Subgroups — C₃²×#4851 #13162268

Order 1728.

728 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#869	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃²×#817	C₂	1	2	yes	maximal
5	C₃²×#816	C₂	1	2	yes	maximal
6	C₃²×#475	—	3	1	no	maximal
7	C₃×#4851	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×Dic₆	—	2	1	no	—
9	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
10	C₂×C₃³×C₈	C₂²	1	2	yes	—
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃²×C₄⋊Dic₃	—	2	1	no	—
13	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₃²×Dic₃⋊C₄	—	2	1	no	—
16	C₃²×C₄×Dic₃	—	2	1	no	—
17	C₃²×C₄⋊Q₈	—	3	1	no	—
18	C₃²×C₄×Q₈	—	3	1	no	—
19	C₃×#869	C₂×C₃	1	4	yes	—
20	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	—
21	C₃²×#106	—	3	2	no	—
22	C₃²×C₄⋊C₈	—	2	4	no	—
23	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
24	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	—
25	C₃²×#103	—	3	2	no	—
26	C₃×#817	C₂×C₃	1	8	yes	—
27	C₃×#816	C₂×C₃	1	8	yes	—
28	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃²×Dic₃	—	2	1	no	—
32	C₂×C₃²×Dic₃	—	2	2	no	—
33	C₂×C₃²×Dic₃	—	2	1	no	—
34	C₃²×Dic₆	—	2	1	no	—
35	C₃²×Dic₆	—	4	2	no	—
36	C₃²×Dic₆	Q₈	1	2	yes	complemented
37	C₃³×C₈	—	2	2	no	—
38	C₃×#475	—	3	4	no	—
39	#4851	C₃²	1	1	yes	characteristic, direct product factor
40	C₂×C₃²×Q₈	—	3	1	no	—
41	C₂×C₃²×Q₈	—	6	1	no	—
42	C₂×C₃×Dic₆	—	2	4	no	—
43	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
44	C₂×C₃²×C₈	—	2	8	no	—
45	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
46	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
47	C₃²×C₄⋊C₄	—	3	1	no	—
48	C₃²×C₄⋊C₄	—	3	2	no	—
49	C₃²×C₄⋊C₄	—	6	1	no	—
50	C₃²×C₄⋊C₄	—	6	1	no	—
51	C₃²×C₄²	—	2	4	no	—
52	C₃²×C₄²	—	6	1	no	—
53	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₃²×C₄²	C₂²×C₃	1	4	yes	—
55	C₃×C₄⋊Dic₃	—	2	4	no	—
56	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
57	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
58	C₃×Dic₃⋊C₄	—	2	4	no	—
59	C₃×C₄×Dic₃	—	2	4	no	—
60	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
61	C₃³×C₄	—	2	1	no	—
62	C₃³×C₄	SD₁₆	1	2	yes	—
63	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
64	C₃³×C₄	#41	1	1	yes	characteristic
65	C₃²×Dic₃	—	2	2	no	—
66	C₃²×Dic₃	—	4	2	no	—
67	C₃²×Dic₃	—	4	1	no	—
68	C₃×C₄⋊Q₈	—	3	4	no	—
69	C₃×C₄×Q₈	—	3	4	no	—
70	#869	C₂×C₃²	1	1	yes	characteristic
71	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic
72	C₃×#106	—	3	8	no	—
73	C₃×C₄⋊C₈	—	2	4	no	—
74	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	—
75	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
76	C₃×#103	—	3	8	no	—
77	#817	C₂×C₃²	1	2	yes	—
78	#816	C₂×C₃²	1	2	yes	—
79	C₃²×Q₈	—	3	2	no	—
80	C₃²×Q₈	—	6	1	no	—
81	C₃²×Q₈	—	12	2	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	—	6	1	no	—
86	C₂×C₃²×C₄	—	6	1	no	—
87	C₂×C₃²×C₄	—	6	2	no	—
88	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
89	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
91	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
92	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
94	C₂×C₃×Dic₃	—	2	4	no	—
95	C₂×C₃×Dic₃	—	2	4	no	—
96	C₂×C₃×Dic₃	—	2	8	no	—
97	C₃×Dic₆	—	2	4	no	—
98	C₃×Dic₆	—	4	8	no	—
99	C₃×Dic₆	C₃×Q₈	1	8	yes	complemented
100	C₃²×C₈	—	2	2	no	—
101	C₃²×C₈	—	2	8	no	—
102	C₃²×C₈	—	4	8	no	—
103	#475	—	3	1	no	Sylow
104	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₃³	#137	1	1	yes	characteristic
107	C₂×C₃×Q₈	—	3	4	no	—
108	C₂×C₃×Q₈	—	6	4	no	—
109	C₂×Dic₆	—	2	1	no	—
110	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
111	C₂×C₃×C₈	—	2	8	no	—
112	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
113	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
114	C₃×C₄⋊C₄	—	3	8	no	—
115	C₃×C₄⋊C₄	—	3	4	no	—
116	C₃×C₄⋊C₄	—	6	4	no	—
117	C₃×C₄⋊C₄	—	6	4	no	—
118	C₃×C₄²	—	2	4	no	—
119	C₃×C₄²	—	6	4	no	—
120	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
121	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
122	C₄⋊Dic₃	—	2	1	no	—
123	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
124	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
125	Dic₃⋊C₄	—	2	1	no	—
126	C₄×Dic₃	—	2	1	no	—
127	C₂²×C₃²	—	2	4	no	—
128	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
129	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
130	C₃²×C₄	—	2	1	no	—
131	C₃²×C₄	—	2	8	no	—
132	C₃²×C₄	—	2	4	no	—
133	C₃²×C₄	—	2	4	no	—
134	C₃²×C₄	—	2	4	no	—
135	C₃²×C₄	—	4	4	no	—
136	C₃²×C₄	—	6	2	no	—
137	C₃²×C₄	—	12	2	no	—
138	C₃²×C₄	—	12	1	no	—
139	C₃²×C₄	#204	1	2	yes	—
140	C₃²×C₄	C₃×SD₁₆	1	8	yes	—
141	C₃²×C₄	#237	1	1	yes	characteristic
142	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
143	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
144	C₃²×C₄	C₃×#41	1	4	yes	—
145	C₃×Dic₃	—	2	8	no	—
146	C₃×Dic₃	—	4	8	no	—
147	C₃×Dic₃	—	4	4	no	—
148	C₄⋊Q₈	—	3	1	no	—
149	C₄×Q₈	—	3	1	no	—
150	#106	—	3	2	no	—
151	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic
152	#103	—	3	2	no	—
153	C₃³	#475	1	1	yes	characteristic, complemented, Sylow
154	C₃×Q₈	—	3	8	no	—
155	C₃×Q₈	—	6	4	no	—
156	C₃×Q₈	—	12	8	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	6	4	no	—
161	C₂×C₃×C₄	—	6	4	no	—
162	C₂×C₃×C₄	—	6	8	no	—
163	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
164	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
165	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
168	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
169	C₂×Dic₃	—	2	1	no	—
170	C₂×Dic₃	—	2	1	no	—
171	C₂×Dic₃	—	2	2	no	—
172	Dic₆	—	2	1	no	—
173	Dic₆	—	4	2	no	—
174	Dic₆	C₃²×Q₈	1	2	yes	complemented
175	C₃×C₈	—	2	8	no	—
176	C₃×C₈	—	2	2	no	—
177	C₃×C₈	—	4	8	no	—
178	C₂×C₃²	—	2	4	no	—
179	C₂×C₃²	—	2	4	no	—
180	C₂×C₃²	—	2	4	no	—
181	C₂×C₃²	#897	1	1	yes	characteristic
182	C₂×C₃²	C₂×#204	1	1	yes	characteristic
183	C₂×C₃²	#909	1	1	yes	characteristic
184	C₂×C₃²	C₃×C₂²⋊Q₈	1	4	yes	—
185	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
186	C₂×C₃²	C₃×#137	1	4	yes	—
187	C₂×Q₈	—	3	1	no	—
188	C₂×Q₈	—	6	1	no	—
189	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
190	C₄⋊C₄	—	3	2	no	—
191	C₄⋊C₄	—	3	1	no	—
192	C₄⋊C₄	—	6	1	no	—
193	C₄⋊C₄	—	6	1	no	—
194	C₄²	—	6	1	no	—
195	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
196	C₂²×C₃	—	2	4	no	—
197	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
198	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	—	2	4	no	—
203	C₃×C₄	—	2	8	no	—
204	C₃×C₄	—	4	4	no	—
205	C₃×C₄	—	6	8	no	—
206	C₃×C₄	—	12	8	no	—
207	C₃×C₄	—	12	4	no	—
208	C₃×C₄	C₃×#204	1	8	yes	—
209	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
210	C₃×C₄	C₃×#237	1	4	yes	—
211	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
212	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
213	C₃×C₄	C₃²×#41	1	1	yes	characteristic
214	Dic₃	—	2	2	no	—
215	Dic₃	—	4	2	no	—
216	Dic₃	—	4	1	no	—
217	C₃²	—	2	4	no	—
218	C₃²	#4851	1	1	yes	characteristic, direct product factor
219	C₃²	C₃×#475	1	4	yes	complemented
220	Q₈	—	3	2	no	—
221	Q₈	—	6	1	no	—
222	Q₈	—	12	2	no	—
223	C₂×C₄	—	6	1	no	—
224	C₂×C₄	—	6	1	no	—
225	C₂×C₄	—	6	2	no	—
226	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
227	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₂×C₃	—	2	4	no	—
231	C₂×C₃	—	2	4	no	—
232	C₂×C₃	—	2	4	no	—
233	C₂×C₃	C₃×#897	1	4	yes	—
234	C₂×C₃	C₂×C₃×#204	1	4	yes	—
235	C₂×C₃	C₃×#909	1	4	yes	—
236	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
238	C₂×C₃	C₃²×#137	1	1	yes	characteristic
239	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	6	2	no	—
242	C₄	—	12	1	no	—
243	C₄	—	12	2	no	—
244	C₄	C₃²×#204	1	2	yes	—
245	C₄	C₃²×#237	1	1	yes	characteristic
246	C₄	C₃²×D₃×Q₈	1	1	yes	characteristic
247	C₃	—	2	4	no	—
248	C₃	C₃×#4851	1	4	yes	direct product factor
249	C₃	C₃²×#475	1	1	yes	characteristic, complemented
250	C₂	C₃²×#897	1	1	yes	characteristic
251	C₂	C₂×C₃²×#204	1	1	yes	characteristic
252	C₂	C₃²×#909	1	1	yes	characteristic

	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, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₃²×C₄⋊C₈	D₃
hypocenter	C₃	C₃²×#475
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
7. C₃×#4851	C₃	direct (N × H)
36. C₃²×Dic₆	Q₈	semidirect
39. #4851	C₃²	direct (N × H)
99. C₃×Dic₆	C₃×Q₈	semidirect
153. C₃³	#475	semidirect
174. Dic₆	C₃²×Q₈	semidirect
218. C₃²	#4851	direct (N × H)
219. C₃²	C₃×#475	semidirect
248. C₃	C₃×#4851	direct (N × H)
249. C₃	C₃²×#475	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#869	2
C₃²×C₄×Dic₆	2
C₃³×C₄⋊C₈	2
C₃²×#817	2
C₃²×#816	2
C₃²×#475	3
C₃×#4851	3

Subgroups — C32×#4851 #13162268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4851 #13162268