Subgroups — C₂×C₃²×#812 #13162533

Order 1728.

844 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	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×#812	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃²×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₃²×M₃₂	—	3	1	no	maximal
5	C₂×C₃×#812	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃³×C₄	C₄	1	1	yes	characteristic
7	C₂×C₃³×C₈	C₄	1	1	yes	characteristic
8	C₂×C₃³×C₈	C₄	1	2	yes	—
9	C₂×C₃³×C₈	C₂²	1	2	yes	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃²×C₃⋊C₁₆	C₂²	1	4	yes	complemented
12	C₂²×C₃²×C₈	—	2	4	no	—
13	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
14	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	—
15	C₃²×M₃₂	—	3	4	no	—
16	C₂×C₃²×C₁₆	—	3	2	no	—
17	C₃×#812	C₂×C₃	1	16	yes	direct product factor
18	C₂×C₃×C₃⋊C₁₆	C₂×C₃	1	8	yes	complemented
19	C₂³×C₃³	C₈	1	1	yes	characteristic
20	C₂×C₃³×C₄	C₈	1	2	yes	—
21	C₂×C₃³×C₄	C₈	1	1	yes	characteristic
22	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
23	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
24	C₃³×C₈	C₂×C₄	1	2	yes	—
25	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃³×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₃×M₃₂	—	3	4	no	—
28	C₂×#812	C₃²	1	1	yes	characteristic, direct product factor
29	C₂²×C₃²×C₄	—	2	4	no	—
30	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₃×C₄	1	4	yes	—
32	C₂×C₃²×C₈	—	2	4	no	—
33	C₂×C₃²×C₈	—	2	8	no	—
34	C₂×C₃²×C₈	—	2	4	no	—
35	C₂×C₃²×C₈	—	2	8	no	—
36	C₂×C₃²×C₈	Dic₃	1	1	yes	characteristic
37	C₂×C₃²×C₈	Dic₃	1	2	yes	—
38	C₂×C₃²×C₈	C₃×C₄	1	8	yes	—
39	C₂×C₃²×C₈	C₃×C₄	1	4	yes	—
40	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
41	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
43	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
44	C₃²×C₁₆	—	3	4	no	—
45	C₃×C₃⋊C₁₆	C₂²×C₃	1	16	yes	complemented
46	C₂²×C₃³	—	2	2	no	—
47	C₂²×C₃³	C₂×C₈	1	2	yes	—
48	C₂²×C₃³	C₂×C₈	1	1	yes	characteristic
49	C₃³×C₄	C₂×C₈	1	2	yes	—
50	C₃³×C₄	C₂×C₈	1	1	yes	characteristic
51	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₃×C₈	—	2	4	no	—
53	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
54	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
55	C₃×M₃₂	—	3	16	no	—
56	C₂×C₃×C₁₆	—	3	8	no	—
57	#812	C₂×C₃²	1	4	yes	direct product factor
58	C₂×C₃⋊C₁₆	C₂×C₃²	1	2	yes	complemented
59	C₂³×C₃²	—	2	4	no	—
60	C₂³×C₃²	C₃⋊C₈	1	1	yes	characteristic
61	C₂³×C₃²	C₃×C₈	1	4	yes	—
62	C₂×C₃²×C₄	—	2	8	no	—
63	C₂×C₃²×C₄	—	2	4	no	—
64	C₂×C₃²×C₄	—	2	8	no	—
65	C₂×C₃²×C₄	—	2	4	no	—
66	C₂×C₃²×C₄	C₃⋊C₈	1	2	yes	—
67	C₂×C₃²×C₄	C₃⋊C₈	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₃×C₈	1	8	yes	—
69	C₂×C₃²×C₄	C₃×C₈	1	4	yes	—
70	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
71	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
73	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
74	C₃²×C₈	—	2	8	no	—
75	C₃²×C₈	—	2	4	no	—
76	C₃²×C₈	—	2	4	no	—
77	C₃²×C₈	C₂×Dic₃	1	2	yes	—
78	C₃²×C₈	C₂×Dic₃	1	1	yes	characteristic
79	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
80	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
81	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
82	C₃²×C₈	C₂³×C₃	1	4	yes	—
83	C₂×M₃₂	—	3	1	no	Sylow
84	C₂×C₃³	—	2	2	no	—
85	C₂×C₃³	M₃₂	1	2	yes	complemented
86	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
87	C₂²×C₃×C₄	—	2	4	no	—
88	C₂²×C₃×C₄	C₃×Dic₃	1	4	yes	—
89	C₂²×C₃×C₄	C₃²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₈	—	2	8	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	—	2	8	no	—
94	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
95	C₂×C₃×C₈	C₃×Dic₃	1	8	yes	—
96	C₂×C₃×C₈	C₃²×C₄	1	2	yes	—
97	C₂×C₃×C₈	C₃²×C₄	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
99	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
100	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
102	C₃×C₁₆	—	3	16	no	—
103	C₃⋊C₁₆	C₂²×C₃²	1	4	yes	complemented
104	C₂²×C₃²	—	2	16	no	—
105	C₂²×C₃²	—	2	2	no	—
106	C₂²×C₃²	—	2	8	no	—
107	C₂²×C₃²	—	2	8	no	—
108	C₂²×C₃²	—	2	4	no	—
109	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
110	C₂²×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
111	C₂²×C₃²	C₂×C₃×C₈	1	8	yes	—
112	C₂²×C₃²	C₂×C₃×C₈	1	4	yes	—
113	C₃²×C₄	—	2	4	no	—
114	C₃²×C₄	—	2	8	no	—
115	C₃²×C₄	—	2	4	no	—
116	C₃²×C₄	C₂×C₃⋊C₈	1	2	yes	—
117	C₃²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
118	C₃²×C₄	C₂×C₃×C₈	1	4	yes	—
119	C₃²×C₄	C₂×C₃×C₈	1	8	yes	—
120	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
121	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
122	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic
123	M₃₂	—	3	4	no	—
124	C₂×C₁₆	—	3	2	no	—
125	C₃³	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₃	—	2	4	no	—
127	C₂³×C₃	C₃×C₃⋊C₈	1	4	yes	—
128	C₂³×C₃	C₃²×C₈	1	1	yes	characteristic
129	C₂×C₃×C₄	—	2	8	no	—
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	—	2	8	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	C₃×C₃⋊C₈	1	8	yes	—
134	C₂×C₃×C₄	C₃×C₃⋊C₈	1	4	yes	—
135	C₂×C₃×C₄	C₃²×C₈	1	2	yes	—
136	C₂×C₃×C₄	C₃²×C₈	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
138	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
139	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
141	C₃×C₈	—	2	4	no	—
142	C₃×C₈	—	2	8	no	—
143	C₃×C₈	—	2	4	no	—
144	C₃×C₈	C₂×C₃×Dic₃	1	8	yes	—
145	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
146	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
147	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
148	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
149	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
150	C₂×C₃²	—	2	16	no	—
151	C₂×C₃²	—	2	8	no	—
152	C₂×C₃²	—	2	2	no	—
153	C₂×C₃²	—	2	4	no	—
154	C₂×C₃²	—	2	8	no	—
155	C₂×C₃²	#812	1	2	yes	direct product factor
156	C₂×C₃²	C₃×M₃₂	1	8	yes	complemented
157	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
158	C₂×C₃²	C₂²×C₃×C₈	1	4	yes	—
159	C₂²×C₄	C₃²×Dic₃	1	1	yes	characteristic
160	C₂×C₈	C₃²×Dic₃	1	2	yes	—
161	C₂×C₈	C₃²×Dic₃	1	1	yes	characteristic
162	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
163	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
164	C₁₆	—	3	4	no	—
165	C₂²×C₃	—	2	16	no	—
166	C₂²×C₃	—	2	8	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	8	no	—
169	C₂²×C₃	—	2	4	no	—
170	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	8	yes	—
171	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	4	yes	—
172	C₂²×C₃	C₂×C₃²×C₈	1	2	yes	—
173	C₂²×C₃	C₂×C₃²×C₈	1	1	yes	characteristic
174	C₃×C₄	—	2	4	no	—
175	C₃×C₄	—	2	4	no	—
176	C₃×C₄	—	2	8	no	—
177	C₃×C₄	C₂×C₃×C₃⋊C₈	1	4	yes	—
178	C₃×C₄	C₂×C₃×C₃⋊C₈	1	8	yes	—
179	C₃×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
180	C₃×C₄	C₂×C₃²×C₈	1	2	yes	—
181	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
182	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
183	C₃²	—	2	4	no	—
184	C₃²	C₂×#812	1	1	yes	characteristic, direct product factor
185	C₃²	C₂×C₃×M₃₂	1	4	yes	complemented
186	C₂³	C₃²×C₃⋊C₈	1	1	yes	characteristic
187	C₂×C₄	C₃²×C₃⋊C₈	1	2	yes	—
188	C₂×C₄	C₃²×C₃⋊C₈	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
190	C₂×C₄	C₂×C₃²×Dic₃	1	1	yes	characteristic
191	C₈	C₂×C₃²×Dic₃	1	1	yes	characteristic
192	C₈	C₂×C₃²×Dic₃	1	2	yes	—
193	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
194	C₂×C₃	—	2	16	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	—	2	8	no	—
197	C₂×C₃	—	2	4	no	—
198	C₂×C₃	—	2	8	no	—
199	C₂×C₃	C₃×#812	1	8	yes	direct product factor
200	C₂×C₃	C₃²×M₃₂	1	2	yes	complemented
201	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	4	yes	—
202	C₂×C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	C₂×C₃²×C₃⋊C₈	1	2	yes	—
205	C₂²	C₂×C₃²×C₃⋊C₈	1	1	yes	characteristic
206	C₄	C₂×C₃²×C₃⋊C₈	1	1	yes	characteristic
207	C₄	C₂×C₃²×C₃⋊C₈	1	2	yes	—
208	C₄	C₂²×C₃²×Dic₃	1	1	yes	characteristic
209	C₃	—	2	4	no	—
210	C₃	C₂×C₃×#812	1	4	yes	direct product factor
211	C₃	C₂×C₃²×M₃₂	1	1	yes	characteristic, complemented
212	C₂	—	2	2	no	—
213	C₂	C₃²×#812	1	2	yes	direct product factor
214	C₂	C₂²×C₃²×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₈	C₂×D₃
commutator	C₂×C₃	C₂²×C₃²×C₈
Frattini	C₈	C₂²×C₃²×D₃
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂²×C₃³	C₂×C₈
prime radical	C₂³×C₃³	C₈
cosocle	C₃×C₈	C₂³×C₃²
hypercenter	C₂²×C₃²×C₈	D₃
hypocenter	C₃	C₂×C₃²×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₈ ◁ C₂²×C₃²×C₈

Complements

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

N	H ≅ G/N	type
2. C₃²×#812	C₂	direct (N × H)
3. C₂×C₃²×C₃⋊C₁₆	C₂	semidirect
5. C₂×C₃×#812	C₃	direct (N × H)
11. C₃²×C₃⋊C₁₆	C₂²	semidirect
17. C₃×#812	C₂×C₃	direct (N × H)
18. C₂×C₃×C₃⋊C₁₆	C₂×C₃	semidirect
28. C₂×#812	C₃²	direct (N × H)
45. C₃×C₃⋊C₁₆	C₂²×C₃	semidirect
57. #812	C₂×C₃²	direct (N × H)
58. C₂×C₃⋊C₁₆	C₂×C₃²	semidirect
85. C₂×C₃³	M₃₂	semidirect
103. C₃⋊C₁₆	C₂²×C₃²	semidirect
125. C₃³	C₂×M₃₂	semidirect
155. C₂×C₃²	#812	direct (N × H)
156. C₂×C₃²	C₃×M₃₂	semidirect
184. C₃²	C₂×#812	direct (N × H)
185. C₃²	C₂×C₃×M₃₂	semidirect
199. C₂×C₃	C₃×#812	direct (N × H)
200. C₂×C₃	C₃²×M₃₂	semidirect
210. C₃	C₂×C₃×#812	direct (N × H)
211. C₃	C₂×C₃²×M₃₂	semidirect
213. C₂	C₃²×#812	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×C₈	2
C₃²×#812	2
C₂×C₃²×C₃⋊C₁₆	2
C₂×C₃²×M₃₂	3
C₂×C₃×#812	3

Subgroups — C2×C32×#812 #13162533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#812 #13162533