	no	—
133	C₃×C₄⋊C₄	—	4	1	no	—
134	C₃×C₄⋊C₄	—	4	1	no	—
135	C₃×C₄⋊C₄	—	4	1	no	—
136	C₃×C₄⋊C₄	—	4	2	no	—
137	C₃×C₄⋊C₄	—	4	2	no	—
138	C₃×C₄⋊C₄	—	4	1	no	—
139	C₃×C₂²⋊C₄	—	4	2	no	—
140	C₃×C₂²⋊C₄	—	4	2	no	—
141	C₃×C₂²⋊C₄	—	4	2	no	—
142	C₃×C₄²	—	4	2	no	—
143	C₃×C₄²	D₈	1	1	yes	characteristic
144	C₃×C₄²	D₈	1	1	yes	characteristic
145	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
146	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
147	C₃×C₁₆	—	2	2	no	—
148	C₃×C₁₆	#41	1	4	yes	complemented
149	C₂×#41	—	4	1	no	—
150	C₂×Q₁₆	—	2	1	no	—
151	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
152	C₂×D₈	—	2	1	no	—
153	C₂×D₈	C₂³×C₃	1	2	yes	—
154	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	1	no	—
157	C₄⋊Q₈	—	2	1	no	—
158	C₄⋊Q₈	—	2	1	no	—
159	#127	—	2	1	no	—
160	#125	—	4	1	no	—
161	#124	—	4	1	no	—
162	C₂²⋊Q₈	—	2	2	no	—
163	C₂²⋊Q₈	—	4	1	no	—
164	#121	—	2	2	no	—
165	#121	—	4	1	no	—
166	C₄×Q₈	—	2	2	no	—
167	C₄×Q₈	—	4	1	no	—
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	4	1	no	—
170	#117	—	4	1	no	—
171	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
172	SD₃₂	—	2	8	no	—
173	M₃₂	—	2	4	no	—
174	C₂×C₁₆	C₂³×C₃	1	2	yes	—
175	C₂×C₁₆	C₂³×C₃	1	2	yes	—
176	#107	—	2	1	no	—
177	#107	—	2	1	no	—
178	#107	C₂³×C₃	1	2	yes	—
179	#107	C₂³×C₃	1	2	yes	—
180	#103	—	2	2	no	—
181	#103	—	2	2	no	—
182	#102	—	2	2	no	—
183	#102	—	2	2	no	—
184	C₄×C₈	C₃×D₄	1	1	yes	characteristic
185	C₄×C₈	C₃×D₄	1	1	yes	characteristic
186	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
188	C₂³×C₃	—	4	2	no	—
189	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
190	C₃×Q₈	—	4	2	no	—
191	C₃×Q₈	—	4	2	no	—
192	C₃×Q₈	—	8	2	no	—
193	C₃×Q₈	—	8	2	no	—
194	C₃×D₄	—	4	2	no	—
195	C₃×D₄	—	4	2	no	—
196	C₃×D₄	—	8	4	no	—
197	C₃×D₄	—	8	2	no	—
198	C₃×D₄	—	8	2	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	4	2	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	—	8	4	no	—
207	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	#135	1	2	yes	—
216	C₃×C₈	C₂×#41	1	2	yes	—
217	#41	—	8	4	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
227	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
229	Q₁₆	—	2	4	no	—
230	Q₁₆	—	4	2	no	—
231	D₈	—	2	4	no	—
232	D₈	—	4	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
236	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
238	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	—	4	2	no	—
251	C₄²	C₃×D₈	1	1	yes	characteristic
252	C₄²	C₃×D₈	1	1	yes	characteristic
253	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
254	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
255	C₁₆	—	2	2	no	—
256	C₁₆	C₃×#41	1	4	yes	complemented
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	8	2	no	—
261	C₂²×C₃	—	8	2	no	—
262	C₂²×C₃	—	8	2	no	—
263	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	8	2	no	—
268	C₃×C₄	—	8	2	no	—
269	C₃×C₄	—	8	2	no	—
270	C₃×C₄	#509	1	2	yes	—
271	C₃×C₄	#510	1	2	yes	—
272	C₃×C₄	#532	1	1	yes	characteristic
273	C₃×C₄	C₂×#135	1	1	yes	characteristic
274	C₂³	—	4	2	no	—
275	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	8	2	no	—
279	Q₈	—	8	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	8	4	no	—
283	D₄	—	8	2	no	—
284	D₄	—	8	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	8	4	no	—
293	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
294	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
295	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
296	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
297	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	C₃×#135	1	2	yes	—
302	C₈	C₂×C₃×#41	1	2	yes	—
303	C₂×C₃	—	4	1	no	—
304	C₂×C₃	—	16	2	no	—
305	C₂×C₃	#3169	1	1	yes	characteristic
306	C₂×C₃	C₂×#509	1	1	yes	characteristic
307	C₂×C₃	C₂×#510	1	1	yes	characteristic
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₃×#509	1	2	yes	—
322	C₄	C₃×#510	1	2	yes	—
323	C₄	C₃×#532	1	1	yes	characteristic
324	C₄	C₂×C₃×#135	1	1	yes	characteristic
325	C₃	#33751	1	1	yes	characteristic, direct product factor, Sylow
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	C₃×#3169	1	1	yes	characteristic
329	C₂	C₂×C₃×#509	1	1	yes	characteristic
330	C₂	C₂×C₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₃×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3170	C₂	semidirect
3. C₃×#2252	C₂	semidirect
4. C₃×#2248	C₂	semidirect
5. C₃×#2241	C₂	semidirect
6. C₃×#2240	C₂	semidirect
7. C₃×#2220	C₂	semidirect
8. C₃×#2175	C₂	semidirect
9. C₃×C₄×M₃₂	C₂	semidirect
10. #33751	C₃	direct (N × H)
36. #3170	C₂×C₃	semidirect
38. #2252	C₂×C₃	semidirect
39. #2248	C₂×C₃	semidirect
40. #2241	C₂×C₃	semidirect
41. #2240	C₂×C₃	semidirect
42. #2220	C₂×C₃	semidirect
43. #2175	C₂×C₃	semidirect
44. C₄×M₃₂	C₂×C₃	semidirect
148. C₃×C₁₆	#41	semidirect
256. C₁₆	C₃×#41	semidirect
325. C₃	#33751	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#3170	2
C₃×#3169	2
C₃×#2252	2
C₃×#2248	2
C₃×#2241	2
C₃×#2240	2
C₃×#2220	2
C₃×#2175	2
C₃×C₄×M₃₂	2
#33751	3

Subgroups — C3×#33751 #10652358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33751 #10652358