Subgroups — C₃×#33782 #10652389

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 342 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×#3147	C₂	1	1	yes	maximal, characteristic
3	C₃×#2251	C₂	1	2	yes	maximal, complemented
4	C₃×#2243	C₂	1	2	yes	maximal, complemented
5	C₃×#2209	C₂	1	2	yes	maximal, complemented
6	C₃×#2207	C₂	1	2	yes	maximal, complemented
7	C₃×#2189	C₂	1	4	yes	maximal, complemented
8	C₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	#33782	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#135	—	2	1	no	—
11	C₂×C₃×#135	C₂²	1	2	yes	—
12	C₃×#533	—	2	1	no	—
13	C₃×#532	—	2	1	no	—
14	C₂×C₃×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
16	C₃×#500	C₂²	1	1	yes	characteristic
17	C₃×#495	—	2	1	no	—
18	C₃×#495	C₂²	1	2	yes	—
19	C₃×#494	C₂²	1	1	yes	characteristic
20	C₃×#494	C₂²	1	1	yes	characteristic
21	C₃×#493	C₂²	1	1	yes	characteristic
22	C₃×#492	—	2	1	no	—
23	C₃×#487	—	2	1	no	—
24	C₃×#467	—	2	1	no	—
25	C₃×#467	C₂²	1	2	yes	—
26	C₃×C₄×Q₁₆	—	2	1	no	—
27	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₃×#358	C₂²	1	4	yes	—
31	C₃×#357	C₂²	1	4	yes	—
32	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₃×#346	C₂²	1	2	yes	—
34	#3170	C₂×C₃	1	1	yes	characteristic, complemented
35	#3147	C₂×C₃	1	1	yes	characteristic
36	#2251	C₂×C₃	1	2	yes	complemented
37	#2243	C₂×C₃	1	2	yes	complemented
38	#2209	C₂×C₃	1	2	yes	complemented
39	#2207	C₂×C₃	1	2	yes	complemented
40	#2189	C₂×C₃	1	4	yes	complemented
41	#2163	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×C₃×#41	—	2	2	no	—
43	C₃×#135	—	4	4	no	—
44	C₃×#135	—	4	4	no	—
45	C₂×C₃×Q₁₆	—	2	1	no	—
46	C₂×C₃×Q₁₆	—	2	1	no	—
47	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
48	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
49	C₂×C₃×SD₁₆	—	2	2	no	—
50	C₂×C₃×SD₁₆	—	2	2	no	—
51	C₂×C₃×D₈	—	2	1	no	—
52	C₂×C₃×D₈	C₂³	1	2	yes	—
53	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
55	C₃×C₄⋊Q₈	—	2	1	no	—
56	C₃×C₄⋊Q₈	—	2	1	no	—
57	C₃×C₄⋊Q₈	—	2	1	no	—
58	C₃×#127	—	2	1	no	—
59	C₃×#125	—	4	1	no	—
60	C₃×#124	—	2	2	no	—
61	C₃×C₂²⋊Q₈	—	2	2	no	—
62	C₃×C₂²⋊Q₈	—	4	1	no	—
63	C₃×#121	—	4	1	no	—
64	C₃×#121	—	4	1	no	—
65	C₃×C₄×Q₈	—	2	2	no	—
66	C₃×C₄×Q₈	—	4	1	no	—
67	C₃×C₄×D₄	—	4	1	no	—
68	C₃×C₄×D₄	—	4	1	no	—
69	C₃×#117	—	4	1	no	—
70	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
71	C₃×Q₃₂	—	4	4	no	—
72	C₃×SD₃₂	—	4	4	no	—
73	C₂×C₃×C₁₆	C₂³	1	4	yes	—
74	C₃×#107	—	2	1	no	—
75	C₃×#107	C₂³	1	2	yes	—
76	C₃×#103	—	2	2	no	—
77	C₃×#103	—	2	2	no	—
78	C₃×C₄×C₈	D₄	1	1	yes	characteristic
79	C₃×C₄×C₈	D₄	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂×#135	—	2	1	no	—
83	C₂×#135	C₂²×C₃	1	2	yes	—
84	#533	—	2	1	no	—
85	#532	—	2	1	no	—
86	C₂×Q₃₂	C₂²×C₃	1	4	yes	—
87	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
88	#500	C₂²×C₃	1	1	yes	characteristic
89	#495	—	2	1	no	—
90	#495	C₂²×C₃	1	2	yes	—
91	#494	C₂²×C₃	1	1	yes	characteristic
92	#494	C₂²×C₃	1	1	yes	characteristic
93	#493	C₂²×C₃	1	1	yes	characteristic
94	#492	—	2	1	no	—
95	#487	—	2	1	no	—
96	#467	—	2	1	no	—
97	#467	C₂²×C₃	1	2	yes	—
98	C₄×Q₁₆	—	2	1	no	—
99	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
101	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
102	#358	C₂²×C₃	1	4	yes	—
103	#357	C₂²×C₃	1	4	yes	—
104	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
105	#346	C₂²×C₃	1	2	yes	—
106	C₃×#41	—	4	4	no	—
107	C₃×#41	—	4	4	no	—
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	—	2	2	no	—
110	C₂×C₃×Q₈	—	4	1	no	—
111	C₂×C₃×D₄	—	2	2	no	—
112	C₂×C₃×D₄	—	4	2	no	—
113	C₂×C₃×D₄	—	4	1	no	—
114	C₂×C₃×D₄	—	4	1	no	—
115	C₂²×C₃×C₄	—	4	2	no	—
116	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
117	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
119	C₃×Q₁₆	—	2	2	no	—
120	C₃×Q₁₆	—	2	2	no	—
121	C₃×Q₁₆	—	2	4	no	—
122	C₃×Q₁₆	—	4	2	no	—
123	C₃×Q₁₆	—	4	2	no	—
124	C₃×SD₁₆	—	4	4	no	—
125	C₃×SD₁₆	—	4	4	no	—
126	C₃×D₈	—	2	4	no	—
127	C₃×D₈	—	4	2	no	—
128	C₂×C₃×C₈	—	2	2	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	4	1	no	—
136	C₃×C₄⋊C₄	—	4	1	no	—
137	C₃×C₄⋊C₄	—	4	1	no	—
138	C₃×C₄⋊C₄	—	4	1	no	—
139	C₃×C₄⋊C₄	—	4	2	no	—
140	C₃×C₄⋊C₄	—	4	2	no	—
141	C₃×C₄⋊C₄	—	4	1	no	—
142	C₃×C₂²⋊C₄	—	4	2	no	—
143	C₃×C₂²⋊C₄	—	4	2	no	—
144	C₃×C₂²⋊C₄	—	4	2	no	—
145	C₃×C₄²	—	4	2	no	—
146	C₃×C₄²	D₈	1	1	yes	characteristic
147	C₃×C₄²	D₈	1	1	yes	characteristic
148	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
149	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
150	C₃×C₁₆	—	2	4	no	—
151	C₂×#41	—	2	2	no	—
152	#135	—	4	4	no	—
153	#135	—	4	4	no	—
154	C₂×Q₁₆	—	2	1	no	—
155	C₂×Q₁₆	—	2	1	no	—
156	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
157	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂³×C₃	1	2	yes	—
162	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	1	no	—
165	C₄⋊Q₈	—	2	1	no	—
166	C₄⋊Q₈	—	2	1	no	—
167	#127	—	2	1	no	—
168	#125	—	4	1	no	—
169	#124	—	2	2	no	—
170	C₂²⋊Q₈	—	2	2	no	—
171	C₂²⋊Q₈	—	4	1	no	—
172	#121	—	4	1	no	—
173	#121	—	4	1	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₄×Q₈	—	4	1	no	—
176	C₄×D₄	—	4	1	no	—
177	C₄×D₄	—	4	1	no	—
178	#117	—	4	1	no	—
179	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
180	Q₃₂	—	4	4	no	—
181	SD₃₂	—	4	4	no	—
182	C₂×C₁₆	C₂³×C₃	1	4	yes	—
183	#107	—	2	1	no	—
184	#107	C₂³×C₃	1	2	yes	—
185	#103	—	2	2	no	—
186	#103	—	2	2	no	—
187	C₄×C₈	C₃×D₄	1	1	yes	characteristic
188	C₄×C₈	C₃×D₄	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
190	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
191	C₂³×C₃	—	4	2	no	—
192	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
193	C₃×Q₈	—	4	2	no	—
194	C₃×Q₈	—	4	2	no	—
195	C₃×Q₈	—	4	2	no	—
196	C₃×Q₈	—	4	2	no	—
197	C₃×Q₈	—	8	2	no	—
198	C₃×D₄	—	4	2	no	—
199	C₃×D₄	—	4	2	no	—
200	C₃×D₄	—	8	2	no	—
201	C₃×D₄	—	8	4	no	—
202	C₃×D₄	—	8	2	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	2	no	—
208	C₂×C₃×C₄	—	4	2	no	—
209	C₂×C₃×C₄	—	4	2	no	—
210	C₂×C₃×C₄	—	8	4	no	—
211	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
217	C₃×C₈	—	2	1	no	—
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
220	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
221	#41	—	4	4	no	—
222	#41	—	4	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
232	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	4	no	—
237	Q₁₆	—	4	2	no	—
238	Q₁₆	—	4	2	no	—
239	SD₁₆	—	4	4	no	—
240	SD₁₆	—	4	4	no	—
241	D₈	—	2	4	no	—
242	D₈	—	4	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
246	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
247	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
248	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	C₃×D₈	1	1	yes	characteristic
262	C₄²	C₃×D₈	1	1	yes	characteristic
263	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
264	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
265	C₁₆	—	2	4	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	8	2	no	—
270	C₂²×C₃	—	8	2	no	—
271	C₂²×C₃	—	8	2	no	—
272	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	8	2	no	—
278	C₃×C₄	—	8	2	no	—
279	C₃×C₄	—	8	2	no	—
280	C₃×C₄	#510	1	2	yes	—
281	C₃×C₄	#535	1	1	yes	characteristic
282	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₂³	—	4	2	no	—
284	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	8	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	8	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	4	no	—
303	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
306	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
307	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
312	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
313	C₂×C₃	—	4	1	no	—
314	C₂×C₃	—	16	2	no	—
315	C₂×C₃	#3205	1	1	yes	characteristic
316	C₂×C₃	C₂×#510	1	1	yes	characteristic
317	C₂×C₃	#3414	1	1	yes	characteristic
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	C₃×#510	1	2	yes	—
333	C₄	C₃×#535	1	1	yes	characteristic
334	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
335	C₃	#33782	1	1	yes	characteristic, direct product factor, Sylow
336	C₂	—	4	1	no	—
337	C₂	—	16	2	no	—
338	C₂	C₃×#3205	1	1	yes	characteristic
339	C₂	C₂×C₃×#510	1	1	yes	characteristic
340	C₂	C₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₃×#3147	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×#2251	C₂	semidirect
4. C₃×#2243	C₂	semidirect
5. C₃×#2209	C₂	semidirect
6. C₃×#2207	C₂	semidirect
7. C₃×#2189	C₂	semidirect
8. C₃×#2163	C₂	semidirect
9. #33782	C₃	direct (N × H)
34. #3170	C₂×C₃	semidirect
36. #2251	C₂×C₃	semidirect
37. #2243	C₂×C₃	semidirect
38. #2209	C₂×C₃	semidirect
39. #2207	C₂×C₃	semidirect
40. #2189	C₂×C₃	semidirect
41. #2163	C₂×C₃	semidirect
335. C₃	#33782	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×#3147	2
C₃×#2251	2
C₃×#2243	2
C₃×#2209	2
C₃×#2207	2
C₃×#2189	2
C₃×#2163	2
#33782	3

Subgroups — C3×#33782 #10652389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33782 #10652389