Subgroups — C₃²×#4849 #13162287

Order 1728.

800 conjugacy classes of subgroups.

The subgroup lattice has 266 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	2	yes	maximal
2	C₃²×#869	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
4	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃²×#816	C₂	1	2	yes	maximal
6	C₃²×#462	—	3	1	no	maximal
7	C₃×#4849	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
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	—	2	1	no	—
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×Dic₁₂	—	2	4	no	—
18	C₂×C₃²×Q₁₆	—	3	2	no	—
19	C₃²×C₄⋊Q₈	—	3	1	no	—
20	C₃²×C₄×Q₈	—	3	1	no	—
21	C₂×C₃×Dic₁₂	C₂×C₃	1	8	yes	—
22	C₃×#869	C₂×C₃	1	4	yes	—
23	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	—
24	C₃²×C₄⋊C₈	—	2	4	no	—
25	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
26	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	—
27	C₃²×#103	—	3	2	no	—
28	C₃×#816	C₂×C₃	1	8	yes	—
29	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃²×Dic₃	—	2	1	no	—
33	C₂×C₃²×Dic₃	—	2	2	no	—
34	C₂×C₃²×Dic₃	—	2	1	no	—
35	C₃²×Dic₆	—	2	1	no	—
36	C₃²×Dic₆	—	2	2	no	—
37	C₃²×Dic₆	—	2	2	no	—
38	C₃²×Dic₆	D₄	1	2	yes	—
39	C₃³×C₈	—	2	2	no	—
40	C₃×#462	—	3	4	no	—
41	#4849	C₃²	1	1	yes	characteristic, direct product factor
42	C₂×C₃²×Q₈	—	3	1	no	—
43	C₂×C₃²×Q₈	—	3	2	no	—
44	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
45	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
46	C₃²×Q₁₆	—	6	4	no	—
47	C₂×C₃²×C₈	—	2	8	no	—
48	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
49	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
50	C₃²×C₄⋊C₄	—	3	1	no	—
51	C₃²×C₄⋊C₄	—	6	1	no	—
52	C₃²×C₄⋊C₄	—	6	1	no	—
53	C₃²×C₄⋊C₄	—	6	1	no	—
54	C₃²×C₄²	—	2	4	no	—
55	C₃²×C₄²	—	6	1	no	—
56	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₃²×C₄²	C₂²×C₃	1	4	yes	—
58	C₃×C₄⋊Dic₃	—	2	4	no	—
59	C₃×C₄⋊Dic₃	—	2	4	no	—
60	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
61	C₃×Dic₃⋊C₄	—	2	4	no	—
62	C₃×C₄×Dic₃	—	2	4	no	—
63	C₃×Dic₁₂	—	2	16	no	—
64	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
65	C₃³×C₄	—	2	1	no	—
66	C₃³×C₄	Q₁₆	1	2	yes	complemented
67	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
68	C₃³×C₄	#41	1	1	yes	characteristic
69	C₃²×Dic₃	—	2	2	no	—
70	C₃²×Dic₃	—	4	2	no	—
71	C₃²×Dic₃	—	4	1	no	—
72	C₂×C₃×Q₁₆	—	3	8	no	—
73	C₃×C₄⋊Q₈	—	3	4	no	—
74	C₃×C₄×Q₈	—	3	4	no	—
75	C₂×Dic₁₂	C₂×C₃²	1	2	yes	—
76	#869	C₂×C₃²	1	1	yes	characteristic
77	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic
78	C₃×C₄⋊C₈	—	2	4	no	—
79	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	—
80	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
81	C₃×#103	—	3	8	no	—
82	#816	C₂×C₃²	1	2	yes	—
83	C₃²×Q₈	—	3	2	no	—
84	C₃²×Q₈	—	6	1	no	—
85	C₃²×Q₈	—	6	2	no	—
86	C₃²×Q₈	—	6	2	no	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	—	2	4	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	6	1	no	—
91	C₂×C₃²×C₄	—	6	1	no	—
92	C₂×C₃²×C₄	—	6	2	no	—
93	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
94	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
96	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
97	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
99	C₂×C₃×Dic₃	—	2	4	no	—
100	C₂×C₃×Dic₃	—	2	4	no	—
101	C₂×C₃×Dic₃	—	2	8	no	—
102	C₃×Dic₆	—	2	4	no	—
103	C₃×Dic₆	—	2	8	no	—
104	C₃×Dic₆	—	2	8	no	—
105	C₃×Dic₆	C₃×D₄	1	8	yes	—
106	C₃²×C₈	—	2	2	no	—
107	C₃²×C₈	—	2	8	no	—
108	C₃²×C₈	—	4	8	no	—
109	#462	—	3	1	no	Sylow
110	C₂×C₃³	#121	1	1	yes	characteristic
111	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₃³	#137	1	1	yes	characteristic
113	C₂×C₃×Q₈	—	3	4	no	—
114	C₂×C₃×Q₈	—	3	8	no	—
115	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
116	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
117	C₃×Q₁₆	—	6	16	no	—
118	C₂×C₃×C₈	—	2	8	no	—
119	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
120	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
121	C₃×C₄⋊C₄	—	3	4	no	—
122	C₃×C₄⋊C₄	—	6	4	no	—
123	C₃×C₄⋊C₄	—	6	4	no	—
124	C₃×C₄⋊C₄	—	6	4	no	—
125	C₃×C₄²	—	2	4	no	—
126	C₃×C₄²	—	6	4	no	—
127	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
128	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
129	C₄⋊Dic₃	—	2	1	no	—
130	C₄⋊Dic₃	—	2	1	no	—
131	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
132	Dic₃⋊C₄	—	2	1	no	—
133	C₄×Dic₃	—	2	1	no	—
134	Dic₁₂	—	2	4	no	—
135	C₂²×C₃²	—	2	4	no	—
136	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
137	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
138	C₃²×C₄	—	2	1	no	—
139	C₃²×C₄	—	2	8	no	—
140	C₃²×C₄	—	2	4	no	—
141	C₃²×C₄	—	2	4	no	—
142	C₃²×C₄	—	2	4	no	—
143	C₃²×C₄	—	4	4	no	—
144	C₃²×C₄	—	6	2	no	—
145	C₃²×C₄	—	12	2	no	—
146	C₃²×C₄	—	12	1	no	—
147	C₃²×C₄	Dic₁₂	1	2	yes	complemented
148	C₃²×C₄	C₃×Q₁₆	1	8	yes	complemented
149	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
150	C₃²×C₄	#239	1	1	yes	characteristic
151	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
152	C₃²×C₄	C₃×#41	1	4	yes	—
153	C₃×Dic₃	—	2	8	no	—
154	C₃×Dic₃	—	4	8	no	—
155	C₃×Dic₃	—	4	4	no	—
156	C₂×Q₁₆	—	3	2	no	—
157	C₄⋊Q₈	—	3	1	no	—
158	C₄×Q₈	—	3	1	no	—
159	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic
160	#103	—	3	2	no	—
161	C₃³	#462	1	1	yes	characteristic, complemented, Sylow
162	C₃×Q₈	—	3	8	no	—
163	C₃×Q₈	—	6	4	no	—
164	C₃×Q₈	—	6	8	no	—
165	C₃×Q₈	—	6	8	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	6	4	no	—
170	C₂×C₃×C₄	—	6	4	no	—
171	C₂×C₃×C₄	—	6	8	no	—
172	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
173	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
174	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
177	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
178	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	2	2	no	—
181	Dic₆	—	2	1	no	—
182	Dic₆	—	2	2	no	—
183	Dic₆	—	2	2	no	—
184	Dic₆	C₃²×D₄	1	2	yes	—
185	C₃×C₈	—	2	8	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	—	4	8	no	—
188	C₂×C₃²	—	2	4	no	—
189	C₂×C₃²	—	2	4	no	—
190	C₂×C₃²	—	2	4	no	—
191	C₂×C₃²	#895	1	1	yes	characteristic
192	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
193	C₂×C₃²	#909	1	1	yes	characteristic
194	C₂×C₃²	C₃×#121	1	4	yes	—
195	C₂×C₃²	C₂×C₃×Q₁₆	1	4	yes	—
196	C₂×C₃²	C₃×#137	1	4	yes	—
197	C₂×Q₈	—	3	1	no	—
198	C₂×Q₈	—	3	2	no	—
199	Q₁₆	—	6	4	no	—
200	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
201	C₄⋊C₄	—	3	1	no	—
202	C₄⋊C₄	—	6	1	no	—
203	C₄⋊C₄	—	6	1	no	—
204	C₄⋊C₄	—	6	1	no	—
205	C₄²	—	6	1	no	—
206	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
209	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	2	8	no	—
215	C₃×C₄	—	4	4	no	—
216	C₃×C₄	—	6	8	no	—
217	C₃×C₄	—	12	8	no	—
218	C₃×C₄	—	12	4	no	—
219	C₃×C₄	C₃×Dic₁₂	1	8	yes	complemented
220	C₃×C₄	C₃²×Q₁₆	1	2	yes	complemented
221	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
222	C₃×C₄	C₃×#239	1	4	yes	—
223	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
224	C₃×C₄	C₃²×#41	1	1	yes	characteristic
225	Dic₃	—	2	2	no	—
226	Dic₃	—	4	2	no	—
227	Dic₃	—	4	1	no	—
228	C₃²	—	2	4	no	—
229	C₃²	#4849	1	1	yes	characteristic, direct product factor
230	C₃²	C₃×#462	1	4	yes	complemented
231	Q₈	—	3	2	no	—
232	Q₈	—	6	1	no	—
233	Q₈	—	6	2	no	—
234	Q₈	—	6	2	no	—
235	C₂×C₄	—	6	1	no	—
236	C₂×C₄	—	6	1	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
239	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₂×C₃	—	2	4	no	—
243	C₂×C₃	—	2	4	no	—
244	C₂×C₃	—	2	4	no	—
245	C₂×C₃	C₃×#895	1	4	yes	—
246	C₂×C₃	C₂×C₃×Dic₁₂	1	4	yes	—
247	C₂×C₃	C₃×#909	1	4	yes	—
248	C₂×C₃	C₃²×#121	1	1	yes	characteristic
249	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
250	C₂×C₃	C₃²×#137	1	1	yes	characteristic
251	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	6	2	no	—
254	C₄	—	12	1	no	—
255	C₄	—	12	2	no	—
256	C₄	C₃²×Dic₁₂	1	2	yes	complemented
257	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
258	C₄	C₃²×#239	1	1	yes	characteristic
259	C₃	—	2	4	no	—
260	C₃	C₃×#4849	1	4	yes	direct product factor
261	C₃	C₃²×#462	1	1	yes	characteristic, complemented
262	C₂	C₃²×#895	1	1	yes	characteristic
263	C₂	C₂×C₃²×Dic₁₂	1	1	yes	characteristic
264	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×#462
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×#4849	C₃	direct (N × H)
41. #4849	C₃²	direct (N × H)
66. C₃³×C₄	Q₁₆	semidirect
147. C₃²×C₄	Dic₁₂	semidirect
148. C₃²×C₄	C₃×Q₁₆	semidirect
161. C₃³	#462	semidirect
219. C₃×C₄	C₃×Dic₁₂	semidirect
220. C₃×C₄	C₃²×Q₁₆	semidirect
229. C₃²	#4849	direct (N × H)
230. C₃²	C₃×#462	semidirect
256. C₄	C₃²×Dic₁₂	semidirect
260. C₃	C₃×#4849	direct (N × H)
261. C₃	C₃²×#462	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
C₃²×#869	2
C₃²×C₄×Dic₆	2
C₃³×C₄⋊C₈	2
C₃²×#816	2
C₃²×#462	3
C₃×#4849	3

Subgroups — C32×#4849 #13162287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4849 #13162287