Subgroups — C₃×#10596048 #13179328

Order 1728.

2152 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×C₃×#3832	C₂	1	2	yes	maximal, complemented
2	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64104	C₂	1	6	yes	maximal, complemented
4	C₃×#64083	C₂	1	3	yes	maximal, complemented
5	C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	1	3	yes	maximal, complemented
6	#10596048	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5577	—	3	4	no	maximal
8	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
9	C₃×#3832	—	2	8	no	—
10	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	3	yes	—
11	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	6	yes	—
12	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	6	yes	—
13	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	6	yes	—
14	C₃³×C₄⋊C₄	C₂²	1	3	yes	—
15	C₃³×C₄²	C₂²	1	3	yes	—
16	C₃×#3752	C₂²	1	6	yes	—
17	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
18	C₂×#3832	C₂×C₃	1	2	yes	complemented
19	C₂×C₃×#239	—	3	8	no	—
20	C₃²×C₄×Q₈	—	2	4	no	—
21	C₃²×C₄×Q₈	D₃	1	4	yes	complemented
22	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
23	#64104	C₂×C₃	1	6	yes	complemented
24	#64083	C₂×C₃	1	3	yes	complemented
25	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	3	yes	complemented
26	C₃×#895	—	3	24	no	—
27	C₃×#874	—	3	12	no	—
28	C₃×C₄×D₁₂	—	3	12	no	—
29	C₂²×C₃×C₃⋊D₃	C₂³	1	6	yes	—
30	C₃³×Q₈	D₄	1	4	yes	complemented
31	C₂×C₃³×C₄	C₂³	1	3	yes	—
32	C₂×C₃³×C₄	C₂³	1	3	yes	—
33	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₃⋊Dic₃	C₂³	1	2	yes	—
35	C₃×C₄⋊(C₃⋊D₃)	—	2	12	no	—
36	C₃×C₄⋊(C₃⋊D₃)	—	2	6	no	—
37	C₃×C₄⋊(C₃⋊D₃)	—	4	6	no	—
38	C₃×C₄×C₃⋊D₃	—	2	12	no	—
39	C₃×#550	—	9	1	no	—
40	#5577	—	3	4	no	—
41	C₂×C₃²×Q₈	—	2	4	no	—
42	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
43	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
44	#3832	—	2	8	no	—
45	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	3	yes	—
46	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	6	yes	—
47	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	6	yes	—
48	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	6	yes	—
49	C₃×#239	—	6	32	no	—
50	C₂×C₃×D₁₂	—	3	24	no	—
51	C₂×C₃×D₁₂	—	3	12	no	—
52	C₂×C₃×D₁₂	—	3	24	no	—
53	C₂×C₃×C₄×D₃	—	3	24	no	—
54	C₃²×C₄⋊C₄	—	2	12	no	—
55	C₃²×C₄⋊C₄	C₂×D₃	1	12	yes	—
56	C₃²×C₄⋊C₄	C₂²×C₃	1	3	yes	—
57	C₃²×C₄²	—	2	12	no	—
58	C₃²×C₄²	C₂×D₃	1	12	yes	—
59	C₃²×C₄²	C₂²×C₃	1	3	yes	—
60	#3752	C₂²×C₃	1	6	yes	—
61	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
62	C₃×(C₂×D₃)⋊C₄	—	3	24	no	—
63	C₃×C₄⋊Dic₃	—	3	4	no	—
64	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
65	C₂×C₃×C₃⋊D₃	—	2	6	no	—
66	C₂×C₃×C₃⋊D₃	—	2	6	no	—
67	C₂×C₃×C₃⋊D₃	—	2	6	no	—
68	C₃³×C₄	—	2	3	no	—
69	C₃³×C₄	C₂×D₄	1	6	yes	—
70	C₃³×C₄	#41	1	2	yes	complemented
71	C₃×C₃⋊Dic₃	—	2	2	no	—
72	C₂×C₃×#41	—	9	2	no	—
73	C₂×#239	—	3	8	no	—
74	C₃×#127	—	9	3	no	—
75	C₃×#121	—	9	6	no	—
76	C₃×C₄×Q₈	—	2	4	no	—
77	C₃×C₄×Q₈	C₃×D₃	1	4	yes	complemented
78	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
79	C₃×C₄×D₄	—	9	3	no	—
80	#895	—	3	24	no	—
81	#874	—	3	12	no	—
82	C₄×D₁₂	—	3	12	no	—
83	C₂²×C₃⋊D₃	C₂³×C₃	1	6	yes	—
84	C₂²×C₃×D₃	—	3	24	no	—
85	C₃²×Q₈	—	2	16	no	—
86	C₃²×Q₈	D₁₂	1	16	yes	complemented
87	C₃²×Q₈	C₃×D₄	1	4	yes	complemented
88	C₂×C₃²×C₄	—	2	4	no	—
89	C₂×C₃²×C₄	—	2	12	no	—
90	C₂×C₃²×C₄	—	2	12	no	—
91	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	—
92	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	—
93	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
94	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
95	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
96	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
98	C₄⋊(C₃⋊D₃)	—	2	12	no	—
99	C₄⋊(C₃⋊D₃)	—	2	6	no	—
100	C₄⋊(C₃⋊D₃)	—	4	6	no	—
101	C₄×C₃⋊D₃	—	2	12	no	—
102	C₂×C₃×Dic₃	—	3	8	no	—
103	C₃×D₁₂	—	6	48	no	—
104	C₃×D₁₂	—	6	24	no	—
105	C₃×D₁₂	—	12	24	no	—
106	C₃×C₄×D₃	—	6	48	no	—
107	#550	—	9	1	no	Sylow
108	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₃³	C₂×#41	1	1	yes	characteristic
110	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
111	C₃×C₃⋊D₃	—	4	6	no	—
112	C₃×#41	—	18	8	no	—
113	C₂×C₃×Q₈	—	2	4	no	—
114	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
115	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
116	C₂×C₃×D₄	—	9	3	no	—
117	C₂×C₃×D₄	—	9	6	no	—
118	C₂×C₃×D₄	—	9	6	no	—
119	C₂²×C₃×C₄	—	9	6	no	—
120	#239	—	6	32	no	—
121	C₂×D₁₂	—	3	12	no	—
122	C₂×D₁₂	—	3	24	no	—
123	C₂×D₁₂	—	3	24	no	—
124	C₂×C₄×D₃	—	3	24	no	—
125	C₃×C₄⋊C₄	—	2	12	no	—
126	C₃×C₄⋊C₄	—	9	1	no	—
127	C₃×C₄⋊C₄	C₂×C₃×D₃	1	12	yes	—
128	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	3	yes	—
129	C₃×C₂²⋊C₄	—	9	6	no	—
130	C₃×C₄²	—	2	12	no	—
131	C₃×C₄²	C₂×C₃×D₃	1	12	yes	—
132	C₃×C₄²	C₂×C₃⋊D₃	1	3	yes	—
133	(C₂×D₃)⋊C₄	—	3	24	no	—
134	C₄⋊Dic₃	—	3	4	no	—
135	C₂²×C₃²	—	2	4	no	—
136	C₂²×C₃²	C₂³×D₃	1	4	yes	—
137	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
138	C₂×C₃⋊D₃	—	2	6	no	—
139	C₂×C₃⋊D₃	—	2	6	no	—
140	C₂×C₃⋊D₃	—	2	6	no	—
141	C₂×C₃×D₃	—	6	24	no	—
142	C₂×C₃×D₃	—	6	24	no	—
143	C₂×C₃×D₃	—	6	24	no	—
144	C₃²×C₄	—	2	3	no	—
145	C₃²×C₄	—	2	24	no	—
146	C₃²×C₄	—	2	8	no	—
147	C₃²×C₄	—	2	12	no	—
148	C₃²×C₄	—	4	12	no	—
149	C₃²×C₄	C₂×D₁₂	1	24	yes	—
150	C₃²×C₄	#239	1	8	yes	complemented
151	C₃²×C₄	C₂×C₃×D₄	1	6	yes	—
152	C₃²×C₄	C₃×#41	1	2	yes	complemented
153	C₃⋊Dic₃	—	2	2	no	—
154	C₃×Dic₃	—	6	8	no	—
155	C₂×#41	—	9	2	no	—
156	#127	—	9	3	no	—
157	#121	—	9	6	no	—
158	C₄×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
159	C₄×D₄	—	9	3	no	—
160	C₃³	#550	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₃	—	9	6	no	—
162	C₂²×D₃	—	3	24	no	—
163	C₃×Q₈	—	2	16	no	—
164	C₃×Q₈	C₃×D₁₂	1	16	yes	complemented
165	C₃×Q₈	C₄⋊(C₃⋊D₃)	1	4	yes	complemented
166	C₃×D₄	—	18	12	no	—
167	C₃×D₄	—	18	6	no	—
168	C₃×D₄	—	36	6	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	12	no	—
171	C₂×C₃×C₄	—	2	12	no	—
172	C₂×C₃×C₄	—	9	2	no	—
173	C₂×C₃×C₄	—	18	12	no	—
174	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
175	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
176	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
177	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	3	yes	—
178	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	3	yes	—
179	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
180	C₂×Dic₃	—	3	8	no	—
181	D₁₂	—	6	48	no	—
182	D₁₂	—	6	24	no	—
183	D₁₂	—	12	24	no	—
184	C₄×D₃	—	6	48	no	—
185	C₂×C₃²	—	2	4	no	—
186	C₂×C₃²	—	2	4	no	—
187	C₂×C₃²	—	2	4	no	—
188	C₂×C₃²	C₂²×D₁₂	1	4	yes	—
189	C₂×C₃²	C₂×#239	1	4	yes	—
190	C₂×C₃²	#1009	1	4	yes	—
191	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
192	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
193	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
194	C₃⋊D₃	—	4	6	no	—
195	C₃×D₃	—	12	24	no	—
196	#41	—	18	8	no	—
197	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
198	C₂×D₄	—	9	3	no	—
199	C₂×D₄	—	9	6	no	—
200	C₂×D₄	—	9	6	no	—
201	C₂²×C₄	—	9	6	no	—
202	C₄⋊C₄	—	9	1	no	—
203	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	3	yes	—
204	C₂²⋊C₄	—	9	6	no	—
205	C₄²	C₂×C₃×C₃⋊D₃	1	3	yes	—
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	—	18	6	no	—
208	C₂²×C₃	—	18	6	no	—
209	C₂²×C₃	—	18	6	no	—
210	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
211	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
212	C₂×D₃	—	6	24	no	—
213	C₂×D₃	—	6	24	no	—
214	C₂×D₃	—	6	24	no	—
215	C₃×C₄	—	2	12	no	—
216	C₃×C₄	—	2	3	no	—
217	C₃×C₄	—	2	24	no	—
218	C₃×C₄	—	2	8	no	—
219	C₃×C₄	—	4	12	no	—
220	C₃×C₄	—	18	2	no	—
221	C₃×C₄	C₂×C₃×D₁₂	1	24	yes	—
222	C₃×C₄	C₃×#239	1	8	yes	complemented
223	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	6	yes	—
224	C₃×C₄	#3832	1	2	yes	complemented
225	Dic₃	—	6	8	no	—
226	C₃²	—	2	4	no	—
227	C₃²	#5577	1	4	yes	complemented
228	C₃²	C₃×#550	1	1	yes	characteristic, complemented
229	C₂³	—	9	6	no	—
230	Q₈	C₃×C₄⋊(C₃⋊D₃)	1	4	yes	complemented
231	D₄	—	18	12	no	—
232	D₄	—	18	6	no	—
233	D₄	—	36	6	no	—
234	C₂×C₄	—	9	2	no	—
235	C₂×C₄	—	18	12	no	—
236	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	3	yes	—
237	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	3	yes	—
238	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
239	C₂×C₃	—	2	4	no	—
240	C₂×C₃	—	2	4	no	—
241	C₂×C₃	—	2	4	no	—
242	C₂×C₃	—	36	6	no	—
243	C₂×C₃	C₂²×C₃×D₁₂	1	4	yes	—
244	C₂×C₃	C₂×C₃×#239	1	4	yes	—
245	C₂×C₃	C₃×#1009	1	4	yes	—
246	C₂×C₃	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
247	C₂×C₃	C₂×#3832	1	1	yes	characteristic
248	C₂×C₃	#64366	1	1	yes	characteristic
249	D₃	—	12	24	no	—
250	C₂²	—	18	6	no	—
251	C₂²	—	18	6	no	—
252	C₂²	—	18	6	no	—
253	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
254	C₄	—	2	3	no	—
255	C₄	—	18	2	no	—
256	C₄	C₂×C₃×C₄⋊(C₃⋊D₃)	1	6	yes	—
257	C₄	C₃×#3832	1	2	yes	complemented
258	C₃	—	2	4	no	—
259	C₃	C₃×#5577	1	4	yes	complemented
260	C₃	#10596048	1	1	yes	characteristic, direct product factor
261	C₂	—	36	6	no	—
262	C₂	C₂²×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
263	C₂	C₂×C₃×#3832	1	1	yes	characteristic
264	C₂	C₃×#64366	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×C₄×Q₈	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃×C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×#550
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3832	C₂	semidirect
2. C₃³×C₄×Q₈	C₂	semidirect
3. C₃×#64104	C₂	semidirect
4. C₃×#64083	C₂	semidirect
5. C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	semidirect
6. #10596048	C₃	direct (N × H)
18. C₂×#3832	C₂×C₃	semidirect
21. C₃²×C₄×Q₈	D₃	semidirect
22. C₃²×C₄×Q₈	C₂×C₃	semidirect
23. #64104	C₂×C₃	semidirect
24. #64083	C₂×C₃	semidirect
25. C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	semidirect
30. C₃³×Q₈	D₄	semidirect
70. C₃³×C₄	#41	semidirect
77. C₃×C₄×Q₈	C₃×D₃	semidirect
78. C₃×C₄×Q₈	C₃⋊D₃	semidirect
86. C₃²×Q₈	D₁₂	semidirect
87. C₃²×Q₈	C₃×D₄	semidirect
150. C₃²×C₄	#239	semidirect
152. C₃²×C₄	C₃×#41	semidirect
158. C₄×Q₈	C₃×C₃⋊D₃	semidirect
160. C₃³	#550	semidirect
164. C₃×Q₈	C₃×D₁₂	semidirect
165. C₃×Q₈	C₄⋊(C₃⋊D₃)	semidirect
222. C₃×C₄	C₃×#239	semidirect
224. C₃×C₄	#3832	semidirect
227. C₃²	#5577	semidirect
228. C₃²	C₃×#550	semidirect
230. Q₈	C₃×C₄⋊(C₃⋊D₃)	semidirect
257. C₄	C₃×#3832	semidirect
259. C₃	C₃×#5577	semidirect
260. C₃	#10596048	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3832	2
C₃³×C₄×Q₈	2
C₃×#64104	2
C₃×#64083	2
C₃×C₄×C₄⋊(C₃⋊D₃)	2
#10596048	3
C₃×#5577	3

Subgroups — C3×#10596048 #13179328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596048 #13179328