Subgroups — C₃×#63092 #10681699

Order 768.

10796 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×2₋¹⁺⁶	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3586	C₂	1	15	yes	maximal, complemented
4	C₃×#3585	C₂	1	15	yes	maximal, complemented
5	C₃×#3584	C₂	1	15	yes	maximal, complemented
6	C₂×C₃×#577	C₂	1	15	yes	maximal, complemented
7	C₃×#3573	C₂	1	1	yes	maximal, characteristic, complemented
8	#63092	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×#585	—	2	6	no	—
10	C₃×#585	—	2	10	no	—
11	C₃×#585	C₂²	1	1	yes	characteristic
12	C₃×#585	C₂²	1	15	yes	complemented
13	C₃×#585	C₂²	1	15	yes	complemented
14	C₂×C₃×2₋¹⁺⁴	—	2	6	no	—
15	C₂×C₃×2₋¹⁺⁴	C₂²	1	15	yes	complemented
16	C₂×C₃×H(5,𝔽₂)	—	2	10	no	—
17	C₂×C₃×H(5,𝔽₂)	C₂²	1	15	yes	complemented
18	C₃×#578	C₂²	1	60	yes	complemented
19	C₃×#577	C₂²	1	60	yes	complemented
20	C₃×#577	C₂²	1	120	yes	complemented
21	C₃×#577	C₂²	1	20	yes	complemented
22	C₃×#576	C₂²	1	60	yes	complemented
23	C₃×#575	C₂²	1	90	yes	complemented
24	C₂×C₃×#137	C₂²	1	45	yes	complemented
25	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	45	yes	complemented
26	C₂×C₃×#135	C₂²	1	45	yes	complemented
27	C₂²×C₃×SD₁₆	C₂²	1	15	yes	complemented
28	C₃×#568	C₂²	1	15	yes	complemented
29	C₂×C₃×#131	C₂²	1	15	yes	complemented
30	2₋¹⁺⁶	C₂×C₃	1	1	yes	characteristic, complemented
31	H(7,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
32	#3586	C₂×C₃	1	15	yes	complemented
33	#3585	C₂×C₃	1	15	yes	complemented
34	#3584	C₂×C₃	1	15	yes	complemented
35	C₂×#577	C₂×C₃	1	15	yes	complemented
36	#3573	C₂×C₃	1	1	yes	characteristic, complemented
37	C₃×2₋¹⁺⁴	—	2	60	no	—
38	C₃×2₋¹⁺⁴	—	2	15	no	—
39	C₃×2₋¹⁺⁴	—	2	15	no	—
40	C₃×2₋¹⁺⁴	D₄	1	6	yes	—
41	C₃×2₋¹⁺⁴	C₂³	1	20	yes	—
42	C₃×2₋¹⁺⁴	C₂³	1	60	yes	complemented
43	C₃×H(5,𝔽₂)	—	2	60	no	—
44	C₃×H(5,𝔽₂)	—	2	45	no	—
45	C₃×H(5,𝔽₂)	—	2	45	no	—
46	C₃×H(5,𝔽₂)	D₄	1	10	yes	—
47	C₃×H(5,𝔽₂)	C₂³	1	60	yes	—
48	C₃×H(5,𝔽₂)	C₂³	1	20	yes	complemented
49	C₂×C₃×#41	—	2	60	no	—
50	C₂×C₃×#41	—	2	60	no	—
51	C₂×C₃×#41	—	2	15	no	—
52	C₂×C₃×#41	—	2	45	no	—
53	C₂×C₃×#41	C₂³	1	45	yes	—
54	C₂×C₃×#41	C₂³	1	15	yes	—
55	C₂×C₃×#41	C₂³	1	45	yes	complemented
56	C₂²×C₃×Q₈	—	2	15	no	—
57	C₂²×C₃×Q₈	C₂³	1	15	yes	complemented
58	C₂²×C₃×D₄	—	2	45	no	—
59	C₂²×C₃×D₄	C₂³	1	15	yes	—
60	C₃×#137	C₂³	1	180	yes	complemented
61	C₃×#137	C₂³	1	60	yes	complemented
62	C₃×AGL(1,ℤ/8)	C₂³	1	180	yes	complemented
63	C₃×AGL(1,ℤ/8)	C₂³	1	60	yes	complemented
64	C₃×#135	C₂³	1	60	yes	complemented
65	C₃×#135	C₂³	1	180	yes	complemented
66	C₂×C₃×Q₁₆	C₂³	1	45	yes	complemented
67	C₂×C₃×SD₁₆	C₂³	1	45	yes	complemented
68	C₂×C₃×SD₁₆	C₂³	1	90	yes	complemented
69	C₂×C₃×SD₁₆	C₂³	1	15	yes	complemented
70	C₂×C₃×D₈	C₂³	1	45	yes	complemented
71	C₃×#131	C₂³	1	20	yes	complemented
72	C₃×#131	C₂³	1	60	yes	complemented
73	C₂×C₃×M₁₆	C₂³	1	45	yes	complemented
74	C₂²×C₃×C₈	C₂³	1	15	yes	complemented
75	#585	—	2	6	no	—
76	#585	—	2	10	no	—
77	#585	C₂²×C₃	1	1	yes	characteristic
78	#585	C₂²×C₃	1	15	yes	complemented
79	#585	C₂²×C₃	1	15	yes	complemented
80	C₂×2₋¹⁺⁴	—	2	6	no	—
81	C₂×2₋¹⁺⁴	C₂²×C₃	1	15	yes	complemented
82	C₂×H(5,𝔽₂)	—	2	10	no	—
83	C₂×H(5,𝔽₂)	C₂²×C₃	1	15	yes	complemented
84	#578	C₂²×C₃	1	60	yes	complemented
85	#577	C₂²×C₃	1	60	yes	complemented
86	#577	C₂²×C₃	1	120	yes	complemented
87	#577	C₂²×C₃	1	20	yes	complemented
88	#576	C₂²×C₃	1	60	yes	complemented
89	#575	C₂²×C₃	1	90	yes	complemented
90	C₂×#137	C₂²×C₃	1	45	yes	complemented
91	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	45	yes	complemented
92	C₂×#135	C₂²×C₃	1	45	yes	complemented
93	C₂²×SD₁₆	C₂²×C₃	1	15	yes	complemented
94	#568	C₂²×C₃	1	15	yes	complemented
95	C₂×#131	C₂²×C₃	1	15	yes	complemented
96	C₂⁴×C₃	—	2	15	no	—
97	C₃×#41	—	2	60	no	—
98	C₃×#41	—	2	20	no	—
99	C₃×#41	—	2	60	no	—
100	C₃×#41	—	2	180	no	—
101	C₃×#41	—	2	90	no	—
102	C₃×#41	—	2	90	no	—
103	C₃×#41	C₂×D₄	1	60	yes	—
104	C₃×#41	C₂⁴	1	60	yes	—
105	C₃×#41	C₂⁴	1	60	yes	—
106	C₃×#41	C₂⁴	1	20	yes	—
107	C₂×C₃×Q₈	—	2	60	no	—
108	C₂×C₃×Q₈	—	2	20	no	—
109	C₂×C₃×Q₈	—	2	45	no	—
110	C₂×C₃×Q₈	C₂×D₄	1	15	yes	—
111	C₂×C₃×Q₈	C₂⁴	1	45	yes	—
112	C₂×C₃×Q₈	C₂⁴	1	15	yes	—
113	C₂×C₃×D₄	—	2	45	no	—
114	C₂×C₃×D₄	—	2	45	no	—
115	C₂×C₃×D₄	—	2	180	no	—
116	C₂×C₃×D₄	—	2	60	no	—
117	C₂×C₃×D₄	—	2	45	no	—
118	C₂×C₃×D₄	C₂×D₄	1	45	yes	—
119	C₂×C₃×D₄	C₂⁴	1	15	yes	—
120	C₂×C₃×D₄	C₂⁴	1	45	yes	—
121	C₂²×C₃×C₄	—	2	15	no	—
122	C₂²×C₃×C₄	—	2	45	no	—
123	C₂²×C₃×C₄	—	2	45	no	—
124	C₂²×C₃×C₄	C₂⁴	1	15	yes	—
125	C₃×Q₁₆	C₂⁴	1	60	yes	—
126	C₃×SD₁₆	C₂⁴	1	10	yes	—
127	C₃×SD₁₆	C₂⁴	1	6	yes	—
128	C₃×SD₁₆	C₂⁴	1	90	yes	—
129	C₃×SD₁₆	C₂⁴	1	30	yes	—
130	C₃×D₈	C₂⁴	1	60	yes	—
131	C₃×M₁₆	C₂⁴	1	60	yes	—
132	C₂×C₃×C₈	C₂⁴	1	15	yes	—
133	C₂×C₃×C₈	C₂⁴	1	45	yes	—
134	2₋¹⁺⁴	—	2	60	no	—
135	2₋¹⁺⁴	—	2	15	no	—
136	2₋¹⁺⁴	—	2	15	no	—
137	2₋¹⁺⁴	C₃×D₄	1	6	yes	—
138	2₋¹⁺⁴	C₂³×C₃	1	20	yes	—
139	2₋¹⁺⁴	C₂³×C₃	1	60	yes	complemented
140	H(5,𝔽₂)	—	2	60	no	—
141	H(5,𝔽₂)	—	2	45	no	—
142	H(5,𝔽₂)	—	2	45	no	—
143	H(5,𝔽₂)	C₃×D₄	1	10	yes	—
144	H(5,𝔽₂)	C₂³×C₃	1	60	yes	—
145	H(5,𝔽₂)	C₂³×C₃	1	20	yes	complemented
146	C₂×#41	—	2	60	no	—
147	C₂×#41	—	2	60	no	—
148	C₂×#41	—	2	15	no	—
149	C₂×#41	—	2	45	no	—
150	C₂×#41	C₂³×C₃	1	45	yes	—
151	C₂×#41	C₂³×C₃	1	15	yes	—
152	C₂×#41	C₂³×C₃	1	45	yes	complemented
153	C₂²×Q₈	—	2	15	no	—
154	C₂²×Q₈	C₂³×C₃	1	15	yes	complemented
155	C₂²×D₄	—	2	45	no	—
156	C₂²×D₄	C₂³×C₃	1	15	yes	—
157	#137	C₂³×C₃	1	180	yes	complemented
158	#137	C₂³×C₃	1	60	yes	complemented
159	AGL(1,ℤ/8)	C₂³×C₃	1	180	yes	complemented
160	AGL(1,ℤ/8)	C₂³×C₃	1	60	yes	complemented
161	#135	C₂³×C₃	1	60	yes	complemented
162	#135	C₂³×C₃	1	180	yes	complemented
163	C₂×Q₁₆	C₂³×C₃	1	45	yes	complemented
164	C₂×SD₁₆	C₂³×C₃	1	45	yes	complemented
165	C₂×SD₁₆	C₂³×C₃	1	90	yes	complemented
166	C₂×SD₁₆	C₂³×C₃	1	15	yes	complemented
167	C₂×D₈	C₂³×C₃	1	45	yes	complemented
168	#131	C₂³×C₃	1	20	yes	complemented
169	#131	C₂³×C₃	1	60	yes	complemented
170	C₂×M₁₆	C₂³×C₃	1	45	yes	complemented
171	C₂²×C₈	C₂³×C₃	1	15	yes	complemented
172	C₂³×C₃	—	2	15	no	—
173	C₂³×C₃	—	2	45	no	—
174	C₂³×C₃	—	16	15	no	—
175	C₂³×C₃	C₂²×D₄	1	15	yes	—
176	C₃×Q₈	—	2	45	no	—
177	C₃×Q₈	—	2	15	no	—
178	C₃×Q₈	C₂²×D₄	1	20	yes	—
179	C₃×Q₈	C₂⁵	1	10	yes	—
180	C₃×Q₈	C₂⁵	1	6	yes	—
181	C₃×D₄	—	2	60	no	—
182	C₃×D₄	—	2	60	no	—
183	C₃×D₄	—	2	45	no	—
184	C₃×D₄	—	2	15	no	—
185	C₃×D₄	C₂²×D₄	1	60	yes	—
186	C₃×D₄	C₂⁵	1	6	yes	—
187	C₃×D₄	C₂⁵	1	10	yes	—
188	C₂×C₃×C₄	—	2	15	no	—
189	C₂×C₃×C₄	—	2	45	no	—
190	C₂×C₃×C₄	—	2	60	no	—
191	C₂×C₃×C₄	—	2	60	no	—
192	C₂×C₃×C₄	C₂²×D₄	1	45	yes	—
193	C₂×C₃×C₄	C₂⁵	1	15	yes	—
194	C₃×C₈	C₂⁵	1	1	yes	characteristic
195	C₃×C₈	C₂⁵	1	15	yes	—
196	C₂⁴	—	2	15	no	—
197	#41	—	2	60	no	—
198	#41	—	2	20	no	—
199	#41	—	2	60	no	—
200	#41	—	2	180	no	—
201	#41	—	2	90	no	—
202	#41	—	2	90	no	—
203	#41	C₂×C₃×D₄	1	60	yes	—
204	#41	C₂⁴×C₃	1	60	yes	—
205	#41	C₂⁴×C₃	1	60	yes	—
206	#41	C₂⁴×C₃	1	20	yes	—
207	C₂×Q₈	—	2	60	no	—
208	C₂×Q₈	—	2	20	no	—
209	C₂×Q₈	—	2	45	no	—
210	C₂×Q₈	C₂×C₃×D₄	1	15	yes	—
211	C₂×Q₈	C₂⁴×C₃	1	45	yes	—
212	C₂×Q₈	C₂⁴×C₃	1	15	yes	—
213	C₂×D₄	—	2	45	no	—
214	C₂×D₄	—	2	45	no	—
215	C₂×D₄	—	2	180	no	—
216	C₂×D₄	—	2	60	no	—
217	C₂×D₄	—	2	45	no	—
218	C₂×D₄	C₂×C₃×D₄	1	45	yes	—
219	C₂×D₄	C₂⁴×C₃	1	15	yes	—
220	C₂×D₄	C₂⁴×C₃	1	45	yes	—
221	C₂²×C₄	—	2	15	no	—
222	C₂²×C₄	—	2	45	no	—
223	C₂²×C₄	—	2	45	no	—
224	C₂²×C₄	C₂⁴×C₃	1	15	yes	—
225	Q₁₆	C₂⁴×C₃	1	60	yes	—
226	SD₁₆	C₂⁴×C₃	1	10	yes	—
227	SD₁₆	C₂⁴×C₃	1	6	yes	—
228	SD₁₆	C₂⁴×C₃	1	90	yes	—
229	SD₁₆	C₂⁴×C₃	1	30	yes	—
230	D₈	C₂⁴×C₃	1	60	yes	—
231	M₁₆	C₂⁴×C₃	1	60	yes	—
232	C₂×C₈	C₂⁴×C₃	1	15	yes	—
233	C₂×C₈	C₂⁴×C₃	1	45	yes	—
234	C₂²×C₃	—	2	6	no	—
235	C₂²×C₃	—	2	10	no	—
236	C₂²×C₃	—	4	15	no	—
237	C₂²×C₃	—	8	15	no	—
238	C₂²×C₃	—	8	45	no	—
239	C₂²×C₃	C₂³×D₄	1	15	yes	—
240	C₃×C₄	—	2	6	no	—
241	C₃×C₄	—	2	10	no	—
242	C₃×C₄	C₂³×D₄	1	15	yes	—
243	C₃×C₄	C₂⁶	1	1	yes	characteristic
244	C₂³	—	2	15	no	—
245	C₂³	—	2	45	no	—
246	C₂³	—	16	15	no	—
247	C₂³	C₂²×C₃×D₄	1	15	yes	—
248	Q₈	—	2	45	no	—
249	Q₈	—	2	15	no	—
250	Q₈	C₂²×C₃×D₄	1	20	yes	—
251	Q₈	C₂⁵×C₃	1	10	yes	—
252	Q₈	C₂⁵×C₃	1	6	yes	—
253	D₄	—	2	60	no	—
254	D₄	—	2	60	no	—
255	D₄	—	2	45	no	—
256	D₄	—	2	15	no	—
257	D₄	C₂²×C₃×D₄	1	60	yes	—
258	D₄	C₂⁵×C₃	1	6	yes	—
259	D₄	C₂⁵×C₃	1	10	yes	—
260	C₂×C₄	—	2	15	no	—
261	C₂×C₄	—	2	45	no	—
262	C₂×C₄	—	2	60	no	—
263	C₂×C₄	—	2	60	no	—
264	C₂×C₄	C₂²×C₃×D₄	1	45	yes	—
265	C₂×C₄	C₂⁵×C₃	1	15	yes	—
266	C₈	C₂⁵×C₃	1	1	yes	characteristic
267	C₈	C₂⁵×C₃	1	15	yes	—
268	C₂×C₃	—	2	15	no	—
269	C₂×C₃	—	4	6	no	—
270	C₂×C₃	—	4	10	no	—
271	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
272	C₂²	—	2	6	no	—
273	C₂²	—	2	10	no	—
274	C₂²	—	4	15	no	—
275	C₂²	—	8	45	no	—
276	C₂²	—	8	15	no	—
277	C₂²	C₂³×C₃×D₄	1	15	yes	—
278	C₄	—	2	6	no	—
279	C₄	—	2	10	no	—
280	C₄	C₂³×C₃×D₄	1	15	yes	—
281	C₄	C₂⁶×C₃	1	1	yes	characteristic
282	C₃	#63092	1	1	yes	characteristic, direct product factor, Sylow
283	C₂	—	2	15	no	—
284	C₂	—	4	10	no	—
285	C₂	—	4	6	no	—
286	C₂	C₂⁴×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂⁴×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#585 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×2₋¹⁺⁶	C₂	semidirect
2. C₃×H(7,𝔽₂)	C₂	semidirect
3. C₃×#3586	C₂	semidirect
4. C₃×#3585	C₂	semidirect
5. C₃×#3584	C₂	semidirect
6. C₂×C₃×#577	C₂	semidirect
7. C₃×#3573	C₂	semidirect
8. #63092	C₃	direct (N × H)
12. C₃×#585	C₂²	semidirect
13. C₃×#585	C₂²	semidirect
15. C₂×C₃×2₋¹⁺⁴	C₂²	semidirect
17. C₂×C₃×H(5,𝔽₂)	C₂²	semidirect
18. C₃×#578	C₂²	semidirect
19. C₃×#577	C₂²	semidirect
20. C₃×#577	C₂²	semidirect
21. C₃×#577	C₂²	semidirect
22. C₃×#576	C₂²	semidirect
23. C₃×#575	C₂²	semidirect
24. C₂×C₃×#137	C₂²	semidirect
25. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
26. C₂×C₃×#135	C₂²	semidirect
27. C₂²×C₃×SD₁₆	C₂²	semidirect
28. C₃×#568	C₂²	semidirect
29. C₂×C₃×#131	C₂²	semidirect
30. 2₋¹⁺⁶	C₂×C₃	semidirect
31. H(7,𝔽₂)	C₂×C₃	semidirect
32. #3586	C₂×C₃	semidirect
33. #3585	C₂×C₃	semidirect
34. #3584	C₂×C₃	semidirect
35. C₂×#577	C₂×C₃	semidirect
36. #3573	C₂×C₃	semidirect
42. C₃×2₋¹⁺⁴	C₂³	semidirect
48. C₃×H(5,𝔽₂)	C₂³	semidirect
55. C₂×C₃×#41	C₂³	semidirect
57. C₂²×C₃×Q₈	C₂³	semidirect
60. C₃×#137	C₂³	semidirect
61. C₃×#137	C₂³	semidirect
62. C₃×AGL(1,ℤ/8)	C₂³	semidirect
63. C₃×AGL(1,ℤ/8)	C₂³	semidirect
64. C₃×#135	C₂³	semidirect
65. C₃×#135	C₂³	semidirect
66. C₂×C₃×Q₁₆	C₂³	semidirect
67. C₂×C₃×SD₁₆	C₂³	semidirect
68. C₂×C₃×SD₁₆	C₂³	semidirect
69. C₂×C₃×SD₁₆	C₂³	semidirect
70. C₂×C₃×D₈	C₂³	semidirect
71. C₃×#131	C₂³	semidirect
72. C₃×#131	C₂³	semidirect
73. C₂×C₃×M₁₆	C₂³	semidirect
74. C₂²×C₃×C₈	C₂³	semidirect
78. #585	C₂²×C₃	semidirect
79. #585	C₂²×C₃	semidirect
81. C₂×2₋¹⁺⁴	C₂²×C₃	semidirect
83. C₂×H(5,𝔽₂)	C₂²×C₃	semidirect
84. #578	C₂²×C₃	semidirect
85. #577	C₂²×C₃	semidirect
86. #577	C₂²×C₃	semidirect
87. #577	C₂²×C₃	semidirect
88. #576	C₂²×C₃	semidirect
89. #575	C₂²×C₃	semidirect
90. C₂×#137	C₂²×C₃	semidirect
91. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
92. C₂×#135	C₂²×C₃	semidirect
93. C₂²×SD₁₆	C₂²×C₃	semidirect
94. #568	C₂²×C₃	semidirect
95. C₂×#131	C₂²×C₃	semidirect
139. 2₋¹⁺⁴	C₂³×C₃	semidirect
145. H(5,𝔽₂)	C₂³×C₃	semidirect
152. C₂×#41	C₂³×C₃	semidirect
154. C₂²×Q₈	C₂³×C₃	semidirect
157. #137	C₂³×C₃	semidirect
158. #137	C₂³×C₃	semidirect
159. AGL(1,ℤ/8)	C₂³×C₃	semidirect
160. AGL(1,ℤ/8)	C₂³×C₃	semidirect
161. #135	C₂³×C₃	semidirect
162. #135	C₂³×C₃	semidirect
163. C₂×Q₁₆	C₂³×C₃	semidirect
164. C₂×SD₁₆	C₂³×C₃	semidirect
165. C₂×SD₁₆	C₂³×C₃	semidirect
166. C₂×SD₁₆	C₂³×C₃	semidirect
167. C₂×D₈	C₂³×C₃	semidirect
168. #131	C₂³×C₃	semidirect
169. #131	C₂³×C₃	semidirect
170. C₂×M₁₆	C₂³×C₃	semidirect
171. C₂²×C₈	C₂³×C₃	semidirect
282. C₃	#63092	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×2₋¹⁺⁶	2
C₃×H(7,𝔽₂)	2
C₃×#3586	2
C₃×#3585	2
C₃×#3584	2
C₂×C₃×#577	2
C₃×#3573	2
#63092	3

Subgroups — C3×#63092 #10681699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#63092 #10681699