Subgroups — C₃³×#534 #13179989

Order 1728.

4844 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×D₄	C₂	1	2	yes	maximal, complemented
2	C₃³×#123	C₂	1	4	yes	maximal, complemented
3	C₃³×#121	C₂	1	4	yes	maximal, complemented
4	C₃³×#120	C₂	1	4	yes	maximal, complemented
5	C₂×C₃³×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#534	C₃	1	13	yes	maximal, direct product factor
7	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
8	C₂⁴×C₃³	C₂²	1	2	yes	complemented
9	C₂×C₃³×D₄	—	2	8	no	—
10	C₂×C₃³×D₄	C₂²	1	4	yes	—
11	C₂×C₃³×D₄	C₂²	1	4	yes	complemented
12	C₂×C₃³×D₄	C₂²	1	4	yes	complemented
13	C₂²×C₃³×C₄	C₂²	1	2	yes	complemented
14	C₂²×C₃³×C₄	C₂²	1	2	yes	complemented
15	C₃³×C₄⋊C₄	C₂²	1	4	yes	complemented
16	C₃³×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₃³×C₂²⋊C₄	C₂²	1	8	yes	complemented
18	C₂²×C₃²×D₄	C₂×C₃	1	26	yes	complemented
19	C₃²×#123	C₂×C₃	1	52	yes	complemented
20	C₃²×#121	C₂×C₃	1	52	yes	complemented
21	C₃²×#120	C₂×C₃	1	52	yes	complemented
22	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	13	yes	complemented
23	C₂³×C₃³	—	2	4	no	—
24	C₂³×C₃³	—	4	2	no	—
25	C₂³×C₃³	—	4	4	no	—
26	C₂³×C₃³	D₄	1	4	yes	complemented
27	C₂³×C₃³	C₂³	1	4	yes	—
28	C₂³×C₃³	C₂³	1	2	yes	—
29	C₂³×C₃³	C₂³	1	1	yes	characteristic
30	C₃³×D₄	—	2	8	no	—
31	C₃³×D₄	—	2	8	no	—
32	C₃³×D₄	—	4	4	no	—
33	C₂×C₃³×C₄	—	2	4	no	—
34	C₂×C₃³×C₄	—	4	2	no	—
35	C₂×C₃³×C₄	C₂³	1	4	yes	—
36	C₂×C₃³×C₄	C₂³	1	4	yes	—
37	C₃×#534	C₃²	1	13	yes	direct product factor
38	C₂⁴×C₃²	C₂²×C₃	1	13	yes	—
39	C₂⁴×C₃²	C₂²×C₃	1	26	yes	complemented
40	C₂×C₃²×D₄	—	2	104	no	—
41	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	—
42	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	complemented
43	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	complemented
44	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	complemented
45	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	complemented
46	C₃²×C₄⋊C₄	C₂²×C₃	1	52	yes	complemented
47	C₃²×C₂²⋊C₄	C₂²×C₃	1	52	yes	complemented
48	C₃²×C₂²⋊C₄	C₂²×C₃	1	104	yes	complemented
49	C₂²×C₃³	—	2	2	no	—
50	C₂²×C₃³	—	2	4	no	—
51	C₂²×C₃³	—	2	4	no	—
52	C₂²×C₃³	—	2	8	no	—
53	C₂²×C₃³	—	4	4	no	—
54	C₂²×C₃³	—	4	8	no	—
55	C₂²×C₃³	C₂×D₄	1	4	yes	—
56	C₂²×C₃³	C₂×D₄	1	2	yes	—
57	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
58	C₃³×C₄	—	2	4	no	—
59	C₃³×C₄	—	2	4	no	—
60	C₂²×C₃×D₄	C₂×C₃²	1	26	yes	complemented
61	C₃×#123	C₂×C₃²	1	52	yes	complemented
62	C₃×#121	C₂×C₃²	1	52	yes	complemented
63	C₃×#120	C₂×C₃²	1	52	yes	complemented
64	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	13	yes	complemented
65	C₂³×C₃²	—	2	52	no	—
66	C₂³×C₃²	—	4	26	no	—
67	C₂³×C₃²	—	4	52	no	—
68	C₂³×C₃²	C₃×D₄	1	52	yes	complemented
69	C₂³×C₃²	C₂³×C₃	1	52	yes	—
70	C₂³×C₃²	C₂³×C₃	1	26	yes	—
71	C₂³×C₃²	C₂³×C₃	1	13	yes	—
72	C₃²×D₄	—	2	104	no	—
73	C₃²×D₄	—	2	104	no	—
74	C₃²×D₄	—	4	52	no	—
75	C₂×C₃²×C₄	—	2	52	no	—
76	C₂×C₃²×C₄	—	4	26	no	—
77	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
78	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
79	#534	C₃³	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×C₃³	—	2	4	no	—
81	C₂×C₃³	—	2	2	no	—
82	C₂×C₃³	—	4	4	no	—
83	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₃³	H(5,𝔽₂)	1	2	yes	—
85	C₂⁴×C₃	C₂²×C₃²	1	13	yes	—
86	C₂⁴×C₃	C₂²×C₃²	1	26	yes	complemented
87	C₂×C₃×D₄	—	2	104	no	—
88	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	—
89	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	complemented
90	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	complemented
91	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	complemented
92	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	complemented
93	C₃×C₄⋊C₄	C₂²×C₃²	1	52	yes	complemented
94	C₃×C₂²⋊C₄	C₂²×C₃²	1	52	yes	complemented
95	C₃×C₂²⋊C₄	C₂²×C₃²	1	104	yes	complemented
96	C₂²×C₃²	—	2	26	no	—
97	C₂²×C₃²	—	2	52	no	—
98	C₂²×C₃²	—	2	52	no	—
99	C₂²×C₃²	—	2	104	no	—
100	C₂²×C₃²	—	4	52	no	—
101	C₂²×C₃²	—	4	104	no	—
102	C₂²×C₃²	C₂×C₃×D₄	1	52	yes	—
103	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
104	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
105	C₃²×C₄	—	2	52	no	—
106	C₃²×C₄	—	2	52	no	—
107	C₂²×D₄	C₂×C₃³	1	2	yes	complemented
108	#123	C₂×C₃³	1	4	yes	complemented
109	#121	C₂×C₃³	1	4	yes	complemented
110	#120	C₂×C₃³	1	4	yes	complemented
111	C₂×C₂²⋊C₄	C₂×C₃³	1	1	yes	characteristic, complemented
112	C₃³	#534	1	1	yes	characteristic, direct product factor, Sylow
113	C₂³×C₃	—	2	52	no	—
114	C₂³×C₃	—	4	26	no	—
115	C₂³×C₃	—	4	52	no	—
116	C₂³×C₃	C₃²×D₄	1	52	yes	complemented
117	C₂³×C₃	C₂³×C₃²	1	52	yes	—
118	C₂³×C₃	C₂³×C₃²	1	26	yes	—
119	C₂³×C₃	C₂³×C₃²	1	13	yes	—
120	C₃×D₄	—	2	104	no	—
121	C₃×D₄	—	2	104	no	—
122	C₃×D₄	—	4	52	no	—
123	C₂×C₃×C₄	—	2	52	no	—
124	C₂×C₃×C₄	—	4	26	no	—
125	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
126	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
127	C₂×C₃²	—	2	52	no	—
128	C₂×C₃²	—	2	26	no	—
129	C₂×C₃²	—	4	52	no	—
130	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
131	C₂×C₃²	C₃×H(5,𝔽₂)	1	26	yes	—
132	C₂⁴	C₂²×C₃³	1	1	yes	characteristic
133	C₂⁴	C₂²×C₃³	1	2	yes	complemented
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	C₂²×C₃³	1	4	yes	—
136	C₂×D₄	C₂²×C₃³	1	4	yes	complemented
137	C₂×D₄	C₂²×C₃³	1	4	yes	complemented
138	C₂²×C₄	C₂²×C₃³	1	2	yes	complemented
139	C₂²×C₄	C₂²×C₃³	1	2	yes	complemented
140	C₄⋊C₄	C₂²×C₃³	1	4	yes	complemented
141	C₂²⋊C₄	C₂²×C₃³	1	4	yes	complemented
142	C₂²⋊C₄	C₂²×C₃³	1	8	yes	complemented
143	C₂²×C₃	—	2	26	no	—
144	C₂²×C₃	—	2	52	no	—
145	C₂²×C₃	—	2	52	no	—
146	C₂²×C₃	—	2	104	no	—
147	C₂²×C₃	—	4	52	no	—
148	C₂²×C₃	—	4	104	no	—
149	C₂²×C₃	C₂×C₃²×D₄	1	52	yes	—
150	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
151	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
152	C₃×C₄	—	2	52	no	—
153	C₃×C₄	—	2	52	no	—
154	C₃²	C₃×#534	1	13	yes	direct product factor
155	C₂³	—	2	4	no	—
156	C₂³	—	4	2	no	—
157	C₂³	—	4	4	no	—
158	C₂³	C₃³×D₄	1	4	yes	complemented
159	C₂³	C₂³×C₃³	1	4	yes	—
160	C₂³	C₂³×C₃³	1	2	yes	—
161	C₂³	C₂³×C₃³	1	1	yes	characteristic
162	D₄	—	2	8	no	—
163	D₄	—	2	8	no	—
164	D₄	—	4	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	C₂³×C₃³	1	4	yes	—
168	C₂×C₄	C₂³×C₃³	1	4	yes	—
169	C₂×C₃	—	2	52	no	—
170	C₂×C₃	—	2	26	no	—
171	C₂×C₃	—	4	52	no	—
172	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
173	C₂×C₃	C₃²×H(5,𝔽₂)	1	26	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	8	no	—
178	C₂²	—	4	4	no	—
179	C₂²	—	4	8	no	—
180	C₂²	C₂×C₃³×D₄	1	4	yes	—
181	C₂²	C₂×C₃³×D₄	1	2	yes	—
182	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₃	C₃²×#534	1	13	yes	direct product factor
186	C₂	—	2	4	no	—
187	C₂	—	2	2	no	—
188	C₂	—	4	4	no	—
189	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic
190	C₂	C₃³×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×D₄	C₂	semidirect
2. C₃³×#123	C₂	semidirect
3. C₃³×#121	C₂	semidirect
4. C₃³×#120	C₂	semidirect
5. C₂×C₃³×C₂²⋊C₄	C₂	semidirect
6. C₃²×#534	C₃	direct (N × H)
8. C₂⁴×C₃³	C₂²	semidirect
11. C₂×C₃³×D₄	C₂²	semidirect
12. C₂×C₃³×D₄	C₂²	semidirect
13. C₂²×C₃³×C₄	C₂²	semidirect
14. C₂²×C₃³×C₄	C₂²	semidirect
15. C₃³×C₄⋊C₄	C₂²	semidirect
16. C₃³×C₂²⋊C₄	C₂²	semidirect
17. C₃³×C₂²⋊C₄	C₂²	semidirect
18. C₂²×C₃²×D₄	C₂×C₃	semidirect
19. C₃²×#123	C₂×C₃	semidirect
20. C₃²×#121	C₂×C₃	semidirect
21. C₃²×#120	C₂×C₃	semidirect
22. C₂×C₃²×C₂²⋊C₄	C₂×C₃	semidirect
26. C₂³×C₃³	D₄	semidirect
37. C₃×#534	C₃²	direct (N × H)
39. C₂⁴×C₃²	C₂²×C₃	semidirect
42. C₂×C₃²×D₄	C₂²×C₃	semidirect
43. C₂×C₃²×D₄	C₂²×C₃	semidirect
44. C₂²×C₃²×C₄	C₂²×C₃	semidirect
45. C₂²×C₃²×C₄	C₂²×C₃	semidirect
46. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
47. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
48. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
60. C₂²×C₃×D₄	C₂×C₃²	semidirect
61. C₃×#123	C₂×C₃²	semidirect
62. C₃×#121	C₂×C₃²	semidirect
63. C₃×#120	C₂×C₃²	semidirect
64. C₂×C₃×C₂²⋊C₄	C₂×C₃²	semidirect
68. C₂³×C₃²	C₃×D₄	semidirect
79. #534	C₃³	direct (N × H)
86. C₂⁴×C₃	C₂²×C₃²	semidirect
89. C₂×C₃×D₄	C₂²×C₃²	semidirect
90. C₂×C₃×D₄	C₂²×C₃²	semidirect
91. C₂²×C₃×C₄	C₂²×C₃²	semidirect
92. C₂²×C₃×C₄	C₂²×C₃²	semidirect
93. C₃×C₄⋊C₄	C₂²×C₃²	semidirect
94. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
95. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
107. C₂²×D₄	C₂×C₃³	semidirect
108. #123	C₂×C₃³	semidirect
109. #121	C₂×C₃³	semidirect
110. #120	C₂×C₃³	semidirect
111. C₂×C₂²⋊C₄	C₂×C₃³	semidirect
112. C₃³	#534	direct (N × H)
116. C₂³×C₃	C₃²×D₄	semidirect
133. C₂⁴	C₂²×C₃³	semidirect
136. C₂×D₄	C₂²×C₃³	semidirect
137. C₂×D₄	C₂²×C₃³	semidirect
138. C₂²×C₄	C₂²×C₃³	semidirect
139. C₂²×C₄	C₂²×C₃³	semidirect
140. C₄⋊C₄	C₂²×C₃³	semidirect
141. C₂²⋊C₄	C₂²×C₃³	semidirect
142. C₂²⋊C₄	C₂²×C₃³	semidirect
154. C₃²	C₃×#534	direct (N × H)
158. C₂³	C₃³×D₄	semidirect
185. C₃	C₃²×#534	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃³×D₄	2
C₃³×#123	2
C₃³×#121	2
C₃³×#120	2
C₂×C₃³×C₂²⋊C₄	2
C₃²×#534	3

Subgroups — C33×#534 #13179989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#534 #13179989