Subgroups — C₃²×#5135 #13162570

Order 1728.

1052 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#930	C₂	1	2	yes	maximal, complemented
3	C₂×C₃²×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#821	C₂	1	2	yes	maximal, complemented
5	C₃²×#817	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#469	—	3	1	no	maximal
7	C₃×#5135	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃²×#67	—	2	2	no	—
10	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
11	C₃³×M₁₆	—	2	2	no	—
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₃²×(C₂×D₃)⋊C₄	—	2	2	no	—
15	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₃²×D₂₄	—	2	2	no	—
17	C₂×C₃²×D₈	—	3	1	no	—
18	C₂×C₃²×M₁₆	—	2	4	no	—
19	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic, complemented
20	C₂×C₃²×M₁₆	C₂×C₃	1	4	yes	complemented
21	C₃²×#121	—	3	2	no	—
22	C₃×#930	C₂×C₃	1	8	yes	complemented
23	C₂×C₃×D₂₄	C₂×C₃	1	4	yes	complemented
24	C₃²×#106	—	3	1	no	—
25	C₃²×#102	—	3	2	no	—
26	C₃×#821	C₂×C₃	1	8	yes	complemented
27	C₃×#817	C₂×C₃	1	4	yes	complemented
28	C₂³×C₃³	D₄	1	1	yes	characteristic
29	C₂²×C₃²×D₃	—	2	2	no	—
30	C₂×C₃³×C₄	—	2	1	no	—
31	C₂×C₃³×C₄	—	2	1	no	—
32	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
33	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
34	C₃²×#67	—	4	4	no	—
35	C₂×C₃²×Dic₃	—	2	2	no	—
36	C₃²×D₁₂	—	2	2	no	—
37	C₃²×D₁₂	—	2	2	no	—
38	C₃³×C₈	—	2	1	no	—
39	C₃³×C₈	D₄	1	2	yes	complemented
40	C₃×#469	—	3	4	no	—
41	#5135	C₃²	1	1	yes	characteristic, direct product factor
42	C₂×C₃²×D₄	—	3	2	no	—
43	C₂×C₃²×D₄	—	6	2	no	—
44	C₂²×C₃²×C₄	—	2	4	no	—
45	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
47	C₂×C₃×#67	—	2	8	no	—
48	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	—
49	C₃²×D₈	—	6	2	no	—
50	C₃²×M₁₆	—	2	16	no	—
51	C₃²×M₁₆	—	2	2	no	—
52	C₃²×M₁₆	—	2	8	no	—
53	C₂×C₃²×C₈	—	2	4	no	—
54	C₂×C₃²×C₈	—	2	4	no	—
55	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
58	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
59	C₃²×C₄⋊C₄	—	3	2	no	—
60	C₃²×C₂²⋊C₄	—	6	2	no	—
61	C₃×(C₂×D₃)⋊C₄	—	2	8	no	—
62	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
63	C₃×D₂₄	—	2	8	no	—
64	C₂²×C₃³	—	2	1	no	—
65	C₂²×C₃³	—	2	2	no	—
66	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×D₃	—	4	2	no	—
68	C₂×C₃²×D₃	—	4	2	no	—
69	C₂×C₃²×D₃	—	4	2	no	—
70	C₃³×C₄	—	2	1	no	—
71	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
72	C₃³×C₄	#41	1	1	yes	characteristic
73	C₃²×Dic₃	—	4	2	no	—
74	C₂×C₃×D₈	—	3	4	no	—
75	C₂×C₃×M₁₆	—	2	4	no	—
76	C₂×C₃×M₁₆	C₃×D₃	1	4	yes	complemented
77	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
78	C₃×#121	—	3	8	no	—
79	#930	C₂×C₃²	1	2	yes	complemented
80	C₂×D₂₄	C₂×C₃²	1	1	yes	characteristic, complemented
81	C₃×#106	—	3	4	no	—
82	C₃×#102	—	3	8	no	—
83	#821	C₂×C₃²	1	2	yes	complemented
84	#817	C₂×C₃²	1	1	yes	characteristic, complemented
85	C₂³×C₃²	—	2	4	no	—
86	C₂³×C₃²	—	6	2	no	—
87	C₂³×C₃²	D₁₂	1	1	yes	characteristic
88	C₂³×C₃²	C₃×D₄	1	4	yes	—
89	C₂²×C₃×D₃	—	2	8	no	—
90	C₃²×D₄	—	6	2	no	—
91	C₃²×D₄	—	6	2	no	—
92	C₃²×D₄	—	12	4	no	—
93	C₂×C₃²×C₄	—	2	8	no	—
94	C₂×C₃²×C₄	—	2	8	no	—
95	C₂×C₃²×C₄	—	2	1	no	—
96	C₂×C₃²×C₄	—	2	1	no	—
97	C₂×C₃²×C₄	—	2	4	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	—	2	4	no	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	—	6	2	no	—
102	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
106	C₃×#67	—	4	16	no	—
107	C₂×C₃×Dic₃	—	2	8	no	—
108	C₃×D₁₂	—	2	8	no	—
109	C₃×D₁₂	—	2	8	no	—
110	C₃²×C₈	—	2	8	no	—
111	C₃²×C₈	—	2	1	no	—
112	C₃²×C₈	—	2	4	no	—
113	C₃²×C₈	—	2	8	no	—
114	C₃²×C₈	#67	1	2	yes	complemented
115	C₃²×C₈	C₃×D₄	1	8	yes	complemented
116	#469	—	3	1	no	Sylow
117	C₂×C₃³	—	4	1	no	—
118	C₂×C₃³	#121	1	1	yes	characteristic
119	C₂×C₃³	AGL(1,ℤ/8)	1	2	yes	—
120	C₃²×D₃	—	8	2	no	—
121	C₂×C₃×D₄	—	3	8	no	—
122	C₂×C₃×D₄	—	6	8	no	—
123	C₂²×C₃×C₄	—	2	4	no	—
124	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
125	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
126	C₂×#67	—	2	2	no	—
127	C₂×D₁₂	C₂²×C₃²	1	2	yes	—
128	C₃×D₈	—	6	8	no	—
129	C₃×M₁₆	—	2	16	no	—
130	C₃×M₁₆	—	2	8	no	—
131	C₃×M₁₆	—	2	2	no	—
132	C₂×C₃×C₈	—	2	4	no	—
133	C₂×C₃×C₈	—	2	4	no	—
134	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
135	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
136	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	3	8	no	—
139	C₃×C₂²⋊C₄	—	6	8	no	—
140	(C₂×D₃)⋊C₄	—	2	2	no	—
141	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
142	D₂₄	—	2	2	no	—
143	C₂²×C₃²	—	2	8	no	—
144	C₂²×C₃²	—	2	2	no	—
145	C₂²×C₃²	—	2	1	no	—
146	C₂²×C₃²	—	2	4	no	—
147	C₂²×C₃²	—	2	4	no	—
148	C₂²×C₃²	—	2	8	no	—
149	C₂²×C₃²	—	4	8	no	—
150	C₂²×C₃²	—	12	2	no	—
151	C₂²×C₃²	—	12	2	no	—
152	C₂²×C₃²	—	12	2	no	—
153	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
154	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
155	C₂×C₃×D₃	—	4	8	no	—
156	C₂×C₃×D₃	—	4	8	no	—
157	C₂×C₃×D₃	—	4	8	no	—
158	C₃²×C₄	—	2	8	no	—
159	C₃²×C₄	—	2	1	no	—
160	C₃²×C₄	—	2	4	no	—
161	C₃²×C₄	—	2	4	no	—
162	C₃²×C₄	—	2	4	no	—
163	C₃²×C₄	—	12	2	no	—
164	C₃²×C₄	#235	1	1	yes	characteristic
165	C₃²×C₄	C₂×#67	1	1	yes	characteristic
166	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
167	C₃²×C₄	C₃×#41	1	4	yes	—
168	C₃×Dic₃	—	4	8	no	—
169	C₂×D₈	—	3	1	no	—
170	C₂×M₁₆	C₃²×D₃	1	1	yes	characteristic, complemented
171	#121	—	3	2	no	—
172	#106	—	3	1	no	—
173	#102	—	3	2	no	—
174	C₃³	#469	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	2	4	no	—
176	C₂³×C₃	—	6	8	no	—
177	C₂³×C₃	C₃×D₁₂	1	4	yes	—
178	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
179	C₂²×D₃	—	2	2	no	—
180	C₃×D₄	—	6	8	no	—
181	C₃×D₄	—	6	8	no	—
182	C₃×D₄	—	12	16	no	—
183	C₂×C₃×C₄	—	2	8	no	—
184	C₂×C₃×C₄	—	2	8	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	6	8	no	—
192	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
193	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
195	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
196	#67	—	4	4	no	—
197	C₂×Dic₃	—	2	2	no	—
198	D₁₂	—	2	2	no	—
199	D₁₂	—	2	2	no	—
200	C₃×C₈	—	2	8	no	—
201	C₃×C₈	—	2	4	no	—
202	C₃×C₈	—	2	1	no	—
203	C₃×C₈	—	2	8	no	—
204	C₃×C₈	C₃×#67	1	8	yes	complemented
205	C₃×C₈	C₃²×D₄	1	2	yes	complemented
206	C₂×C₃²	—	2	8	no	—
207	C₂×C₃²	—	2	4	no	—
208	C₂×C₃²	—	4	8	no	—
209	C₂×C₃²	—	4	1	no	—
210	C₂×C₃²	—	4	4	no	—
211	C₂×C₃²	—	24	2	no	—
212	C₂×C₃²	#908	1	2	yes	—
213	C₂×C₃²	#930	1	1	yes	characteristic
214	C₂×C₃²	C₃×#121	1	4	yes	—
215	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	8	yes	—
216	C₃×D₃	—	8	8	no	—
217	C₂×D₄	—	3	2	no	—
218	C₂×D₄	—	6	2	no	—
219	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
220	D₈	—	6	2	no	—
221	M₁₆	—	2	2	no	—
222	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
223	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
224	C₄⋊C₄	—	3	2	no	—
225	C₂²⋊C₄	—	6	2	no	—
226	C₂²×C₃	—	2	8	no	—
227	C₂²×C₃	—	2	8	no	—
228	C₂²×C₃	—	2	4	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	4	no	—
232	C₂²×C₃	—	4	8	no	—
233	C₂²×C₃	—	12	8	no	—
234	C₂²×C₃	—	12	8	no	—
235	C₂²×C₃	—	12	8	no	—
236	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
237	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
238	C₂×D₃	—	4	2	no	—
239	C₂×D₃	—	4	2	no	—
240	C₂×D₃	—	4	2	no	—
241	C₃×C₄	—	2	8	no	—
242	C₃×C₄	—	2	4	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	4	no	—
245	C₃×C₄	—	2	4	no	—
246	C₃×C₄	—	12	8	no	—
247	C₃×C₄	C₃×#235	1	4	yes	—
248	C₃×C₄	C₂×C₃×#67	1	4	yes	—
249	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
250	C₃×C₄	C₃²×#41	1	1	yes	characteristic
251	Dic₃	—	4	2	no	—
252	C₃²	—	2	4	no	—
253	C₃²	#5135	1	1	yes	characteristic, direct product factor
254	C₃²	C₃×#469	1	4	yes	complemented
255	C₂³	—	6	2	no	—
256	C₂³	C₃²×D₁₂	1	1	yes	characteristic
257	D₄	—	6	2	no	—
258	D₄	—	6	2	no	—
259	D₄	—	12	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	6	2	no	—
263	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	C₃²×#67	1	2	yes	complemented
267	C₂×C₃	—	2	8	no	—
268	C₂×C₃	—	2	4	no	—
269	C₂×C₃	—	4	8	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	—	4	4	no	—
272	C₂×C₃	—	24	8	no	—
273	C₂×C₃	C₃×#908	1	8	yes	—
274	C₂×C₃	C₃×#930	1	4	yes	—
275	C₂×C₃	C₃²×#121	1	1	yes	characteristic
276	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	2	yes	—
277	D₃	—	8	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	12	2	no	—
281	C₂²	—	12	2	no	—
282	C₂²	—	12	2	no	—
283	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	12	2	no	—
286	C₄	C₃²×#235	1	1	yes	characteristic
287	C₄	C₂×C₃²×#67	1	1	yes	characteristic
288	C₃	—	2	4	no	—
289	C₃	C₃×#5135	1	4	yes	direct product factor
290	C₃	C₃²×#469	1	1	yes	characteristic, complemented
291	C₂	—	4	1	no	—
292	C₂	—	24	2	no	—
293	C₂	C₃²×#908	1	2	yes	—
294	C₂	C₃²×#930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₂×C₃³×M₁₆	C₂
socle	C₂²×C₃³	C₂×D₄
hypercenter	C₂×C₃²×M₁₆	D₃
hypocenter	C₃	C₃²×#469
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂×C₃²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×M₁₆	C₂	semidirect
2. C₃²×#930	C₂	semidirect
3. C₂×C₃²×D₂₄	C₂	semidirect
4. C₃²×#821	C₂	semidirect
5. C₃²×#817	C₂	semidirect
7. C₃×#5135	C₃	direct (N × H)
19. C₂×C₃²×M₁₆	D₃	semidirect
20. C₂×C₃²×M₁₆	C₂×C₃	semidirect
22. C₃×#930	C₂×C₃	semidirect
23. C₂×C₃×D₂₄	C₂×C₃	semidirect
26. C₃×#821	C₂×C₃	semidirect
27. C₃×#817	C₂×C₃	semidirect
39. C₃³×C₈	D₄	semidirect
41. #5135	C₃²	direct (N × H)
76. C₂×C₃×M₁₆	C₃×D₃	semidirect
77. C₂×C₃×M₁₆	C₂×C₃²	semidirect
79. #930	C₂×C₃²	semidirect
80. C₂×D₂₄	C₂×C₃²	semidirect
83. #821	C₂×C₃²	semidirect
84. #817	C₂×C₃²	semidirect
114. C₃²×C₈	#67	semidirect
115. C₃²×C₈	C₃×D₄	semidirect
170. C₂×M₁₆	C₃²×D₃	semidirect
174. C₃³	#469	semidirect
204. C₃×C₈	C₃×#67	semidirect
205. C₃×C₈	C₃²×D₄	semidirect
253. C₃²	#5135	direct (N × H)
254. C₃²	C₃×#469	semidirect
266. C₈	C₃²×#67	semidirect
289. C₃	C₃×#5135	direct (N × H)
290. C₃	C₃²×#469	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×M₁₆	2
C₃²×#930	2
C₂×C₃²×D₂₄	2
C₃²×#821	2
C₃²×#817	2
C₃²×#469	3
C₃×#5135	3

Subgroups — C32×#5135 #13162570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5135 #13162570