Subgroups — C₃³×#525 #13179980

Order 1728.

4620 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	C₂×C₃³×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#123	C₂	1	2	yes	maximal, complemented
4	C₃³×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₃³×#121	C₂	1	2	yes	maximal, complemented
6	C₃³×#120	C₂	1	2	yes	maximal, complemented
7	C₃³×C₄×D₄	C₂	1	4	yes	maximal, complemented
8	C₃³×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃²×#525	C₃	1	13	yes	maximal, direct product factor
10	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
11	C₃³×#41	—	2	4	no	—
12	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃³×D₄	C₂²	1	2	yes	—
14	C₂×C₃³×D₄	C₂²	1	4	yes	—
15	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₃³×C₄	—	2	4	no	—
17	C₂²×C₃³×C₄	C₂²	1	2	yes	—
18	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
19	C₂²×C₃³×C₄	C₂²	1	2	yes	—
20	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
21	C₂²×C₃³×C₄	C₂²	1	2	yes	complemented
22	C₃³×C₄⋊C₄	C₂²	1	2	yes	complemented
23	C₃³×C₄⋊C₄	C₂²	1	4	yes	complemented
24	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₃³×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₃³×C₂²⋊C₄	C₂²	1	4	yes	complemented
27	C₃³×C₄²	C₂²	1	2	yes	complemented
28	C₂×C₃²×#41	C₂×C₃	1	13	yes	complemented
29	C₂³×C₃²×C₄	C₂×C₃	1	13	yes	complemented
30	C₃²×#123	C₂×C₃	1	26	yes	complemented
31	C₃²×C₂²⋊Q₈	C₂×C₃	1	26	yes	complemented
32	C₃²×#121	C₂×C₃	1	26	yes	complemented
33	C₃²×#120	C₂×C₃	1	26	yes	complemented
34	C₃²×C₄×D₄	C₂×C₃	1	52	yes	complemented
35	C₃²×#117	C₂×C₃	1	13	yes	complemented
36	C₂³×C₃³	—	2	2	no	—
37	C₂³×C₃³	—	2	4	no	—
38	C₂³×C₃³	C₂³	1	2	yes	—
39	C₂³×C₃³	C₂³	1	1	yes	characteristic
40	C₂³×C₃³	C₂³	1	2	yes	—
41	C₃³×Q₈	—	2	2	no	—
42	C₃³×D₄	—	2	4	no	—
43	C₃³×D₄	—	2	8	no	—
44	C₃³×D₄	—	2	2	no	—
45	C₂×C₃³×C₄	—	2	2	no	—
46	C₂×C₃³×C₄	—	2	4	no	—
47	C₂×C₃³×C₄	—	2	4	no	—
48	C₂×C₃³×C₄	—	2	4	no	—
49	C₂×C₃³×C₄	D₄	1	4	yes	complemented
50	C₂×C₃³×C₄	C₂³	1	4	yes	—
51	C₂×C₃³×C₄	C₂³	1	2	yes	—
52	C₂×C₃³×C₄	C₂³	1	2	yes	—
53	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
54	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
55	C₃×#525	C₃²	1	13	yes	direct product factor
56	C₂⁴×C₃²	C₂²×C₃	1	13	yes	—
57	C₃²×#41	—	2	52	no	—
58	C₂×C₃²×Q₈	C₂²×C₃	1	13	yes	complemented
59	C₂×C₃²×D₄	C₂²×C₃	1	26	yes	—
60	C₂×C₃²×D₄	C₂²×C₃	1	52	yes	—
61	C₂×C₃²×D₄	C₂²×C₃	1	13	yes	complemented
62	C₂²×C₃²×C₄	—	2	52	no	—
63	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
64	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
65	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
66	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
67	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	complemented
68	C₃²×C₄⋊C₄	C₂²×C₃	1	26	yes	complemented
69	C₃²×C₄⋊C₄	C₂²×C₃	1	52	yes	complemented
70	C₃²×C₂²⋊C₄	C₂²×C₃	1	26	yes	—
71	C₃²×C₂²⋊C₄	C₂²×C₃	1	52	yes	—
72	C₃²×C₂²⋊C₄	C₂²×C₃	1	52	yes	complemented
73	C₃²×C₄²	C₂²×C₃	1	26	yes	complemented
74	C₂²×C₃³	—	2	8	no	—
75	C₂²×C₃³	—	2	2	no	—
76	C₂²×C₃³	—	2	2	no	—
77	C₂²×C₃³	—	2	2	no	—
78	C₂²×C₃³	—	2	2	no	—
79	C₂²×C₃³	—	2	4	no	—
80	C₂²×C₃³	C₂×D₄	1	2	yes	—
81	C₂²×C₃³	#41	1	4	yes	complemented
82	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
83	C₃³×C₄	—	2	2	no	—
84	C₃³×C₄	—	2	2	no	—
85	C₃³×C₄	—	2	4	no	—
86	C₃³×C₄	C₂×D₄	1	2	yes	—
87	C₃³×C₄	C₂×D₄	1	2	yes	—
88	C₂×C₃×#41	C₂×C₃²	1	13	yes	complemented
89	C₂³×C₃×C₄	C₂×C₃²	1	13	yes	complemented
90	C₃×#123	C₂×C₃²	1	26	yes	complemented
91	C₃×C₂²⋊Q₈	C₂×C₃²	1	26	yes	complemented
92	C₃×#121	C₂×C₃²	1	26	yes	complemented
93	C₃×#120	C₂×C₃²	1	26	yes	complemented
94	C₃×C₄×D₄	C₂×C₃²	1	52	yes	complemented
95	C₃×#117	C₂×C₃²	1	13	yes	complemented
96	C₂³×C₃²	—	2	26	no	—
97	C₂³×C₃²	—	2	52	no	—
98	C₂³×C₃²	C₂³×C₃	1	26	yes	—
99	C₂³×C₃²	C₂³×C₃	1	13	yes	—
100	C₂³×C₃²	C₂³×C₃	1	26	yes	—
101	C₃²×Q₈	—	2	26	no	—
102	C₃²×D₄	—	2	52	no	—
103	C₃²×D₄	—	2	104	no	—
104	C₃²×D₄	—	2	26	no	—
105	C₂×C₃²×C₄	—	2	26	no	—
106	C₂×C₃²×C₄	—	2	52	no	—
107	C₂×C₃²×C₄	—	2	52	no	—
108	C₂×C₃²×C₄	—	2	52	no	—
109	C₂×C₃²×C₄	C₃×D₄	1	52	yes	complemented
110	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
111	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
112	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
113	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
114	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
115	#525	C₃³	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×C₃³	—	2	2	no	—
117	C₂×C₃³	—	2	4	no	—
118	C₂×C₃³	—	4	2	no	—
119	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₃³	C₂×#41	1	2	yes	—
121	C₂⁴×C₃	C₂²×C₃²	1	13	yes	—
122	C₃×#41	—	2	52	no	—
123	C₂×C₃×Q₈	C₂²×C₃²	1	13	yes	complemented
124	C₂×C₃×D₄	C₂²×C₃²	1	52	yes	—
125	C₂×C₃×D₄	C₂²×C₃²	1	26	yes	—
126	C₂×C₃×D₄	C₂²×C₃²	1	13	yes	complemented
127	C₂²×C₃×C₄	—	2	52	no	—
128	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
129	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
130	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
131	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
132	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	complemented
133	C₃×C₄⋊C₄	C₂²×C₃²	1	26	yes	complemented
134	C₃×C₄⋊C₄	C₂²×C₃²	1	52	yes	complemented
135	C₃×C₂²⋊C₄	C₂²×C₃²	1	26	yes	—
136	C₃×C₂²⋊C₄	C₂²×C₃²	1	52	yes	—
137	C₃×C₂²⋊C₄	C₂²×C₃²	1	52	yes	complemented
138	C₃×C₄²	C₂²×C₃²	1	26	yes	complemented
139	C₂²×C₃²	—	2	104	no	—
140	C₂²×C₃²	—	2	26	no	—
141	C₂²×C₃²	—	2	26	no	—
142	C₂²×C₃²	—	2	26	no	—
143	C₂²×C₃²	—	2	26	no	—
144	C₂²×C₃²	—	2	52	no	—
145	C₂²×C₃²	C₂×C₃×D₄	1	26	yes	—
146	C₂²×C₃²	C₃×#41	1	52	yes	complemented
147	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
148	C₃²×C₄	—	2	26	no	—
149	C₃²×C₄	—	2	26	no	—
150	C₃²×C₄	—	2	52	no	—
151	C₃²×C₄	C₂×C₃×D₄	1	26	yes	—
152	C₃²×C₄	C₂×C₃×D₄	1	26	yes	—
153	C₂×#41	C₂×C₃³	1	1	yes	characteristic, complemented
154	C₂³×C₄	C₂×C₃³	1	1	yes	characteristic, complemented
155	#123	C₂×C₃³	1	2	yes	complemented
156	C₂²⋊Q₈	C₂×C₃³	1	2	yes	complemented
157	#121	C₂×C₃³	1	2	yes	complemented
158	#120	C₂×C₃³	1	2	yes	complemented
159	C₄×D₄	C₂×C₃³	1	4	yes	complemented
160	#117	C₂×C₃³	1	1	yes	characteristic, complemented
161	C₃³	#525	1	1	yes	characteristic, direct product factor, Sylow
162	C₂³×C₃	—	2	26	no	—
163	C₂³×C₃	—	2	52	no	—
164	C₂³×C₃	C₂³×C₃²	1	26	yes	—
165	C₂³×C₃	C₂³×C₃²	1	13	yes	—
166	C₂³×C₃	C₂³×C₃²	1	26	yes	—
167	C₃×Q₈	—	2	26	no	—
168	C₃×D₄	—	2	52	no	—
169	C₃×D₄	—	2	104	no	—
170	C₃×D₄	—	2	26	no	—
171	C₂×C₃×C₄	—	2	26	no	—
172	C₂×C₃×C₄	—	2	52	no	—
173	C₂×C₃×C₄	—	2	52	no	—
174	C₂×C₃×C₄	—	2	52	no	—
175	C₂×C₃×C₄	C₃²×D₄	1	52	yes	complemented
176	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
177	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
178	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
179	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
180	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
181	C₂×C₃²	—	2	26	no	—
182	C₂×C₃²	—	2	52	no	—
183	C₂×C₃²	—	4	26	no	—
184	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
185	C₂×C₃²	C₂×C₃×#41	1	26	yes	—
186	C₂⁴	C₂²×C₃³	1	1	yes	characteristic
187	#41	—	2	4	no	—
188	C₂×Q₈	C₂²×C₃³	1	1	yes	characteristic, complemented
189	C₂×D₄	C₂²×C₃³	1	4	yes	—
190	C₂×D₄	C₂²×C₃³	1	2	yes	—
191	C₂×D₄	C₂²×C₃³	1	1	yes	characteristic, complemented
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	C₂²×C₃³	1	2	yes	—
194	C₂²×C₄	C₂²×C₃³	1	2	yes	—
195	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₃³	1	2	yes	complemented
198	C₄⋊C₄	C₂²×C₃³	1	2	yes	complemented
199	C₄⋊C₄	C₂²×C₃³	1	4	yes	complemented
200	C₂²⋊C₄	C₂²×C₃³	1	2	yes	—
201	C₂²⋊C₄	C₂²×C₃³	1	4	yes	—
202	C₂²⋊C₄	C₂²×C₃³	1	4	yes	complemented
203	C₄²	C₂²×C₃³	1	2	yes	complemented
204	C₂²×C₃	—	2	104	no	—
205	C₂²×C₃	—	2	26	no	—
206	C₂²×C₃	—	2	26	no	—
207	C₂²×C₃	—	2	26	no	—
208	C₂²×C₃	—	2	26	no	—
209	C₂²×C₃	—	2	52	no	—
210	C₂²×C₃	C₂×C₃²×D₄	1	26	yes	—
211	C₂²×C₃	C₃²×#41	1	52	yes	complemented
212	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
213	C₃×C₄	—	2	26	no	—
214	C₃×C₄	—	2	26	no	—
215	C₃×C₄	—	2	52	no	—
216	C₃×C₄	C₂×C₃²×D₄	1	26	yes	—
217	C₃×C₄	C₂×C₃²×D₄	1	26	yes	—
218	C₃²	C₃×#525	1	13	yes	direct product factor
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	C₂³×C₃³	1	2	yes	—
222	C₂³	C₂³×C₃³	1	1	yes	characteristic
223	C₂³	C₂³×C₃³	1	2	yes	—
224	Q₈	—	2	2	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	8	no	—
227	D₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	C₃³×D₄	1	4	yes	complemented
233	C₂×C₄	C₂³×C₃³	1	4	yes	—
234	C₂×C₄	C₂³×C₃³	1	2	yes	—
235	C₂×C₄	C₂³×C₃³	1	2	yes	—
236	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
237	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
238	C₂×C₃	—	2	26	no	—
239	C₂×C₃	—	2	52	no	—
240	C₂×C₃	—	4	26	no	—
241	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
242	C₂×C₃	C₂×C₃²×#41	1	26	yes	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	4	no	—
249	C₂²	C₂×C₃³×D₄	1	2	yes	—
250	C₂²	C₃³×#41	1	4	yes	complemented
251	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	C₂×C₃³×D₄	1	2	yes	—
256	C₄	C₂×C₃³×D₄	1	2	yes	—
257	C₃	C₃²×#525	1	13	yes	direct product factor
258	C₂	—	2	2	no	—
259	C₂	—	2	4	no	—
260	C₂	—	4	2	no	—
261	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic
262	C₂	C₂×C₃³×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₃³×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
socle	C₂²×C₃³	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×#41	C₂	semidirect
2. C₂³×C₃³×C₄	C₂	semidirect
3. C₃³×#123	C₂	semidirect
4. C₃³×C₂²⋊Q₈	C₂	semidirect
5. C₃³×#121	C₂	semidirect
6. C₃³×#120	C₂	semidirect
7. C₃³×C₄×D₄	C₂	semidirect
8. C₃³×#117	C₂	semidirect
9. C₃²×#525	C₃	direct (N × H)
12. C₂×C₃³×Q₈	C₂²	semidirect
15. C₂×C₃³×D₄	C₂²	semidirect
21. C₂²×C₃³×C₄	C₂²	semidirect
22. C₃³×C₄⋊C₄	C₂²	semidirect
23. C₃³×C₄⋊C₄	C₂²	semidirect
26. C₃³×C₂²⋊C₄	C₂²	semidirect
27. C₃³×C₄²	C₂²	semidirect
28. C₂×C₃²×#41	C₂×C₃	semidirect
29. C₂³×C₃²×C₄	C₂×C₃	semidirect
30. C₃²×#123	C₂×C₃	semidirect
31. C₃²×C₂²⋊Q₈	C₂×C₃	semidirect
32. C₃²×#121	C₂×C₃	semidirect
33. C₃²×#120	C₂×C₃	semidirect
34. C₃²×C₄×D₄	C₂×C₃	semidirect
35. C₃²×#117	C₂×C₃	semidirect
49. C₂×C₃³×C₄	D₄	semidirect
55. C₃×#525	C₃²	direct (N × H)
58. C₂×C₃²×Q₈	C₂²×C₃	semidirect
61. C₂×C₃²×D₄	C₂²×C₃	semidirect
67. C₂²×C₃²×C₄	C₂²×C₃	semidirect
68. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
69. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
72. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
73. C₃²×C₄²	C₂²×C₃	semidirect
81. C₂²×C₃³	#41	semidirect
88. C₂×C₃×#41	C₂×C₃²	semidirect
89. C₂³×C₃×C₄	C₂×C₃²	semidirect
90. C₃×#123	C₂×C₃²	semidirect
91. C₃×C₂²⋊Q₈	C₂×C₃²	semidirect
92. C₃×#121	C₂×C₃²	semidirect
93. C₃×#120	C₂×C₃²	semidirect
94. C₃×C₄×D₄	C₂×C₃²	semidirect
95. C₃×#117	C₂×C₃²	semidirect
109. C₂×C₃²×C₄	C₃×D₄	semidirect
115. #525	C₃³	direct (N × H)
123. C₂×C₃×Q₈	C₂²×C₃²	semidirect
126. C₂×C₃×D₄	C₂²×C₃²	semidirect
132. C₂²×C₃×C₄	C₂²×C₃²	semidirect
133. C₃×C₄⋊C₄	C₂²×C₃²	semidirect
134. C₃×C₄⋊C₄	C₂²×C₃²	semidirect
137. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
138. C₃×C₄²	C₂²×C₃²	semidirect
146. C₂²×C₃²	C₃×#41	semidirect
153. C₂×#41	C₂×C₃³	semidirect
154. C₂³×C₄	C₂×C₃³	semidirect
155. #123	C₂×C₃³	semidirect
156. C₂²⋊Q₈	C₂×C₃³	semidirect
157. #121	C₂×C₃³	semidirect
158. #120	C₂×C₃³	semidirect
159. C₄×D₄	C₂×C₃³	semidirect
160. #117	C₂×C₃³	semidirect
161. C₃³	#525	direct (N × H)
175. C₂×C₃×C₄	C₃²×D₄	semidirect
188. C₂×Q₈	C₂²×C₃³	semidirect
191. C₂×D₄	C₂²×C₃³	semidirect
197. C₂²×C₄	C₂²×C₃³	semidirect
198. C₄⋊C₄	C₂²×C₃³	semidirect
199. C₄⋊C₄	C₂²×C₃³	semidirect
202. C₂²⋊C₄	C₂²×C₃³	semidirect
203. C₄²	C₂²×C₃³	semidirect
211. C₂²×C₃	C₃²×#41	semidirect
218. C₃²	C₃×#525	direct (N × H)
232. C₂×C₄	C₃³×D₄	semidirect
250. C₂²	C₃³×#41	semidirect
257. C₃	C₃²×#525	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×C₃³×#41	2
C₂³×C₃³×C₄	2
C₃³×#123	2
C₃³×C₂²⋊Q₈	2
C₃³×#121	2
C₃³×#120	2
C₃³×C₄×D₄	2
C₃³×#117	2
C₃²×#525	3

Subgroups — C33×#525 #13179980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#525 #13179980