Subgroups — C₃×#33781 #10652388

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 352 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×#495	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×#2211	C₂	1	4	yes	maximal, complemented
6	C₃×#2188	C₂	1	2	yes	maximal, complemented
7	C₃×#2187	C₂	1	2	yes	maximal
8	C₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	#33781	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×SD₁₆	—	2	1	no	—
12	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
13	C₃×#533	—	2	1	no	—
14	C₂×C₃×#124	—	2	1	no	—
15	C₂×C₃×Q₃₂	C₂²	1	4	yes	complemented
16	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
17	C₃×#500	C₂²	1	1	yes	characteristic, complemented
18	C₃×#495	—	2	1	no	—
19	C₃×#495	—	2	1	no	—
20	C₃×#495	C₂²	1	2	yes	—
21	C₃×#495	C₂²	1	2	yes	—
22	C₃×#494	C₂²	1	1	yes	characteristic, complemented
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	complemented
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
30	C₃×#358	C₂²	1	4	yes	complemented
31	C₃×#357	C₂²	1	4	yes	—
32	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₃×#346	C₂²	1	2	yes	complemented
34	#3170	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×#495	C₂×C₃	1	1	yes	characteristic
36	#2251	C₂×C₃	1	2	yes	complemented
37	#2243	C₂×C₃	1	2	yes	complemented
38	#2211	C₂×C₃	1	4	yes	complemented
39	#2188	C₂×C₃	1	2	yes	complemented
40	#2187	C₂×C₃	1	2	yes	—
41	#2163	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂²×C₃×Q₈	—	2	1	no	—
43	C₂²×C₃×D₄	—	2	1	no	—
44	C₂×C₃×Q₁₆	—	2	1	no	—
45	C₂×C₃×Q₁₆	—	2	1	no	—
46	C₂×C₃×Q₁₆	—	4	2	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×SD₁₆	—	4	4	no	—
52	C₂×C₃×D₈	—	2	1	no	—
53	C₂×C₃×D₈	—	4	2	no	—
54	C₂×C₃×D₈	C₂³	1	2	yes	—
55	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₃×C₄⋊Q₈	—	2	1	no	—
58	C₃×C₄⋊Q₈	—	2	1	no	—
59	C₃×#125	—	4	1	no	—
60	C₃×#124	—	2	2	no	—
61	C₃×#124	—	2	2	no	—
62	C₃×#124	—	4	1	no	—
63	C₃×#124	—	4	1	no	—
64	C₃×C₂²⋊Q₈	—	2	2	no	—
65	C₃×C₂²⋊Q₈	—	4	1	no	—
66	C₃×C₄×Q₈	—	2	2	no	—
67	C₃×C₄×Q₈	—	4	1	no	—
68	C₃×#117	—	4	1	no	—
69	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
70	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
71	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
72	C₃×Q₃₂	—	4	4	no	—
73	C₃×SD₃₂	—	4	4	no	—
74	C₂×C₃×C₁₆	C₂³	1	4	yes	—
75	C₃×#107	—	2	1	no	—
76	C₃×#107	C₂³	1	2	yes	—
77	C₃×#103	—	2	2	no	—
78	C₃×#103	—	2	2	no	—
79	C₃×C₄×C₈	D₄	1	1	yes	characteristic
80	C₃×C₄×C₈	D₄	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
82	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
83	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
84	C₂²×SD₁₆	—	2	1	no	—
85	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
86	#533	—	2	1	no	—
87	C₂×#124	—	2	1	no	—
88	C₂×Q₃₂	C₂²×C₃	1	4	yes	complemented
89	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
90	#500	C₂²×C₃	1	1	yes	characteristic, complemented
91	#495	—	2	1	no	—
92	#495	—	2	1	no	—
93	#495	C₂²×C₃	1	2	yes	—
94	#495	C₂²×C₃	1	2	yes	—
95	#494	C₂²×C₃	1	1	yes	characteristic, complemented
96	#487	—	2	1	no	—
97	#467	—	2	1	no	—
98	#467	C₂²×C₃	1	2	yes	—
99	C₄×Q₁₆	—	2	1	no	—
100	C₄×Q₁₆	C₂²×C₃	1	2	yes	complemented
101	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
102	C₄⋊C₁₆	C₂²×C₃	1	2	yes	complemented
103	#358	C₂²×C₃	1	4	yes	complemented
104	#357	C₂²×C₃	1	4	yes	—
105	C₂²⋊C₁₆	C₂²×C₃	1	2	yes	—
106	#346	C₂²×C₃	1	2	yes	complemented
107	C₂⁴×C₃	—	4	1	no	—
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	—	2	2	no	—
110	C₂×C₃×Q₈	—	4	2	no	—
111	C₂×C₃×Q₈	—	4	2	no	—
112	C₂×C₃×Q₈	—	4	1	no	—
113	C₂×C₃×Q₈	—	4	1	no	—
114	C₂×C₃×D₄	—	2	2	no	—
115	C₂×C₃×D₄	—	4	2	no	—
116	C₂×C₃×D₄	—	4	2	no	—
117	C₂×C₃×D₄	—	4	1	no	—
118	C₂²×C₃×C₄	—	4	1	no	—
119	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
120	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₃×Q₁₆	—	2	4	no	—
123	C₃×Q₁₆	—	2	4	no	—
124	C₃×Q₁₆	—	4	2	no	—
125	C₃×Q₁₆	—	4	2	no	—
126	C₃×SD₁₆	—	4	4	no	—
127	C₃×SD₁₆	—	4	4	no	—
128	C₃×D₈	—	2	4	no	—
129	C₃×D₈	—	4	2	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
136	C₃×C₄⋊C₄	—	2	2	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×C₂²⋊C₄	—	4	2	no	—
146	C₃×C₂²⋊C₄	—	4	2	no	—
147	C₃×C₄²	—	4	2	no	—
148	C₃×C₄²	D₈	1	1	yes	characteristic
149	C₃×C₄²	D₈	1	1	yes	characteristic
150	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
151	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
152	C₃×C₁₆	—	2	4	no	—
153	C₂²×Q₈	—	2	1	no	—
154	C₂²×D₄	—	2	1	no	—
155	C₂×Q₁₆	—	2	1	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	—	4	2	no	—
158	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
159	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	4	4	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	4	2	no	—
165	C₂×D₈	C₂³×C₃	1	2	yes	—
166	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	1	no	—
169	C₄⋊Q₈	—	2	1	no	—
170	#125	—	4	1	no	—
171	#124	—	2	2	no	—
172	#124	—	2	2	no	—
173	#124	—	4	1	no	—
174	#124	—	4	1	no	—
175	C₂²⋊Q₈	—	2	2	no	—
176	C₂²⋊Q₈	—	4	1	no	—
177	C₄×Q₈	—	2	2	no	—
178	C₄×Q₈	—	4	1	no	—
179	#117	—	4	1	no	—
180	C₂×C₂²⋊C₄	—	4	1	no	—
181	C₂×C₂²⋊C₄	—	4	1	no	—
182	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
183	Q₃₂	—	4	4	no	—
184	SD₃₂	—	4	4	no	—
185	C₂×C₁₆	C₂³×C₃	1	4	yes	—
186	#107	—	2	1	no	—
187	#107	C₂³×C₃	1	2	yes	—
188	#103	—	2	2	no	—
189	#103	—	2	2	no	—
190	C₄×C₈	C₃×D₄	1	1	yes	characteristic
191	C₄×C₈	C₃×D₄	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
194	C₂³×C₃	—	4	2	no	—
195	C₂³×C₃	—	8	2	no	—
196	C₂³×C₃	—	8	2	no	—
197	C₂³×C₃	—	8	2	no	—
198	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
199	C₃×Q₈	—	4	2	no	—
200	C₃×Q₈	—	4	2	no	—
201	C₃×Q₈	—	4	2	no	—
202	C₃×Q₈	—	4	2	no	—
203	C₃×Q₈	—	8	2	no	—
204	C₃×Q₈	—	8	2	no	—
205	C₃×D₄	—	4	2	no	—
206	C₃×D₄	—	4	2	no	—
207	C₃×D₄	—	8	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	4	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	—	8	2	no	—
216	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
222	C₃×C₈	—	2	1	no	—
223	C₃×C₈	—	2	1	no	—
224	C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
225	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
226	C₂⁴	—	4	1	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
239	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	4	no	—
243	Q₁₆	—	4	2	no	—
244	Q₁₆	—	4	2	no	—
245	SD₁₆	—	4	4	no	—
246	SD₁₆	—	4	4	no	—
247	D₈	—	2	4	no	—
248	D₈	—	4	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
252	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
253	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
254	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₃×D₈	1	1	yes	characteristic
268	C₄²	C₃×D₈	1	1	yes	characteristic
269	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
270	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
271	C₁₆	—	2	4	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	8	2	no	—
275	C₂²×C₃	—	8	2	no	—
276	C₂²×C₃	—	8	2	no	—
277	C₂²×C₃	—	16	4	no	—
278	C₂²×C₃	#510	1	2	yes	—
279	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	8	2	no	—
286	C₃×C₄	—	8	2	no	—
287	C₃×C₄	—	8	2	no	—
288	C₃×C₄	#535	1	1	yes	characteristic
289	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂³	—	4	2	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	4	2	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	8	2	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	8	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
313	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
314	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
315	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
316	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	C₃×AGL(1,ℤ/8)	1	2	yes	—
321	C₈	C₃×H(5,𝔽₂)	1	2	yes	—
322	C₂×C₃	—	2	2	no	—
323	C₂×C₃	—	16	2	no	—
324	C₂×C₃	#3205	1	1	yes	characteristic
325	C₂×C₃	C₂×#510	1	1	yes	characteristic
326	C₂×C₃	#3414	1	1	yes	characteristic
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	16	4	no	—
333	C₂²	C₃×#510	1	2	yes	—
334	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₃×#535	1	1	yes	characteristic
344	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
345	C₃	#33781	1	1	yes	characteristic, direct product factor, Sylow
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₃×#3205	1	1	yes	characteristic
349	C₂	C₂×C₃×#510	1	1	yes	characteristic
350	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	C₂²×C₃×D₈	C₂²
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×#2211	C₂	semidirect
6. C₃×#2188	C₂	semidirect
8. C₃×#2163	C₂	semidirect
9. #33781	C₃	direct (N × H)
15. C₂×C₃×Q₃₂	C₂²	semidirect
17. C₃×#500	C₂²	semidirect
22. C₃×#494	C₂²	semidirect
27. C₃×C₄×Q₁₆	C₂²	semidirect
29. C₃×C₄⋊C₁₆	C₂²	semidirect
30. C₃×#358	C₂²	semidirect
33. C₃×#346	C₂²	semidirect
34. #3170	C₂×C₃	semidirect
36. #2251	C₂×C₃	semidirect
37. #2243	C₂×C₃	semidirect
38. #2211	C₂×C₃	semidirect
39. #2188	C₂×C₃	semidirect
41. #2163	C₂×C₃	semidirect
88. C₂×Q₃₂	C₂²×C₃	semidirect
90. #500	C₂²×C₃	semidirect
95. #494	C₂²×C₃	semidirect
100. C₄×Q₁₆	C₂²×C₃	semidirect
102. C₄⋊C₁₆	C₂²×C₃	semidirect
103. #358	C₂²×C₃	semidirect
106. #346	C₂²×C₃	semidirect
345. C₃	#33781	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×#495	2
C₃×#2251	2
C₃×#2243	2
C₃×#2211	2
C₃×#2188	2
C₃×#2187	2
C₃×#2163	2
#33781	3

Subgroups — C3×#33781 #10652388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33781 #10652388