Subgroups — C₃×#33774 #10652381

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 320 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	C₂×C₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₃×#2248	C₂	1	2	yes	maximal, complemented
4	C₃×#2240	C₂	1	2	yes	maximal, complemented
5	C₃×#2207	C₂	1	4	yes	maximal, complemented
6	C₃×#2187	C₂	1	4	yes	maximal, complemented
7	C₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	#33774	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×D₈	—	2	1	no	—
10	C₂²×C₃×D₈	C₂²	1	2	yes	—
11	C₃×#533	—	2	1	no	—
12	C₂×C₃×#127	—	2	1	no	—
13	C₂×C₃×SD₃₂	C₂²	1	8	yes	complemented
14	C₃×#500	C₂²	1	1	yes	characteristic, complemented
15	C₃×#494	C₂²	1	1	yes	characteristic, complemented
16	C₃×#493	—	2	1	no	—
17	C₃×#493	—	2	1	no	—
18	C₃×#493	C₂²	1	2	yes	—
19	C₃×#493	C₂²	1	2	yes	—
20	C₃×#487	—	2	1	no	—
21	C₃×#467	—	2	1	no	—
22	C₃×#467	C₂²	1	2	yes	—
23	C₃×C₄×Q₁₆	—	2	1	no	—
24	C₃×C₄×Q₁₆	C₂²	1	2	yes	complemented
25	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
27	C₃×#357	C₂²	1	8	yes	complemented
28	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₃×C₄×C₁₆	C₂²	1	2	yes	complemented
30	#3170	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#493	C₂×C₃	1	1	yes	characteristic
32	#2248	C₂×C₃	1	2	yes	complemented
33	#2240	C₂×C₃	1	2	yes	complemented
34	#2207	C₂×C₃	1	4	yes	complemented
35	#2187	C₂×C₃	1	4	yes	complemented
36	#2162	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂²×C₃×D₄	—	2	2	no	—
38	C₂²×C₃×D₄	—	4	1	no	—
39	C₂²×C₃×D₄	—	4	1	no	—
40	C₂×C₃×Q₁₆	—	2	1	no	—
41	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₃×D₈	—	2	2	no	—
43	C₂×C₃×D₈	—	2	2	no	—
44	C₂×C₃×D₈	—	2	2	no	—
45	C₂×C₃×D₈	—	4	4	no	—
46	C₂×C₃×D₈	—	4	4	no	—
47	C₂×C₃×D₈	C₂³	1	4	yes	—
48	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
50	C₃×C₄⋊Q₈	—	2	1	no	—
51	C₃×C₄⋊Q₈	—	2	1	no	—
52	C₃×#127	—	2	2	no	—
53	C₃×#127	—	2	2	no	—
54	C₃×#127	—	4	1	no	—
55	C₃×#127	—	4	1	no	—
56	C₃×#125	—	4	1	no	—
57	C₃×C₂²⋊Q₈	—	2	2	no	—
58	C₃×C₂²⋊Q₈	—	4	1	no	—
59	C₃×C₄×Q₈	—	2	2	no	—
60	C₃×C₄×Q₈	—	4	1	no	—
61	C₃×#117	—	4	1	no	—
62	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
63	C₃×SD₃₂	—	4	8	no	—
64	C₂×C₃×C₁₆	C₂³	1	4	yes	—
65	C₃×#107	—	2	1	no	—
66	C₃×#107	C₂³	1	2	yes	—
67	C₃×#103	—	2	2	no	—
68	C₃×#103	—	2	2	no	—
69	C₃×C₄×C₈	D₄	1	1	yes	characteristic
70	C₃×C₄×C₈	D₄	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂²×D₈	—	2	1	no	—
74	C₂²×D₈	C₂²×C₃	1	2	yes	—
75	#533	—	2	1	no	—
76	C₂×#127	—	2	1	no	—
77	C₂×SD₃₂	C₂²×C₃	1	8	yes	complemented
78	#500	C₂²×C₃	1	1	yes	characteristic, complemented
79	#494	C₂²×C₃	1	1	yes	characteristic, complemented
80	#493	—	2	1	no	—
81	#493	—	2	1	no	—
82	#493	C₂²×C₃	1	2	yes	—
83	#493	C₂²×C₃	1	2	yes	—
84	#487	—	2	1	no	—
85	#467	—	2	1	no	—
86	#467	C₂²×C₃	1	2	yes	—
87	C₄×Q₁₆	—	2	1	no	—
88	C₄×Q₁₆	C₂²×C₃	1	2	yes	complemented
89	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊C₁₆	C₂²×C₃	1	2	yes	complemented
91	#357	C₂²×C₃	1	8	yes	complemented
92	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
93	C₄×C₁₆	C₂²×C₃	1	2	yes	complemented
94	C₂⁴×C₃	—	4	2	no	—
95	C₂×C₃×Q₈	—	2	2	no	—
96	C₂×C₃×Q₈	—	4	1	no	—
97	C₂×C₃×D₄	—	2	4	no	—
98	C₂×C₃×D₄	—	4	4	no	—
99	C₂×C₃×D₄	—	4	4	no	—
100	C₂×C₃×D₄	—	4	2	no	—
101	C₂×C₃×D₄	—	4	2	no	—
102	C₂×C₃×D₄	—	4	2	no	—
103	C₂×C₃×D₄	—	4	2	no	—
104	C₂×C₃×D₄	—	4	2	no	—
105	C₂×C₃×D₄	—	8	4	no	—
106	C₂×C₃×D₄	—	8	4	no	—
107	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
108	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
110	C₃×Q₁₆	—	2	4	no	—
111	C₃×Q₁₆	—	4	2	no	—
112	C₃×D₈	—	2	8	no	—
113	C₃×D₈	—	4	4	no	—
114	C₃×D₈	—	4	4	no	—
115	C₃×D₈	—	4	4	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	4	1	no	—
124	C₃×C₄⋊C₄	—	4	1	no	—
125	C₃×C₄⋊C₄	—	4	2	no	—
126	C₃×C₄⋊C₄	—	4	2	no	—
127	C₃×C₄⋊C₄	—	4	1	no	—
128	C₃×C₂²⋊C₄	—	4	2	no	—
129	C₃×C₄²	—	4	2	no	—
130	C₃×C₄²	D₈	1	1	yes	characteristic
131	C₃×C₄²	D₈	1	1	yes	characteristic
132	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
133	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
134	C₃×C₁₆	—	2	4	no	—
135	C₂²×D₄	—	2	2	no	—
136	C₂²×D₄	—	4	1	no	—
137	C₂²×D₄	—	4	1	no	—
138	C₂×Q₁₆	—	2	1	no	—
139	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	4	4	no	—
144	C₂×D₈	—	4	4	no	—
145	C₂×D₈	C₂³×C₃	1	4	yes	—
146	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
148	C₄⋊Q₈	—	2	1	no	—
149	C₄⋊Q₈	—	2	1	no	—
150	#127	—	2	2	no	—
151	#127	—	2	2	no	—
152	#127	—	4	1	no	—
153	#127	—	4	1	no	—
154	#125	—	4	1	no	—
155	C₂²⋊Q₈	—	2	2	no	—
156	C₂²⋊Q₈	—	4	1	no	—
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	—	4	1	no	—
159	#117	—	4	1	no	—
160	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
161	SD₃₂	—	4	8	no	—
162	C₂×C₁₆	C₂³×C₃	1	4	yes	—
163	#107	—	2	1	no	—
164	#107	C₂³×C₃	1	2	yes	—
165	#103	—	2	2	no	—
166	#103	—	2	2	no	—
167	C₄×C₈	C₃×D₄	1	1	yes	characteristic
168	C₄×C₈	C₃×D₄	1	1	yes	characteristic
169	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
170	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
171	C₂³×C₃	—	4	4	no	—
172	C₂³×C₃	—	8	4	no	—
173	C₂³×C₃	—	8	4	no	—
174	C₂³×C₃	—	8	4	no	—
175	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
176	C₃×Q₈	—	4	2	no	—
177	C₃×Q₈	—	4	2	no	—
178	C₃×Q₈	—	8	2	no	—
179	C₃×D₄	—	4	4	no	—
180	C₃×D₄	—	4	4	no	—
181	C₃×D₄	—	8	4	no	—
182	C₃×D₄	—	8	4	no	—
183	C₃×D₄	—	8	4	no	—
184	C₃×D₄	—	8	4	no	—
185	C₃×D₄	—	8	4	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	SD₃₂	1	4	yes	complemented
192	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
198	C₃×C₈	—	2	1	no	—
199	C₃×C₈	—	2	1	no	—
200	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
201	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
202	C₂⁴	—	4	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	8	4	no	—
214	C₂×D₄	—	8	4	no	—
215	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
216	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	4	2	no	—
220	D₈	—	2	8	no	—
221	D₈	—	4	4	no	—
222	D₈	—	4	4	no	—
223	D₈	—	4	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
227	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
228	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	C₃×D₈	1	1	yes	characteristic
239	C₄²	C₃×D₈	1	1	yes	characteristic
240	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
241	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
242	C₁₆	—	2	4	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	8	4	no	—
246	C₂²×C₃	—	8	4	no	—
247	C₂²×C₃	—	8	4	no	—
248	C₂²×C₃	—	16	8	no	—
249	C₂²×C₃	C₂×SD₃₂	1	2	yes	—
250	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	8	2	no	—
254	C₃×C₄	—	8	2	no	—
255	C₃×C₄	C₂×SD₃₂	1	2	yes	—
256	C₃×C₄	C₂×SD₃₂	1	2	yes	—
257	C₃×C₄	#509	1	2	yes	—
258	C₃×C₄	#535	1	1	yes	characteristic
259	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₂³	—	4	4	no	—
261	C₂³	—	8	4	no	—
262	C₂³	—	8	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	Q₈	—	8	2	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	4	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₃×SD₃₂	1	4	yes	complemented
281	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
285	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
290	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	16	4	no	—
293	C₂×C₃	#3205	1	1	yes	characteristic
294	C₂×C₃	C₂²×SD₃₂	1	1	yes	characteristic
295	C₂×C₃	C₂×#509	1	1	yes	characteristic
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	16	8	no	—
302	C₂²	C₂×C₃×SD₃₂	1	2	yes	—
303	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	C₂×C₃×SD₃₂	1	2	yes	—
309	C₄	C₂×C₃×SD₃₂	1	2	yes	—
310	C₄	C₃×#509	1	2	yes	—
311	C₄	C₃×#535	1	1	yes	characteristic
312	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₃	#33774	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	16	4	no	—
316	C₂	C₃×#3205	1	1	yes	characteristic
317	C₂	C₂²×C₃×SD₃₂	1	1	yes	characteristic
318	C₂	C₂×C₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#493	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×#2248	C₂	semidirect
4. C₃×#2240	C₂	semidirect
5. C₃×#2207	C₂	semidirect
6. C₃×#2187	C₂	semidirect
7. C₃×#2162	C₂	semidirect
8. #33774	C₃	direct (N × H)
13. C₂×C₃×SD₃₂	C₂²	semidirect
14. C₃×#500	C₂²	semidirect
15. C₃×#494	C₂²	semidirect
24. C₃×C₄×Q₁₆	C₂²	semidirect
26. C₃×C₄⋊C₁₆	C₂²	semidirect
27. C₃×#357	C₂²	semidirect
29. C₃×C₄×C₁₆	C₂²	semidirect
30. #3170	C₂×C₃	semidirect
32. #2248	C₂×C₃	semidirect
33. #2240	C₂×C₃	semidirect
34. #2207	C₂×C₃	semidirect
35. #2187	C₂×C₃	semidirect
36. #2162	C₂×C₃	semidirect
77. C₂×SD₃₂	C₂²×C₃	semidirect
78. #500	C₂²×C₃	semidirect
79. #494	C₂²×C₃	semidirect
88. C₄×Q₁₆	C₂²×C₃	semidirect
90. C₄⋊C₁₆	C₂²×C₃	semidirect
91. #357	C₂²×C₃	semidirect
93. C₄×C₁₆	C₂²×C₃	semidirect
191. C₂×C₃×C₄	SD₃₂	semidirect
280. C₂×C₄	C₃×SD₃₂	semidirect
313. C₃	#33774	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
C₂×C₃×#493	2
C₃×#2248	2
C₃×#2240	2
C₃×#2207	2
C₃×#2187	2
C₃×#2162	2
#33774	3

Subgroups — C3×#33774 #10652381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33774 #10652381