Subgroups — C₃²×#5137 #13162573

Order 1728.

932 conjugacy classes of subgroups.

The subgroup lattice has 272 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
2	C₃²×#924	C₂	1	2	yes	maximal
3	C₂×C₃²×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#817	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#816	C₂	1	2	yes	maximal, complemented
6	C₃²×#470	—	3	1	no	maximal
7	C₃×#5137	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
10	C₃³×M₁₆	—	2	2	no	—
11	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃²×#217	—	2	2	no	—
14	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₃²×Dic₃⋊C₄	—	2	2	no	—
16	C₃²×Dic₁₂	—	2	2	no	—
17	C₂×C₃²×Q₁₆	—	3	1	no	—
18	C₂×C₃²×M₁₆	—	2	4	no	—
19	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
20	C₂×C₃²×M₁₆	C₂×C₃	1	4	yes	—
21	C₃²×C₂²⋊Q₈	—	3	2	no	—
22	C₃×#924	C₂×C₃	1	8	yes	—
23	C₂×C₃×Dic₁₂	C₂×C₃	1	4	yes	complemented
24	C₃²×#106	—	3	1	no	—
25	C₃²×#103	—	3	2	no	—
26	C₃×#817	C₂×C₃	1	4	yes	complemented
27	C₃×#816	C₂×C₃	1	8	yes	complemented
28	C₂³×C₃³	D₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	—	2	1	no	—
30	C₂×C₃³×C₄	—	2	1	no	—
31	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃²×Dic₃	—	2	2	no	—
34	C₂×C₃²×Dic₃	—	2	2	no	—
35	C₃²×Dic₆	—	2	2	no	—
36	C₃²×Dic₆	—	2	2	no	—
37	C₃³×C₈	—	2	1	no	—
38	C₃³×C₈	D₄	1	2	yes	—
39	C₃×#470	—	3	4	no	—
40	#5137	C₃²	1	1	yes	characteristic, direct product factor
41	C₂×C₃²×Q₈	—	3	2	no	—
42	C₂²×C₃²×C₄	—	2	4	no	—
43	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
45	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
46	C₃²×Q₁₆	—	6	2	no	—
47	C₃²×M₁₆	—	2	16	no	—
48	C₃²×M₁₆	—	2	2	no	—
49	C₃²×M₁₆	—	2	8	no	—
50	C₂×C₃²×C₈	—	2	4	no	—
51	C₂×C₃²×C₈	—	2	4	no	—
52	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
55	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
56	C₃²×C₄⋊C₄	—	3	2	no	—
57	C₃²×C₄⋊C₄	—	6	2	no	—
58	C₃²×C₂²⋊C₄	—	6	2	no	—
59	C₃×#217	—	2	8	no	—
60	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
61	C₃×Dic₃⋊C₄	—	2	8	no	—
62	C₃×Dic₁₂	—	2	8	no	—
63	C₂²×C₃³	—	2	1	no	—
64	C₂²×C₃³	—	2	2	no	—
65	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
66	C₃³×C₄	—	2	1	no	—
67	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
68	C₃³×C₄	#41	1	1	yes	characteristic
69	C₃²×Dic₃	—	4	2	no	—
70	C₃²×Dic₃	—	4	2	no	—
71	C₂×C₃×Q₁₆	—	3	4	no	—
72	C₂×C₃×M₁₆	—	2	4	no	—
73	C₂×C₃×M₁₆	C₃×D₃	1	4	yes	—
74	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
75	C₃×C₂²⋊Q₈	—	3	8	no	—
76	#924	C₂×C₃²	1	2	yes	—
77	C₂×Dic₁₂	C₂×C₃²	1	1	yes	characteristic, complemented
78	C₃×#106	—	3	4	no	—
79	C₃×#103	—	3	8	no	—
80	#817	C₂×C₃²	1	1	yes	characteristic, complemented
81	#816	C₂×C₃²	1	2	yes	complemented
82	C₂³×C₃²	—	2	4	no	—
83	C₂³×C₃²	D₁₂	1	1	yes	characteristic
84	C₂³×C₃²	C₃×D₄	1	4	yes	—
85	C₃²×Q₈	—	6	2	no	—
86	C₃²×Q₈	—	6	2	no	—
87	C₂×C₃²×C₄	—	2	8	no	—
88	C₂×C₃²×C₄	—	2	8	no	—
89	C₂×C₃²×C₄	—	2	1	no	—
90	C₂×C₃²×C₄	—	2	1	no	—
91	C₂×C₃²×C₄	—	2	4	no	—
92	C₂×C₃²×C₄	—	2	4	no	—
93	C₂×C₃²×C₄	—	2	4	no	—
94	C₂×C₃²×C₄	—	2	4	no	—
95	C₂×C₃²×C₄	—	6	2	no	—
96	C₂×C₃²×C₄	—	6	2	no	—
97	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
99	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
101	C₂×C₃×Dic₃	—	2	8	no	—
102	C₂×C₃×Dic₃	—	2	8	no	—
103	C₃×Dic₆	—	2	8	no	—
104	C₃×Dic₆	—	2	8	no	—
105	C₃²×C₈	—	2	8	no	—
106	C₃²×C₈	—	2	1	no	—
107	C₃²×C₈	—	2	4	no	—
108	C₃²×C₈	—	2	8	no	—
109	C₃²×C₈	#67	1	2	yes	—
110	C₃²×C₈	C₃×D₄	1	8	yes	—
111	#470	—	3	1	no	Sylow
112	C₂×C₃³	—	4	1	no	—
113	C₂×C₃³	#121	1	1	yes	characteristic
114	C₂×C₃³	#137	1	2	yes	—
115	C₂×C₃×Q₈	—	3	8	no	—
116	C₂²×C₃×C₄	—	2	4	no	—
117	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
118	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
119	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
120	C₃×Q₁₆	—	6	8	no	—
121	C₃×M₁₆	—	2	16	no	—
122	C₃×M₁₆	—	2	8	no	—
123	C₃×M₁₆	—	2	2	no	—
124	C₂×C₃×C₈	—	2	4	no	—
125	C₂×C₃×C₈	—	2	4	no	—
126	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
127	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
128	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
129	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	3	8	no	—
131	C₃×C₄⋊C₄	—	6	8	no	—
132	C₃×C₂²⋊C₄	—	6	8	no	—
133	#217	—	2	2	no	—
134	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
135	Dic₃⋊C₄	—	2	2	no	—
136	Dic₁₂	—	2	2	no	—
137	C₂²×C₃²	—	2	8	no	—
138	C₂²×C₃²	—	2	2	no	—
139	C₂²×C₃²	—	2	1	no	—
140	C₂²×C₃²	—	2	4	no	—
141	C₂²×C₃²	—	2	4	no	—
142	C₂²×C₃²	—	2	8	no	—
143	C₂²×C₃²	—	4	8	no	—
144	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
145	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
146	C₃²×C₄	—	2	8	no	—
147	C₃²×C₄	—	2	1	no	—
148	C₃²×C₄	—	2	4	no	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	—	2	4	no	—
151	C₃²×C₄	—	12	2	no	—
152	C₃²×C₄	—	12	2	no	—
153	C₃²×C₄	#235	1	1	yes	characteristic
154	C₃²×C₄	C₂×#67	1	1	yes	characteristic
155	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
156	C₃²×C₄	C₃×#41	1	4	yes	—
157	C₃×Dic₃	—	4	8	no	—
158	C₃×Dic₃	—	4	8	no	—
159	C₂×Q₁₆	—	3	1	no	—
160	C₂×M₁₆	C₃²×D₃	1	1	yes	characteristic
161	C₂²⋊Q₈	—	3	2	no	—
162	#106	—	3	1	no	—
163	#103	—	3	2	no	—
164	C₃³	#470	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₃	—	2	4	no	—
166	C₂³×C₃	C₃×D₁₂	1	4	yes	—
167	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
168	C₃×Q₈	—	6	8	no	—
169	C₃×Q₈	—	6	8	no	—
170	C₂×C₃×C₄	—	2	8	no	—
171	C₂×C₃×C₄	—	2	8	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	6	8	no	—
179	C₂×C₃×C₄	—	6	8	no	—
180	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
181	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
183	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
184	C₂×Dic₃	—	2	2	no	—
185	C₂×Dic₃	—	2	2	no	—
186	Dic₆	—	2	2	no	—
187	Dic₆	—	2	2	no	—
188	C₃×C₈	—	2	8	no	—
189	C₃×C₈	—	2	4	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	—	2	8	no	—
192	C₃×C₈	C₃×#67	1	8	yes	—
193	C₃×C₈	C₃²×D₄	1	2	yes	—
194	C₂×C₃²	—	2	8	no	—
195	C₂×C₃²	—	2	4	no	—
196	C₂×C₃²	—	4	8	no	—
197	C₂×C₃²	—	4	1	no	—
198	C₂×C₃²	—	4	4	no	—
199	C₂×C₃²	#909	1	2	yes	—
200	C₂×C₃²	#930	1	1	yes	characteristic
201	C₂×C₃²	C₃×#121	1	4	yes	—
202	C₂×C₃²	C₃×#137	1	8	yes	—
203	C₂×Q₈	—	3	2	no	—
204	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
205	Q₁₆	—	6	2	no	—
206	M₁₆	—	2	2	no	—
207	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
209	C₄⋊C₄	—	3	2	no	—
210	C₄⋊C₄	—	6	2	no	—
211	C₂²⋊C₄	—	6	2	no	—
212	C₂²×C₃	—	2	8	no	—
213	C₂²×C₃	—	2	8	no	—
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	4	no	—
218	C₂²×C₃	—	4	8	no	—
219	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
220	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
221	C₃×C₄	—	2	8	no	—
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	12	8	no	—
227	C₃×C₄	—	12	8	no	—
228	C₃×C₄	C₃×#235	1	4	yes	—
229	C₃×C₄	C₂×C₃×#67	1	4	yes	—
230	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
231	C₃×C₄	C₃²×#41	1	1	yes	characteristic
232	Dic₃	—	4	2	no	—
233	Dic₃	—	4	2	no	—
234	C₃²	—	2	4	no	—
235	C₃²	#5137	1	1	yes	characteristic, direct product factor
236	C₃²	C₃×#470	1	4	yes	complemented
237	C₂³	C₃²×D₁₂	1	1	yes	characteristic
238	Q₈	—	6	2	no	—
239	Q₈	—	6	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	6	2	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	C₃²×#67	1	2	yes	—
248	C₂×C₃	—	2	8	no	—
249	C₂×C₃	—	2	4	no	—
250	C₂×C₃	—	4	8	no	—
251	C₂×C₃	—	4	1	no	—
252	C₂×C₃	—	4	4	no	—
253	C₂×C₃	C₃×#909	1	8	yes	—
254	C₂×C₃	C₃×#930	1	4	yes	—
255	C₂×C₃	C₃²×#121	1	1	yes	characteristic
256	C₂×C₃	C₃²×#137	1	2	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	12	2	no	—
262	C₄	—	12	2	no	—
263	C₄	C₃²×#235	1	1	yes	characteristic
264	C₄	C₂×C₃²×#67	1	1	yes	characteristic
265	C₃	—	2	4	no	—
266	C₃	C₃×#5137	1	4	yes	direct product factor
267	C₃	C₃²×#470	1	1	yes	characteristic, complemented
268	C₂	—	4	1	no	—
269	C₂	C₃²×#909	1	2	yes	—
270	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
prime radical	C₂³×C₃³	D₄
hypercenter	C₂×C₃²×M₁₆	D₃
hypocenter	C₃	C₃²×#470
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
3. C₂×C₃²×Dic₁₂	C₂	semidirect
4. C₃²×#817	C₂	semidirect
5. C₃²×#816	C₂	semidirect
7. C₃×#5137	C₃	direct (N × H)
23. C₂×C₃×Dic₁₂	C₂×C₃	semidirect
26. C₃×#817	C₂×C₃	semidirect
27. C₃×#816	C₂×C₃	semidirect
40. #5137	C₃²	direct (N × H)
77. C₂×Dic₁₂	C₂×C₃²	semidirect
80. #817	C₂×C₃²	semidirect
81. #816	C₂×C₃²	semidirect
164. C₃³	#470	semidirect
235. C₃²	#5137	direct (N × H)
236. C₃²	C₃×#470	semidirect
266. C₃	C₃×#5137	direct (N × H)
267. C₃	C₃²×#470	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×#924	2
C₂×C₃²×Dic₁₂	2
C₃²×#817	2
C₃²×#816	2
C₃²×#470	3
C₃×#5137	3

Subgroups — C32×#5137 #13162573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5137 #13162573