Subgroups — C₃×#33743 #10652350

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3176	C₂	1	1	yes	maximal, characteristic
3	C₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2245	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2244	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2220	C₂	1	2	yes	maximal, complemented
8	C₃×#2217	C₂	1	2	yes	maximal, complemented
9	C₃×#2175	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×SD₃₂	C₂	1	2	yes	maximal, complemented
11	C₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	#33743	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃×#541	—	2	1	no	—
14	C₃×#540	—	2	1	no	—
15	C₂×C₃×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₃×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₃×#499	C₂²	1	1	yes	characteristic
20	C₃×#499	C₂²	1	1	yes	characteristic
21	C₃×#494	C₂²	1	1	yes	characteristic
22	C₃×#493	C₂²	1	1	yes	characteristic
23	C₃×#491	—	2	1	no	—
24	C₃×#490	—	2	1	no	—
25	C₃×#470	—	2	1	no	—
26	C₃×#469	—	2	1	no	—
27	C₃×#467	C₂²	1	2	yes	—
28	C₃×#466	C₂²	1	2	yes	—
29	C₃×#442	—	2	1	no	—
30	C₃×#441	—	2	1	no	—
31	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
32	C₃×C₄×D₈	C₂²	1	2	yes	—
33	C₃×#405	C₂²	1	1	yes	characteristic
34	C₃×#367	C₂²	1	2	yes	—
35	C₃×#366	C₂²	1	2	yes	—
36	C₃×#358	C₂²	1	2	yes	—
37	C₃×#358	C₂²	1	2	yes	—
38	C₃×#357	C₂²	1	2	yes	—
39	C₃×#357	C₂²	1	2	yes	—
40	C₃×#346	C₂²	1	1	yes	characteristic
41	C₃×#346	C₂²	1	1	yes	characteristic
42	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
43	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
44	#3177	C₂×C₃	1	1	yes	characteristic, complemented
45	#3176	C₂×C₃	1	1	yes	characteristic
46	#2253	C₂×C₃	1	1	yes	characteristic, complemented
47	#2248	C₂×C₃	1	1	yes	characteristic, complemented
48	#2245	C₂×C₃	1	1	yes	characteristic, complemented
49	#2244	C₂×C₃	1	1	yes	characteristic, complemented
50	#2220	C₂×C₃	1	2	yes	complemented
51	#2217	C₂×C₃	1	2	yes	complemented
52	#2175	C₂×C₃	1	2	yes	complemented
53	C₄×SD₃₂	C₂×C₃	1	2	yes	complemented
54	#2108	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×C₃×Q₁₆	—	2	1	no	—
56	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
57	C₂×C₃×D₈	—	2	1	no	—
58	C₂×C₃×D₈	C₂³	1	2	yes	—
59	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
60	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₃×C₄⋊Q₈	—	2	1	no	—
62	C₃×#127	—	2	1	no	—
63	C₃×#126	—	4	2	no	—
64	C₃×#125	—	2	1	no	—
65	C₃×#125	—	2	1	no	—
66	C₃×#125	—	4	2	no	—
67	C₃×#123	—	4	2	no	—
68	C₃×C₂²⋊Q₈	—	2	2	no	—
69	C₃×#121	—	2	2	no	—
70	C₃×#121	—	4	2	no	—
71	C₃×C₄×Q₈	—	2	2	no	—
72	C₃×C₄×D₄	—	2	2	no	—
73	C₃×#117	D₄	1	1	yes	characteristic
74	C₃×SD₃₂	—	2	4	no	—
75	C₃×SD₃₂	—	2	4	no	—
76	C₃×M₃₂	—	2	2	no	—
77	C₂×C₃×C₁₆	—	2	2	no	—
78	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
82	C₃×#107	C₂³	1	2	yes	—
83	C₃×#107	C₂³	1	2	yes	—
84	C₃×#106	—	2	1	no	—
85	C₃×#106	—	2	1	no	—
86	C₃×#103	—	2	2	no	—
87	C₃×#103	—	2	2	no	—
88	C₃×#102	—	2	2	no	—
89	C₃×#102	—	2	2	no	—
90	C₃×#97	D₄	1	1	yes	characteristic
91	C₃×#97	D₄	1	1	yes	characteristic
92	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
94	#541	—	2	1	no	—
95	#540	—	2	1	no	—
96	C₂×SD₃₂	C₂²×C₃	1	2	yes	—
97	C₂×SD₃₂	C₂²×C₃	1	2	yes	—
98	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
100	#499	C₂²×C₃	1	1	yes	characteristic
101	#499	C₂²×C₃	1	1	yes	characteristic
102	#494	C₂²×C₃	1	1	yes	characteristic
103	#493	C₂²×C₃	1	1	yes	characteristic
104	#491	—	2	1	no	—
105	#490	—	2	1	no	—
106	#470	—	2	1	no	—
107	#469	—	2	1	no	—
108	#467	C₂²×C₃	1	2	yes	—
109	#466	C₂²×C₃	1	2	yes	—
110	#442	—	2	1	no	—
111	#441	—	2	1	no	—
112	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
113	C₄×D₈	C₂²×C₃	1	2	yes	—
114	#405	C₂²×C₃	1	1	yes	characteristic
115	#367	C₂²×C₃	1	2	yes	—
116	#366	C₂²×C₃	1	2	yes	—
117	#358	C₂²×C₃	1	2	yes	—
118	#358	C₂²×C₃	1	2	yes	—
119	#357	C₂²×C₃	1	2	yes	—
120	#357	C₂²×C₃	1	2	yes	—
121	#346	C₂²×C₃	1	1	yes	characteristic
122	#346	C₂²×C₃	1	1	yes	characteristic
123	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
124	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
125	C₂×C₃×Q₈	—	2	2	no	—
126	C₂×C₃×D₄	—	2	2	no	—
127	C₂×C₃×D₄	—	4	2	no	—
128	C₂×C₃×D₄	—	4	2	no	—
129	C₂²×C₃×C₄	—	4	2	no	—
130	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
131	C₃×Q₁₆	—	2	2	no	—
132	C₃×Q₁₆	—	2	2	no	—
133	C₃×Q₁₆	—	4	2	no	—
134	C₃×D₈	—	2	2	no	—
135	C₃×D₈	—	2	2	no	—
136	C₃×D₈	—	4	2	no	—
137	C₃×M₁₆	—	2	2	no	—
138	C₂×C₃×C₈	—	2	1	no	—
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄⋊C₄	—	4	2	no	—
147	C₃×C₄⋊C₄	—	4	2	no	—
148	C₃×C₄⋊C₄	—	4	2	no	—
149	C₃×C₄⋊C₄	—	4	2	no	—
150	C₃×C₄⋊C₄	—	4	2	no	—
151	C₃×C₄⋊C₄	D₈	1	2	yes	—
152	C₃×C₂²⋊C₄	—	4	2	no	—
153	C₃×C₂²⋊C₄	—	4	2	no	—
154	C₃×C₂²⋊C₄	—	4	2	no	—
155	C₃×C₂²⋊C₄	D₈	1	2	yes	—
156	C₃×C₄²	—	4	2	no	—
157	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
158	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
159	C₃×C₁₆	—	2	1	no	—
160	C₃×C₁₆	—	2	1	no	—
161	C₃×C₁₆	#41	1	2	yes	—
162	C₃×C₁₆	#41	1	2	yes	—
163	C₂×Q₁₆	—	2	1	no	—
164	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	C₂³×C₃	1	2	yes	—
167	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	1	no	—
170	#127	—	2	1	no	—
171	#126	—	4	2	no	—
172	#125	—	2	1	no	—
173	#125	—	2	1	no	—
174	#125	—	4	2	no	—
175	#123	—	4	2	no	—
176	C₂²⋊Q₈	—	2	2	no	—
177	#121	—	2	2	no	—
178	#121	—	4	2	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×D₄	—	2	2	no	—
181	#117	C₃×D₄	1	1	yes	characteristic
182	SD₃₂	—	2	4	no	—
183	SD₃₂	—	2	4	no	—
184	M₃₂	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
187	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
188	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
189	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
190	#107	C₂³×C₃	1	2	yes	—
191	#107	C₂³×C₃	1	2	yes	—
192	#106	—	2	1	no	—
193	#106	—	2	1	no	—
194	#103	—	2	2	no	—
195	#103	—	2	2	no	—
196	#102	—	2	2	no	—
197	#102	—	2	2	no	—
198	#97	C₃×D₄	1	1	yes	characteristic
199	#97	C₃×D₄	1	1	yes	characteristic
200	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
202	C₂³×C₃	—	4	2	no	—
203	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
204	C₃×Q₈	—	4	2	no	—
205	C₃×Q₈	—	4	2	no	—
206	C₃×D₄	—	4	2	no	—
207	C₃×D₄	—	4	2	no	—
208	C₃×D₄	—	16	2	no	—
209	C₃×D₄	—	16	2	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	—	4	2	no	—
215	C₂×C₃×C₄	—	16	2	no	—
216	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
217	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
218	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
220	C₃×C₈	—	2	1	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	#135	1	2	yes	—
223	C₃×C₈	C₂×#41	1	1	yes	characteristic
224	C₃×C₈	C₂×#41	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	4	2	no	—
234	D₈	—	2	2	no	—
235	D₈	—	2	2	no	—
236	D₈	—	4	2	no	—
237	M₁₆	—	2	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
241	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
243	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	C₃×D₈	1	2	yes	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	C₃×D₈	1	2	yes	—
256	C₄²	—	4	2	no	—
257	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
258	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
259	C₁₆	—	2	1	no	—
260	C₁₆	—	2	1	no	—
261	C₁₆	C₃×#41	1	2	yes	—
262	C₁₆	C₃×#41	1	2	yes	—
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	8	2	no	—
266	C₂²×C₃	—	8	2	no	—
267	C₂²×C₃	—	8	2	no	—
268	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	8	2	no	—
273	C₃×C₄	—	8	2	no	—
274	C₃×C₄	—	8	2	no	—
275	C₃×C₄	#532	1	1	yes	characteristic
276	C₃×C₄	C₂×#135	1	1	yes	characteristic
277	C₂³	—	4	2	no	—
278	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	16	2	no	—
284	D₄	—	16	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	16	2	no	—
291	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
292	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
293	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	C₃×#135	1	2	yes	—
298	C₈	C₂×C₃×#41	1	1	yes	characteristic
299	C₈	C₂×C₃×#41	1	1	yes	characteristic
300	C₂×C₃	—	4	1	no	—
301	C₂×C₃	—	16	2	no	—
302	C₂×C₃	#3169	1	1	yes	characteristic
303	C₂×C₃	#3416	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	C₃×#532	1	1	yes	characteristic
317	C₄	C₂×C₃×#135	1	1	yes	characteristic
318	C₃	#33743	1	1	yes	characteristic, direct product factor, Sylow
319	C₂	—	4	1	no	—
320	C₂	—	16	2	no	—
321	C₂	C₃×#3169	1	1	yes	characteristic
322	C₂	C₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₃×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ C₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3177	C₂	semidirect
3. C₃×#2253	C₂	semidirect
4. C₃×#2248	C₂	semidirect
5. C₃×#2245	C₂	semidirect
6. C₃×#2244	C₂	semidirect
7. C₃×#2220	C₂	semidirect
8. C₃×#2217	C₂	semidirect
9. C₃×#2175	C₂	semidirect
10. C₃×C₄×SD₃₂	C₂	semidirect
11. C₃×#2108	C₂	semidirect
12. #33743	C₃	direct (N × H)
44. #3177	C₂×C₃	semidirect
46. #2253	C₂×C₃	semidirect
47. #2248	C₂×C₃	semidirect
48. #2245	C₂×C₃	semidirect
49. #2244	C₂×C₃	semidirect
50. #2220	C₂×C₃	semidirect
51. #2217	C₂×C₃	semidirect
52. #2175	C₂×C₃	semidirect
53. C₄×SD₃₂	C₂×C₃	semidirect
54. #2108	C₂×C₃	semidirect
318. C₃	#33743	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×#3177	2
C₃×#3176	2
C₃×#2253	2
C₃×#2248	2
C₃×#2245	2
C₃×#2244	2
C₃×#2220	2
C₃×#2217	2
C₃×#2175	2
C₃×C₄×SD₃₂	2
C₃×#2108	2
#33743	3

Subgroups — C3×#33743 #10652350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33743 #10652350