Subgroups — C₃²×#5125 #13162561

Order 1728.

884 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#208	C₂	1	2	yes	maximal
3	C₃²×#819	C₂	1	4	yes	maximal, complemented
4	C₃²×#430	—	3	1	no	maximal
5	C₃×#5125	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₃³×M₁₆	C₄	1	4	yes	—
8	C₂×C₃³×C₈	C₂²	1	2	yes	—
9	C₃²×#208	—	2	2	no	—
10	C₃²×#208	C₂²	1	4	yes	—
11	C₂×C₃²×C₃⋊C₈	—	2	2	no	—
12	C₂×C₃²×M₁₆	—	2	4	no	—
13	C₂×C₃²×M₁₆	—	3	2	no	—
14	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
15	C₂×C₃²×M₁₆	C₂×C₃	1	4	yes	—
16	C₂×C₃×#208	C₂×C₃	1	8	yes	—
17	C₃²×#108	—	3	4	no	—
18	C₃×#819	C₂×C₃	1	16	yes	complemented
19	C₂³×C₃³	Q₈	1	1	yes	characteristic
20	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
21	C₂×C₃³×C₄	D₄	1	2	yes	—
22	C₂×C₃³×C₄	Q₈	1	1	yes	characteristic
23	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
24	C₃³×C₈	C₂×C₄	1	4	yes	—
25	C₃²×C₃⋊C₈	—	2	4	no	—
26	C₃×#430	—	3	4	no	—
27	#5125	C₃²	1	1	yes	characteristic, direct product factor
28	C₂²×C₃²×C₄	—	2	4	no	—
29	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
31	C₃²×M₁₆	—	2	16	no	—
32	C₃²×M₁₆	—	3	4	no	—
33	C₃²×M₁₆	—	6	2	no	—
34	C₃²×M₁₆	Dic₃	1	4	yes	—
35	C₃²×M₁₆	C₃×C₄	1	16	yes	—
36	C₂×C₃²×C₈	—	2	8	no	—
37	C₂×C₃²×C₈	—	6	2	no	—
38	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
39	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
40	C₃×#208	—	2	8	no	—
41	C₃×#208	C₂²×C₃	1	16	yes	—
42	C₂×C₃×C₃⋊C₈	—	2	8	no	—
43	C₂²×C₃³	—	4	1	no	—
44	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
45	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
46	C₃³×C₄	C₄⋊C₄	1	2	yes	—
47	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×M₁₆	—	2	4	no	—
50	C₂×C₃×M₁₆	—	3	8	no	—
51	C₂×C₃×M₁₆	C₃×D₃	1	4	yes	—
52	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
53	C₂×#208	C₂×C₃²	1	2	yes	—
54	C₃×#108	—	3	16	no	—
55	#819	C₂×C₃²	1	4	yes	complemented
56	C₂³×C₃²	—	2	4	no	—
57	C₂³×C₃²	Dic₆	1	1	yes	characteristic
58	C₂³×C₃²	C₃×Q₈	1	4	yes	—
59	C₂×C₃²×C₄	—	2	4	no	—
60	C₂×C₃²×C₄	—	2	8	no	—
61	C₂×C₃²×C₄	—	2	8	no	—
62	C₂×C₃²×C₄	—	2	4	no	—
63	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
64	C₂×C₃²×C₄	D₁₂	1	2	yes	—
65	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
66	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
67	C₂×C₃²×C₄	C₃×D₄	1	8	yes	—
68	C₂×C₃²×C₄	C₃×Q₈	1	4	yes	—
69	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
71	C₃²×C₈	—	2	16	no	—
72	C₃²×C₈	—	6	4	no	—
73	C₃²×C₈	C₂×Dic₃	1	4	yes	—
74	C₃²×C₈	C₂×C₃×C₄	1	16	yes	—
75	C₃×C₃⋊C₈	—	2	16	no	—
76	#430	—	3	1	no	Sylow
77	C₂×C₃³	—	2	1	no	—
78	C₂×C₃³	—	2	2	no	—
79	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₃×C₄	—	2	4	no	—
81	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
82	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
83	C₃×M₁₆	—	2	16	no	—
84	C₃×M₁₆	—	3	16	no	—
85	C₃×M₁₆	—	6	8	no	—
86	C₃×M₁₆	C₃×Dic₃	1	16	yes	—
87	C₃×M₁₆	C₃²×C₄	1	4	yes	—
88	C₂×C₃×C₈	—	2	8	no	—
89	C₂×C₃×C₈	—	6	8	no	—
90	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
91	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
92	#208	—	2	2	no	—
93	#208	C₂²×C₃²	1	4	yes	—
94	C₂×C₃⋊C₈	—	2	2	no	—
95	C₂²×C₃²	—	2	8	no	—
96	C₂²×C₃²	—	2	4	no	—
97	C₂²×C₃²	—	4	1	no	—
98	C₂²×C₃²	—	4	4	no	—
99	C₂²×C₃²	—	4	8	no	—
100	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
101	C₂²×C₃²	C₃×C₄⋊C₄	1	8	yes	—
102	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
103	C₂²×C₃²	C₂×C₃×Q₈	1	4	yes	—
104	C₃²×C₄	—	2	8	no	—
105	C₃²×C₄	—	2	4	no	—
106	C₃²×C₄	—	2	4	no	—
107	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
108	C₃²×C₄	C₃×C₄⋊C₄	1	8	yes	—
109	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
110	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
111	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
112	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
113	C₂×M₁₆	—	3	2	no	—
114	C₂×M₁₆	C₃²×D₃	1	1	yes	characteristic
115	#108	—	3	4	no	—
116	C₃³	#430	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₃	—	2	4	no	—
118	C₂³×C₃	C₃×Dic₆	1	4	yes	—
119	C₂³×C₃	C₃²×Q₈	1	1	yes	characteristic
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	2	8	no	—
122	C₂×C₃×C₄	—	2	8	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	C₃×Dic₆	1	4	yes	—
125	C₂×C₃×C₄	C₃×D₁₂	1	8	yes	—
126	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
127	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
128	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
129	C₂×C₃×C₄	C₃²×Q₈	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
131	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
132	C₃×C₈	—	2	16	no	—
133	C₃×C₈	—	6	16	no	—
134	C₃×C₈	C₂×C₃×Dic₃	1	16	yes	—
135	C₃×C₈	C₂×C₃²×C₄	1	4	yes	—
136	C₃⋊C₈	—	2	4	no	—
137	C₂×C₃²	—	2	8	no	—
138	C₂×C₃²	—	2	2	no	—
139	C₂×C₃²	—	2	1	no	—
140	C₂×C₃²	—	2	4	no	—
141	C₂×C₃²	—	2	4	no	—
142	C₂×C₃²	—	2	8	no	—
143	C₂×C₃²	—	4	8	no	—
144	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
145	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	4	yes	—
146	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
147	M₁₆	—	3	4	no	—
148	M₁₆	—	6	2	no	—
149	M₁₆	C₃²×Dic₃	1	4	yes	—
150	C₂×C₈	—	6	2	no	—
151	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
152	C₂²×C₃	—	2	4	no	—
153	C₂²×C₃	—	2	8	no	—
154	C₂²×C₃	—	4	4	no	—
155	C₂²×C₃	—	4	1	no	—
156	C₂²×C₃	—	4	8	no	—
157	C₂²×C₃	C₃×C₄⋊Dic₃	1	8	yes	—
158	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
159	C₂²×C₃	C₂×C₃×Dic₆	1	4	yes	—
160	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
161	C₃×C₄	—	2	4	no	—
162	C₃×C₄	—	2	4	no	—
163	C₃×C₄	—	2	8	no	—
164	C₃×C₄	C₃×C₄⋊Dic₃	1	8	yes	—
165	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
166	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
167	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
168	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
169	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
170	C₃²	—	2	4	no	—
171	C₃²	#5125	1	1	yes	characteristic, direct product factor
172	C₃²	C₃×#430	1	4	yes	complemented
173	C₂³	C₃²×Dic₆	1	1	yes	characteristic
174	C₂×C₄	C₃²×Dic₆	1	1	yes	characteristic
175	C₂×C₄	C₃²×D₁₂	1	2	yes	—
176	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
177	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
178	C₈	—	6	4	no	—
179	C₈	C₂×C₃²×Dic₃	1	4	yes	—
180	C₂×C₃	—	2	8	no	—
181	C₂×C₃	—	2	8	no	—
182	C₂×C₃	—	2	4	no	—
183	C₂×C₃	—	2	2	no	—
184	C₂×C₃	—	2	1	no	—
185	C₂×C₃	—	2	4	no	—
186	C₂×C₃	—	4	8	no	—
187	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	4	yes	—
188	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
189	C₂²	—	4	1	no	—
190	C₂²	C₃²×C₄⋊Dic₃	1	2	yes	—
191	C₂²	C₂×C₃²×Dic₆	1	1	yes	characteristic
192	C₄	C₃²×C₄⋊Dic₃	1	2	yes	—
193	C₄	C₂×C₃²×D₁₂	1	1	yes	characteristic
194	C₄	C₂²×C₃²×Dic₃	1	1	yes	characteristic
195	C₃	—	2	4	no	—
196	C₃	C₃×#5125	1	4	yes	direct product factor
197	C₃	C₃²×#430	1	1	yes	characteristic, complemented
198	C₂	—	2	1	no	—
199	C₂	—	2	2	no	—
200	C₂	C₂×C₃²×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₂×C₃³×M₁₆	C₂
socle	C₂×C₃³	C₂×C₄⋊C₄
prime radical	C₂³×C₃³	Q₈
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₂×C₃²×M₁₆	D₃
hypocenter	C₃	C₃²×#430
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ C₂×C₃²×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₃²×#819	C₂	semidirect
5. C₃×#5125	C₃	direct (N × H)
18. C₃×#819	C₂×C₃	semidirect
27. #5125	C₃²	direct (N × H)
55. #819	C₂×C₃²	semidirect
116. C₃³	#430	semidirect
171. C₃²	#5125	direct (N × H)
172. C₃²	C₃×#430	semidirect
196. C₃	C₃×#5125	direct (N × H)
197. C₃	C₃²×#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×M₁₆	2
C₂×C₃²×#208	2
C₃²×#819	2
C₃²×#430	3
C₃×#5125	3

Subgroups — C32×#5125 #13162561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5125 #13162561