Subgroups — C₃²×#5070 #13162506

Order 1728.

1140 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#127	C₂	1	1	yes	maximal, characteristic
2	C₃²×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#832	C₂	1	4	yes	maximal, complemented
4	C₃²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#486	—	3	1	no	maximal
6	C₃×#5070	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃³×D₄	—	2	1	no	—
8	C₂×C₃³×D₄	—	2	1	no	—
9	C₂×C₃³×D₄	C₂²	1	2	yes	—
10	C₂×C₃²×Dic₆	—	2	1	no	—
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃²×C₄⋊Dic₃	—	2	1	no	—
13	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃²×C₄⋊Q₈	—	3	1	no	—
16	C₃²×#127	—	2	4	no	—
17	C₃²×#127	D₃	1	1	yes	characteristic
18	C₃²×#127	C₂×C₃	1	4	yes	—
19	C₃×#869	C₂×C₃	1	4	yes	complemented
20	C₃²×#102	—	3	4	no	—
21	C₃²×C₄×C₈	—	3	1	no	—
22	C₃×#832	C₂×C₃	1	16	yes	complemented
23	C₃×C₄×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
24	C₂³×C₃³	—	2	2	no	—
25	C₃³×D₄	—	2	4	no	—
26	C₃³×D₄	—	4	2	no	—
27	C₃³×D₄	—	4	2	no	—
28	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃²×Dic₃	—	2	2	no	—
32	C₃²×Dic₆	—	4	2	no	—
33	C₃²×C₃⋊C₈	—	2	2	no	—
34	C₃×#486	—	3	4	no	—
35	#5070	C₃²	1	1	yes	characteristic, direct product factor
36	C₂×C₃²×Q₈	—	6	1	no	—
37	C₂×C₃²×D₄	—	2	8	no	—
38	C₂×C₃²×D₄	—	2	8	no	—
39	C₂×C₃²×D₄	—	2	8	no	—
40	C₂×C₃²×D₄	—	2	1	no	—
41	C₂×C₃²×D₄	—	2	1	no	—
42	C₂×C₃²×D₄	—	2	4	no	—
43	C₂×C₃²×D₄	—	2	4	no	—
44	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
45	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	—
46	C₂×C₃×Dic₆	—	2	4	no	—
47	C₂×C₃²×C₈	—	3	2	no	—
48	C₃²×C₄⋊C₄	—	3	2	no	—
49	C₃²×C₄⋊C₄	—	6	1	no	—
50	C₃²×C₄²	—	2	4	no	—
51	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₃²×C₄²	C₂²×C₃	1	4	yes	—
53	C₃×C₄⋊Dic₃	—	2	4	no	—
54	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
55	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
56	C₂²×C₃³	—	4	2	no	—
57	C₂²×C₃³	—	4	2	no	—
58	C₂²×C₃³	—	4	2	no	—
59	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
60	C₃³×C₄	D₈	1	2	yes	—
61	C₃³×C₄	SD₁₆	1	2	yes	—
62	C₃³×C₄	#41	1	2	yes	—
63	C₃²×Dic₃	—	4	2	no	—
64	C₃×C₄⋊Q₈	—	3	4	no	—
65	C₃×#127	—	2	4	no	—
66	C₃×#127	C₃×D₃	1	4	yes	—
67	C₃×#127	C₂×C₃²	1	1	yes	characteristic
68	#869	C₂×C₃²	1	1	yes	characteristic, complemented
69	C₃×#102	—	3	16	no	—
70	C₃×C₄×C₈	—	3	4	no	—
71	#832	C₂×C₃²	1	4	yes	complemented
72	C₄×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
73	C₂³×C₃²	—	2	16	no	—
74	C₂³×C₃²	—	2	2	no	—
75	C₂³×C₃²	—	2	8	no	—
76	C₃²×Q₈	—	12	2	no	—
77	C₃²×D₄	—	2	4	no	—
78	C₃²×D₄	—	2	16	no	—
79	C₃²×D₄	—	4	16	no	—
80	C₃²×D₄	—	4	16	no	—
81	C₃²×D₄	—	4	16	no	—
82	C₃²×D₄	—	4	2	no	—
83	C₃²×D₄	—	4	2	no	—
84	C₃²×D₄	—	4	8	no	—
85	C₃²×D₄	—	4	8	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	—	2	4	no	—
89	C₂×C₃²×C₄	—	6	2	no	—
90	C₂×C₃²×C₄	#67	1	1	yes	characteristic
91	C₂×C₃²×C₄	#67	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
93	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
94	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
96	C₂×C₃×Dic₃	—	2	8	no	—
97	C₃×Dic₆	—	4	8	no	—
98	C₃²×C₈	—	6	2	no	—
99	C₃×C₃⋊C₈	—	2	8	no	—
100	#486	—	3	1	no	Sylow
101	C₂×C₃³	—	8	2	no	—
102	C₂×C₃³	#124	1	1	yes	characteristic
103	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
104	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₃×Q₈	—	6	4	no	—
106	C₂×C₃×D₄	—	2	8	no	—
107	C₂×C₃×D₄	—	2	8	no	—
108	C₂×C₃×D₄	—	2	4	no	—
109	C₂×C₃×D₄	—	2	4	no	—
110	C₂×C₃×D₄	—	2	1	no	—
111	C₂×C₃×D₄	—	2	1	no	—
112	C₂×C₃×D₄	—	2	8	no	—
113	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
114	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
115	C₂×Dic₆	—	2	1	no	—
116	C₂×C₃×C₈	—	3	8	no	—
117	C₃×C₄⋊C₄	—	3	8	no	—
118	C₃×C₄⋊C₄	—	6	4	no	—
119	C₃×C₄²	—	2	4	no	—
120	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
121	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
122	C₄⋊Dic₃	—	2	1	no	—
123	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
124	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
125	C₂²×C₃²	—	2	4	no	—
126	C₂²×C₃²	—	4	16	no	—
127	C₂²×C₃²	—	4	16	no	—
128	C₂²×C₃²	—	4	16	no	—
129	C₂²×C₃²	—	4	2	no	—
130	C₂²×C₃²	—	4	2	no	—
131	C₂²×C₃²	—	4	2	no	—
132	C₂²×C₃²	—	4	8	no	—
133	C₂²×C₃²	—	4	8	no	—
134	C₂²×C₃²	—	4	8	no	—
135	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
136	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
137	C₃²×C₄	—	2	8	no	—
138	C₃²×C₄	—	2	8	no	—
139	C₃²×C₄	—	2	8	no	—
140	C₃²×C₄	—	12	2	no	—
141	C₃²×C₄	#213	1	2	yes	—
142	C₃²×C₄	#214	1	2	yes	—
143	C₃²×C₄	C₃×D₈	1	8	yes	—
144	C₃²×C₄	C₃×SD₁₆	1	8	yes	—
145	C₃²×C₄	#237	1	2	yes	—
146	C₃²×C₄	C₃×#41	1	8	yes	—
147	C₃×Dic₃	—	4	8	no	—
148	C₄⋊Q₈	—	3	1	no	—
149	#127	C₃²×D₃	1	1	yes	characteristic
150	#102	—	3	4	no	—
151	C₄×C₈	—	3	1	no	—
152	C₃³	#486	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₃	—	2	16	no	—
154	C₂³×C₃	—	2	8	no	—
155	C₂³×C₃	—	2	2	no	—
156	C₃×Q₈	—	12	8	no	—
157	C₃×D₄	—	2	16	no	—
158	C₃×D₄	—	2	4	no	—
159	C₃×D₄	—	4	16	no	—
160	C₃×D₄	—	4	16	no	—
161	C₃×D₄	—	4	16	no	—
162	C₃×D₄	—	4	8	no	—
163	C₃×D₄	—	4	8	no	—
164	C₃×D₄	—	4	2	no	—
165	C₃×D₄	—	4	2	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	6	8	no	—
170	C₂×C₃×C₄	C₃×#67	1	4	yes	—
171	C₂×C₃×C₄	C₃×#67	1	4	yes	—
172	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
175	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
176	C₂×Dic₃	—	2	2	no	—
177	Dic₆	—	4	2	no	—
178	C₃×C₈	—	6	8	no	—
179	C₃⋊C₈	—	2	2	no	—
180	C₂×C₃²	—	2	4	no	—
181	C₂×C₃²	—	2	4	no	—
182	C₂×C₃²	—	2	4	no	—
183	C₂×C₃²	—	8	16	no	—
184	C₂×C₃²	—	8	2	no	—
185	C₂×C₃²	—	8	8	no	—
186	C₂×C₃²	C₂×#213	1	1	yes	characteristic
187	C₂×C₃²	C₂×#214	1	1	yes	characteristic
188	C₂×C₃²	#936	1	1	yes	characteristic
189	C₂×C₃²	C₃×#124	1	4	yes	—
190	C₂×C₃²	C₂×C₃×D₈	1	4	yes	—
191	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
192	C₂×Q₈	—	6	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	C₂×C₃²×D₃	1	2	yes	—
196	C₂×C₈	—	3	2	no	—
197	C₄⋊C₄	—	3	2	no	—
198	C₄⋊C₄	—	6	1	no	—
199	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	4	16	no	—
202	C₂²×C₃	—	4	16	no	—
203	C₂²×C₃	—	4	16	no	—
204	C₂²×C₃	—	4	8	no	—
205	C₂²×C₃	—	4	8	no	—
206	C₂²×C₃	—	4	8	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	—	4	2	no	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
211	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
212	C₃×C₄	—	2	8	no	—
213	C₃×C₄	—	2	8	no	—
214	C₃×C₄	—	2	8	no	—
215	C₃×C₄	—	12	8	no	—
216	C₃×C₄	C₃×#213	1	8	yes	—
217	C₃×C₄	C₃×#214	1	8	yes	—
218	C₃×C₄	C₃²×D₈	1	2	yes	—
219	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
220	C₃×C₄	C₃×#237	1	8	yes	—
221	C₃×C₄	C₃²×#41	1	2	yes	—
222	Dic₃	—	4	2	no	—
223	C₃²	—	2	4	no	—
224	C₃²	#5070	1	1	yes	characteristic, direct product factor
225	C₃²	C₃×#486	1	4	yes	complemented
226	C₂³	—	2	2	no	—
227	Q₈	—	12	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	6	2	no	—
232	C₂×C₄	C₃²×#67	1	1	yes	characteristic
233	C₂×C₄	C₃²×#67	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
235	C₈	—	6	2	no	—
236	C₂×C₃	—	2	4	no	—
237	C₂×C₃	—	2	4	no	—
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	—	8	16	no	—
240	C₂×C₃	—	8	2	no	—
241	C₂×C₃	—	8	8	no	—
242	C₂×C₃	C₂×C₃×#213	1	4	yes	—
243	C₂×C₃	C₂×C₃×#214	1	4	yes	—
244	C₂×C₃	C₃×#936	1	4	yes	—
245	C₂×C₃	C₃²×#124	1	1	yes	characteristic
246	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
247	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	C₂×C₃²×#67	1	1	yes	characteristic
252	C₄	—	12	2	no	—
253	C₄	C₃²×#213	1	2	yes	—
254	C₄	C₃²×#214	1	2	yes	—
255	C₄	C₃²×#237	1	2	yes	—
256	C₃	—	2	4	no	—
257	C₃	C₃×#5070	1	4	yes	direct product factor
258	C₃	C₃²×#486	1	1	yes	characteristic, complemented
259	C₂	—	8	2	no	—
260	C₂	C₂×C₃²×#213	1	1	yes	characteristic
261	C₂	C₂×C₃²×#214	1	1	yes	characteristic
262	C₂	C₃²×#936	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting, prime radical	C₃³×#127	C₂
socle	C₂²×C₃³	C₂×D₄
hypercenter	C₃²×#127	D₃
hypocenter	C₃	C₃²×#486
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×#127

Complements

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

N	H ≅ G/N	type
2. C₃²×#869	C₂	semidirect
3. C₃²×#832	C₂	semidirect
4. C₃²×C₄×C₃⋊C₈	C₂	semidirect
6. C₃×#5070	C₃	direct (N × H)
19. C₃×#869	C₂×C₃	semidirect
22. C₃×#832	C₂×C₃	semidirect
23. C₃×C₄×C₃⋊C₈	C₂×C₃	semidirect
35. #5070	C₃²	direct (N × H)
68. #869	C₂×C₃²	semidirect
71. #832	C₂×C₃²	semidirect
72. C₄×C₃⋊C₈	C₂×C₃²	semidirect
152. C₃³	#486	semidirect
224. C₃²	#5070	direct (N × H)
225. C₃²	C₃×#486	semidirect
257. C₃	C₃×#5070	direct (N × H)
258. C₃	C₃²×#486	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#127	2
C₃²×#869	2
C₃²×#832	2
C₃²×C₄×C₃⋊C₈	2
C₃²×#486	3
C₃×#5070	3

Subgroups — C32×#5070 #13162506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5070 #13162506