Subgroups — C₃²×#4873 #13162327

Order 1728.

804 conjugacy classes of subgroups.

The subgroup lattice has 232 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
2	C₃²×#887	C₂	1	2	yes	maximal
3	C₃³×#107	C₂	1	1	yes	maximal, characteristic
4	C₃²×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₃²×#810	C₂	1	2	yes	maximal
6	C₃²×C₄×Q₁₆	—	3	1	no	maximal
7	C₃×#4873	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
10	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
11	C₃²×Dic₃⋊C₄	—	2	2	no	—
12	C₃²×C₄×Dic₃	—	2	2	no	—
13	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₃²×Dic₁₂	C₄	1	4	yes	complemented
16	C₂×C₃²×Q₁₆	—	3	1	no	—
17	C₃²×C₄×Q₈	—	3	2	no	—
18	C₂×C₃×Dic₁₂	C₂×C₃	1	4	yes	—
19	C₃×#887	C₂×C₃	1	8	yes	—
20	C₃²×#107	—	2	4	no	—
21	C₃²×#107	D₃	1	1	yes	characteristic
22	C₃²×#107	C₂×C₃	1	4	yes	—
23	C₃²×#103	—	3	2	no	—
24	C₃²×C₄×C₈	—	3	1	no	—
25	C₃×C₈×Dic₃	C₂×C₃	1	4	yes	—
26	C₃×#810	C₂×C₃	1	8	yes	—
27	C₂×C₃³×C₄	—	2	2	no	—
28	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃²×Dic₃	—	2	2	no	—
30	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
31	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
32	C₃²×Dic₆	—	2	2	no	—
33	C₃²×Dic₆	C₂×C₄	1	4	yes	—
34	C₃³×C₈	C₂×C₄	1	2	yes	—
35	C₃²×C₃⋊C₈	—	2	1	no	—
36	C₃×C₄×Q₁₆	—	3	4	no	—
37	#4873	C₃²	1	1	yes	characteristic, direct product factor
38	C₂×C₃²×Q₈	—	3	2	no	—
39	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
40	C₃²×Q₁₆	—	3	4	no	—
41	C₂×C₃²×C₈	—	2	4	no	—
42	C₂×C₃²×C₈	—	3	1	no	—
43	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
45	C₃²×C₄⋊C₄	—	2	8	no	—
46	C₃²×C₄⋊C₄	—	6	2	no	—
47	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
48	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
49	C₃²×C₄²	—	3	1	no	—
50	C₃²×C₄²	—	6	2	no	—
51	C₃×Dic₃⋊C₄	—	2	8	no	—
52	C₃×C₄×Dic₃	—	2	8	no	—
53	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
54	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
55	C₃×Dic₁₂	C₃×C₄	1	16	yes	complemented
56	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
57	C₃³×C₄	—	4	2	no	—
58	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₃³×C₄	#41	1	1	yes	characteristic
60	C₃²×Dic₃	—	2	4	no	—
61	C₃²×Dic₃	—	2	1	no	—
62	C₃²×Dic₃	Q₁₆	1	2	yes	complemented
63	C₂×C₃×Q₁₆	—	3	4	no	—
64	C₃×C₄×Q₈	—	3	8	no	—
65	C₂×Dic₁₂	C₂×C₃²	1	1	yes	characteristic
66	#887	C₂×C₃²	1	2	yes	—
67	C₃×#107	—	2	4	no	—
68	C₃×#107	C₃×D₃	1	4	yes	—
69	C₃×#107	C₂×C₃²	1	1	yes	characteristic
70	C₃×#103	—	3	8	no	—
71	C₃×C₄×C₈	—	3	4	no	—
72	C₈×Dic₃	C₂×C₃²	1	1	yes	characteristic
73	#810	C₂×C₃²	1	2	yes	—
74	C₃²×Q₈	—	3	4	no	—
75	C₃²×Q₈	—	6	2	no	—
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	—	2	4	no	—
78	C₂×C₃²×C₄	—	2	8	no	—
79	C₂×C₃²×C₄	—	3	1	no	—
80	C₂×C₃²×C₄	—	3	1	no	—
81	C₂×C₃²×C₄	—	4	8	no	—
82	C₂×C₃²×C₄	—	6	2	no	—
83	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
85	C₂×C₃×Dic₃	—	2	8	no	—
86	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
87	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
88	C₃×Dic₆	—	2	8	no	—
89	C₃×Dic₆	C₂×C₃×C₄	1	16	yes	—
90	C₃²×C₈	—	2	8	no	—
91	C₃²×C₈	—	6	1	no	—
92	C₃²×C₈	C₄×D₃	1	2	yes	—
93	C₃²×C₈	C₂×C₃×C₄	1	8	yes	—
94	C₃×C₃⋊C₈	—	2	4	no	—
95	C₄×Q₁₆	—	3	1	no	Sylow
96	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
97	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₃³	#135	1	1	yes	characteristic
99	C₂×C₃×Q₈	—	3	8	no	—
100	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
101	C₃×Q₁₆	—	3	16	no	—
102	C₂×C₃×C₈	—	2	4	no	—
103	C₂×C₃×C₈	—	3	4	no	—
104	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
105	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	2	8	no	—
107	C₃×C₄⋊C₄	—	6	8	no	—
108	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
109	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
110	C₃×C₄²	—	3	4	no	—
111	C₃×C₄²	—	6	8	no	—
112	Dic₃⋊C₄	—	2	2	no	—
113	C₄×Dic₃	—	2	2	no	—
114	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
115	C₂×C₃⋊C₈	C₂²×C₃²	1	1	yes	characteristic
116	Dic₁₂	C₃²×C₄	1	4	yes	complemented
117	C₂²×C₃²	—	2	4	no	—
118	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
119	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
120	C₃²×C₄	—	2	4	no	—
121	C₃²×C₄	—	2	4	no	—
122	C₃²×C₄	—	3	2	no	—
123	C₃²×C₄	—	4	2	no	—
124	C₃²×C₄	—	4	8	no	—
125	C₃²×C₄	—	6	4	no	—
126	C₃²×C₄	—	6	1	no	—
127	C₃²×C₄	—	8	8	no	—
128	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
129	C₃²×C₄	#239	1	1	yes	characteristic
130	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
131	C₃²×C₄	C₃×#41	1	4	yes	—
132	C₃×Dic₃	—	2	16	no	—
133	C₃×Dic₃	—	2	4	no	—
134	C₃×Dic₃	C₃×Q₁₆	1	8	yes	complemented
135	C₂×Q₁₆	—	3	1	no	—
136	C₄×Q₈	—	3	2	no	—
137	#107	C₃²×D₃	1	1	yes	characteristic
138	#103	—	3	2	no	—
139	C₄×C₈	—	3	1	no	—
140	C₃³	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
141	C₃×Q₈	—	3	16	no	—
142	C₃×Q₈	—	6	8	no	—
143	C₂×C₃×C₄	—	2	8	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	3	4	no	—
147	C₂×C₃×C₄	—	3	4	no	—
148	C₂×C₃×C₄	—	4	8	no	—
149	C₂×C₃×C₄	—	6	8	no	—
150	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
151	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
152	C₂×Dic₃	—	2	2	no	—
153	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
154	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
155	Dic₆	—	2	2	no	—
156	Dic₆	C₂×C₃²×C₄	1	4	yes	—
157	C₃×C₈	—	2	8	no	—
158	C₃×C₈	—	6	4	no	—
159	C₃×C₈	C₃×C₄×D₃	1	8	yes	—
160	C₃×C₈	C₂×C₃²×C₄	1	2	yes	—
161	C₃⋊C₈	—	2	1	no	—
162	C₂×C₃²	—	2	4	no	—
163	C₂×C₃²	—	2	4	no	—
164	C₂×C₃²	—	2	4	no	—
165	C₂×C₃²	#893	1	1	yes	characteristic
166	C₂×C₃²	#912	1	1	yes	characteristic
167	C₂×C₃²	D₃×Q₁₆	1	1	yes	characteristic
168	C₂×C₃²	C₃×C₄×D₄	1	4	yes	—
169	C₂×C₃²	C₂×C₃×Q₁₆	1	4	yes	—
170	C₂×C₃²	C₃×#135	1	4	yes	—
171	C₂×Q₈	—	3	2	no	—
172	Q₁₆	—	3	4	no	—
173	C₂×C₈	—	3	1	no	—
174	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
175	C₄⋊C₄	—	6	2	no	—
176	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
177	C₄²	—	3	1	no	—
178	C₄²	—	6	2	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
181	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
182	C₃×C₄	—	2	4	no	—
183	C₃×C₄	—	2	4	no	—
184	C₃×C₄	—	3	8	no	—
185	C₃×C₄	—	4	8	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	—	6	16	no	—
188	C₃×C₄	—	6	4	no	—
189	C₃×C₄	—	8	8	no	—
190	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
191	C₃×C₄	C₃×#239	1	4	yes	—
192	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
193	C₃×C₄	C₃²×#41	1	1	yes	characteristic
194	Dic₃	—	2	4	no	—
195	Dic₃	—	2	1	no	—
196	Dic₃	C₃²×Q₁₆	1	2	yes	complemented
197	C₃²	—	2	4	no	—
198	C₃²	#4873	1	1	yes	characteristic, direct product factor
199	C₃²	C₃×C₄×Q₁₆	1	4	yes	complemented
200	Q₈	—	3	4	no	—
201	Q₈	—	6	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	3	1	no	—
204	C₂×C₄	—	3	1	no	—
205	C₂×C₄	—	6	2	no	—
206	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
207	C₈	—	6	1	no	—
208	C₈	C₃²×C₄×D₃	1	2	yes	—
209	C₂×C₃	—	2	4	no	—
210	C₂×C₃	—	2	4	no	—
211	C₂×C₃	—	2	4	no	—
212	C₂×C₃	C₃×#893	1	4	yes	—
213	C₂×C₃	C₃×#912	1	4	yes	—
214	C₂×C₃	C₃×D₃×Q₁₆	1	4	yes	—
215	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
216	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
217	C₂×C₃	C₃²×#135	1	1	yes	characteristic
218	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
219	C₄	—	3	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	6	1	no	—
222	C₄	—	6	4	no	—
223	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
224	C₄	C₃²×#239	1	1	yes	characteristic
225	C₃	—	2	4	no	—
226	C₃	C₃×#4873	1	4	yes	direct product factor
227	C₃	C₃²×C₄×Q₁₆	1	1	yes	characteristic, complemented
228	C₂	C₃²×#893	1	1	yes	characteristic
229	C₂	C₃²×#912	1	1	yes	characteristic
230	C₂	C₃²×D₃×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	D₃×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#107	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×#107	D₃
hypocenter	C₃	C₃²×C₄×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#107

Complements

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

N	H ≅ G/N	type
7. C₃×#4873	C₃	direct (N × H)
15. C₃²×Dic₁₂	C₄	semidirect
37. #4873	C₃²	direct (N × H)
55. C₃×Dic₁₂	C₃×C₄	semidirect
62. C₃²×Dic₃	Q₁₆	semidirect
116. Dic₁₂	C₃²×C₄	semidirect
134. C₃×Dic₃	C₃×Q₁₆	semidirect
140. C₃³	C₄×Q₁₆	semidirect
196. Dic₃	C₃²×Q₁₆	semidirect
198. C₃²	#4873	direct (N × H)
199. C₃²	C₃×C₄×Q₁₆	semidirect
226. C₃	C₃×#4873	direct (N × H)
227. C₃	C₃²×C₄×Q₁₆	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×#887	2
C₃³×#107	2
C₃²×C₈×Dic₃	2
C₃²×#810	2
C₃²×C₄×Q₁₆	3
C₃×#4873	3

Subgroups — C32×#4873 #13162327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4873 #13162327