Subgroups — C₂×C₃×#2736 #10670778

Order 768.

3382 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2736	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#397	C₂	1	3	yes	maximal, complemented
3	C₂×C₃×#393	C₂	1	3	yes	maximal, complemented
4	C₂×C₃×#388	C₂	1	6	yes	maximal, complemented
5	C₂×C₃×#379	C₂	1	3	yes	maximal
6	C₂×#2736	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁶×C₃	C₂²	1	1	yes	characteristic
8	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
9	C₂²×C₃×C₄⋊C₄	C₂²	1	3	yes	complemented
10	C₂²×C₃×C₂²⋊C₄	C₂²	1	6	yes	—
11	C₂²×C₃×C₂²⋊C₄	C₂²	1	3	yes	—
12	C₂²×C₃×C₂²⋊C₄	C₂²	1	6	yes	—
13	C₂²×C₃×C₂²⋊C₄	C₂²	1	3	yes	—
14	C₂²×C₃×C₄²	C₂²	1	2	yes	complemented
15	C₃×#397	C₂²	1	24	yes	complemented
16	C₃×#393	C₂²	1	24	yes	complemented
17	C₃×#388	C₂²	1	48	yes	complemented
18	C₃×#379	C₂²	1	24	yes	—
19	C₂×C₃×#95	C₂²	1	6	yes	complemented
20	C₂×C₃×#95	C₂²	1	3	yes	complemented
21	C₂×C₃×#95	C₂²	1	1	yes	characteristic, complemented
22	#2736	C₂×C₃	1	16	yes	direct product factor
23	C₂×#397	C₂×C₃	1	3	yes	complemented
24	C₂×#393	C₂×C₃	1	3	yes	complemented
25	C₂×#388	C₂×C₃	1	6	yes	complemented
26	C₂×#379	C₂×C₃	1	3	yes	—
27	C₂⁵×C₃	—	2	12	no	—
28	C₂⁵×C₃	—	4	6	no	—
29	C₂⁵×C₃	—	4	2	no	—
30	C₂⁵×C₃	C₂³	1	3	yes	—
31	C₂⁵×C₃	C₂³	1	4	yes	—
32	C₂²×C₃×Q₈	—	4	6	no	—
33	C₂²×C₃×Q₈	C₂³	1	4	yes	complemented
34	C₂³×C₃×C₄	C₂³	1	3	yes	—
35	C₂³×C₃×C₄	C₂³	1	3	yes	—
36	C₂³×C₃×C₄	C₂³	1	6	yes	—
37	C₂×C₃×C₄⋊C₄	—	4	6	no	—
38	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	complemented
39	C₂×C₃×C₂²⋊C₄	—	2	24	no	—
40	C₂×C₃×C₂²⋊C₄	—	4	6	no	—
41	C₂×C₃×C₂²⋊C₄	—	4	6	no	—
42	C₂×C₃×C₂²⋊C₄	—	4	12	no	—
43	C₂×C₃×C₂²⋊C₄	C₂³	1	24	yes	—
44	C₂×C₃×C₂²⋊C₄	C₂³	1	12	yes	—
45	C₂×C₃×C₂²⋊C₄	C₂³	1	24	yes	—
46	C₂×C₃×C₂²⋊C₄	C₂³	1	12	yes	—
47	C₂×C₃×C₄²	—	4	4	no	—
48	C₂×C₃×C₄²	C₂³	1	8	yes	complemented
49	C₃×#95	C₂³	1	12	yes	complemented
50	C₃×#95	C₂³	1	24	yes	complemented
51	C₃×#95	C₂³	1	4	yes	complemented
52	C₂⁶	C₂²×C₃	1	1	yes	characteristic
53	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
54	C₂²×C₄⋊C₄	C₂²×C₃	1	3	yes	complemented
55	C₂²×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
56	C₂²×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
57	C₂²×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
58	C₂²×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
59	C₂²×C₄²	C₂²×C₃	1	2	yes	complemented
60	#397	C₂²×C₃	1	24	yes	complemented
61	#393	C₂²×C₃	1	24	yes	complemented
62	#388	C₂²×C₃	1	48	yes	complemented
63	#379	C₂²×C₃	1	24	yes	—
64	C₂×#95	C₂²×C₃	1	6	yes	complemented
65	C₂×#95	C₂²×C₃	1	3	yes	complemented
66	C₂×#95	C₂²×C₃	1	1	yes	characteristic, complemented
67	C₂⁴×C₃	—	2	12	no	—
68	C₂⁴×C₃	—	2	6	no	—
69	C₂⁴×C₃	—	2	6	no	—
70	C₂⁴×C₃	—	2	12	no	—
71	C₂⁴×C₃	—	2	24	no	—
72	C₂⁴×C₃	—	4	12	no	—
73	C₂⁴×C₃	—	4	4	no	—
74	C₂⁴×C₃	—	4	48	no	—
75	C₂⁴×C₃	—	4	48	no	—
76	C₂⁴×C₃	—	4	16	no	—
77	C₂⁴×C₃	#41	1	12	yes	—
78	C₂⁴×C₃	C₂⁴	1	6	yes	—
79	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₃×Q₈	—	4	12	no	—
81	C₂×C₃×Q₈	—	4	16	no	—
82	C₂²×C₃×C₄	—	2	12	no	—
83	C₂²×C₃×C₄	—	2	12	no	—
84	C₂²×C₃×C₄	—	2	12	no	—
85	C₂²×C₃×C₄	—	2	12	no	—
86	C₂²×C₃×C₄	—	2	6	no	—
87	C₂²×C₃×C₄	—	2	6	no	—
88	C₂²×C₃×C₄	—	2	12	no	—
89	C₂²×C₃×C₄	C₂⁴	1	6	yes	—
90	C₂²×C₃×C₄	C₂⁴	1	6	yes	—
91	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
92	C₃×C₄⋊C₄	—	4	12	no	—
93	C₃×C₂²⋊C₄	—	2	48	no	—
94	C₃×C₂²⋊C₄	—	4	12	no	—
95	C₃×C₂²⋊C₄	—	4	24	no	—
96	C₃×C₂²⋊C₄	—	4	12	no	—
97	C₃×C₄²	—	4	8	no	—
98	C₂⁵	—	2	12	no	—
99	C₂⁵	—	4	6	no	—
100	C₂⁵	—	4	2	no	—
101	C₂⁵	C₂³×C₃	1	3	yes	—
102	C₂⁵	C₂³×C₃	1	4	yes	—
103	C₂²×Q₈	—	4	6	no	—
104	C₂²×Q₈	C₂³×C₃	1	4	yes	complemented
105	C₂³×C₄	C₂³×C₃	1	3	yes	—
106	C₂³×C₄	C₂³×C₃	1	6	yes	—
107	C₂³×C₄	C₂³×C₃	1	3	yes	—
108	C₂×C₄⋊C₄	—	4	6	no	—
109	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	complemented
110	C₂×C₂²⋊C₄	—	2	24	no	—
111	C₂×C₂²⋊C₄	—	4	6	no	—
112	C₂×C₂²⋊C₄	—	4	6	no	—
113	C₂×C₂²⋊C₄	—	4	12	no	—
114	C₂×C₂²⋊C₄	C₂³×C₃	1	24	yes	—
115	C₂×C₂²⋊C₄	C₂³×C₃	1	12	yes	—
116	C₂×C₂²⋊C₄	C₂³×C₃	1	24	yes	—
117	C₂×C₂²⋊C₄	C₂³×C₃	1	12	yes	—
118	C₂×C₄²	—	4	4	no	—
119	C₂×C₄²	C₂³×C₃	1	8	yes	complemented
120	#95	C₂³×C₃	1	12	yes	complemented
121	#95	C₂³×C₃	1	24	yes	complemented
122	#95	C₂³×C₃	1	4	yes	complemented
123	C₂³×C₃	—	2	24	no	—
124	C₂³×C₃	—	2	48	no	—
125	C₂³×C₃	—	2	12	no	—
126	C₂³×C₃	—	2	6	no	—
127	C₂³×C₃	—	2	6	no	—
128	C₂³×C₃	—	2	12	no	—
129	C₂³×C₃	—	4	48	no	—
130	C₂³×C₃	—	4	48	no	—
131	C₂³×C₃	—	4	128	no	—
132	C₂³×C₃	—	4	16	no	—
133	C₂³×C₃	—	4	24	no	—
134	C₂³×C₃	—	4	12	no	—
135	C₂³×C₃	—	4	12	no	—
136	C₂³×C₃	C₂×#41	1	12	yes	—
137	C₂³×C₃	C₂×#41	1	6	yes	—
138	C₂³×C₃	H(5,𝔽₂)	1	2	yes	—
139	C₂³×C₃	H(5,𝔽₂)	1	6	yes	—
140	C₂³×C₃	C₂⁵	1	1	yes	characteristic
141	C₃×Q₈	—	4	16	no	—
142	C₂×C₃×C₄	—	2	12	no	—
143	C₂×C₃×C₄	—	2	12	no	—
144	C₂×C₃×C₄	—	2	12	no	—
145	C₂×C₃×C₄	—	2	12	no	—
146	C₂×C₃×C₄	—	2	6	no	—
147	C₂×C₃×C₄	—	2	12	no	—
148	C₂×C₃×C₄	—	2	6	no	—
149	C₂×C₃×C₄	—	4	24	no	—
150	C₂×C₃×C₄	—	4	12	no	—
151	C₂×C₃×C₄	—	4	12	no	—
152	C₂⁴	—	2	12	no	—
153	C₂⁴	—	2	6	no	—
154	C₂⁴	—	2	6	no	—
155	C₂⁴	—	2	12	no	—
156	C₂⁴	—	2	24	no	—
157	C₂⁴	—	4	12	no	—
158	C₂⁴	—	4	4	no	—
159	C₂⁴	—	4	48	no	—
160	C₂⁴	—	4	48	no	—
161	C₂⁴	—	4	16	no	—
162	C₂⁴	C₃×#41	1	12	yes	—
163	C₂⁴	C₂⁴×C₃	1	6	yes	—
164	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
165	C₂×Q₈	—	4	12	no	—
166	C₂×Q₈	—	4	16	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	2	12	no	—
169	C₂²×C₄	—	2	12	no	—
170	C₂²×C₄	—	2	12	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	C₂⁴×C₃	1	6	yes	—
175	C₂²×C₄	C₂⁴×C₃	1	6	yes	—
176	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
177	C₄⋊C₄	—	4	12	no	—
178	C₂²⋊C₄	—	2	48	no	—
179	C₂²⋊C₄	—	4	12	no	—
180	C₂²⋊C₄	—	4	24	no	—
181	C₂²⋊C₄	—	4	12	no	—
182	C₄²	—	4	8	no	—
183	C₂²×C₃	—	2	12	no	—
184	C₂²×C₃	—	2	24	no	—
185	C₂²×C₃	—	4	64	no	—
186	C₂²×C₃	—	4	12	no	—
187	C₂²×C₃	—	4	6	no	—
188	C₂²×C₃	—	4	48	no	—
189	C₂²×C₃	—	4	6	no	—
190	C₂²×C₃	#538	1	12	yes	—
191	C₂²×C₃	#551	1	12	yes	—
192	C₂²×C₃	H(3,𝔽₄)	1	4	yes	—
193	C₂²×C₃	C₂²×#41	1	3	yes	—
194	C₂²×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
195	C₂²×C₃	C₂×H(5,𝔽₂)	1	3	yes	—
196	C₃×C₄	—	4	12	no	—
197	C₃×C₄	—	4	6	no	—
198	C₃×C₄	—	4	6	no	—
199	C₂³	—	2	12	no	—
200	C₂³	—	2	6	no	—
201	C₂³	—	2	6	no	—
202	C₂³	—	2	12	no	—
203	C₂³	—	2	24	no	—
204	C₂³	—	2	48	no	—
205	C₂³	—	4	48	no	—
206	C₂³	—	4	48	no	—
207	C₂³	—	4	16	no	—
208	C₂³	—	4	128	no	—
209	C₂³	—	4	12	no	—
210	C₂³	—	4	24	no	—
211	C₂³	—	4	12	no	—
212	C₂³	C₂×C₃×#41	1	12	yes	—
213	C₂³	C₂×C₃×#41	1	6	yes	—
214	C₂³	C₃×H(5,𝔽₂)	1	6	yes	—
215	C₂³	C₃×H(5,𝔽₂)	1	2	yes	—
216	C₂³	C₂⁵×C₃	1	1	yes	characteristic
217	Q₈	—	4	16	no	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	12	no	—
220	C₂×C₄	—	2	12	no	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	12	no	—
224	C₂×C₄	—	2	6	no	—
225	C₂×C₄	—	4	12	no	—
226	C₂×C₄	—	4	24	no	—
227	C₂×C₄	—	4	12	no	—
228	C₂×C₃	—	4	12	no	—
229	C₂×C₃	#2736	1	8	yes	direct product factor
230	C₂×C₃	C₂×#538	1	3	yes	—
231	C₂×C₃	C₂×#551	1	3	yes	—
232	C₂×C₃	C₂×H(3,𝔽₄)	1	1	yes	characteristic
233	C₂²	—	2	12	no	—
234	C₂²	—	2	24	no	—
235	C₂²	—	4	64	no	—
236	C₂²	—	4	12	no	—
237	C₂²	—	4	6	no	—
238	C₂²	—	4	6	no	—
239	C₂²	—	4	48	no	—
240	C₂²	C₃×#538	1	12	yes	—
241	C₂²	C₃×#551	1	12	yes	—
242	C₂²	C₃×H(3,𝔽₄)	1	4	yes	—
243	C₂²	C₂²×C₃×#41	1	3	yes	—
244	C₂²	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
245	C₂²	C₂×C₃×H(5,𝔽₂)	1	3	yes	—
246	C₄	—	4	12	no	—
247	C₄	—	4	6	no	—
248	C₄	—	4	6	no	—
249	C₃	C₂×#2736	1	1	yes	characteristic, direct product factor, Sylow
250	C₂	—	4	12	no	—
251	C₂	C₃×#2736	1	8	yes	direct product factor
252	C₂	C₂×C₃×#538	1	3	yes	—
253	C₂	C₂×C₃×#551	1	3	yes	—
254	C₂	C₂×C₃×H(3,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂⁶×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#2736	C₂	direct (N × H)
2. C₂×C₃×#397	C₂	semidirect
3. C₂×C₃×#393	C₂	semidirect
4. C₂×C₃×#388	C₂	semidirect
6. C₂×#2736	C₃	direct (N × H)
8. C₂³×C₃×Q₈	C₂²	semidirect
9. C₂²×C₃×C₄⋊C₄	C₂²	semidirect
14. C₂²×C₃×C₄²	C₂²	semidirect
15. C₃×#397	C₂²	semidirect
16. C₃×#393	C₂²	semidirect
17. C₃×#388	C₂²	semidirect
19. C₂×C₃×#95	C₂²	semidirect
20. C₂×C₃×#95	C₂²	semidirect
21. C₂×C₃×#95	C₂²	semidirect
22. #2736	C₂×C₃	direct (N × H)
23. C₂×#397	C₂×C₃	semidirect
24. C₂×#393	C₂×C₃	semidirect
25. C₂×#388	C₂×C₃	semidirect
33. C₂²×C₃×Q₈	C₂³	semidirect
38. C₂×C₃×C₄⋊C₄	C₂³	semidirect
48. C₂×C₃×C₄²	C₂³	semidirect
49. C₃×#95	C₂³	semidirect
50. C₃×#95	C₂³	semidirect
51. C₃×#95	C₂³	semidirect
53. C₂³×Q₈	C₂²×C₃	semidirect
54. C₂²×C₄⋊C₄	C₂²×C₃	semidirect
59. C₂²×C₄²	C₂²×C₃	semidirect
60. #397	C₂²×C₃	semidirect
61. #393	C₂²×C₃	semidirect
62. #388	C₂²×C₃	semidirect
64. C₂×#95	C₂²×C₃	semidirect
65. C₂×#95	C₂²×C₃	semidirect
66. C₂×#95	C₂²×C₃	semidirect
104. C₂²×Q₈	C₂³×C₃	semidirect
109. C₂×C₄⋊C₄	C₂³×C₃	semidirect
119. C₂×C₄²	C₂³×C₃	semidirect
120. #95	C₂³×C₃	semidirect
121. #95	C₂³×C₃	semidirect
122. #95	C₂³×C₃	semidirect
229. C₂×C₃	#2736	direct (N × H)
249. C₃	C₂×#2736	direct (N × H)
251. C₂	C₃×#2736	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×#2736	2
C₂×C₃×#397	2
C₂×C₃×#393	2
C₂×C₃×#388	2
C₂×C₃×#379	2
C₂×#2736	3

Subgroups — C2×C3×#2736 #10670778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2736 #10670778