Subgroups — C₃×#10592723 #13163062

Order 1728.

844 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	2	yes	maximal
3	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
4	C₃×#63630	C₂	1	2	yes	maximal
5	C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#10592723	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5024	—	3	4	no	maximal
8	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic
9	C₃³×C₄⋊C₄	C₄	1	2	yes	—
10	C₃³×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃³×C₄²	C₂²	1	2	yes	—
13	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₃²×C₄×Q₈	—	2	4	no	—
17	C₃²×C₄×Q₈	D₃	1	4	yes	—
18	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	2	yes	—
20	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	1	yes	characteristic
21	#63630	C₂×C₃	1	2	yes	—
22	C₄×C₃⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
23	C₃×C₄⋊(C₃⋊C₈)	—	3	8	no	—
24	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
25	C₃×#803	—	3	8	no	—
26	C₃×C₄×C₃⋊C₈	—	3	4	no	—
27	C₃³×Q₈	—	2	2	no	—
28	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
30	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
31	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
34	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
35	C₃×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	complemented
36	C₃×#446	—	9	1	no	—
37	#5024	—	3	4	no	—
38	C₂×C₃²×Q₈	—	2	4	no	—
39	C₂×C₃²×Q₈	Dic₃	1	4	yes	—
40	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic
41	C₃²×C₄⋊C₄	—	2	8	no	—
42	C₃²×C₄⋊C₄	—	2	4	no	—
43	C₃²×C₄⋊C₄	Dic₃	1	8	yes	—
44	C₃²×C₄⋊C₄	Dic₃	1	4	yes	—
45	C₃²×C₄⋊C₄	C₃×C₄	1	2	yes	—
46	C₃²×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
47	C₃²×C₄²	—	2	4	no	—
48	C₃²×C₄²	—	2	8	no	—
49	C₃²×C₄²	C₂×D₃	1	4	yes	—
50	C₃²×C₄²	C₂×D₃	1	8	yes	—
51	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₃²×C₄²	C₂²×C₃	1	2	yes	—
53	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
55	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	—	3	4	no	—
57	C₂×C₃×C₃⋊C₈	—	3	4	no	—
58	C₂×C₃×C₃⋊C₈	—	3	8	no	—
59	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
60	C₃³×C₄	—	2	2	no	—
61	C₃³×C₄	—	2	1	no	—
62	C₃³×C₄	—	2	2	no	—
63	C₃³×C₄	M₁₆	1	2	yes	—
64	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₃³×C₄	#41	1	1	yes	characteristic
66	C₃×C₄×Q₈	—	2	4	no	—
67	C₃×C₄×Q₈	C₃×D₃	1	4	yes	—
68	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
69	C₃×C₄⋊C₈	—	9	1	no	—
70	C₃×C₄⋊C₈	—	9	2	no	—
71	C₃×#97	—	9	2	no	—
72	C₃×C₄×C₈	—	9	1	no	—
73	C₄⋊(C₃⋊C₈)	—	3	8	no	—
74	C₄⋊(C₃⋊C₈)	—	3	4	no	—
75	#803	—	3	8	no	—
76	C₄×C₃⋊C₈	—	3	4	no	—
77	C₃²×Q₈	—	2	16	no	—
78	C₃²×Q₈	—	2	8	no	—
79	C₃²×Q₈	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	8	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	8	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
86	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
87	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
88	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
89	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
94	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
95	C₃⋊(C₃⋊C₈)	—	2	1	no	—
96	C₃⋊(C₃⋊C₈)	—	2	2	no	—
97	C₃⋊(C₃⋊C₈)	C₃×Q₈	1	2	yes	complemented
98	C₃×C₃⋊C₈	—	3	8	no	—
99	C₃×C₃⋊C₈	—	6	4	no	—
100	C₃×C₃⋊C₈	—	6	8	no	—
101	#446	—	9	1	no	Sylow
102	C₂×C₃³	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₃³	#131	1	1	yes	characteristic
105	C₂×C₃×Q₈	—	2	4	no	—
106	C₂×C₃×Q₈	C₃×Dic₃	1	4	yes	—
107	C₂×C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
108	C₂×C₃×C₈	—	9	1	no	—
109	C₂×C₃×C₈	—	9	1	no	—
110	C₂×C₃×C₈	—	9	2	no	—
111	C₃×C₄⋊C₄	—	2	8	no	—
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄⋊C₄	C₃×Dic₃	1	8	yes	—
114	C₃×C₄⋊C₄	C₃×Dic₃	1	4	yes	—
115	C₃×C₄⋊C₄	C₃⋊Dic₃	1	2	yes	—
116	C₃×C₄⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
117	C₃×C₄²	—	2	4	no	—
118	C₃×C₄²	—	2	8	no	—
119	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
120	C₃×C₄²	C₂×C₃×D₃	1	8	yes	—
121	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
122	C₃×C₄²	C₂×C₃⋊D₃	1	2	yes	—
123	C₂×C₃⋊C₈	—	3	4	no	—
124	C₂×C₃⋊C₈	—	3	4	no	—
125	C₂×C₃⋊C₈	—	3	8	no	—
126	C₂²×C₃²	—	2	4	no	—
127	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
128	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
129	C₃²×C₄	—	2	16	no	—
130	C₃²×C₄	—	2	8	no	—
131	C₃²×C₄	—	2	2	no	—
132	C₃²×C₄	—	2	1	no	—
133	C₃²×C₄	—	2	2	no	—
134	C₃²×C₄	—	2	8	no	—
135	C₃²×C₄	—	2	4	no	—
136	C₃²×C₄	—	2	4	no	—
137	C₃²×C₄	—	2	4	no	—
138	C₃²×C₄	—	2	8	no	—
139	C₃²×C₄	—	4	4	no	—
140	C₃²×C₄	—	4	8	no	—
141	C₃²×C₄	#208	1	8	yes	—
142	C₃²×C₄	C₃×M₁₆	1	2	yes	—
143	C₃²×C₄	D₃×Q₈	1	4	yes	—
144	C₃²×C₄	#239	1	4	yes	—
145	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
146	C₃²×C₄	C₃×#41	1	1	yes	characteristic
147	C₄×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
148	C₄⋊C₈	—	9	1	no	—
149	C₄⋊C₈	—	9	2	no	—
150	#97	—	9	2	no	—
151	C₄×C₈	—	9	1	no	—
152	C₃³	#446	1	1	yes	characteristic, complemented, Sylow
153	C₃×Q₈	—	2	16	no	—
154	C₃×Q₈	—	2	8	no	—
155	C₃×Q₈	—	2	2	no	—
156	C₂×C₃×C₄	—	2	8	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	8	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
162	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
163	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
164	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
165	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
167	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
168	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
170	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
171	C₃×C₈	—	9	2	no	—
172	C₃×C₈	—	18	1	no	—
173	C₃×C₈	—	18	2	no	—
174	C₃⋊C₈	—	3	8	no	—
175	C₃⋊C₈	—	6	4	no	—
176	C₃⋊C₈	—	6	8	no	—
177	C₂×C₃²	—	2	4	no	—
178	C₂×C₃²	—	2	4	no	—
179	C₂×C₃²	—	2	4	no	—
180	C₂×C₃²	C₂×#208	1	4	yes	—
181	C₂×C₃²	Q₈×Dic₃	1	4	yes	—
182	C₂×C₃²	#948	1	4	yes	—
183	C₂×C₃²	C₃×C₄×Q₈	1	1	yes	characteristic
184	C₂×C₃²	C₂×C₃×M₁₆	1	1	yes	characteristic
185	C₂×C₃²	C₃×#131	1	1	yes	characteristic
186	C₂×Q₈	C₃×C₃⋊Dic₃	1	1	yes	characteristic
187	C₂×C₈	—	9	1	no	—
188	C₂×C₈	—	9	1	no	—
189	C₂×C₈	—	9	2	no	—
190	C₄⋊C₄	C₃×C₃⋊Dic₃	1	2	yes	—
191	C₄⋊C₄	C₃×C₃⋊Dic₃	1	1	yes	characteristic
192	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
193	C₄²	C₂×C₃×C₃⋊D₃	1	2	yes	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	C₂²×C₃×Dic₃	1	4	yes	—
196	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
197	C₃×C₄	—	2	16	no	—
198	C₃×C₄	—	2	8	no	—
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	—	2	8	no	—
201	C₃×C₄	—	2	2	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	8	no	—
205	C₃×C₄	—	2	4	no	—
206	C₃×C₄	—	2	4	no	—
207	C₃×C₄	—	4	4	no	—
208	C₃×C₄	—	4	8	no	—
209	C₃×C₄	C₃×#208	1	8	yes	—
210	C₃×C₄	#3748	1	2	yes	—
211	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
212	C₃×C₄	C₃×#239	1	4	yes	—
213	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
214	C₃×C₄	#3832	1	1	yes	characteristic
215	C₃²	—	2	4	no	—
216	C₃²	#5024	1	4	yes	complemented
217	C₃²	C₃×#446	1	1	yes	characteristic, complemented
218	Q₈	—	2	2	no	—
219	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
220	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
221	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
222	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
224	C₈	—	9	2	no	—
225	C₈	—	18	1	no	—
226	C₈	—	18	2	no	—
227	C₂×C₃	—	2	4	no	—
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	C₂×C₃×#208	1	4	yes	—
231	C₂×C₃	C₃×Q₈×Dic₃	1	4	yes	—
232	C₂×C₃	C₃×#948	1	4	yes	—
233	C₂×C₃	C₂×#3748	1	1	yes	characteristic
234	C₂×C₃	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
235	C₂×C₃	#64157	1	1	yes	characteristic
236	C₂²	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	C₃×#3748	1	2	yes	—
241	C₄	C₃×Q₈×C₃⋊D₃	1	1	yes	characteristic
242	C₄	C₃×#3832	1	1	yes	characteristic
243	C₃	—	2	4	no	—
244	C₃	C₃×#5024	1	4	yes	complemented
245	C₃	#10592723	1	1	yes	characteristic, direct product factor
246	C₂	C₂×C₃×#3748	1	1	yes	characteristic
247	C₂	C₃×Q₈×C₃⋊Dic₃	1	1	yes	characteristic
248	C₂	C₃×#64157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄×Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×#446
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
6. #10592723	C₃	direct (N × H)
35. C₃×C₃⋊(C₃⋊C₈)	Q₈	semidirect
97. C₃⋊(C₃⋊C₈)	C₃×Q₈	semidirect
152. C₃³	#446	semidirect
216. C₃²	#5024	semidirect
217. C₃²	C₃×#446	semidirect
244. C₃	C₃×#5024	semidirect
245. C₃	#10592723	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄×Q₈	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×#63630	2
C₃×C₄×C₃⋊(C₃⋊C₈)	2
#10592723	3
C₃×#5024	3

Subgroups — C3×#10592723 #13163062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592723 #13163062