Subgroups — C₃×#33631 #10652238

Order 768.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 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×#2948	C₂	1	1	yes	maximal, characteristic
3	C₃×#2228	C₂	1	2	yes	maximal, complemented
4	C₃×#2228	C₂	1	2	yes	maximal, complemented
5	C₃×#2226	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	C₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#2140	C₂	1	1	yes	maximal, characteristic
11	C₂×C₃×#357	C₂	1	1	yes	maximal, characteristic
12	#33631	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
14	C₃×#544	—	2	1	no	—
15	C₃×#518	—	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	complemented
20	C₃×#499	C₂²	1	2	yes	complemented
21	C₃×#442	—	2	2	no	—
22	C₃×C₄×D₈	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	C₃×#427	C₂²	1	1	yes	characteristic
27	C₂×C₃×#107	C₂²	1	1	yes	characteristic
28	C₃×#418	—	2	1	no	—
29	C₃×#405	C₂²	1	1	yes	characteristic
30	C₃×#367	C₂²	1	1	yes	characteristic, complemented
31	C₃×#367	C₂²	1	1	yes	characteristic, complemented
32	C₃×#367	C₂²	1	2	yes	complemented
33	C₃×#366	C₂²	1	1	yes	characteristic, complemented
34	C₃×#366	C₂²	1	1	yes	characteristic, complemented
35	C₃×#366	C₂²	1	2	yes	complemented
36	C₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
37	C₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
38	C₃×#357	C₂²	1	2	yes	—
39	C₃×#357	C₂²	1	2	yes	—
40	C₃×#357	C₂²	1	2	yes	—
41	C₃×#357	C₂²	1	2	yes	—
42	#3164	C₂×C₃	1	1	yes	characteristic, complemented
43	#2948	C₂×C₃	1	1	yes	characteristic
44	#2228	C₂×C₃	1	2	yes	complemented
45	#2228	C₂×C₃	1	2	yes	complemented
46	#2226	C₂×C₃	1	2	yes	complemented
47	#2224	C₂×C₃	1	2	yes	complemented
48	#2159	C₂×C₃	1	1	yes	characteristic, complemented
49	#2158	C₂×C₃	1	1	yes	characteristic, complemented
50	#2151	C₂×C₃	1	1	yes	characteristic, complemented
51	#2140	C₂×C₃	1	1	yes	characteristic
52	C₂×#357	C₂×C₃	1	1	yes	characteristic
53	C₂²×C₃×D₄	—	2	1	no	—
54	C₂×C₃×D₈	—	2	1	no	—
55	C₂×C₃×D₈	—	2	2	no	—
56	C₂×C₃×D₈	Q₈	1	4	yes	complemented
57	C₂×C₃×D₈	C₂³	1	2	yes	—
58	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
60	C₃×C₄⋊Q₈	—	2	2	no	—
61	C₃×C₄⋊Q₈	—	4	1	no	—
62	C₃×#125	—	2	2	no	—
63	C₃×#125	—	4	1	no	—
64	C₃×C₂²⋊Q₈	—	4	2	no	—
65	C₃×C₄×D₄	—	2	2	no	—
66	C₃×C₄×D₄	—	2	2	no	—
67	C₃×C₄×D₄	—	4	1	no	—
68	C₃×C₄×D₄	—	4	1	no	—
69	C₃×#117	—	2	1	no	—
70	C₃×#117	—	2	1	no	—
71	C₃×#117	D₄	1	1	yes	characteristic
72	C₂×C₃×C₄⋊C₄	—	2	1	no	—
73	C₂×C₃×C₂²⋊C₄	—	4	2	no	—
74	C₃×M₃₂	—	2	2	no	—
75	C₂×C₃×C₁₆	—	2	2	no	—
76	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
78	C₂×C₃×C₁₆	C₂³	1	2	yes	—
79	C₃×#107	—	2	1	no	—
80	C₃×#107	—	2	2	no	—
81	C₃×#107	C₂³	1	1	yes	characteristic
82	C₃×#107	C₂³	1	1	yes	characteristic
83	C₃×#107	C₂³	1	2	yes	—
84	C₃×#107	C₂³	1	1	yes	characteristic
85	C₃×#107	C₂³	1	1	yes	characteristic
86	C₃×#106	—	2	1	no	—
87	C₃×#106	—	2	1	no	—
88	C₃×#106	—	2	2	no	—
89	C₃×#102	—	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	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
94	#544	—	2	1	no	—
95	#518	—	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	complemented
100	#499	C₂²×C₃	1	2	yes	complemented
101	#442	—	2	2	no	—
102	C₄×D₈	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	#427	C₂²×C₃	1	1	yes	characteristic
107	C₂×#107	C₂²×C₃	1	1	yes	characteristic
108	#418	—	2	1	no	—
109	#405	C₂²×C₃	1	1	yes	characteristic
110	#367	C₂²×C₃	1	1	yes	characteristic, complemented
111	#367	C₂²×C₃	1	1	yes	characteristic, complemented
112	#367	C₂²×C₃	1	2	yes	complemented
113	#366	C₂²×C₃	1	1	yes	characteristic, complemented
114	#366	C₂²×C₃	1	1	yes	characteristic, complemented
115	#366	C₂²×C₃	1	2	yes	complemented
116	C₄⋊C₁₆	C₂²×C₃	1	2	yes	complemented
117	C₄⋊C₁₆	C₂²×C₃	1	2	yes	complemented
118	#357	C₂²×C₃	1	2	yes	—
119	#357	C₂²×C₃	1	2	yes	—
120	#357	C₂²×C₃	1	2	yes	—
121	#357	C₂²×C₃	1	2	yes	—
122	C₂⁴×C₃	—	4	1	no	—
123	C₂×C₃×Q₈	—	4	2	no	—
124	C₂×C₃×D₄	—	2	4	no	—
125	C₂×C₃×D₄	—	2	2	no	—
126	C₂×C₃×D₄	—	4	1	no	—
127	C₂×C₃×D₄	—	4	2	no	—
128	C₂²×C₃×C₄	—	4	1	no	—
129	C₂²×C₃×C₄	—	4	2	no	—
130	C₂²×C₃×C₄	—	4	2	no	—
131	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
132	C₃×D₈	—	2	4	no	—
133	C₃×D₈	—	2	2	no	—
134	C₃×D₈	C₂×Q₈	1	4	yes	—
135	C₃×M₁₆	—	2	2	no	—
136	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
137	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
139	C₂×C₃×C₈	#41	1	2	yes	—
140	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
141	C₃×C₄⋊C₄	—	2	1	no	—
142	C₃×C₄⋊C₄	—	2	1	no	—
143	C₃×C₄⋊C₄	—	2	1	no	—
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	4	1	no	—
147	C₃×C₄⋊C₄	—	4	2	no	—
148	C₃×C₄⋊C₄	—	4	2	no	—
149	C₃×C₄⋊C₄	—	4	2	no	—
150	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
151	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
152	C₃×C₂²⋊C₄	—	4	1	no	—
153	C₃×C₂²⋊C₄	—	4	1	no	—
154	C₃×C₂²⋊C₄	—	4	2	no	—
155	C₃×C₂²⋊C₄	—	4	2	no	—
156	C₃×C₂²⋊C₄	D₈	1	2	yes	—
157	C₃×C₄²	—	4	1	no	—
158	C₃×C₄²	—	4	1	no	—
159	C₃×C₄²	C₂×D₄	1	2	yes	—
160	C₃×C₁₆	—	2	1	no	—
161	C₃×C₁₆	—	2	1	no	—
162	C₃×C₁₆	—	2	2	no	—
163	C₂²×D₄	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	2	no	—
166	C₂×D₈	C₃×Q₈	1	4	yes	complemented
167	C₂×D₈	C₂³×C₃	1	2	yes	—
168	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	—	4	1	no	—
172	#125	—	2	2	no	—
173	#125	—	4	1	no	—
174	C₂²⋊Q₈	—	4	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	4	1	no	—
178	C₄×D₄	—	4	1	no	—
179	#117	—	2	1	no	—
180	#117	—	2	1	no	—
181	#117	C₃×D₄	1	1	yes	characteristic
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₂²⋊C₄	—	4	2	no	—
184	M₃₂	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
187	C₂×C₁₆	C₂³×C₃	1	2	yes	—
188	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
189	#107	—	2	1	no	—
190	#107	—	2	2	no	—
191	#107	C₂³×C₃	1	2	yes	—
192	#107	C₂³×C₃	1	1	yes	characteristic
193	#107	C₂³×C₃	1	1	yes	characteristic
194	#107	C₂³×C₃	1	1	yes	characteristic
195	#107	C₂³×C₃	1	1	yes	characteristic
196	#106	—	2	1	no	—
197	#106	—	2	1	no	—
198	#106	—	2	2	no	—
199	#102	—	2	2	no	—
200	#102	—	2	2	no	—
201	#97	C₃×D₄	1	2	yes	—
202	C₄×C₈	C₂³×C₃	1	2	yes	—
203	C₂³×C₃	—	4	2	no	—
204	C₂³×C₃	—	4	4	no	—
205	C₂³×C₃	—	16	1	no	—
206	C₂³×C₃	—	16	1	no	—
207	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
208	C₃×Q₈	—	16	2	no	—
209	C₃×D₄	—	2	4	no	—
210	C₃×D₄	—	4	2	no	—
211	C₃×D₄	—	4	4	no	—
212	C₂×C₃×C₄	—	4	1	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	—	4	1	no	—
216	C₂×C₃×C₄	—	4	2	no	—
217	C₂×C₃×C₄	—	16	2	no	—
218	C₂×C₃×C₄	—	16	2	no	—
219	C₂×C₃×C₄	—	16	1	no	—
220	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
221	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
222	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
223	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
224	C₂×C₃×C₄	#137	1	2	yes	—
225	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
226	C₃×C₈	—	2	2	no	—
227	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
228	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
229	C₃×C₈	C₂×#41	1	1	yes	characteristic
230	C₂⁴	—	4	1	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	D₈	—	2	4	no	—
241	D₈	—	2	2	no	—
242	D₈	C₂×C₃×Q₈	1	4	yes	—
243	M₁₆	—	2	2	no	—
244	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
245	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
246	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
247	C₂×C₈	C₃×#41	1	2	yes	—
248	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	C₃×D₈	1	1	yes	characteristic
259	C₄⋊C₄	C₃×D₈	1	1	yes	characteristic
260	C₂²⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	C₃×D₈	1	2	yes	—
265	C₄²	—	4	1	no	—
266	C₄²	—	4	1	no	—
267	C₄²	C₂×C₃×D₄	1	2	yes	—
268	C₁₆	—	2	1	no	—
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	2	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	4	4	no	—
274	C₂²×C₃	—	8	2	no	—
275	C₂²×C₃	—	8	2	no	—
276	C₂²×C₃	—	16	4	no	—
277	C₂²×C₃	#474	1	2	yes	—
278	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	8	1	no	—
282	C₃×C₄	—	8	2	no	—
283	C₃×C₄	—	8	1	no	—
284	C₃×C₄	—	8	1	no	—
285	C₃×C₄	—	8	1	no	—
286	C₃×C₄	#474	1	2	yes	—
287	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
288	C₃×C₄	C₂×#137	1	1	yes	characteristic
289	C₂³	—	4	2	no	—
290	C₂³	—	4	4	no	—
291	C₂³	—	16	1	no	—
292	C₂³	—	16	1	no	—
293	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
294	Q₈	—	16	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	4	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	16	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	—	16	1	no	—
306	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
307	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
308	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
309	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
310	C₂×C₄	C₃×#137	1	2	yes	—
311	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
314	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
315	C₈	C₂×C₃×#41	1	1	yes	characteristic
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	—	8	4	no	—
318	C₂×C₃	C₂×#474	1	1	yes	characteristic
319	C₂×C₃	#3415	1	1	yes	characteristic
320	C₂×C₃	#3416	1	1	yes	characteristic
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	4	4	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	16	4	no	—
327	C₂²	C₃×#474	1	2	yes	—
328	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	C₃×#474	1	2	yes	—
337	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
338	C₄	C₂×C₃×#137	1	1	yes	characteristic
339	C₃	#33631	1	1	yes	characteristic, direct product factor, Sylow
340	C₂	—	2	2	no	—
341	C₂	—	8	4	no	—
342	C₂	C₂×C₃×#474	1	1	yes	characteristic
343	C₂	C₃×#3415	1	1	yes	characteristic
344	C₂	C₃×#3416	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 ◁ 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×#2228	C₂	semidirect
4. C₃×#2228	C₂	semidirect
5. C₃×#2226	C₂	semidirect
6. C₃×#2224	C₂	semidirect
7. C₃×#2159	C₂	semidirect
8. C₃×#2158	C₂	semidirect
9. C₃×#2151	C₂	semidirect
12. #33631	C₃	direct (N × H)
19. C₃×#500	C₂²	semidirect
20. C₃×#499	C₂²	semidirect
30. C₃×#367	C₂²	semidirect
31. C₃×#367	C₂²	semidirect
32. C₃×#367	C₂²	semidirect
33. C₃×#366	C₂²	semidirect
34. C₃×#366	C₂²	semidirect
35. C₃×#366	C₂²	semidirect
36. C₃×C₄⋊C₁₆	C₂²	semidirect
37. C₃×C₄⋊C₁₆	C₂²	semidirect
42. #3164	C₂×C₃	semidirect
44. #2228	C₂×C₃	semidirect
45. #2228	C₂×C₃	semidirect
46. #2226	C₂×C₃	semidirect
47. #2224	C₂×C₃	semidirect
48. #2159	C₂×C₃	semidirect
49. #2158	C₂×C₃	semidirect
50. #2151	C₂×C₃	semidirect
56. C₂×C₃×D₈	Q₈	semidirect
99. #500	C₂²×C₃	semidirect
100. #499	C₂²×C₃	semidirect
110. #367	C₂²×C₃	semidirect
111. #367	C₂²×C₃	semidirect
112. #367	C₂²×C₃	semidirect
113. #366	C₂²×C₃	semidirect
114. #366	C₂²×C₃	semidirect
115. #366	C₂²×C₃	semidirect
116. C₄⋊C₁₆	C₂²×C₃	semidirect
117. C₄⋊C₁₆	C₂²×C₃	semidirect
166. C₂×D₈	C₃×Q₈	semidirect
339. C₃	#33631	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×#2948	2
C₃×#2228	2
C₃×#2226	2
C₃×#2224	2
C₃×#2159	2
C₃×#2158	2
C₃×#2151	2
C₃×#2140	2
C₂×C₃×#357	2
#33631	3

Subgroups — C3×#33631 #10652238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33631 #10652238