Subgroups — C₃×#10589417 #13145975

Order 1728.

988 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	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal
3	#10589417	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×#4551	—	3	4	no	maximal
5	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₃⋊Dic₃	—	2	2	no	—
7	C₂×C₃³×C₈	—	2	2	no	—
8	C₂×C₃³×C₈	C₄	1	2	yes	complemented
9	C₃×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
10	C₃×C₄⋊(C₃⋊Dic₃)	C₄	1	4	yes	complemented
11	C₂²×C₃²×C₈	—	2	4	no	—
12	C₂²×C₃²×C₈	D₃	1	4	yes	—
13	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊Dic₃)	C₂×C₃	1	2	yes	—
15	C₂×C₃×C₄⋊Dic₃	—	3	8	no	—
16	C₂³×C₃³	D₄	1	1	yes	characteristic
17	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
19	C₂×C₃³×C₄	D₄	1	2	yes	—
20	C₂×C₃³×C₄	Q₈	1	1	yes	characteristic
21	C₂×C₃×C₃⋊Dic₃	—	2	4	no	—
22	C₂×C₃×C₃⋊Dic₃	—	4	2	no	—
23	C₂×C₃×C₃⋊Dic₃	—	4	2	no	—
24	C₃³×C₈	—	2	2	no	—
25	C₃×#340	—	9	1	no	—
26	#4551	—	3	4	no	—
27	C₂²×C₃²×C₄	—	2	4	no	—
28	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
29	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	—	2	2	no	—
31	C₂²×C₃×Dic₃	—	6	8	no	—
32	C₂×C₃²×C₈	—	2	16	no	—
33	C₂×C₃²×C₈	—	2	8	no	—
34	C₂×C₃²×C₈	—	2	8	no	—
35	C₂×C₃²×C₈	—	2	2	no	—
36	C₂×C₃²×C₈	Dic₃	1	8	yes	complemented
37	C₂×C₃²×C₈	C₃×C₄	1	2	yes	complemented
38	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
39	C₄⋊(C₃⋊Dic₃)	C₃×C₄	1	4	yes	complemented
40	C₃×C₄⋊Dic₃	—	3	16	no	—
41	C₃×C₄⋊Dic₃	—	6	8	no	—
42	C₂²×C₃³	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₃³	D₈	1	1	yes	characteristic
45	C₂²×C₃³	SD₁₆	1	2	yes	—
46	C₂²×C₃³	Q₁₆	1	1	yes	characteristic
47	C₃³×C₄	C₄²	1	1	yes	characteristic
48	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₃³×C₄	C₄⋊C₄	1	2	yes	—
50	C₃×C₃⋊Dic₃	—	4	4	no	—
51	C₂²×C₃×C₈	—	2	4	no	—
52	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
53	C₂²×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	—	9	2	no	—
55	C₂×C₄⋊Dic₃	—	3	8	no	—
56	C₂³×C₃²	—	2	4	no	—
57	C₂³×C₃²	D₁₂	1	4	yes	—
58	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	4	no	—
60	C₂×C₃²×C₄	—	2	8	no	—
61	C₂×C₃²×C₄	—	2	4	no	—
62	C₂×C₃²×C₄	—	2	8	no	—
63	C₂×C₃²×C₄	Dic₆	1	4	yes	—
64	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
65	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
66	C₂×C₃²×C₄	#67	1	8	yes	—
67	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
68	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
70	C₂×C₃²×C₄	C₃×Q₈	1	1	yes	characteristic
71	C₂×C₃⋊Dic₃	—	2	4	no	—
72	C₂×C₃⋊Dic₃	—	4	2	no	—
73	C₂×C₃⋊Dic₃	—	4	2	no	—
74	C₂×C₃×Dic₃	—	6	16	no	—
75	C₂×C₃×Dic₃	—	12	8	no	—
76	C₂×C₃×Dic₃	—	12	8	no	—
77	C₃²×C₈	—	2	16	no	—
78	C₃²×C₈	—	2	8	no	—
79	C₃²×C₈	—	2	2	no	—
80	#340	—	9	1	no	Sylow
81	C₂×C₃³	#95	1	1	yes	characteristic
82	C₂×C₃³	#102	1	2	yes	—
83	C₂×C₃³	#103	1	2	yes	—
84	C₂×C₃³	#106	1	1	yes	characteristic
85	C₂×C₃³	#107	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	4	no	—
87	C₂²×C₃×C₄	—	18	2	no	—
88	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
89	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
90	C₂²×Dic₃	—	6	8	no	—
91	C₂×C₃×C₈	—	2	16	no	—
92	C₂×C₃×C₈	—	2	8	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	8	no	—
95	C₂×C₃×C₈	C₃×Dic₃	1	8	yes	complemented
96	C₂×C₃×C₈	C₃⋊Dic₃	1	2	yes	complemented
97	C₃×C₄⋊C₄	—	9	4	no	—
98	C₃×C₄⋊C₄	—	18	2	no	—
99	C₄⋊Dic₃	—	3	16	no	—
100	C₄⋊Dic₃	—	6	8	no	—
101	C₂²×C₃²	—	2	8	no	—
102	C₂²×C₃²	—	2	4	no	—
103	C₂²×C₃²	—	2	4	no	—
104	C₂²×C₃²	—	2	4	no	—
105	C₂²×C₃²	—	2	8	no	—
106	C₂²×C₃²	#204	1	8	yes	—
107	C₂²×C₃²	D₂₄	1	4	yes	—
108	C₂²×C₃²	Dic₁₂	1	4	yes	—
109	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
110	C₂²×C₃²	(C₂×D₃)⋊C₄	1	8	yes	—
111	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
112	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₃²	C₃×D₈	1	1	yes	characteristic
114	C₂²×C₃²	C₃×SD₁₆	1	2	yes	—
115	C₂²×C₃²	C₃×Q₁₆	1	1	yes	characteristic
116	C₃²×C₄	—	2	4	no	—
117	C₃²×C₄	—	2	8	no	—
118	C₃²×C₄	—	2	4	no	—
119	C₃²×C₄	C₄×Dic₃	1	4	yes	—
120	C₃²×C₄	Dic₃⋊C₄	1	8	yes	—
121	C₃²×C₄	#217	1	4	yes	—
122	C₃²×C₄	C₃×C₄²	1	1	yes	characteristic
123	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
124	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
125	C₃⋊Dic₃	—	4	4	no	—
126	C₃×Dic₃	—	12	16	no	—
127	C₂²×C₈	C₃×C₃⋊D₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	9	2	no	—
129	C₃³	#340	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₃	—	2	4	no	—
131	C₂³×C₃	C₃×D₁₂	1	4	yes	—
132	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	8	no	—
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	2	8	no	—
137	C₂×C₃×C₄	—	18	4	no	—
138	C₂×C₃×C₄	—	36	2	no	—
139	C₂×C₃×C₄	—	36	2	no	—
140	C₂×C₃×C₄	C₃×Dic₆	1	4	yes	—
141	C₂×C₃×C₄	C₃×C₄×D₃	1	8	yes	—
142	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
143	C₂×C₃×C₄	C₃×#67	1	8	yes	—
144	C₂×C₃×C₄	#634	1	1	yes	characteristic
145	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
147	C₂×C₃×C₄	#638	1	2	yes	—
148	C₂×Dic₃	—	6	16	no	—
149	C₂×Dic₃	—	12	8	no	—
150	C₂×Dic₃	—	12	8	no	—
151	C₃×C₈	—	2	16	no	—
152	C₃×C₈	—	2	8	no	—
153	C₃×C₈	—	2	2	no	—
154	C₂×C₃²	—	2	8	no	—
155	C₂×C₃²	—	2	4	no	—
156	C₂×C₃²	—	2	4	no	—
157	C₂×C₃²	—	2	8	no	—
158	C₂×C₃²	—	2	4	no	—
159	C₂×C₃²	#816	1	8	yes	—
160	C₂×C₃²	#817	1	4	yes	—
161	C₂×C₃²	#818	1	4	yes	—
162	C₂×C₃²	#821	1	8	yes	—
163	C₂×C₃²	C₃⋊#95	1	4	yes	—
164	C₂×C₃²	C₃×#95	1	1	yes	characteristic
165	C₂×C₃²	C₃×#102	1	2	yes	—
166	C₂×C₃²	C₃×#103	1	2	yes	—
167	C₂×C₃²	C₃×#106	1	1	yes	characteristic
168	C₂×C₃²	C₃×#107	1	1	yes	characteristic
169	C₂²×C₄	—	18	2	no	—
170	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	C₃×C₃⋊Dic₃	1	2	yes	complemented
173	C₄⋊C₄	—	9	4	no	—
174	C₄⋊C₄	—	18	2	no	—
175	C₂²×C₃	—	2	8	no	—
176	C₂²×C₃	—	2	4	no	—
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	2	8	no	—
180	C₂²×C₃	C₃×#204	1	8	yes	—
181	C₂²×C₃	C₃×D₂₄	1	4	yes	—
182	C₂²×C₃	C₃×Dic₁₂	1	4	yes	—
183	C₂²×C₃	C₃×C₄⋊Dic₃	1	4	yes	—
184	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
185	C₂²×C₃	#3744	1	2	yes	—
186	C₂²×C₃	#3745	1	1	yes	characteristic
187	C₂²×C₃	#3746	1	1	yes	characteristic
188	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
189	C₂²×C₃	#3752	1	2	yes	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	2	8	no	—
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	36	4	no	—
194	C₃×C₄	C₃×C₄×Dic₃	1	4	yes	—
195	C₃×C₄	C₃×Dic₃⋊C₄	1	8	yes	—
196	C₃×C₄	C₃×#217	1	4	yes	—
197	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
198	C₃×C₄	#3750	1	2	yes	—
199	C₃×C₄	#3757	1	1	yes	characteristic
200	Dic₃	—	12	16	no	—
201	C₃²	—	2	4	no	—
202	C₃²	#4551	1	4	yes	complemented
203	C₃²	C₃×#340	1	1	yes	characteristic, complemented
204	C₂³	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
205	C₂×C₄	—	18	4	no	—
206	C₂×C₄	—	36	2	no	—
207	C₂×C₄	—	36	2	no	—
208	C₂×C₄	C₃×#634	1	1	yes	characteristic
209	C₂×C₄	C₃×C₄×C₃⋊D₃	1	2	yes	—
210	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
211	C₂×C₄	C₃×#638	1	2	yes	—
212	C₈	—	2	2	no	—
213	C₂×C₃	—	2	8	no	—
214	C₂×C₃	—	2	4	no	—
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	2	8	no	—
217	C₂×C₃	—	2	4	no	—
218	C₂×C₃	C₃×#816	1	8	yes	—
219	C₂×C₃	C₃×#817	1	4	yes	—
220	C₂×C₃	C₃×#818	1	4	yes	—
221	C₂×C₃	C₃×#821	1	8	yes	—
222	C₂×C₃	C₃×C₃⋊#95	1	4	yes	—
223	C₂×C₃	#63643	1	2	yes	—
224	C₂×C₃	#63644	1	1	yes	characteristic
225	C₂×C₃	#63645	1	1	yes	characteristic
226	C₂×C₃	#63648	1	2	yes	—
227	C₂×C₃	#63658	1	1	yes	characteristic
228	C₂²	C₃×#3744	1	2	yes	—
229	C₂²	C₃×#3745	1	1	yes	characteristic
230	C₂²	C₃×#3746	1	1	yes	characteristic
231	C₂²	C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
232	C₂²	C₃×#3752	1	2	yes	—
233	C₄	—	36	4	no	—
234	C₄	C₃×C₄×C₃⋊Dic₃	1	1	yes	characteristic
235	C₄	C₃×#3750	1	2	yes	—
236	C₄	C₃×#3757	1	1	yes	characteristic
237	C₃	—	2	4	no	—
238	C₃	C₃×#4551	1	4	yes	complemented
239	C₃	#10589417	1	1	yes	characteristic, direct product factor
240	C₂	C₃×#63643	1	2	yes	—
241	C₂	C₃×#63644	1	1	yes	characteristic
242	C₂	C₃×#63645	1	1	yes	characteristic
243	C₂	C₃×#63648	1	2	yes	—
244	C₂	C₃×#63658	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₃×C₄²
Frattini	C₂²×C₄	C₂×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₈	C₂
socle, prime radical	C₂³×C₃³	D₄
cosocle	C₂²×C₃²×C₄	C₂²×C₃
hypercenter	C₂²×C₃×C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#340
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. #10589417	C₃	direct (N × H)
8. C₂×C₃³×C₈	C₄	semidirect
10. C₃×C₄⋊(C₃⋊Dic₃)	C₄	semidirect
36. C₂×C₃²×C₈	Dic₃	semidirect
37. C₂×C₃²×C₈	C₃×C₄	semidirect
39. C₄⋊(C₃⋊Dic₃)	C₃×C₄	semidirect
95. C₂×C₃×C₈	C₃×Dic₃	semidirect
96. C₂×C₃×C₈	C₃⋊Dic₃	semidirect
129. C₃³	#340	semidirect
172. C₂×C₈	C₃×C₃⋊Dic₃	semidirect
202. C₃²	#4551	semidirect
203. C₃²	C₃×#340	semidirect
238. C₃	C₃×#4551	semidirect
239. C₃	#10589417	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
C₂×C₃×C₄⋊(C₃⋊Dic₃)	2
#10589417	3
C₃×#4551	3

Subgroups — C3×#10589417 #13145975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589417 #13145975