Subgroups — C₃×#33771 #10652378

Order 768.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2246	C₂	1	2	yes	maximal, complemented
4	C₃×#2240	C₂	1	2	yes	maximal, complemented
5	C₃×#2206	C₂	1	4	yes	maximal, complemented
6	C₃×#2184	C₂	1	4	yes	maximal, complemented
7	C₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	#33771	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×#532	—	2	1	no	—
12	C₂×C₃×#127	—	2	1	no	—
13	C₂×C₃×D₁₆	C₂²	1	8	yes	complemented
14	C₃×#500	C₂²	1	1	yes	characteristic, complemented
15	C₃×#493	—	2	1	no	—
16	C₃×#493	—	2	1	no	—
17	C₃×#493	C₂²	1	2	yes	—
18	C₃×#493	C₂²	1	2	yes	—
19	C₃×#493	C₂²	1	1	yes	characteristic, complemented
20	C₃×#486	—	2	1	no	—
21	C₃×#466	—	2	1	no	—
22	C₃×#466	C₂²	1	2	yes	—
23	C₃×C₄×D₈	—	2	1	no	—
24	C₃×C₄×D₈	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	#3169	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#493	C₂×C₃	1	1	yes	characteristic, complemented
32	#2246	C₂×C₃	1	2	yes	complemented
33	#2240	C₂×C₃	1	2	yes	complemented
34	#2206	C₂×C₃	1	4	yes	complemented
35	#2184	C₂×C₃	1	4	yes	complemented
36	#2162	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×C₃×#41	—	4	1	no	—
38	C₂²×C₃×D₄	—	2	2	no	—
39	C₂²×C₃×D₄	—	4	1	no	—
40	C₂²×C₃×D₄	—	4	1	no	—
41	C₂×C₃×D₈	—	2	2	no	—
42	C₂×C₃×D₈	—	2	2	no	—
43	C₂×C₃×D₈	—	2	2	no	—
44	C₂×C₃×D₈	—	2	1	no	—
45	C₂×C₃×D₈	—	4	4	no	—
46	C₂×C₃×D₈	—	4	4	no	—
47	C₂×C₃×D₈	C₂³	1	2	yes	—
48	C₂×C₃×D₈	C₂³	1	4	yes	—
49	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
51	C₃×C₄⋊Q₈	—	2	1	no	—
52	C₃×#127	—	2	2	no	—
53	C₃×#127	—	2	2	no	—
54	C₃×#127	—	2	1	no	—
55	C₃×#127	—	4	1	no	—
56	C₃×#127	—	4	1	no	—
57	C₃×#124	—	4	1	no	—
58	C₃×#121	—	2	2	no	—
59	C₃×#121	—	4	1	no	—
60	C₃×C₄×D₄	—	2	2	no	—
61	C₃×C₄×D₄	—	4	1	no	—
62	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
63	C₃×D₁₆	—	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×#102	—	2	2	no	—
68	C₃×#102	—	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	#532	—	2	1	no	—
76	C₂×#127	—	2	1	no	—
77	C₂×D₁₆	C₂²×C₃	1	8	yes	complemented
78	#500	C₂²×C₃	1	1	yes	characteristic, complemented
79	#493	—	2	1	no	—
80	#493	—	2	1	no	—
81	#493	C₂²×C₃	1	2	yes	—
82	#493	C₂²×C₃	1	2	yes	—
83	#493	C₂²×C₃	1	1	yes	characteristic, complemented
84	#486	—	2	1	no	—
85	#466	—	2	1	no	—
86	#466	C₂²×C₃	1	2	yes	—
87	C₄×D₈	—	2	1	no	—
88	C₄×D₈	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	C₃×#41	—	8	4	no	—
96	C₂×C₃×Q₈	—	4	1	no	—
97	C₂×C₃×D₄	—	2	2	no	—
98	C₂×C₃×D₄	—	2	4	no	—
99	C₂×C₃×D₄	—	4	4	no	—
100	C₂×C₃×D₄	—	4	4	no	—
101	C₂×C₃×D₄	—	4	2	no	—
102	C₂×C₃×D₄	—	4	1	no	—
103	C₂×C₃×D₄	—	4	1	no	—
104	C₂×C₃×D₄	—	4	2	no	—
105	C₂×C₃×D₄	—	4	2	no	—
106	C₂×C₃×D₄	—	4	2	no	—
107	C₂×C₃×D₄	—	4	2	no	—
108	C₂×C₃×D₄	—	4	2	no	—
109	C₂×C₃×D₄	—	8	4	no	—
110	C₂×C₃×D₄	—	8	4	no	—
111	C₂²×C₃×C₄	—	4	2	no	—
112	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
113	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
115	C₃×D₈	—	2	4	no	—
116	C₃×D₈	—	2	8	no	—
117	C₃×D₈	—	4	4	no	—
118	C₃×D₈	—	4	4	no	—
119	C₃×D₈	—	4	2	no	—
120	C₃×D₈	—	4	4	no	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
127	C₃×C₄⋊C₄	—	2	2	no	—
128	C₃×C₄⋊C₄	—	4	1	no	—
129	C₃×C₂²⋊C₄	—	4	2	no	—
130	C₃×C₂²⋊C₄	—	4	2	no	—
131	C₃×C₄²	D₈	1	1	yes	characteristic
132	C₃×C₄²	D₈	1	1	yes	characteristic
133	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
134	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
135	C₃×C₁₆	—	2	4	no	—
136	C₂×#41	—	4	1	no	—
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	—	4	1	no	—
139	C₂²×D₄	—	4	1	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	1	no	—
144	C₂×D₈	—	4	4	no	—
145	C₂×D₈	—	4	4	no	—
146	C₂×D₈	C₂³×C₃	1	4	yes	—
147	C₂×D₈	C₂³×C₃	1	2	yes	—
148	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	1	no	—
151	#127	—	2	2	no	—
152	#127	—	2	2	no	—
153	#127	—	2	1	no	—
154	#127	—	4	1	no	—
155	#127	—	4	1	no	—
156	#124	—	4	1	no	—
157	#121	—	2	2	no	—
158	#121	—	4	1	no	—
159	C₄×D₄	—	2	2	no	—
160	C₄×D₄	—	4	1	no	—
161	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
162	D₁₆	—	4	8	no	—
163	C₂×C₁₆	C₂³×C₃	1	4	yes	—
164	#107	—	2	1	no	—
165	#107	C₂³×C₃	1	2	yes	—
166	#102	—	2	2	no	—
167	#102	—	2	2	no	—
168	C₄×C₈	C₃×D₄	1	1	yes	characteristic
169	C₄×C₈	C₃×D₄	1	1	yes	characteristic
170	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
171	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
172	C₂³×C₃	—	4	2	no	—
173	C₂³×C₃	—	4	4	no	—
174	C₂³×C₃	—	8	4	no	—
175	C₂³×C₃	—	8	4	no	—
176	C₂³×C₃	—	8	4	no	—
177	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
178	C₃×Q₈	—	8	2	no	—
179	C₃×D₄	—	4	4	no	—
180	C₃×D₄	—	4	2	no	—
181	C₃×D₄	—	4	4	no	—
182	C₃×D₄	—	4	2	no	—
183	C₃×D₄	—	8	4	no	—
184	C₃×D₄	—	8	4	no	—
185	C₃×D₄	—	8	4	no	—
186	C₃×D₄	—	8	4	no	—
187	C₃×D₄	—	8	4	no	—
188	C₃×D₄	—	8	2	no	—
189	C₃×D₄	—	8	2	no	—
190	C₃×D₄	—	8	4	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	4	2	no	—
195	C₂×C₃×C₄	—	8	4	no	—
196	C₂×C₃×C₄	D₁₆	1	4	yes	complemented
197	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
203	C₃×C₈	—	2	1	no	—
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
206	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
207	C₂⁴	—	4	2	no	—
208	#41	—	8	4	no	—
209	C₂×Q₈	—	4	1	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	8	4	no	—
223	C₂×D₄	—	8	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
226	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
228	D₈	—	2	4	no	—
229	D₈	—	2	8	no	—
230	D₈	—	4	4	no	—
231	D₈	—	4	4	no	—
232	D₈	—	4	4	no	—
233	D₈	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
238	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
239	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₄²	C₃×D₈	1	1	yes	characteristic
245	C₄²	C₃×D₈	1	1	yes	characteristic
246	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
247	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
248	C₁₆	—	2	4	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	8	2	no	—
252	C₂²×C₃	—	8	4	no	—
253	C₂²×C₃	—	8	4	no	—
254	C₂²×C₃	—	8	2	no	—
255	C₂²×C₃	—	8	4	no	—
256	C₂²×C₃	—	8	2	no	—
257	C₂²×C₃	—	16	8	no	—
258	C₂²×C₃	C₂×D₁₆	1	2	yes	—
259	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	8	2	no	—
263	C₃×C₄	C₂×D₁₆	1	2	yes	—
264	C₃×C₄	C₂×D₁₆	1	2	yes	—
265	C₃×C₄	#509	1	2	yes	—
266	C₃×C₄	#535	1	1	yes	characteristic
267	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₂³	—	4	2	no	—
269	C₂³	—	4	4	no	—
270	C₂³	—	8	4	no	—
271	C₂³	—	8	4	no	—
272	C₂³	—	8	4	no	—
273	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
274	Q₈	—	8	2	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	2	no	—
279	D₄	—	8	4	no	—
280	D₄	—	8	4	no	—
281	D₄	—	8	4	no	—
282	D₄	—	8	4	no	—
283	D₄	—	8	4	no	—
284	D₄	—	8	2	no	—
285	D₄	—	8	4	no	—
286	D₄	—	8	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	C₃×D₁₆	1	4	yes	complemented
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×AGL(1,ℤ/8)	1	2	yes	—
302	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	16	2	no	—
305	C₂×C₃	—	16	4	no	—
306	C₂×C₃	#3205	1	1	yes	characteristic
307	C₂×C₃	C₂²×D₁₆	1	1	yes	characteristic
308	C₂×C₃	C₂×#509	1	1	yes	characteristic
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	16	8	no	—
318	C₂²	C₂×C₃×D₁₆	1	2	yes	—
319	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	C₂×C₃×D₁₆	1	2	yes	—
324	C₄	C₂×C₃×D₁₆	1	2	yes	—
325	C₄	C₃×#509	1	2	yes	—
326	C₄	C₃×#535	1	1	yes	characteristic
327	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₃	#33771	1	1	yes	characteristic, direct product factor, Sylow
329	C₂	—	2	2	no	—
330	C₂	—	16	4	no	—
331	C₂	—	16	2	no	—
332	C₂	C₃×#3205	1	1	yes	characteristic
333	C₂	C₂²×C₃×D₁₆	1	1	yes	characteristic
334	C₂	C₂×C₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
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×#3169	C₂	semidirect
2. C₂×C₃×#493	C₂	semidirect
3. C₃×#2246	C₂	semidirect
4. C₃×#2240	C₂	semidirect
5. C₃×#2206	C₂	semidirect
6. C₃×#2184	C₂	semidirect
7. C₃×#2162	C₂	semidirect
8. #33771	C₃	direct (N × H)
13. C₂×C₃×D₁₆	C₂²	semidirect
14. C₃×#500	C₂²	semidirect
19. C₃×#493	C₂²	semidirect
24. C₃×C₄×D₈	C₂²	semidirect
26. C₃×C₄⋊C₁₆	C₂²	semidirect
27. C₃×#357	C₂²	semidirect
29. C₃×C₄×C₁₆	C₂²	semidirect
30. #3169	C₂×C₃	semidirect
31. C₂×#493	C₂×C₃	semidirect
32. #2246	C₂×C₃	semidirect
33. #2240	C₂×C₃	semidirect
34. #2206	C₂×C₃	semidirect
35. #2184	C₂×C₃	semidirect
36. #2162	C₂×C₃	semidirect
77. C₂×D₁₆	C₂²×C₃	semidirect
78. #500	C₂²×C₃	semidirect
83. #493	C₂²×C₃	semidirect
88. C₄×D₈	C₂²×C₃	semidirect
90. C₄⋊C₁₆	C₂²×C₃	semidirect
91. #357	C₂²×C₃	semidirect
93. C₄×C₁₆	C₂²×C₃	semidirect
196. C₂×C₃×C₄	D₁₆	semidirect
292. C₂×C₄	C₃×D₁₆	semidirect
328. C₃	#33771	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×#3169	2
C₂×C₃×#493	2
C₃×#2246	2
C₃×#2240	2
C₃×#2206	2
C₃×#2184	2
C₃×#2162	2
#33771	3

Subgroups — C3×#33771 #10652378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33771 #10652378