Subgroups — C₃³×#577 #13180032

Order 1728.

3612 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃³×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#137	C₂	1	3	yes	maximal, complemented
4	C₃³×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
5	C₃³×#135	C₂	1	3	yes	maximal, complemented
6	C₂×C₃³×SD₁₆	C₂	1	3	yes	maximal, complemented
7	C₃³×#131	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#577	C₃	1	13	yes	maximal, direct product factor
9	C₃³×#41	—	2	1	no	—
10	C₃³×#41	—	2	3	no	—
11	C₃³×#41	C₂²	1	3	yes	—
12	C₃³×#41	C₂²	1	1	yes	characteristic
13	C₃³×#41	C₂²	1	3	yes	complemented
14	C₂×C₃³×Q₈	—	2	1	no	—
15	C₂×C₃³×Q₈	C₂²	1	3	yes	complemented
16	C₂×C₃³×D₄	—	2	3	no	—
17	C₂×C₃³×D₄	C₂²	1	3	yes	—
18	C₃³×Q₁₆	C₂²	1	3	yes	complemented
19	C₃³×SD₁₆	C₂²	1	3	yes	complemented
20	C₃³×SD₁₆	C₂²	1	6	yes	complemented
21	C₃³×SD₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₃³×D₈	C₂²	1	3	yes	complemented
23	C₃³×M₁₆	C₂²	1	3	yes	complemented
24	C₂×C₃³×C₈	C₂²	1	3	yes	complemented
25	C₃²×2₋¹⁺⁴	C₂×C₃	1	13	yes	complemented
26	C₃²×H(5,𝔽₂)	C₂×C₃	1	13	yes	complemented
27	C₃²×#137	C₂×C₃	1	39	yes	complemented
28	C₃²×AGL(1,ℤ/8)	C₂×C₃	1	39	yes	complemented
29	C₃²×#135	C₂×C₃	1	39	yes	complemented
30	C₂×C₃²×SD₁₆	C₂×C₃	1	39	yes	complemented
31	C₃²×#131	C₂×C₃	1	13	yes	complemented
32	C₂³×C₃³	—	2	3	no	—
33	C₃³×Q₈	—	2	3	no	—
34	C₃³×Q₈	D₄	1	1	yes	characteristic
35	C₃³×Q₈	C₂³	1	3	yes	—
36	C₃³×Q₈	C₂³	1	1	yes	characteristic
37	C₃³×D₄	—	2	3	no	—
38	C₃³×D₄	—	2	3	no	—
39	C₃³×D₄	—	2	3	no	—
40	C₃³×D₄	D₄	1	3	yes	—
41	C₃³×D₄	C₂³	1	1	yes	characteristic
42	C₃³×D₄	C₂³	1	3	yes	—
43	C₂×C₃³×C₄	—	2	3	no	—
44	C₂×C₃³×C₄	—	2	3	no	—
45	C₂×C₃³×C₄	—	2	3	no	—
46	C₂×C₃³×C₄	C₂³	1	3	yes	—
47	C₃³×C₈	C₂³	1	1	yes	characteristic
48	C₃³×C₈	C₂³	1	3	yes	—
49	C₃×#577	C₃²	1	13	yes	direct product factor
50	C₃²×#41	—	2	13	no	—
51	C₃²×#41	—	2	39	no	—
52	C₃²×#41	C₂²×C₃	1	39	yes	—
53	C₃²×#41	C₂²×C₃	1	13	yes	—
54	C₃²×#41	C₂²×C₃	1	39	yes	complemented
55	C₂×C₃²×Q₈	—	2	13	no	—
56	C₂×C₃²×Q₈	C₂²×C₃	1	39	yes	complemented
57	C₂×C₃²×D₄	—	2	39	no	—
58	C₂×C₃²×D₄	C₂²×C₃	1	39	yes	—
59	C₃²×Q₁₆	C₂²×C₃	1	39	yes	complemented
60	C₃²×SD₁₆	C₂²×C₃	1	39	yes	complemented
61	C₃²×SD₁₆	C₂²×C₃	1	78	yes	complemented
62	C₃²×SD₁₆	C₂²×C₃	1	13	yes	complemented
63	C₃²×D₈	C₂²×C₃	1	39	yes	complemented
64	C₃²×M₁₆	C₂²×C₃	1	39	yes	complemented
65	C₂×C₃²×C₈	C₂²×C₃	1	39	yes	complemented
66	C₂²×C₃³	—	2	1	no	—
67	C₂²×C₃³	—	2	3	no	—
68	C₂²×C₃³	—	8	3	no	—
69	C₂²×C₃³	C₂×D₄	1	3	yes	—
70	C₃³×C₄	—	2	1	no	—
71	C₃³×C₄	—	2	3	no	—
72	C₃³×C₄	C₂×D₄	1	3	yes	—
73	C₃³×C₄	C₂⁴	1	1	yes	characteristic
74	C₃×2₋¹⁺⁴	C₂×C₃²	1	13	yes	complemented
75	C₃×H(5,𝔽₂)	C₂×C₃²	1	13	yes	complemented
76	C₃×#137	C₂×C₃²	1	39	yes	complemented
77	C₃×AGL(1,ℤ/8)	C₂×C₃²	1	39	yes	complemented
78	C₃×#135	C₂×C₃²	1	39	yes	complemented
79	C₂×C₃×SD₁₆	C₂×C₃²	1	39	yes	complemented
80	C₃×#131	C₂×C₃²	1	13	yes	complemented
81	C₂³×C₃²	—	2	39	no	—
82	C₃²×Q₈	—	2	39	no	—
83	C₃²×Q₈	C₃×D₄	1	13	yes	—
84	C₃²×Q₈	C₂³×C₃	1	39	yes	—
85	C₃²×Q₈	C₂³×C₃	1	13	yes	—
86	C₃²×D₄	—	2	39	no	—
87	C₃²×D₄	—	2	39	no	—
88	C₃²×D₄	—	2	39	no	—
89	C₃²×D₄	C₃×D₄	1	39	yes	—
90	C₃²×D₄	C₂³×C₃	1	13	yes	—
91	C₃²×D₄	C₂³×C₃	1	39	yes	—
92	C₂×C₃²×C₄	—	2	39	no	—
93	C₂×C₃²×C₄	—	2	39	no	—
94	C₂×C₃²×C₄	—	2	39	no	—
95	C₂×C₃²×C₄	C₂³×C₃	1	39	yes	—
96	C₃²×C₈	C₂³×C₃	1	13	yes	—
97	C₃²×C₈	C₂³×C₃	1	39	yes	—
98	#577	C₃³	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×C₃³	—	2	3	no	—
100	C₂×C₃³	—	4	1	no	—
101	C₂×C₃³	—	4	3	no	—
102	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
103	C₃×#41	—	2	13	no	—
104	C₃×#41	—	2	39	no	—
105	C₃×#41	C₂²×C₃²	1	39	yes	—
106	C₃×#41	C₂²×C₃²	1	13	yes	—
107	C₃×#41	C₂²×C₃²	1	39	yes	complemented
108	C₂×C₃×Q₈	—	2	13	no	—
109	C₂×C₃×Q₈	C₂²×C₃²	1	39	yes	complemented
110	C₂×C₃×D₄	—	2	39	no	—
111	C₂×C₃×D₄	C₂²×C₃²	1	39	yes	—
112	C₃×Q₁₆	C₂²×C₃²	1	39	yes	complemented
113	C₃×SD₁₆	C₂²×C₃²	1	39	yes	complemented
114	C₃×SD₁₆	C₂²×C₃²	1	78	yes	complemented
115	C₃×SD₁₆	C₂²×C₃²	1	13	yes	complemented
116	C₃×D₈	C₂²×C₃²	1	39	yes	complemented
117	C₃×M₁₆	C₂²×C₃²	1	39	yes	complemented
118	C₂×C₃×C₈	C₂²×C₃²	1	39	yes	complemented
119	C₂²×C₃²	—	2	13	no	—
120	C₂²×C₃²	—	2	39	no	—
121	C₂²×C₃²	—	8	39	no	—
122	C₂²×C₃²	C₂×C₃×D₄	1	39	yes	—
123	C₃²×C₄	—	2	13	no	—
124	C₃²×C₄	—	2	39	no	—
125	C₃²×C₄	C₂×C₃×D₄	1	39	yes	—
126	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
127	2₋¹⁺⁴	C₂×C₃³	1	1	yes	characteristic, complemented
128	H(5,𝔽₂)	C₂×C₃³	1	1	yes	characteristic, complemented
129	#137	C₂×C₃³	1	3	yes	complemented
130	AGL(1,ℤ/8)	C₂×C₃³	1	3	yes	complemented
131	#135	C₂×C₃³	1	3	yes	complemented
132	C₂×SD₁₆	C₂×C₃³	1	3	yes	complemented
133	#131	C₂×C₃³	1	1	yes	characteristic, complemented
134	C₃³	#577	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³×C₃	—	2	39	no	—
136	C₃×Q₈	—	2	39	no	—
137	C₃×Q₈	C₃²×D₄	1	13	yes	—
138	C₃×Q₈	C₂³×C₃²	1	39	yes	—
139	C₃×Q₈	C₂³×C₃²	1	13	yes	—
140	C₃×D₄	—	2	39	no	—
141	C₃×D₄	—	2	39	no	—
142	C₃×D₄	—	2	39	no	—
143	C₃×D₄	C₃²×D₄	1	39	yes	—
144	C₃×D₄	C₂³×C₃²	1	13	yes	—
145	C₃×D₄	C₂³×C₃²	1	39	yes	—
146	C₂×C₃×C₄	—	2	39	no	—
147	C₂×C₃×C₄	—	2	39	no	—
148	C₂×C₃×C₄	—	2	39	no	—
149	C₂×C₃×C₄	C₂³×C₃²	1	39	yes	—
150	C₃×C₈	C₂³×C₃²	1	13	yes	—
151	C₃×C₈	C₂³×C₃²	1	39	yes	—
152	C₂×C₃²	—	2	39	no	—
153	C₂×C₃²	—	4	13	no	—
154	C₂×C₃²	—	4	39	no	—
155	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
156	#41	—	2	1	no	—
157	#41	—	2	3	no	—
158	#41	C₂²×C₃³	1	3	yes	—
159	#41	C₂²×C₃³	1	1	yes	characteristic
160	#41	C₂²×C₃³	1	3	yes	complemented
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	C₂²×C₃³	1	3	yes	complemented
163	C₂×D₄	—	2	3	no	—
164	C₂×D₄	C₂²×C₃³	1	3	yes	—
165	Q₁₆	C₂²×C₃³	1	3	yes	complemented
166	SD₁₆	C₂²×C₃³	1	3	yes	complemented
167	SD₁₆	C₂²×C₃³	1	6	yes	complemented
168	SD₁₆	C₂²×C₃³	1	1	yes	characteristic, complemented
169	D₈	C₂²×C₃³	1	3	yes	complemented
170	M₁₆	C₂²×C₃³	1	3	yes	complemented
171	C₂×C₈	C₂²×C₃³	1	3	yes	complemented
172	C₂²×C₃	—	2	13	no	—
173	C₂²×C₃	—	2	39	no	—
174	C₂²×C₃	—	8	39	no	—
175	C₂²×C₃	C₂×C₃²×D₄	1	39	yes	—
176	C₃×C₄	—	2	13	no	—
177	C₃×C₄	—	2	39	no	—
178	C₃×C₄	C₂×C₃²×D₄	1	39	yes	—
179	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
180	C₃²	C₃×#577	1	13	yes	direct product factor
181	C₂³	—	2	3	no	—
182	Q₈	—	2	3	no	—
183	Q₈	C₃³×D₄	1	1	yes	characteristic
184	Q₈	C₂³×C₃³	1	3	yes	—
185	Q₈	C₂³×C₃³	1	1	yes	characteristic
186	D₄	—	2	3	no	—
187	D₄	—	2	3	no	—
188	D₄	—	2	3	no	—
189	D₄	C₃³×D₄	1	3	yes	—
190	D₄	C₂³×C₃³	1	1	yes	characteristic
191	D₄	C₂³×C₃³	1	3	yes	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	C₂³×C₃³	1	3	yes	—
196	C₈	C₂³×C₃³	1	1	yes	characteristic
197	C₈	C₂³×C₃³	1	3	yes	—
198	C₂×C₃	—	2	39	no	—
199	C₂×C₃	—	4	39	no	—
200	C₂×C₃	—	4	13	no	—
201	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	3	no	—
204	C₂²	—	8	3	no	—
205	C₂²	C₂×C₃³×D₄	1	3	yes	—
206	C₄	—	2	1	no	—
207	C₄	—	2	3	no	—
208	C₄	C₂×C₃³×D₄	1	3	yes	—
209	C₄	C₂⁴×C₃³	1	1	yes	characteristic
210	C₃	C₃²×#577	1	13	yes	direct product factor
211	C₂	—	2	3	no	—
212	C₂	—	4	3	no	—
213	C₂	—	4	1	no	—
214	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃³	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃³ ◁ C₃³×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃³×2₋¹⁺⁴	C₂	semidirect
2. C₃³×H(5,𝔽₂)	C₂	semidirect
3. C₃³×#137	C₂	semidirect
4. C₃³×AGL(1,ℤ/8)	C₂	semidirect
5. C₃³×#135	C₂	semidirect
6. C₂×C₃³×SD₁₆	C₂	semidirect
7. C₃³×#131	C₂	semidirect
8. C₃²×#577	C₃	direct (N × H)
13. C₃³×#41	C₂²	semidirect
15. C₂×C₃³×Q₈	C₂²	semidirect
18. C₃³×Q₁₆	C₂²	semidirect
19. C₃³×SD₁₆	C₂²	semidirect
20. C₃³×SD₁₆	C₂²	semidirect
21. C₃³×SD₁₆	C₂²	semidirect
22. C₃³×D₈	C₂²	semidirect
23. C₃³×M₁₆	C₂²	semidirect
24. C₂×C₃³×C₈	C₂²	semidirect
25. C₃²×2₋¹⁺⁴	C₂×C₃	semidirect
26. C₃²×H(5,𝔽₂)	C₂×C₃	semidirect
27. C₃²×#137	C₂×C₃	semidirect
28. C₃²×AGL(1,ℤ/8)	C₂×C₃	semidirect
29. C₃²×#135	C₂×C₃	semidirect
30. C₂×C₃²×SD₁₆	C₂×C₃	semidirect
31. C₃²×#131	C₂×C₃	semidirect
49. C₃×#577	C₃²	direct (N × H)
54. C₃²×#41	C₂²×C₃	semidirect
56. C₂×C₃²×Q₈	C₂²×C₃	semidirect
59. C₃²×Q₁₆	C₂²×C₃	semidirect
60. C₃²×SD₁₆	C₂²×C₃	semidirect
61. C₃²×SD₁₆	C₂²×C₃	semidirect
62. C₃²×SD₁₆	C₂²×C₃	semidirect
63. C₃²×D₈	C₂²×C₃	semidirect
64. C₃²×M₁₆	C₂²×C₃	semidirect
65. C₂×C₃²×C₈	C₂²×C₃	semidirect
74. C₃×2₋¹⁺⁴	C₂×C₃²	semidirect
75. C₃×H(5,𝔽₂)	C₂×C₃²	semidirect
76. C₃×#137	C₂×C₃²	semidirect
77. C₃×AGL(1,ℤ/8)	C₂×C₃²	semidirect
78. C₃×#135	C₂×C₃²	semidirect
79. C₂×C₃×SD₁₆	C₂×C₃²	semidirect
80. C₃×#131	C₂×C₃²	semidirect
98. #577	C₃³	direct (N × H)
107. C₃×#41	C₂²×C₃²	semidirect
109. C₂×C₃×Q₈	C₂²×C₃²	semidirect
112. C₃×Q₁₆	C₂²×C₃²	semidirect
113. C₃×SD₁₆	C₂²×C₃²	semidirect
114. C₃×SD₁₆	C₂²×C₃²	semidirect
115. C₃×SD₁₆	C₂²×C₃²	semidirect
116. C₃×D₈	C₂²×C₃²	semidirect
117. C₃×M₁₆	C₂²×C₃²	semidirect
118. C₂×C₃×C₈	C₂²×C₃²	semidirect
127. 2₋¹⁺⁴	C₂×C₃³	semidirect
128. H(5,𝔽₂)	C₂×C₃³	semidirect
129. #137	C₂×C₃³	semidirect
130. AGL(1,ℤ/8)	C₂×C₃³	semidirect
131. #135	C₂×C₃³	semidirect
132. C₂×SD₁₆	C₂×C₃³	semidirect
133. #131	C₂×C₃³	semidirect
134. C₃³	#577	direct (N × H)
160. #41	C₂²×C₃³	semidirect
162. C₂×Q₈	C₂²×C₃³	semidirect
165. Q₁₆	C₂²×C₃³	semidirect
166. SD₁₆	C₂²×C₃³	semidirect
167. SD₁₆	C₂²×C₃³	semidirect
168. SD₁₆	C₂²×C₃³	semidirect
169. D₈	C₂²×C₃³	semidirect
170. M₁₆	C₂²×C₃³	semidirect
171. C₂×C₈	C₂²×C₃³	semidirect
180. C₃²	C₃×#577	direct (N × H)
210. C₃	C₃²×#577	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×2₋¹⁺⁴	2
C₃³×H(5,𝔽₂)	2
C₃³×#137	2
C₃³×AGL(1,ℤ/8)	2
C₃³×#135	2
C₂×C₃³×SD₁₆	2
C₃³×#131	2
C₃²×#577	3

Subgroups — C33×#577 #13180032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#577 #13180032