Subgroups — C₃²×#4696 #13162132

Order 1728.

956 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×D₂₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×#204	C₂	1	2	yes	maximal, complemented
4	C₃²×#875	C₂	1	2	yes	maximal, complemented
5	C₃³×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#495	—	3	1	no	maximal
7	C₃×#4696	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
9	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃²×(C₂×D₃)⋊C₄	—	2	4	no	—
14	C₃²×Dic₁₂	—	2	2	no	—
15	C₃²×D₂₄	—	2	2	no	—
16	C₃²×#204	—	2	4	no	—
17	C₂×C₃²×Q₁₆	—	3	1	no	—
18	C₂×C₃²×SD₁₆	—	3	2	no	—
19	C₂×C₃²×D₈	—	3	1	no	—
20	C₃²×#124	—	3	2	no	—
21	C₂×C₃×Dic₁₂	C₂×C₃	1	4	yes	complemented
22	C₂×C₃×D₂₄	C₂×C₃	1	4	yes	—
23	C₂×C₃×#204	C₂×C₃	1	8	yes	complemented
24	C₃×#875	C₂×C₃	1	8	yes	complemented
25	C₃²×C₄×C₈	—	2	4	no	—
26	C₃²×C₄×C₈	D₃	1	1	yes	characteristic, complemented
27	C₃²×C₄×C₈	C₂×C₃	1	4	yes	complemented
28	C₂²×C₃²×D₃	—	2	2	no	—
29	C₂×C₃³×C₄	D₄	1	2	yes	—
30	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃²×Dic₃	—	2	2	no	—
32	C₃²×D₁₂	—	2	2	no	—
33	C₃²×D₁₂	—	2	2	no	—
34	C₃²×Dic₆	—	2	2	no	—
35	C₃²×Dic₆	—	2	2	no	—
36	C₃³×C₈	D₄	1	2	yes	—
37	C₃³×C₈	D₄	1	2	yes	—
38	C₃×#495	—	3	4	no	—
39	#4696	C₃²	1	1	yes	characteristic, direct product factor
40	C₂×C₃²×Q₈	—	3	2	no	—
41	C₂×C₃²×D₄	—	3	2	no	—
42	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
43	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
44	C₃²×Q₁₆	—	6	2	no	—
45	C₃²×SD₁₆	—	6	4	no	—
46	C₃²×D₈	—	6	2	no	—
47	C₂×C₃²×C₈	—	2	4	no	—
48	C₂×C₃²×C₈	—	2	4	no	—
49	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
52	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
53	C₃²×C₂²⋊C₄	—	6	4	no	—
54	C₃²×C₄²	—	2	4	no	—
55	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₃²×C₄²	C₂²×C₃	1	4	yes	—
57	C₃×(C₂×D₃)⋊C₄	—	2	16	no	—
58	C₃×Dic₁₂	—	2	8	no	—
59	C₃×D₂₄	—	2	8	no	—
60	C₃×#204	—	2	16	no	—
61	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃²×D₃	—	4	2	no	—
63	C₂×C₃²×D₃	—	4	4	no	—
64	C₃³×C₄	—	2	2	no	—
65	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
66	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
67	C₃²×Dic₃	—	4	2	no	—
68	C₂×C₃×Q₁₆	—	3	4	no	—
69	C₂×C₃×SD₁₆	—	3	8	no	—
70	C₂×C₃×D₈	—	3	4	no	—
71	C₃×#124	—	3	8	no	—
72	C₂×Dic₁₂	C₂×C₃²	1	1	yes	characteristic, complemented
73	C₂×D₂₄	C₂×C₃²	1	1	yes	characteristic
74	C₂×#204	C₂×C₃²	1	2	yes	complemented
75	#875	C₂×C₃²	1	2	yes	complemented
76	C₃×C₄×C₈	—	2	4	no	—
77	C₃×C₄×C₈	C₃×D₃	1	4	yes	complemented
78	C₃×C₄×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
79	C₂³×C₃²	—	6	2	no	—
80	C₂²×C₃×D₃	—	2	8	no	—
81	C₃²×Q₈	—	6	2	no	—
82	C₃²×Q₈	—	6	2	no	—
83	C₃²×D₄	—	6	2	no	—
84	C₃²×D₄	—	6	2	no	—
85	C₂×C₃²×C₄	—	2	8	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	—	6	2	no	—
88	C₂×C₃²×C₄	D₁₂	1	2	yes	—
89	C₂×C₃²×C₄	C₃×D₄	1	8	yes	—
90	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
92	C₂×C₃×Dic₃	—	2	8	no	—
93	C₃×D₁₂	—	2	8	no	—
94	C₃×D₁₂	—	2	8	no	—
95	C₃×Dic₆	—	2	8	no	—
96	C₃×Dic₆	—	2	8	no	—
97	C₃²×C₈	—	2	8	no	—
98	C₃²×C₈	—	2	8	no	—
99	C₃²×C₈	D₁₂	1	2	yes	—
100	C₃²×C₈	D₁₂	1	2	yes	—
101	C₃²×C₈	C₃×D₄	1	8	yes	—
102	C₃²×C₈	C₃×D₄	1	8	yes	—
103	#495	—	3	1	no	Sylow
104	C₂×C₃³	#127	1	1	yes	characteristic
105	C₂×C₃³	#135	1	2	yes	—
106	C₃²×D₃	—	8	2	no	—
107	C₂×C₃×Q₈	—	3	8	no	—
108	C₂×C₃×D₄	—	3	8	no	—
109	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
110	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
111	C₃×Q₁₆	—	6	8	no	—
112	C₃×SD₁₆	—	6	16	no	—
113	C₃×D₈	—	6	8	no	—
114	C₂×C₃×C₈	—	2	4	no	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
117	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
118	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
120	C₃×C₂²⋊C₄	—	6	16	no	—
121	C₃×C₄²	—	2	4	no	—
122	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
123	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
124	(C₂×D₃)⋊C₄	—	2	4	no	—
125	Dic₁₂	—	2	2	no	—
126	D₂₄	—	2	2	no	—
127	#204	—	2	4	no	—
128	C₂²×C₃²	—	2	4	no	—
129	C₂²×C₃²	—	12	2	no	—
130	C₂²×C₃²	—	12	4	no	—
131	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
132	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
133	C₂×C₃×D₃	—	4	8	no	—
134	C₂×C₃×D₃	—	4	16	no	—
135	C₃²×C₄	—	2	16	no	—
136	C₃²×C₄	—	2	2	no	—
137	C₃²×C₄	—	2	8	no	—
138	C₃²×C₄	—	2	4	no	—
139	C₃²×C₄	—	2	4	no	—
140	C₃²×C₄	—	12	2	no	—
141	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
142	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
143	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
144	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
145	C₃×Dic₃	—	4	8	no	—
146	C₂×Q₁₆	—	3	1	no	—
147	C₂×SD₁₆	—	3	2	no	—
148	C₂×D₈	—	3	1	no	—
149	#124	—	3	2	no	—
150	C₄×C₈	C₃²×D₃	1	1	yes	characteristic, complemented
151	C₃³	#495	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₃	—	6	8	no	—
153	C₂²×D₃	—	2	2	no	—
154	C₃×Q₈	—	6	8	no	—
155	C₃×Q₈	—	6	8	no	—
156	C₃×D₄	—	6	8	no	—
157	C₃×D₄	—	6	8	no	—
158	C₂×C₃×C₄	—	2	8	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	6	8	no	—
161	C₂×C₃×C₄	C₃×D₁₂	1	8	yes	—
162	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
163	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
164	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
165	C₂×Dic₃	—	2	2	no	—
166	D₁₂	—	2	2	no	—
167	D₁₂	—	2	2	no	—
168	Dic₆	—	2	2	no	—
169	Dic₆	—	2	2	no	—
170	C₃×C₈	—	2	8	no	—
171	C₃×C₈	—	2	8	no	—
172	C₃×C₈	C₃×D₁₂	1	8	yes	—
173	C₃×C₈	C₃×D₁₂	1	8	yes	—
174	C₃×C₈	C₃²×D₄	1	2	yes	—
175	C₃×C₈	C₃²×D₄	1	2	yes	—
176	C₂×C₃²	—	2	8	no	—
177	C₂×C₃²	—	2	4	no	—
178	C₂×C₃²	—	24	2	no	—
179	C₂×C₃²	#874	1	1	yes	characteristic
180	C₂×C₃²	#904	1	2	yes	—
181	C₂×C₃²	C₃×#127	1	4	yes	—
182	C₂×C₃²	C₃×#135	1	8	yes	—
183	C₃×D₃	—	8	8	no	—
184	C₂×Q₈	—	3	2	no	—
185	C₂×D₄	—	3	2	no	—
186	Q₁₆	—	6	2	no	—
187	SD₁₆	—	6	4	no	—
188	D₈	—	6	2	no	—
189	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
190	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
191	C₂²⋊C₄	—	6	4	no	—
192	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
193	C₂²×C₃	—	2	4	no	—
194	C₂²×C₃	—	12	16	no	—
195	C₂²×C₃	—	12	8	no	—
196	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
197	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
198	C₂×D₃	—	4	2	no	—
199	C₂×D₃	—	4	4	no	—
200	C₃×C₄	—	2	16	no	—
201	C₃×C₄	—	2	8	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	2	4	no	—
205	C₃×C₄	—	12	8	no	—
206	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
207	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
208	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
209	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
210	Dic₃	—	4	2	no	—
211	C₃²	—	2	4	no	—
212	C₃²	#4696	1	1	yes	characteristic, direct product factor
213	C₃²	C₃×#495	1	4	yes	complemented
214	C₂³	—	6	2	no	—
215	Q₈	—	6	2	no	—
216	Q₈	—	6	2	no	—
217	D₄	—	6	2	no	—
218	D₄	—	6	2	no	—
219	C₂×C₄	—	6	2	no	—
220	C₂×C₄	C₃²×D₁₂	1	2	yes	—
221	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
222	C₈	C₃²×D₁₂	1	2	yes	—
223	C₈	C₃²×D₁₂	1	2	yes	—
224	C₂×C₃	—	2	4	no	—
225	C₂×C₃	—	2	8	no	—
226	C₂×C₃	—	24	8	no	—
227	C₂×C₃	C₃×#874	1	4	yes	—
228	C₂×C₃	C₃×#904	1	8	yes	—
229	C₂×C₃	C₃²×#127	1	1	yes	characteristic
230	C₂×C₃	C₃²×#135	1	2	yes	—
231	D₃	—	8	2	no	—
232	C₂²	—	12	2	no	—
233	C₂²	—	12	4	no	—
234	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	12	2	no	—
237	C₄	C₂×C₃²×D₁₂	1	1	yes	characteristic
238	C₄	C₂×C₃²×D₁₂	1	1	yes	characteristic
239	C₃	—	2	4	no	—
240	C₃	C₃×#4696	1	4	yes	direct product factor
241	C₃	C₃²×#495	1	1	yes	characteristic, complemented
242	C₂	—	24	2	no	—
243	C₂	C₃²×#874	1	1	yes	characteristic
244	C₂	C₃²×#904	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄×C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₂₄	C₂
hypercenter	C₃²×C₄×C₈	D₃
hypocenter	C₃	C₃²×#495
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×Dic₁₂	C₂	semidirect
3. C₂×C₃²×#204	C₂	semidirect
4. C₃²×#875	C₂	semidirect
5. C₃³×C₄×C₈	C₂	semidirect
7. C₃×#4696	C₃	direct (N × H)
21. C₂×C₃×Dic₁₂	C₂×C₃	semidirect
23. C₂×C₃×#204	C₂×C₃	semidirect
24. C₃×#875	C₂×C₃	semidirect
26. C₃²×C₄×C₈	D₃	semidirect
27. C₃²×C₄×C₈	C₂×C₃	semidirect
39. #4696	C₃²	direct (N × H)
72. C₂×Dic₁₂	C₂×C₃²	semidirect
74. C₂×#204	C₂×C₃²	semidirect
75. #875	C₂×C₃²	semidirect
77. C₃×C₄×C₈	C₃×D₃	semidirect
78. C₃×C₄×C₈	C₂×C₃²	semidirect
150. C₄×C₈	C₃²×D₃	semidirect
151. C₃³	#495	semidirect
212. C₃²	#4696	direct (N × H)
213. C₃²	C₃×#495	semidirect
240. C₃	C₃×#4696	direct (N × H)
241. C₃	C₃²×#495	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×Dic₁₂	2
C₂×C₃²×D₂₄	2
C₂×C₃²×#204	2
C₃²×#875	2
C₃³×C₄×C₈	2
C₃²×#495	3
C₃×#4696	3

Subgroups — C32×#4696 #13162132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4696 #13162132