Subgroups — C₃×#33633 #10652240

Order 768.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#2951	C₂	1	1	yes	maximal, characteristic
3	C₃×#2229	C₂	1	2	yes	maximal, complemented
4	C₃×#2228	C₂	1	2	yes	maximal, complemented
5	C₃×#2227	C₂	1	2	yes	maximal, complemented
6	C₃×#2224	C₂	1	2	yes	maximal, complemented
7	C₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#2139	C₂	1	1	yes	maximal, characteristic
11	C₃×#2138	C₂	1	1	yes	maximal, characteristic
12	#33633	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#135	C₂²	1	1	yes	characteristic
14	C₃×#544	—	2	1	no	—
15	C₃×#520	—	2	1	no	—
16	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₃×#501	—	2	2	no	—
19	C₃×#500	C₂²	1	2	yes	—
20	C₃×#499	C₂²	1	2	yes	—
21	C₃×#440	—	2	2	no	—
22	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
23	C₃×C₄×D₈	C₂²	1	2	yes	—
24	C₃×#428	—	2	1	no	—
25	C₃×#427	C₂²	1	1	yes	characteristic
26	C₂×C₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₃×#107	C₂²	1	1	yes	characteristic
28	C₃×#417	—	2	1	no	—
29	C₃×#405	C₂²	1	1	yes	characteristic
30	C₃×#367	C₂²	1	2	yes	—
31	C₃×#366	C₂²	1	2	yes	—
32	C₃×#366	C₂²	1	2	yes	—
33	C₃×#366	C₂²	1	2	yes	—
34	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
36	C₃×#358	C₂²	1	1	yes	characteristic
37	C₃×#358	C₂²	1	1	yes	characteristic
38	C₃×#358	C₂²	1	2	yes	—
39	C₃×#357	C₂²	1	1	yes	characteristic
40	C₃×#357	C₂²	1	1	yes	characteristic
41	C₃×#357	C₂²	1	2	yes	—
42	#3164	C₂×C₃	1	1	yes	characteristic, complemented
43	#2951	C₂×C₃	1	1	yes	characteristic
44	#2229	C₂×C₃	1	2	yes	complemented
45	#2228	C₂×C₃	1	2	yes	complemented
46	#2227	C₂×C₃	1	2	yes	complemented
47	#2224	C₂×C₃	1	2	yes	complemented
48	#2159	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂×#366	C₂×C₃	1	1	yes	characteristic, complemented
50	#2151	C₂×C₃	1	1	yes	characteristic, complemented
51	#2139	C₂×C₃	1	1	yes	characteristic
52	#2138	C₂×C₃	1	1	yes	characteristic
53	C₂×C₃×#41	—	2	1	no	—
54	C₃×#135	—	2	2	no	—
55	C₃×#135	Q₈	1	4	yes	complemented
56	C₂×C₃×Q₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₃×SD₁₆	—	2	1	no	—
58	C₂×C₃×D₈	C₂³	1	1	yes	characteristic
59	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
60	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₃×C₄⋊Q₈	—	2	2	no	—
62	C₃×C₄⋊Q₈	—	4	1	no	—
63	C₃×#125	—	2	2	no	—
64	C₃×#125	—	4	1	no	—
65	C₃×C₂²⋊Q₈	—	4	2	no	—
66	C₃×C₄×Q₈	—	2	2	no	—
67	C₃×C₄×D₄	—	2	2	no	—
68	C₃×C₄×D₄	—	4	2	no	—
69	C₃×#117	—	2	1	no	—
70	C₃×#117	—	4	1	no	—
71	C₃×#117	D₄	1	1	yes	characteristic
72	C₂×C₃×C₄⋊C₄	—	2	1	no	—
73	C₂×C₃×C₄⋊C₄	—	2	1	no	—
74	C₂×C₃×C₄⋊C₄	—	4	1	no	—
75	C₃×M₃₂	—	2	2	no	—
76	C₂×C₃×C₁₆	—	2	2	no	—
77	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
78	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₁₆	C₂³	1	2	yes	—
80	C₃×#107	—	2	1	no	—
81	C₃×#107	—	2	1	no	—
82	C₃×#107	—	2	2	no	—
83	C₃×#107	C₂³	1	2	yes	—
84	C₃×#107	C₂³	1	2	yes	—
85	C₃×#107	C₂³	1	1	yes	characteristic
86	C₃×#107	C₂³	1	1	yes	characteristic
87	C₃×#106	—	2	1	no	—
88	C₃×#106	—	2	2	no	—
89	C₃×#103	—	2	2	no	—
90	C₃×#102	—	2	2	no	—
91	C₃×#97	D₄	1	2	yes	—
92	C₃×C₄×C₈	C₂³	1	2	yes	—
93	C₂×#135	C₂²×C₃	1	1	yes	characteristic
94	#544	—	2	1	no	—
95	#520	—	2	1	no	—
96	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
97	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
98	#501	—	2	2	no	—
99	#500	C₂²×C₃	1	2	yes	—
100	#499	C₂²×C₃	1	2	yes	—
101	#440	—	2	2	no	—
102	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
103	C₄×D₈	C₂²×C₃	1	2	yes	—
104	#428	—	2	1	no	—
105	#427	C₂²×C₃	1	1	yes	characteristic
106	C₂×#107	C₂²×C₃	1	1	yes	characteristic
107	C₂×#107	C₂²×C₃	1	1	yes	characteristic
108	#417	—	2	1	no	—
109	#405	C₂²×C₃	1	1	yes	characteristic
110	#367	C₂²×C₃	1	2	yes	—
111	#366	C₂²×C₃	1	2	yes	—
112	#366	C₂²×C₃	1	2	yes	—
113	#366	C₂²×C₃	1	2	yes	—
114	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
115	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
116	#358	C₂²×C₃	1	1	yes	characteristic
117	#358	C₂²×C₃	1	1	yes	characteristic
118	#358	C₂²×C₃	1	2	yes	—
119	#357	C₂²×C₃	1	1	yes	characteristic
120	#357	C₂²×C₃	1	1	yes	characteristic
121	#357	C₂²×C₃	1	2	yes	—
122	C₃×#41	—	2	4	no	—
123	C₃×#41	—	4	2	no	—
124	C₂×C₃×Q₈	—	2	1	no	—
125	C₂×C₃×Q₈	—	4	2	no	—
126	C₂×C₃×D₄	—	2	1	no	—
127	C₂×C₃×D₄	—	4	1	no	—
128	C₂²×C₃×C₄	—	4	1	no	—
129	C₂²×C₃×C₄	—	4	1	no	—
130	C₂²×C₃×C₄	—	4	1	no	—
131	C₂²×C₃×C₄	—	4	2	no	—
132	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
133	C₃×Q₁₆	—	2	1	no	—
134	C₃×Q₁₆	C₂×Q₈	1	2	yes	—
135	C₃×SD₁₆	—	2	4	no	—
136	C₃×D₈	—	2	1	no	—
137	C₃×D₈	C₂×Q₈	1	2	yes	—
138	C₃×M₁₆	—	2	2	no	—
139	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
140	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
142	C₂×C₃×C₈	#41	1	2	yes	—
143	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	4	1	no	—
149	C₃×C₄⋊C₄	—	4	1	no	—
150	C₃×C₄⋊C₄	—	4	2	no	—
151	C₃×C₄⋊C₄	—	4	2	no	—
152	C₃×C₄⋊C₄	—	4	2	no	—
153	C₃×C₄⋊C₄	—	4	2	no	—
154	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
155	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
156	C₃×C₂²⋊C₄	—	4	1	no	—
157	C₃×C₂²⋊C₄	—	4	2	no	—
158	C₃×C₂²⋊C₄	D₈	1	2	yes	—
159	C₃×C₄²	—	4	1	no	—
160	C₃×C₄²	—	4	2	no	—
161	C₃×C₄²	C₂×D₄	1	2	yes	—
162	C₃×C₁₆	—	2	1	no	—
163	C₃×C₁₆	—	2	1	no	—
164	C₃×C₁₆	—	2	2	no	—
165	C₂×#41	—	2	1	no	—
166	#135	—	2	2	no	—
167	#135	C₃×Q₈	1	4	yes	complemented
168	C₂×Q₁₆	C₂³×C₃	1	1	yes	characteristic
169	C₂×SD₁₆	—	2	1	no	—
170	C₂×D₈	C₂³×C₃	1	1	yes	characteristic
171	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	—	4	1	no	—
175	#125	—	2	2	no	—
176	#125	—	4	1	no	—
177	C₂²⋊Q₈	—	4	2	no	—
178	C₄×Q₈	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	—	4	2	no	—
181	#117	—	2	1	no	—
182	#117	—	4	1	no	—
183	#117	C₃×D₄	1	1	yes	characteristic
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	4	1	no	—
187	M₃₂	—	2	2	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
190	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
191	C₂×C₁₆	C₂³×C₃	1	2	yes	—
192	#107	—	2	1	no	—
193	#107	—	2	1	no	—
194	#107	—	2	2	no	—
195	#107	C₂³×C₃	1	2	yes	—
196	#107	C₂³×C₃	1	1	yes	characteristic
197	#107	C₂³×C₃	1	1	yes	characteristic
198	#107	C₂³×C₃	1	2	yes	—
199	#106	—	2	1	no	—
200	#106	—	2	2	no	—
201	#103	—	2	2	no	—
202	#102	—	2	2	no	—
203	#97	C₃×D₄	1	2	yes	—
204	C₄×C₈	C₂³×C₃	1	2	yes	—
205	C₂³×C₃	—	4	1	no	—
206	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
207	C₃×Q₈	—	2	2	no	—
208	C₃×Q₈	—	4	1	no	—
209	C₃×Q₈	—	16	2	no	—
210	C₃×D₄	—	2	2	no	—
211	C₃×D₄	—	4	4	no	—
212	C₃×D₄	—	4	1	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	4	no	—
215	C₂×C₃×C₄	—	4	1	no	—
216	C₂×C₃×C₄	—	4	2	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	—	4	2	no	—
219	C₂×C₃×C₄	—	16	2	no	—
220	C₂×C₃×C₄	—	16	1	no	—
221	C₂×C₃×C₄	—	16	1	no	—
222	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
223	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
224	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
225	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
226	C₂×C₃×C₄	#137	1	2	yes	—
227	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
228	C₃×C₈	—	2	2	no	—
229	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
230	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
231	C₃×C₈	C₂×#41	1	1	yes	characteristic
232	#41	—	2	4	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
243	Q₁₆	—	2	1	no	—
244	Q₁₆	C₂×C₃×Q₈	1	2	yes	—
245	SD₁₆	—	2	4	no	—
246	D₈	—	2	1	no	—
247	D₈	C₂×C₃×Q₈	1	2	yes	—
248	M₁₆	—	2	2	no	—
249	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
250	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
251	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
252	C₂×C₈	C₃×#41	1	2	yes	—
253	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	C₃×D₈	1	1	yes	characteristic
265	C₄⋊C₄	C₃×D₈	1	1	yes	characteristic
266	C₂²⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	C₃×D₈	1	2	yes	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	2	no	—
271	C₄²	C₂×C₃×D₄	1	2	yes	—
272	C₁₆	—	2	1	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	—	2	2	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	4	2	no	—
278	C₂²×C₃	—	8	1	no	—
279	C₂²×C₃	—	8	1	no	—
280	C₂²×C₃	#474	1	2	yes	—
281	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	4	2	no	—
285	C₃×C₄	—	8	2	no	—
286	C₃×C₄	—	8	2	no	—
287	C₃×C₄	—	8	1	no	—
288	C₃×C₄	—	8	1	no	—
289	C₃×C₄	#474	1	2	yes	—
290	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
291	C₃×C₄	C₂×#137	1	1	yes	characteristic
292	C₂³	—	4	1	no	—
293	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	Q₈	—	4	1	no	—
296	Q₈	—	16	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	16	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	16	1	no	—
309	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
310	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
311	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
312	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
313	C₂×C₄	C₃×#137	1	2	yes	—
314	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
317	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
318	C₈	C₂×C₃×#41	1	1	yes	characteristic
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	—	8	2	no	—
321	C₂×C₃	C₂×#474	1	1	yes	characteristic
322	C₂×C₃	#3415	1	1	yes	characteristic
323	C₂×C₃	#3417	1	1	yes	characteristic
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	C₃×#474	1	2	yes	—
330	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	1	no	—
338	C₄	C₃×#474	1	2	yes	—
339	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
340	C₄	C₂×C₃×#137	1	1	yes	characteristic
341	C₃	#33633	1	1	yes	characteristic, direct product factor, Sylow
342	C₂	—	2	2	no	—
343	C₂	—	8	2	no	—
344	C₂	C₂×C₃×#474	1	1	yes	characteristic
345	C₂	C₃×#3415	1	1	yes	characteristic
346	C₂	C₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#135	C₂²
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×#3164	C₂	semidirect
3. C₃×#2229	C₂	semidirect
4. C₃×#2228	C₂	semidirect
5. C₃×#2227	C₂	semidirect
6. C₃×#2224	C₂	semidirect
7. C₃×#2159	C₂	semidirect
8. C₂×C₃×#366	C₂	semidirect
9. C₃×#2151	C₂	semidirect
12. #33633	C₃	direct (N × H)
42. #3164	C₂×C₃	semidirect
44. #2229	C₂×C₃	semidirect
45. #2228	C₂×C₃	semidirect
46. #2227	C₂×C₃	semidirect
47. #2224	C₂×C₃	semidirect
48. #2159	C₂×C₃	semidirect
49. C₂×#366	C₂×C₃	semidirect
50. #2151	C₂×C₃	semidirect
55. C₃×#135	Q₈	semidirect
167. #135	C₃×Q₈	semidirect
341. C₃	#33633	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×#3164	2
C₃×#2951	2
C₃×#2229	2
C₃×#2228	2
C₃×#2227	2
C₃×#2224	2
C₃×#2159	2
C₂×C₃×#366	2
C₃×#2151	2
C₃×#2139	2
C₃×#2138	2
#33633	3

Subgroups — C3×#33633 #10652240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33633 #10652240