Subgroups — C₃×#33638 #10652245

Order 768.

514 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×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#2957	C₂	1	1	yes	maximal, characteristic
3	C₃×#2231	C₂	1	4	yes	maximal, complemented
4	C₃×#2230	C₂	1	4	yes	maximal, complemented
5	C₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2145	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃×#360	C₂	1	1	yes	maximal, characteristic
10	#33638	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#135	C₂²	1	1	yes	characteristic
12	C₃×#568	—	2	1	no	—
13	C₃×#544	—	2	1	no	—
14	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₃×#501	—	2	2	no	—
17	C₃×#499	C₂²	1	2	yes	complemented
18	C₃×#498	C₂²	1	2	yes	complemented
19	C₃×#444	—	2	2	no	—
20	C₃×#443	C₂²	1	4	yes	—
21	C₂×C₃×#108	C₂²	1	1	yes	characteristic
22	C₃×#428	—	2	1	no	—
23	C₃×#427	C₂²	1	1	yes	characteristic
24	C₂×C₃×#106	C₂²	1	1	yes	characteristic
25	C₃×#421	—	2	1	no	—
26	C₃×#405	C₂²	1	1	yes	characteristic
27	C₃×#365	C₂²	1	4	yes	complemented
28	C₃×#365	C₂²	1	2	yes	complemented
29	C₃×#365	C₂²	1	2	yes	complemented
30	C₃×#364	C₂²	1	4	yes	complemented
31	C₃×#360	C₂²	1	4	yes	—
32	C₃×#360	C₂²	1	2	yes	—
33	C₃×#360	C₂²	1	2	yes	—
34	#3163	C₂×C₃	1	1	yes	characteristic, complemented
35	#2957	C₂×C₃	1	1	yes	characteristic
36	#2231	C₂×C₃	1	4	yes	complemented
37	#2230	C₂×C₃	1	4	yes	complemented
38	#2155	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×#365	C₂×C₃	1	1	yes	characteristic, complemented
40	#2153	C₂×C₃	1	1	yes	characteristic, complemented
41	#2145	C₂×C₃	1	1	yes	characteristic
42	C₂×#360	C₂×C₃	1	1	yes	characteristic
43	C₂×C₃×#41	—	2	1	no	—
44	C₃×#135	—	2	1	no	—
45	C₃×#135	—	2	1	no	—
46	C₃×#135	Q₈	1	2	yes	complemented
47	C₃×#135	C₂³	1	2	yes	—
48	C₂×C₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
49	C₂×C₃×SD₁₆	—	2	1	no	—
50	C₂×C₃×D₈	Q₈	1	1	yes	characteristic, complemented
51	C₃×#131	—	2	4	no	—
52	C₃×#131	—	4	2	no	—
53	C₂×C₃×M₁₆	—	2	1	no	—
54	C₂×C₃×M₁₆	—	4	2	no	—
55	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₃×C₄⋊Q₈	—	2	2	no	—
58	C₃×C₄⋊Q₈	—	4	1	no	—
59	C₃×#125	—	2	2	no	—
60	C₃×#125	—	4	1	no	—
61	C₃×C₂²⋊Q₈	—	4	2	no	—
62	C₃×#117	—	2	1	no	—
63	C₃×#117	D₄	1	1	yes	characteristic
64	C₂×C₃×C₄⋊C₄	—	2	1	no	—
65	C₃×M₃₂	—	2	1	no	—
66	C₃×M₃₂	—	2	1	no	—
67	C₃×M₃₂	C₂³	1	2	yes	—
68	C₃×M₃₂	C₂³	1	2	yes	—
69	C₂×C₃×C₁₆	—	2	1	no	—
70	C₂×C₃×C₁₆	—	2	1	no	—
71	C₃×#108	—	2	1	no	—
72	C₃×#108	C₂³	1	2	yes	—
73	C₃×#107	—	2	1	no	—
74	C₃×#107	—	2	2	no	—
75	C₃×#106	—	2	1	no	—
76	C₃×#106	—	2	2	no	—
77	C₃×#106	C₂³	1	2	yes	—
78	C₃×#106	C₂³	1	1	yes	characteristic
79	C₃×#106	C₂³	1	1	yes	characteristic
80	C₃×#104	—	2	4	no	—
81	C₃×#97	D₄	1	2	yes	—
82	C₃×C₄×C₈	C₂³	1	2	yes	—
83	C₂×#135	C₂²×C₃	1	1	yes	characteristic
84	#568	—	2	1	no	—
85	#544	—	2	1	no	—
86	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
87	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
88	#501	—	2	2	no	—
89	#499	C₂²×C₃	1	2	yes	complemented
90	#498	C₂²×C₃	1	2	yes	complemented
91	#444	—	2	2	no	—
92	#443	C₂²×C₃	1	4	yes	—
93	C₂×#108	C₂²×C₃	1	1	yes	characteristic
94	#428	—	2	1	no	—
95	#427	C₂²×C₃	1	1	yes	characteristic
96	C₂×#106	C₂²×C₃	1	1	yes	characteristic
97	#421	—	2	1	no	—
98	#405	C₂²×C₃	1	1	yes	characteristic
99	#365	C₂²×C₃	1	4	yes	complemented
100	#365	C₂²×C₃	1	2	yes	complemented
101	#365	C₂²×C₃	1	2	yes	complemented
102	#364	C₂²×C₃	1	4	yes	complemented
103	#360	C₂²×C₃	1	4	yes	—
104	#360	C₂²×C₃	1	2	yes	—
105	#360	C₂²×C₃	1	2	yes	—
106	C₃×#41	—	2	2	no	—
107	C₃×#41	—	2	2	no	—
108	C₃×#41	—	4	1	no	—
109	C₃×#41	—	4	1	no	—
110	C₂×C₃×Q₈	—	2	1	no	—
111	C₂×C₃×Q₈	—	4	2	no	—
112	C₂×C₃×D₄	—	2	1	no	—
113	C₂×C₃×D₄	—	4	1	no	—
114	C₂²×C₃×C₄	—	4	1	no	—
115	C₂²×C₃×C₄	—	4	1	no	—
116	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
117	C₃×Q₁₆	—	2	1	no	—
118	C₃×Q₁₆	C₂×Q₈	1	2	yes	—
119	C₃×SD₁₆	—	2	2	no	—
120	C₃×SD₁₆	—	2	2	no	—
121	C₃×D₈	—	2	1	no	—
122	C₃×D₈	C₂×Q₈	1	2	yes	—
123	C₃×M₁₆	—	2	2	no	—
124	C₃×M₁₆	—	4	1	no	—
125	C₃×M₁₆	—	4	4	no	—
126	C₃×M₁₆	SD₁₆	1	2	yes	—
127	C₃×M₁₆	SD₁₆	1	2	yes	—
128	C₂×C₃×C₈	—	4	1	no	—
129	C₂×C₃×C₈	—	4	4	no	—
130	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
134	C₂×C₃×C₈	#41	1	1	yes	characteristic
135	C₂×C₃×C₈	#41	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	—	2	1	no	—
141	C₃×C₄⋊C₄	—	4	2	no	—
142	C₃×C₄⋊C₄	—	4	2	no	—
143	C₃×C₄⋊C₄	—	4	2	no	—
144	C₃×C₄⋊C₄	—	4	1	no	—
145	C₃×C₂²⋊C₄	—	2	1	no	—
146	C₃×C₂²⋊C₄	—	4	1	no	—
147	C₃×C₄²	—	4	1	no	—
148	C₃×C₄²	C₂×D₄	1	2	yes	—
149	C₃×C₁₆	—	2	2	no	—
150	C₃×C₁₆	—	2	2	no	—
151	C₂×#41	—	2	1	no	—
152	#135	—	2	1	no	—
153	#135	—	2	1	no	—
154	#135	C₃×Q₈	1	2	yes	complemented
155	#135	C₂³×C₃	1	2	yes	—
156	C₂×Q₁₆	C₃×Q₈	1	1	yes	characteristic, complemented
157	C₂×SD₁₆	—	2	1	no	—
158	C₂×D₈	C₃×Q₈	1	1	yes	characteristic, complemented
159	#131	—	2	4	no	—
160	#131	—	4	2	no	—
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	—	4	2	no	—
163	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	4	1	no	—
167	#125	—	2	2	no	—
168	#125	—	4	1	no	—
169	C₂²⋊Q₈	—	4	2	no	—
170	#117	—	2	1	no	—
171	#117	C₃×D₄	1	1	yes	characteristic
172	C₂×C₄⋊C₄	—	2	1	no	—
173	M₃₂	—	2	1	no	—
174	M₃₂	—	2	1	no	—
175	M₃₂	C₂³×C₃	1	2	yes	—
176	M₃₂	C₂³×C₃	1	2	yes	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	#108	—	2	1	no	—
180	#108	C₂³×C₃	1	2	yes	—
181	#107	—	2	1	no	—
182	#107	—	2	2	no	—
183	#106	—	2	1	no	—
184	#106	—	2	2	no	—
185	#106	C₂³×C₃	1	2	yes	—
186	#106	C₂³×C₃	1	1	yes	characteristic
187	#106	C₂³×C₃	1	1	yes	characteristic
188	#104	—	2	4	no	—
189	#97	C₃×D₄	1	2	yes	—
190	C₄×C₈	C₂³×C₃	1	2	yes	—
191	C₂³×C₃	—	4	1	no	—
192	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₃×Q₈	—	2	2	no	—
194	C₃×Q₈	—	4	1	no	—
195	C₃×Q₈	—	16	2	no	—
196	C₃×D₄	—	2	2	no	—
197	C₃×D₄	—	4	2	no	—
198	C₃×D₄	—	4	2	no	—
199	C₃×D₄	—	4	1	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	4	1	no	—
202	C₂×C₃×C₄	—	4	2	no	—
203	C₂×C₃×C₄	—	4	2	no	—
204	C₂×C₃×C₄	—	4	1	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	2	no	—
207	C₂×C₃×C₄	—	16	1	no	—
208	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
212	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
214	C₃×C₈	—	4	2	no	—
215	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
216	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
217	C₃×C₈	C₂×SD₁₆	1	4	yes	—
218	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
219	C₃×C₈	C₂×#41	1	1	yes	characteristic
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	#41	—	4	1	no	—
223	#41	—	4	1	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
231	Q₁₆	—	2	1	no	—
232	Q₁₆	C₂×C₃×Q₈	1	2	yes	—
233	SD₁₆	—	2	2	no	—
234	SD₁₆	—	2	2	no	—
235	D₈	—	2	1	no	—
236	D₈	C₂×C₃×Q₈	1	2	yes	—
237	M₁₆	—	2	2	no	—
238	M₁₆	—	4	1	no	—
239	M₁₆	—	4	4	no	—
240	M₁₆	C₃×SD₁₆	1	2	yes	—
241	M₁₆	C₃×SD₁₆	1	2	yes	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
245	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
246	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
247	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
248	C₂×C₈	C₃×#41	1	1	yes	characteristic
249	C₂×C₈	C₃×#41	1	1	yes	characteristic
250	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	C₂×C₃×D₄	1	2	yes	—
263	C₁₆	—	2	2	no	—
264	C₁₆	—	2	2	no	—
265	C₂²×C₃	—	4	1	no	—
266	C₂²×C₃	—	4	2	no	—
267	C₂²×C₃	—	16	1	no	—
268	C₂²×C₃	#476	1	1	yes	characteristic
269	C₂²×C₃	#476	1	1	yes	characteristic
270	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	—	8	1	no	—
274	C₃×C₄	—	8	2	no	—
275	C₃×C₄	—	8	1	no	—
276	C₃×C₄	#476	1	1	yes	characteristic
277	C₃×C₄	#476	1	1	yes	characteristic
278	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
279	C₃×C₄	C₂²×SD₁₆	1	1	yes	characteristic
280	C₂³	—	4	1	no	—
281	C₂³	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	16	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	16	1	no	—
297	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
298	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
299	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
300	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
301	C₂×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	C₃×C₂²⋊Q₈	1	1	yes	characteristic
305	C₈	C₃×C₂²⋊Q₈	1	1	yes	characteristic
306	C₈	C₂×C₃×SD₁₆	1	4	yes	—
307	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
308	C₈	C₂×C₃×#41	1	1	yes	characteristic
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	8	2	no	—
313	C₂×C₃	C₂×#476	1	1	yes	characteristic
314	C₂²	—	4	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	16	1	no	—
317	C₂²	C₃×#476	1	1	yes	characteristic
318	C₂²	C₃×#476	1	1	yes	characteristic
319	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	C₃×#476	1	1	yes	characteristic
326	C₄	C₃×#476	1	1	yes	characteristic
327	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
328	C₄	C₂²×C₃×SD₁₆	1	1	yes	characteristic
329	C₃	#33638	1	1	yes	characteristic, direct product factor, Sylow
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×C₃×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂×#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#405 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×#3163	C₂	semidirect
3. C₃×#2231	C₂	semidirect
4. C₃×#2230	C₂	semidirect
5. C₃×#2155	C₂	semidirect
6. C₂×C₃×#365	C₂	semidirect
7. C₃×#2153	C₂	semidirect
10. #33638	C₃	direct (N × H)
17. C₃×#499	C₂²	semidirect
18. C₃×#498	C₂²	semidirect
27. C₃×#365	C₂²	semidirect
28. C₃×#365	C₂²	semidirect
29. C₃×#365	C₂²	semidirect
30. C₃×#364	C₂²	semidirect
34. #3163	C₂×C₃	semidirect
36. #2231	C₂×C₃	semidirect
37. #2230	C₂×C₃	semidirect
38. #2155	C₂×C₃	semidirect
39. C₂×#365	C₂×C₃	semidirect
40. #2153	C₂×C₃	semidirect
46. C₃×#135	Q₈	semidirect
48. C₂×C₃×Q₁₆	Q₈	semidirect
50. C₂×C₃×D₈	Q₈	semidirect
89. #499	C₂²×C₃	semidirect
90. #498	C₂²×C₃	semidirect
99. #365	C₂²×C₃	semidirect
100. #365	C₂²×C₃	semidirect
101. #365	C₂²×C₃	semidirect
102. #364	C₂²×C₃	semidirect
154. #135	C₃×Q₈	semidirect
156. C₂×Q₁₆	C₃×Q₈	semidirect
158. C₂×D₈	C₃×Q₈	semidirect
329. C₃	#33638	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×#3163	2
C₃×#2957	2
C₃×#2231	2
C₃×#2230	2
C₃×#2155	2
C₂×C₃×#365	2
C₃×#2153	2
C₃×#2145	2
C₂×C₃×#360	2
#33638	3

Subgroups — C3×#33638 #10652245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33638 #10652245