Subgroups — #13144428

Order 1728.

450 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	#11721707	C₂	1	1	yes	maximal, characteristic
3	#11720582	C₂	1	1	yes	maximal, characteristic
4	#11720582	C₂	1	1	yes	maximal, characteristic
5	#11720526	C₂	1	1	yes	maximal, characteristic
6	#11720525	C₂	1	1	yes	maximal, characteristic
7	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic
8	#10592721	—	3	1	no	maximal
9	#10588704	—	3	3	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₉×C₄⋊C₄	—	2	1	no	—
12	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	—	2	1	no	—
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
17	#88534	—	2	1	no	—
18	#88534	—	2	1	no	—
19	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
21	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
22	#64079	—	3	1	no	—
23	#63662	—	3	1	no	—
24	#63662	—	3	1	no	—
25	#63635	—	3	1	no	—
26	#63634	—	3	1	no	—
27	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
28	C₄×Q₈×C₉	D₃	1	3	yes	—
29	#63432	—	3	3	no	—
30	#63395	—	3	3	no	—
31	#63395	—	3	3	no	—
32	#63367	—	3	3	no	—
33	#63366	—	3	3	no	—
34	C₄⋊(C₉⋊C₈)	—	3	3	no	—
35	C₃×Q₈×C₉	—	2	1	no	—
36	C₃×Q₈×C₉	Q₈	1	2	yes	—
37	C₂×C₃×C₄×C₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₃⋊Dic₉	—	2	1	no	—
43	C₂×C₃⋊Dic₉	—	2	1	no	—
44	C₃⋊(C₉⋊C₈)	—	2	1	no	—
45	C₃⋊(C₉⋊C₈)	—	2	1	no	—
46	#5022	—	9	3	no	—
47	#5022	—	9	1	no	—
48	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
49	C₃²×C₄⋊C₄	—	2	1	no	—
50	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
51	C₃²×C₄²	—	2	1	no	—
52	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
54	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
55	#3750	—	6	1	no	—
56	#3750	—	6	1	no	—
57	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
58	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
59	C₂×Q₈×C₉	C₂×D₃	1	3	yes	—
60	C₉×C₄⋊C₄	—	2	3	no	—
61	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
62	C₄²×C₉	—	2	3	no	—
63	C₄²×C₉	C₂×D₃	1	3	yes	—
64	C₄⋊Dic₉	—	3	3	no	—
65	C₄⋊Dic₉	—	3	3	no	—
66	Dic₉⋊C₄	—	6	3	no	—
67	Dic₉⋊C₄	—	6	3	no	—
68	C₂×C₉⋊C₈	—	3	3	no	—
69	C₂×C₉⋊C₈	—	3	3	no	—
70	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
71	C₃×C₄×C₉	—	2	2	no	—
72	C₃×C₄×C₉	—	2	1	no	—
73	C₃×C₄×C₉	—	2	1	no	—
74	C₃×C₄×C₉	—	4	1	no	—
75	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
76	C₃×C₄×C₉	#41	1	1	yes	characteristic
77	C₃⋊Dic₉	—	4	1	no	—
78	C₃⋊Dic₉	—	4	1	no	—
79	C₃×C₄×Q₈	D₉	1	3	yes	—
80	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
81	#870	—	9	3	no	—
82	#870	—	9	1	no	—
83	#835	—	9	3	no	—
84	#835	—	9	3	no	—
85	#835	—	9	1	no	—
86	#835	—	9	1	no	—
87	#808	—	9	3	no	—
88	#808	—	9	1	no	—
89	#807	—	9	3	no	—
90	#807	—	9	1	no	—
91	C₄⋊(C₃⋊C₈)	—	9	3	no	—
92	C₄⋊(C₃⋊C₈)	—	9	1	no	—
93	C₃²×Q₈	—	2	1	no	—
94	C₃²×Q₈	Dic₆	1	2	yes	—
95	C₂×C₃²×C₄	—	2	1	no	—
96	C₂×C₃²×C₄	—	2	1	no	—
97	C₂×C₃²×C₄	#67	1	1	yes	characteristic
98	C₂×C₃²×C₄	#67	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₃⋊Dic₃	—	6	1	no	—
101	C₂×C₃⋊Dic₃	—	6	1	no	—
102	C₃⋊(C₃⋊C₈)	—	6	1	no	—
103	C₃⋊(C₃⋊C₈)	—	6	1	no	—
104	Q₈×C₉	—	2	3	no	—
105	Q₈×C₉	Dic₆	1	6	yes	—
106	C₂×C₄×C₉	—	2	3	no	—
107	C₂×C₄×C₉	—	2	3	no	—
108	C₂×C₄×C₉	#67	1	3	yes	—
109	C₂×C₄×C₉	#67	1	3	yes	—
110	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
111	C₂×Dic₉	—	6	3	no	—
112	C₂×Dic₉	—	6	3	no	—
113	C₉⋊C₈	—	6	3	no	—
114	C₉⋊C₈	—	6	3	no	—
115	#479	—	27	1	no	Sylow
116	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₃×C₉	#135	1	1	yes	characteristic
118	C₂×C₃×C₉	#137	1	1	yes	characteristic
119	C₂×C₃×Q₈	C₂×D₉	1	3	yes	—
120	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
121	C₃×C₄⋊C₄	—	2	3	no	—
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
124	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
125	C₃×C₄²	—	2	3	no	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	C₂×D₉	1	3	yes	—
128	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
129	C₄⋊Dic₃	—	9	3	no	—
130	C₄⋊Dic₃	—	9	3	no	—
131	C₄⋊Dic₃	—	9	1	no	—
132	C₄⋊Dic₃	—	9	1	no	—
133	Dic₃⋊C₄	—	18	3	no	—
134	Dic₃⋊C₄	—	18	3	no	—
135	Dic₃⋊C₄	—	18	1	no	—
136	Dic₃⋊C₄	—	18	1	no	—
137	C₂×C₃⋊C₈	—	9	3	no	—
138	C₂×C₃⋊C₈	—	9	3	no	—
139	C₂×C₃⋊C₈	—	9	1	no	—
140	C₂×C₃⋊C₈	—	9	1	no	—
141	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
142	C₃²×C₄	—	2	2	no	—
143	C₃²×C₄	—	2	1	no	—
144	C₃²×C₄	—	2	1	no	—
145	C₃²×C₄	—	4	1	no	—
146	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
147	C₃²×C₄	#235	1	1	yes	characteristic
148	C₃⋊Dic₃	—	12	1	no	—
149	C₃⋊Dic₃	—	12	1	no	—
150	C₂²×C₉	C₂×#67	1	3	yes	—
151	C₄×C₉	—	2	6	no	—
152	C₄×C₉	—	2	3	no	—
153	C₄×C₉	—	2	3	no	—
154	C₄×C₉	—	4	3	no	—
155	C₄×C₉	C₂×Dic₆	1	3	yes	—
156	C₄×C₉	#235	1	3	yes	—
157	Dic₉	—	12	3	no	—
158	Dic₉	—	12	3	no	—
159	#125	—	27	1	no	—
160	C₄×Q₈	C₃⋊D₉	1	1	yes	characteristic
161	#107	—	27	1	no	—
162	#106	—	27	1	no	—
163	C₄⋊C₈	—	27	1	no	—
164	#103	—	27	1	no	—
165	#103	—	27	1	no	—
166	C₃×C₉	#479	1	1	yes	characteristic, complemented, Sylow
167	C₃×Q₈	—	2	3	no	—
168	C₃×Q₈	—	2	1	no	—
169	C₃×Q₈	Dic₁₈	1	6	yes	—
170	C₃×Q₈	#634	1	2	yes	—
171	C₂×C₃×C₄	—	2	3	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	3	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	#611	1	3	yes	—
176	C₂×C₃×C₄	#611	1	3	yes	—
177	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
178	C₂×C₃×C₄	#638	1	1	yes	characteristic
179	C₂×C₃×C₄	#638	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
181	C₂×Dic₃	—	18	3	no	—
182	C₂×Dic₃	—	18	3	no	—
183	C₂×Dic₃	—	18	1	no	—
184	C₂×Dic₃	—	18	1	no	—
185	C₃⋊C₈	—	18	3	no	—
186	C₃⋊C₈	—	18	3	no	—
187	C₃⋊C₈	—	18	1	no	—
188	C₃⋊C₈	—	18	1	no	—
189	C₂×C₃²	#924	1	1	yes	characteristic
190	C₂×C₃²	#942	1	1	yes	characteristic
191	C₂×C₃²	#950	1	1	yes	characteristic
192	C₂×C₉	#924	1	3	yes	—
193	C₂×C₉	#942	1	3	yes	—
194	C₂×C₉	#950	1	3	yes	—
195	C₂×Q₈	C₂×C₃⋊D₉	1	1	yes	characteristic
196	C₂×C₈	—	27	1	no	—
197	C₂×C₈	—	27	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	27	1	no	—
200	C₄⋊C₄	—	27	1	no	—
201	C₄⋊C₄	—	54	1	no	—
202	C₄⋊C₄	—	54	1	no	—
203	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
204	C₄²	—	2	1	no	—
205	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
206	C₂²×C₃	C₂×#611	1	3	yes	—
207	C₂²×C₃	C₂×#638	1	1	yes	characteristic
208	C₃×C₄	—	2	3	no	—
209	C₃×C₄	—	2	3	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	6	no	—
214	C₃×C₄	—	4	3	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
217	C₃×C₄	#3697	1	3	yes	—
218	C₃×C₄	C₂×#634	1	1	yes	characteristic
219	C₃×C₄	#3828	1	1	yes	characteristic
220	Dic₃	—	36	3	no	—
221	Dic₃	—	36	3	no	—
222	Dic₃	—	36	1	no	—
223	Dic₃	—	36	1	no	—
224	C₃²	#5022	1	1	yes	characteristic
225	C₉	#5022	1	3	yes	complemented
226	Q₈	—	2	1	no	—
227	Q₈	#6226	1	2	yes	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	54	1	no	—
231	C₂×C₄	—	54	1	no	—
232	C₂×C₄	#6233	1	1	yes	characteristic
233	C₂×C₄	#6233	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
235	C₈	—	54	1	no	—
236	C₈	—	54	1	no	—
237	C₂×C₃	#63486	1	3	yes	—
238	C₂×C₃	#63505	1	3	yes	—
239	C₂×C₃	#63513	1	3	yes	—
240	C₂×C₃	#64133	1	1	yes	characteristic
241	C₂×C₃	#64151	1	1	yes	characteristic
242	C₂×C₃	#64159	1	1	yes	characteristic
243	C₂²	C₂×#6233	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	4	1	no	—
248	C₄	—	108	1	no	—
249	C₄	—	108	1	no	—
250	C₄	C₂×#6226	1	1	yes	characteristic
251	C₄	#88736	1	1	yes	characteristic
252	C₃	#10588704	1	3	yes	complemented
253	C₃	#10592721	1	1	yes	characteristic
254	C₂	#11721813	1	1	yes	characteristic
255	C₂	#11721853	1	1	yes	characteristic
256	C₂	#11721868	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6233
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×#67
hypercenter	C₄×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#479
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
166. C₃×C₉	#479	semidirect
225. C₉	#5022	semidirect
252. C₃	#10588704	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
#11721707	2
#11720582	2
#11720526	2
#11720525	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592721	3
#10588704	3