Subgroups — C₃³×#532 #13179987

Order 1728.

4340 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×#41	C₂	1	2	yes	maximal, complemented
2	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×#124	C₂	1	2	yes	maximal, complemented
5	C₃³×#121	C₂	1	4	yes	maximal, complemented
6	C₃³×C₄×D₄	C₂	1	4	yes	maximal, complemented
7	C₂×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#532	C₃	1	13	yes	maximal, direct product factor
9	C₃³×#41	—	2	8	no	—
10	C₂×C₃³×Q₈	C₂²	1	2	yes	—
11	C₂×C₃³×D₄	C₂²	1	4	yes	—
12	C₂×C₃³×D₄	C₂²	1	2	yes	—
13	C₂×C₃³×D₄	C₂²	1	4	yes	—
14	C₂²×C₃³×C₄	C₂²	1	2	yes	—
15	C₂²×C₃³×C₄	C₂²	1	4	yes	—
16	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
17	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
18	C₃³×C₂²⋊C₄	C₂²	1	8	yes	—
19	C₃³×C₄²	C₂²	1	1	yes	characteristic
20	C₃³×C₄²	C₂²	1	1	yes	characteristic
21	C₃³×C₄²	C₂²	1	2	yes	—
22	C₂×C₃²×#41	C₂×C₃	1	26	yes	complemented
23	C₃²×C₄⋊Q₈	C₂×C₃	1	13	yes	complemented
24	C₃²×#127	C₂×C₃	1	13	yes	complemented
25	C₃²×#124	C₂×C₃	1	26	yes	complemented
26	C₃²×#121	C₂×C₃	1	52	yes	complemented
27	C₃²×C₄×D₄	C₂×C₃	1	52	yes	complemented
28	C₂×C₃²×C₄²	C₂×C₃	1	13	yes	complemented
29	C₂³×C₃³	C₂³	1	4	yes	—
30	C₂³×C₃³	C₂³	1	1	yes	characteristic
31	C₃³×Q₈	—	2	4	no	—
32	C₃³×D₄	—	2	8	no	—
33	C₃³×D₄	—	2	4	no	—
34	C₃³×D₄	—	2	8	no	—
35	C₂×C₃³×C₄	—	2	8	no	—
36	C₂×C₃³×C₄	—	2	4	no	—
37	C₂×C₃³×C₄	D₄	1	4	yes	complemented
38	C₂×C₃³×C₄	C₂³	1	2	yes	—
39	C₂×C₃³×C₄	C₂³	1	4	yes	—
40	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
41	C₂×C₃³×C₄	C₂³	1	2	yes	—
42	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
43	C₃×#532	C₃²	1	13	yes	direct product factor
44	C₃²×#41	—	2	104	no	—
45	C₂×C₃²×Q₈	C₂²×C₃	1	26	yes	—
46	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	—
47	C₂×C₃²×D₄	C₂²×C₃	1	26	yes	—
48	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	—
49	C₂²×C₃²×C₄	C₂²×C₃	1	52	yes	—
50	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
51	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
52	C₃²×C₄⋊C₄	C₂²×C₃	1	52	yes	—
53	C₃²×C₂²⋊C₄	C₂²×C₃	1	104	yes	—
54	C₃²×C₄²	C₂²×C₃	1	26	yes	—
55	C₃²×C₄²	C₂²×C₃	1	13	yes	—
56	C₃²×C₄²	C₂²×C₃	1	13	yes	—
57	C₂²×C₃³	—	2	2	no	—
58	C₂²×C₃³	—	2	4	no	—
59	C₂²×C₃³	—	2	8	no	—
60	C₂²×C₃³	C₂×D₄	1	2	yes	—
61	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
62	C₃³×C₄	—	2	4	no	—
63	C₃³×C₄	—	2	2	no	—
64	C₃³×C₄	C₂×D₄	1	2	yes	—
65	C₃³×C₄	C₂×D₄	1	2	yes	—
66	C₃³×C₄	#41	1	4	yes	complemented
67	C₂×C₃×#41	C₂×C₃²	1	26	yes	complemented
68	C₃×C₄⋊Q₈	C₂×C₃²	1	13	yes	complemented
69	C₃×#127	C₂×C₃²	1	13	yes	complemented
70	C₃×#124	C₂×C₃²	1	26	yes	complemented
71	C₃×#121	C₂×C₃²	1	52	yes	complemented
72	C₃×C₄×D₄	C₂×C₃²	1	52	yes	complemented
73	C₂×C₃×C₄²	C₂×C₃²	1	13	yes	complemented
74	C₂³×C₃²	C₂³×C₃	1	52	yes	—
75	C₂³×C₃²	C₂³×C₃	1	13	yes	—
76	C₃²×Q₈	—	2	52	no	—
77	C₃²×D₄	—	2	104	no	—
78	C₃²×D₄	—	2	52	no	—
79	C₃²×D₄	—	2	104	no	—
80	C₂×C₃²×C₄	—	2	104	no	—
81	C₂×C₃²×C₄	—	2	52	no	—
82	C₂×C₃²×C₄	C₃×D₄	1	52	yes	complemented
83	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
84	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
85	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
86	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
87	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
88	#532	C₃³	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×C₃³	—	2	2	no	—
90	C₂×C₃³	—	4	4	no	—
91	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₃³	C₂×#41	1	2	yes	—
93	C₃×#41	—	2	104	no	—
94	C₂×C₃×Q₈	C₂²×C₃²	1	26	yes	—
95	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	—
96	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	—
97	C₂×C₃×D₄	C₂²×C₃²	1	26	yes	—
98	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
99	C₂²×C₃×C₄	C₂²×C₃²	1	52	yes	—
100	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
101	C₃×C₄⋊C₄	C₂²×C₃²	1	52	yes	—
102	C₃×C₂²⋊C₄	C₂²×C₃²	1	104	yes	—
103	C₃×C₄²	C₂²×C₃²	1	26	yes	—
104	C₃×C₄²	C₂²×C₃²	1	13	yes	—
105	C₃×C₄²	C₂²×C₃²	1	13	yes	—
106	C₂²×C₃²	—	2	26	no	—
107	C₂²×C₃²	—	2	52	no	—
108	C₂²×C₃²	—	2	104	no	—
109	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
110	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
111	C₃²×C₄	—	2	52	no	—
112	C₃²×C₄	—	2	26	no	—
113	C₃²×C₄	C₂×C₃×D₄	1	26	yes	—
114	C₃²×C₄	C₂×C₃×D₄	1	26	yes	—
115	C₃²×C₄	C₃×#41	1	52	yes	complemented
116	C₂×#41	C₂×C₃³	1	2	yes	complemented
117	C₄⋊Q₈	C₂×C₃³	1	1	yes	characteristic, complemented
118	#127	C₂×C₃³	1	1	yes	characteristic, complemented
119	#124	C₂×C₃³	1	2	yes	complemented
120	#121	C₂×C₃³	1	4	yes	complemented
121	C₄×D₄	C₂×C₃³	1	4	yes	complemented
122	C₂×C₄²	C₂×C₃³	1	1	yes	characteristic, complemented
123	C₃³	#532	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₃	C₂³×C₃²	1	52	yes	—
125	C₂³×C₃	C₂³×C₃²	1	13	yes	—
126	C₃×Q₈	—	2	52	no	—
127	C₃×D₄	—	2	104	no	—
128	C₃×D₄	—	2	52	no	—
129	C₃×D₄	—	2	104	no	—
130	C₂×C₃×C₄	—	2	104	no	—
131	C₂×C₃×C₄	—	2	52	no	—
132	C₂×C₃×C₄	C₃²×D₄	1	52	yes	complemented
133	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
134	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
135	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
136	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
137	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
138	C₂×C₃²	—	2	26	no	—
139	C₂×C₃²	—	4	52	no	—
140	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
141	C₂×C₃²	C₂×C₃×#41	1	26	yes	—
142	#41	—	2	8	no	—
143	C₂×Q₈	C₂²×C₃³	1	2	yes	—
144	C₂×D₄	C₂²×C₃³	1	4	yes	—
145	C₂×D₄	C₂²×C₃³	1	4	yes	—
146	C₂×D₄	C₂²×C₃³	1	2	yes	—
147	C₂²×C₄	C₂²×C₃³	1	2	yes	—
148	C₂²×C₄	C₂²×C₃³	1	4	yes	—
149	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
150	C₄⋊C₄	C₂²×C₃³	1	4	yes	—
151	C₂²⋊C₄	C₂²×C₃³	1	8	yes	—
152	C₄²	C₂²×C₃³	1	2	yes	—
153	C₄²	C₂²×C₃³	1	1	yes	characteristic
154	C₄²	C₂²×C₃³	1	1	yes	characteristic
155	C₂²×C₃	—	2	26	no	—
156	C₂²×C₃	—	2	52	no	—
157	C₂²×C₃	—	2	104	no	—
158	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
159	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
160	C₃×C₄	—	2	52	no	—
161	C₃×C₄	—	2	26	no	—
162	C₃×C₄	C₂×C₃²×D₄	1	26	yes	—
163	C₃×C₄	C₂×C₃²×D₄	1	26	yes	—
164	C₃×C₄	C₃²×#41	1	52	yes	complemented
165	C₃²	C₃×#532	1	13	yes	direct product factor
166	C₂³	C₂³×C₃³	1	4	yes	—
167	C₂³	C₂³×C₃³	1	1	yes	characteristic
168	Q₈	—	2	4	no	—
169	D₄	—	2	8	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	8	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	C₃³×D₄	1	4	yes	complemented
175	C₂×C₄	C₂³×C₃³	1	2	yes	—
176	C₂×C₄	C₂³×C₃³	1	4	yes	—
177	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
178	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
179	C₂×C₄	C₂³×C₃³	1	2	yes	—
180	C₂×C₃	—	2	26	no	—
181	C₂×C₃	—	4	52	no	—
182	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
183	C₂×C₃	C₂×C₃²×#41	1	26	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	8	no	—
187	C₂²	C₂×C₃³×D₄	1	2	yes	—
188	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	C₂×C₃³×D₄	1	2	yes	—
192	C₄	C₂×C₃³×D₄	1	2	yes	—
193	C₄	C₃³×#41	1	4	yes	complemented
194	C₃	C₃²×#532	1	13	yes	direct product factor
195	C₂	—	2	2	no	—
196	C₂	—	4	4	no	—
197	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic
198	C₂	C₂×C₃³×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₃³×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
socle	C₂²×C₃³	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×#41	C₂	semidirect
2. C₃³×C₄⋊Q₈	C₂	semidirect
3. C₃³×#127	C₂	semidirect
4. C₃³×#124	C₂	semidirect
5. C₃³×#121	C₂	semidirect
6. C₃³×C₄×D₄	C₂	semidirect
7. C₂×C₃³×C₄²	C₂	semidirect
8. C₃²×#532	C₃	direct (N × H)
22. C₂×C₃²×#41	C₂×C₃	semidirect
23. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
24. C₃²×#127	C₂×C₃	semidirect
25. C₃²×#124	C₂×C₃	semidirect
26. C₃²×#121	C₂×C₃	semidirect
27. C₃²×C₄×D₄	C₂×C₃	semidirect
28. C₂×C₃²×C₄²	C₂×C₃	semidirect
37. C₂×C₃³×C₄	D₄	semidirect
43. C₃×#532	C₃²	direct (N × H)
66. C₃³×C₄	#41	semidirect
67. C₂×C₃×#41	C₂×C₃²	semidirect
68. C₃×C₄⋊Q₈	C₂×C₃²	semidirect
69. C₃×#127	C₂×C₃²	semidirect
70. C₃×#124	C₂×C₃²	semidirect
71. C₃×#121	C₂×C₃²	semidirect
72. C₃×C₄×D₄	C₂×C₃²	semidirect
73. C₂×C₃×C₄²	C₂×C₃²	semidirect
82. C₂×C₃²×C₄	C₃×D₄	semidirect
88. #532	C₃³	direct (N × H)
115. C₃²×C₄	C₃×#41	semidirect
116. C₂×#41	C₂×C₃³	semidirect
117. C₄⋊Q₈	C₂×C₃³	semidirect
118. #127	C₂×C₃³	semidirect
119. #124	C₂×C₃³	semidirect
120. #121	C₂×C₃³	semidirect
121. C₄×D₄	C₂×C₃³	semidirect
122. C₂×C₄²	C₂×C₃³	semidirect
123. C₃³	#532	direct (N × H)
132. C₂×C₃×C₄	C₃²×D₄	semidirect
164. C₃×C₄	C₃²×#41	semidirect
165. C₃²	C₃×#532	direct (N × H)
174. C₂×C₄	C₃³×D₄	semidirect
193. C₄	C₃³×#41	semidirect
194. C₃	C₃²×#532	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃³×#41	2
C₃³×C₄⋊Q₈	2
C₃³×#127	2
C₃³×#124	2
C₃³×#121	2
C₃³×C₄×D₄	2
C₂×C₃³×C₄²	2
C₃²×#532	3

Subgroups — C33×#532 #13179987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#532 #13179987