Subgroups — C₃×#10596188 #13179469

Order 1728.

3000 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₄×C₃⋊D₃	C₂	1	4	yes	maximal, complemented
2	C₃³×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64147	C₂	1	4	yes	maximal, complemented
4	C₃×#64146	C₂	1	4	yes	maximal, complemented
5	C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
6	#10596188	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5717	—	3	4	no	maximal
8	C₂³×C₃×C₃⋊D₃	C₂²	1	4	yes	complemented
9	C₂×C₃³×D₄	C₂²	1	4	yes	complemented
10	C₂×C₃³×D₄	C₂²	1	2	yes	complemented
11	C₂×C₃×#638	C₂²	1	8	yes	complemented
12	C₃×D₄×C₃⋊D₃	—	2	16	no	—
13	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
14	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	4	yes	complemented
15	C₃³×C₄²	C₂²	1	1	yes	characteristic, complemented
16	C₃×#3757	C₂²	1	4	yes	complemented
17	C₃×#3752	C₂²	1	4	yes	complemented
18	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	complemented
19	C₂×D₄×C₃⋊D₃	C₂×C₃	1	4	yes	complemented
20	C₂×C₃×D₃×D₄	—	3	16	no	—
21	C₃²×#127	—	2	4	no	—
22	C₃²×#127	D₃	1	4	yes	complemented
23	C₃²×#127	C₂×C₃	1	1	yes	characteristic, complemented
24	#64147	C₂×C₃	1	4	yes	complemented
25	#64146	C₂×C₃	1	4	yes	complemented
26	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	2	yes	complemented
27	C₃×#938	—	3	16	no	—
28	C₃×#937	—	3	16	no	—
29	C₃×C₄×D₁₂	—	3	8	no	—
30	C₂³×C₃³	C₂³	1	4	yes	—
31	C₂²×C₃×C₃⋊D₃	—	2	16	no	—
32	C₂²×C₃×C₃⋊D₃	—	4	4	no	—
33	C₂²×C₃×C₃⋊D₃	C₂³	1	4	yes	—
34	C₃³×D₄	—	2	8	no	—
35	C₃³×D₄	—	4	2	no	—
36	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂³	1	2	yes	—
38	C₃×#638	—	2	16	no	—
39	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
40	C₃×C₄⋊(C₃⋊D₃)	D₄	1	8	yes	complemented
41	C₃×C₄×C₃⋊D₃	—	2	8	no	—
42	C₃×D₄²	—	9	1	no	—
43	#5717	—	3	4	no	—
44	C₂³×C₃⋊D₃	C₂²×C₃	1	4	yes	complemented
45	C₂³×C₃×D₃	—	3	16	no	—
46	C₂×C₃²×D₄	—	2	16	no	—
47	C₂×C₃²×D₄	—	2	8	no	—
48	C₂×C₃²×D₄	C₂×D₃	1	16	yes	complemented
49	C₂×C₃²×D₄	C₂×D₃	1	8	yes	complemented
50	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	complemented
51	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	complemented
52	C₂×#638	C₂²×C₃	1	8	yes	complemented
53	D₄×C₃⋊D₃	—	2	16	no	—
54	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	2	yes	—
55	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	4	yes	complemented
56	C₂×C₃×#67	—	3	32	no	—
57	C₃×D₃×D₄	—	6	64	no	—
58	C₂×C₃×D₁₂	—	3	8	no	—
59	C₂×C₃×C₄×D₃	—	3	16	no	—
60	C₃²×C₄²	—	2	4	no	—
61	C₃²×C₄²	C₂×D₃	1	4	yes	complemented
62	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
63	#3757	C₂²×C₃	1	4	yes	complemented
64	#3752	C₂²×C₃	1	4	yes	complemented
65	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	complemented
66	C₃×#217	—	3	16	no	—
67	C₃×(C₂×D₃)⋊C₄	—	3	16	no	—
68	C₃×C₄⋊Dic₃	—	3	8	no	—
69	C₂²×C₃³	—	2	8	no	—
70	C₂²×C₃³	—	2	4	no	—
71	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
72	C₂×C₃×C₃⋊D₃	—	2	4	no	—
73	C₂×C₃×C₃⋊D₃	—	2	4	no	—
74	C₂×C₃×C₃⋊D₃	—	4	16	no	—
75	C₂×C₃×C₃⋊D₃	C₂×D₄	1	8	yes	complemented
76	C₃³×C₄	—	2	1	no	—
77	C₃³×C₄	C₂×D₄	1	4	yes	complemented
78	C₃×C₃⋊Dic₃	—	2	4	no	—
79	C₂²×C₃×D₄	—	9	4	no	—
80	C₂×D₃×D₄	—	3	16	no	—
81	C₃×#127	—	2	4	no	—
82	C₃×#127	C₃×D₃	1	4	yes	complemented
83	C₃×#127	C₃⋊D₃	1	1	yes	characteristic, complemented
84	C₃×#121	—	9	4	no	—
85	C₃×#120	—	9	4	no	—
86	C₃×C₄×D₄	—	9	2	no	—
87	#938	—	3	16	no	—
88	#937	—	3	16	no	—
89	C₄×D₁₂	—	3	8	no	—
90	C₂³×C₃²	—	2	16	no	—
91	C₂³×C₃²	C₂²×D₃	1	16	yes	—
92	C₂³×C₃²	C₂³×C₃	1	4	yes	—
93	C₂²×C₃⋊D₃	—	2	16	no	—
94	C₂²×C₃⋊D₃	—	4	4	no	—
95	C₂²×C₃⋊D₃	C₂³×C₃	1	4	yes	—
96	C₂²×C₃×D₃	—	3	16	no	—
97	C₂²×C₃×D₃	—	6	64	no	—
98	C₂²×C₃×D₃	—	12	16	no	—
99	C₃²×D₄	—	2	8	no	—
100	C₃²×D₄	—	2	32	no	—
101	C₃²×D₄	—	4	32	no	—
102	C₃²×D₄	—	4	8	no	—
103	C₃²×D₄	—	4	2	no	—
104	C₃²×D₄	—	4	16	no	—
105	C₂×C₃²×C₄	—	2	4	no	—
106	C₂×C₃²×C₄	—	2	8	no	—
107	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
108	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
109	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
111	#638	—	2	16	no	—
112	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
113	C₄⋊(C₃⋊D₃)	C₃×D₄	1	8	yes	complemented
114	C₄×C₃⋊D₃	—	2	8	no	—
115	C₃×#67	—	6	64	no	—
116	C₂×C₃×Dic₃	—	3	16	no	—
117	C₃×D₁₂	—	3	32	no	—
118	C₃×C₄×D₃	—	6	32	no	—
119	D₄²	—	9	1	no	Sylow
120	C₂×C₃³	—	4	4	no	—
121	C₂×C₃³	C₂²×D₄	1	2	yes	—
122	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
123	C₃×C₃⋊D₃	—	2	8	no	—
124	C₂⁴×C₃	—	9	4	no	—
125	C₂³×D₃	—	3	16	no	—
126	C₂×C₃×D₄	—	2	16	no	—
127	C₂×C₃×D₄	—	2	8	no	—
128	C₂×C₃×D₄	—	9	8	no	—
129	C₂×C₃×D₄	—	9	2	no	—
130	C₂×C₃×D₄	—	18	16	no	—
131	C₂×C₃×D₄	C₂×C₃×D₃	1	16	yes	complemented
132	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	complemented
133	C₂×C₃×D₄	C₂×C₃⋊D₃	1	4	yes	complemented
134	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	complemented
135	C₂²×C₃×C₄	—	9	4	no	—
136	C₂×#67	—	3	32	no	—
137	D₃×D₄	—	6	64	no	—
138	C₂×D₁₂	—	3	8	no	—
139	C₂×C₄×D₃	—	3	16	no	—
140	C₃×C₄⋊C₄	—	9	2	no	—
141	C₃×C₂²⋊C₄	—	9	4	no	—
142	C₃×C₂²⋊C₄	—	9	4	no	—
143	C₃×C₄²	—	2	4	no	—
144	C₃×C₄²	C₂×C₃×D₃	1	4	yes	complemented
145	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
146	#217	—	3	16	no	—
147	(C₂×D₃)⋊C₄	—	3	16	no	—
148	C₄⋊Dic₃	—	3	8	no	—
149	C₂²×C₃²	—	2	8	no	—
150	C₂²×C₃²	—	2	4	no	—
151	C₂²×C₃²	—	2	4	no	—
152	C₂²×C₃²	—	2	32	no	—
153	C₂²×C₃²	—	2	16	no	—
154	C₂²×C₃²	—	4	32	no	—
155	C₂²×C₃²	—	4	16	no	—
156	C₂²×C₃²	C₂³×D₃	1	4	yes	—
157	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
158	C₂×C₃⋊D₃	—	2	4	no	—
159	C₂×C₃⋊D₃	—	2	4	no	—
160	C₂×C₃⋊D₃	—	4	16	no	—
161	C₂×C₃⋊D₃	C₂×C₃×D₄	1	8	yes	complemented
162	C₂×C₃×D₃	—	3	32	no	—
163	C₂×C₃×D₃	—	6	16	no	—
164	C₂×C₃×D₃	—	6	16	no	—
165	C₂×C₃×D₃	—	12	64	no	—
166	C₃²×C₄	—	2	8	no	—
167	C₃²×C₄	—	2	4	no	—
168	C₃²×C₄	—	2	1	no	—
169	C₃²×C₄	—	2	16	no	—
170	C₃²×C₄	D₃×D₄	1	16	yes	complemented
171	C₃²×C₄	C₂×C₃×D₄	1	4	yes	complemented
172	C₃⋊Dic₃	—	2	4	no	—
173	C₃×Dic₃	—	6	16	no	—
174	C₂²×D₄	—	9	4	no	—
175	#127	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
176	#121	—	9	4	no	—
177	#120	—	9	4	no	—
178	C₄×D₄	—	9	2	no	—
179	C₃³	D₄²	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₃	—	2	16	no	—
181	C₂³×C₃	—	9	4	no	—
182	C₂³×C₃	—	18	16	no	—
183	C₂³×C₃	—	36	4	no	—
184	C₂³×C₃	C₂²×C₃×D₃	1	16	yes	—
185	C₂³×C₃	C₂²×C₃⋊D₃	1	4	yes	—
186	C₂²×D₃	—	3	16	no	—
187	C₂²×D₃	—	6	64	no	—
188	C₂²×D₃	—	12	16	no	—
189	C₃×D₄	—	2	32	no	—
190	C₃×D₄	—	2	8	no	—
191	C₃×D₄	—	4	32	no	—
192	C₃×D₄	—	4	8	no	—
193	C₃×D₄	—	4	2	no	—
194	C₃×D₄	—	4	16	no	—
195	C₃×D₄	—	9	8	no	—
196	C₃×D₄	—	18	16	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	8	no	—
199	C₂×C₃×C₄	—	9	4	no	—
200	C₂×C₃×C₄	—	18	8	no	—
201	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
202	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
203	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
205	#67	—	6	64	no	—
206	C₂×Dic₃	—	3	16	no	—
207	D₁₂	—	3	32	no	—
208	C₄×D₃	—	6	32	no	—
209	C₂×C₃²	—	2	8	no	—
210	C₂×C₃²	—	2	4	no	—
211	C₂×C₃²	—	4	4	no	—
212	C₂×C₃²	—	4	16	no	—
213	C₂×C₃²	—	8	16	no	—
214	C₂×C₃²	C₂×D₃×D₄	1	8	yes	—
215	C₂×C₃²	#1004	1	4	yes	—
216	C₂×C₃²	C₂²×C₃×D₄	1	2	yes	—
217	C₂×C₃²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
218	C₃⋊D₃	—	2	8	no	—
219	C₃×D₃	—	6	32	no	—
220	C₂⁴	—	9	4	no	—
221	C₂×D₄	—	9	8	no	—
222	C₂×D₄	—	9	2	no	—
223	C₂×D₄	—	18	16	no	—
224	C₂×D₄	C₂×C₃×C₃⋊D₃	1	4	yes	complemented
225	C₂×D₄	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
226	C₂²×C₄	—	9	4	no	—
227	C₄⋊C₄	—	9	2	no	—
228	C₂²⋊C₄	—	9	4	no	—
229	C₂²⋊C₄	—	9	4	no	—
230	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
231	C₂²×C₃	—	2	32	no	—
232	C₂²×C₃	—	2	16	no	—
233	C₂²×C₃	—	2	8	no	—
234	C₂²×C₃	—	2	4	no	—
235	C₂²×C₃	—	2	4	no	—
236	C₂²×C₃	—	4	32	no	—
237	C₂²×C₃	—	4	16	no	—
238	C₂²×C₃	—	9	8	no	—
239	C₂²×C₃	—	18	4	no	—
240	C₂²×C₃	—	18	4	no	—
241	C₂²×C₃	—	36	16	no	—
242	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
243	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
244	C₂×D₃	—	3	32	no	—
245	C₂×D₃	—	6	16	no	—
246	C₂×D₃	—	6	16	no	—
247	C₂×D₃	—	12	64	no	—
248	C₃×C₄	—	2	8	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	16	no	—
252	C₃×C₄	—	18	4	no	—
253	C₃×C₄	C₃×D₃×D₄	1	16	yes	complemented
254	C₃×C₄	D₄×C₃⋊D₃	1	4	yes	complemented
255	Dic₃	—	6	16	no	—
256	C₃²	—	2	4	no	—
257	C₃²	#5717	1	4	yes	complemented
258	C₃²	C₃×D₄²	1	1	yes	characteristic, complemented
259	C₂³	—	9	4	no	—
260	C₂³	—	18	16	no	—
261	C₂³	—	36	4	no	—
262	C₂³	C₂²×C₃×C₃⋊D₃	1	4	yes	—
263	D₄	—	2	8	no	—
264	D₄	—	4	2	no	—
265	D₄	—	9	8	no	—
266	D₄	—	18	16	no	—
267	C₂×C₄	—	9	4	no	—
268	C₂×C₄	—	18	8	no	—
269	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
271	C₂×C₃	—	2	8	no	—
272	C₂×C₃	—	2	4	no	—
273	C₂×C₃	—	4	4	no	—
274	C₂×C₃	—	4	16	no	—
275	C₂×C₃	—	8	16	no	—
276	C₂×C₃	—	18	8	no	—
277	C₂×C₃	C₂×C₃×D₃×D₄	1	8	yes	—
278	C₂×C₃	C₃×#1004	1	4	yes	—
279	C₂×C₃	C₂×D₄×C₃⋊D₃	1	2	yes	—
280	C₂×C₃	#64361	1	1	yes	characteristic
281	D₃	—	6	32	no	—
282	C₂²	—	2	8	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	9	8	no	—
285	C₂²	—	18	4	no	—
286	C₂²	—	18	4	no	—
287	C₂²	—	36	16	no	—
288	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	18	4	no	—
291	C₄	C₃×D₄×C₃⋊D₃	1	4	yes	complemented
292	C₃	—	2	4	no	—
293	C₃	C₃×#5717	1	4	yes	complemented
294	C₃	#10596188	1	1	yes	characteristic, direct product factor
295	C₂	—	4	4	no	—
296	C₂	—	18	8	no	—
297	C₂	C₂×C₃×D₄×C₃⋊D₃	1	2	yes	—
298	C₂	C₃×#64361	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×#127	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃×#127	C₃⋊D₃
hypocenter	C₃²	C₃×D₄²
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₄×C₃⋊D₃	C₂	semidirect
2. C₃³×#127	C₂	semidirect
3. C₃×#64147	C₂	semidirect
4. C₃×#64146	C₂	semidirect
5. C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	semidirect
6. #10596188	C₃	direct (N × H)
8. C₂³×C₃×C₃⋊D₃	C₂²	semidirect
9. C₂×C₃³×D₄	C₂²	semidirect
10. C₂×C₃³×D₄	C₂²	semidirect
11. C₂×C₃×#638	C₂²	semidirect
14. C₂×C₃×C₄×C₃⋊D₃	C₂²	semidirect
15. C₃³×C₄²	C₂²	semidirect
16. C₃×#3757	C₂²	semidirect
17. C₃×#3752	C₂²	semidirect
18. C₃×C₄⋊(C₃⋊Dic₃)	C₂²	semidirect
19. C₂×D₄×C₃⋊D₃	C₂×C₃	semidirect
22. C₃²×#127	D₃	semidirect
23. C₃²×#127	C₂×C₃	semidirect
24. #64147	C₂×C₃	semidirect
25. #64146	C₂×C₃	semidirect
26. C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	semidirect
40. C₃×C₄⋊(C₃⋊D₃)	D₄	semidirect
44. C₂³×C₃⋊D₃	C₂²×C₃	semidirect
48. C₂×C₃²×D₄	C₂×D₃	semidirect
49. C₂×C₃²×D₄	C₂×D₃	semidirect
50. C₂×C₃²×D₄	C₂²×C₃	semidirect
51. C₂×C₃²×D₄	C₂²×C₃	semidirect
52. C₂×#638	C₂²×C₃	semidirect
55. C₂×C₄×C₃⋊D₃	C₂²×C₃	semidirect
61. C₃²×C₄²	C₂×D₃	semidirect
62. C₃²×C₄²	C₂²×C₃	semidirect
63. #3757	C₂²×C₃	semidirect
64. #3752	C₂²×C₃	semidirect
65. C₄⋊(C₃⋊Dic₃)	C₂²×C₃	semidirect
75. C₂×C₃×C₃⋊D₃	C₂×D₄	semidirect
77. C₃³×C₄	C₂×D₄	semidirect
82. C₃×#127	C₃×D₃	semidirect
83. C₃×#127	C₃⋊D₃	semidirect
113. C₄⋊(C₃⋊D₃)	C₃×D₄	semidirect
131. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
132. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
133. C₂×C₃×D₄	C₂×C₃⋊D₃	semidirect
134. C₂×C₃×D₄	C₂×C₃⋊D₃	semidirect
144. C₃×C₄²	C₂×C₃×D₃	semidirect
145. C₃×C₄²	C₂×C₃⋊D₃	semidirect
161. C₂×C₃⋊D₃	C₂×C₃×D₄	semidirect
170. C₃²×C₄	D₃×D₄	semidirect
171. C₃²×C₄	C₂×C₃×D₄	semidirect
175. #127	C₃×C₃⋊D₃	semidirect
179. C₃³	D₄²	semidirect
224. C₂×D₄	C₂×C₃×C₃⋊D₃	semidirect
225. C₂×D₄	C₂×C₃×C₃⋊D₃	semidirect
230. C₄²	C₂×C₃×C₃⋊D₃	semidirect
253. C₃×C₄	C₃×D₃×D₄	semidirect
254. C₃×C₄	D₄×C₃⋊D₃	semidirect
257. C₃²	#5717	semidirect
258. C₃²	C₃×D₄²	semidirect
291. C₄	C₃×D₄×C₃⋊D₃	semidirect
293. C₃	C₃×#5717	semidirect
294. C₃	#10596188	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×D₄×C₃⋊D₃	2
C₃³×#127	2
C₃×#64147	2
C₃×#64146	2
C₃×C₄×C₄⋊(C₃⋊D₃)	2
#10596188	3
C₃×#5717	3

Subgroups — C3×#10596188 #13179469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596188 #13179469