Finite Group Databasebeta

← #13160256

Subgroups — #13160256

Order 1728.

1172 conjugacy classes of subgroups.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic
2C2×C3×#3750C211yesmaximal, characteristic
3C32×C3⋊#95C211yesmaximal, characteristic
4#11722061C212yesmaximal
5#11722061C212yesmaximal
6#10592358—31nomaximal
7#10591226—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C4⋊(C3⋊(C3⋊Dic3))—22no—
14C3×#3750—22no—
15C2×C4⋊(C3⋊Dic3)—31no—
16C2×C4⋊(C3⋊Dic3)—34no—
17C2×C4⋊(C3⋊Dic3)—64no—
18C2×#3750D311yescharacteristic
19C2×C3×Dic3⋊C4—34no—
20C32×#95—31no—
21#63658—32no—
22#63658—32no—
23C3×C3⋊#95D314yes—
24#63579—38no—
25#63579—38no—
26C23×C33C2311yescharacteristic
27C2×C33×C4—22no—
28C2×C33×C4D412yes—
29C2×C3⋊(C3⋊Dic3)—22no—
30C2×C3⋊(C3⋊Dic3)—24no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3Q812yes—
36C2×C32×Dic3—22no—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—22no—
39#4986—91no—
40#4659—94no—
41C22×C32×C4—24no—
42C22×C32×C4—32no—
43C22×C32×C4C2×D314yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3—32no—
46C22×C3⋊Dic3—38no—
47C22×C3⋊Dic3—68no—
48C22×C3⋊Dic3C2×D311yescharacteristic
49C22×C3⋊Dic3C2×D311yescharacteristic
50C22×C3×Dic3—34no—
51C22×C3×Dic3—34no—
52C22×C3×Dic3C2×D318yes—
53C4⋊(C3⋊Dic3)—616no—
54C4⋊(C3⋊Dic3)—68no—
55C4⋊(C3⋊Dic3)—62no—
56#3750—22no—
57C3×Dic3⋊C4—68no—
58C22×C33C2×D411yescharacteristic
59C22×C33C2×Q811yescharacteristic
60C22×C33#4111yescharacteristic
61C22×C33#4112yes—
62C22×C33#4112yes—
63C33×C4—22no—
64C3⋊(C3⋊Dic3)—42no—
65C3×C3⋊Dic3—22no—
66C3×C3⋊Dic3—41no—
67C32×Dic3—42no—
68C2×C3×C4⋊C4—91no—
69C2×C4⋊Dic3—94no—
70C2×C4⋊Dic3—91no—
71C2×C4⋊Dic3—184no—
72C2×Dic3⋊C4—34no—
73C3×#95—34no—
74C3⋊#95—98no—
75C3⋊#95—98no—
76C3⋊#95—92no—
77C3⋊#95—92no—
78C3⋊#95C3⋊D311yescharacteristic
79C23×C32—24no—
80C23×C32C22×D314yes—
81C23×C32C22×D311yescharacteristic
82C2×C32×C4—216no—
83C2×C32×C4—28no—
84C2×C32×C4—22no—
85C2×C32×C4—28no—
86C2×C32×C4—62no—
87C2×C32×C4—62no—
88C2×C32×C4—62no—
89C2×C32×C4D1218yes—
90C2×C32×C4#6712yes—
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—21no—
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3—21no—
95C2×C3⋊Dic3—616no—
96C2×C3⋊Dic3—68no—
97C2×C3⋊Dic3—64no—
98C2×C3⋊Dic3—62no—
99C2×C3⋊Dic3—616no—
100C2×C3⋊Dic3—1216no—
101C2×C3⋊Dic3Dic612yes—
102C2×C3×Dic3—28no—
103C2×C3×Dic3—28no—
104C2×C3×Dic3—28no—
105C2×C3×Dic3—38no—
106C2×C3×Dic3—64no—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—68no—
110#400—271noSylow
111C2×C33C22⋊Q811yescharacteristic
112C2×C33#12311yescharacteristic
113C2×C33#12411yescharacteristic
114C2×C33#12512yes—
115C2×C33#12612yes—
116C22×C3×C4—24no—
117C22×C3×C4—38no—
118C22×C3×C4—91no—
119C22×C3×C4—91no—
120C22×C3×C4D3214yes—
121C22×C3×C4C2×C3⋊D311yescharacteristic
122C22×Dic3—34no—
123C22×Dic3—34no—
124C22×Dic3—98no—
125C22×Dic3—92no—
126C22×Dic3—188no—
127C22×Dic3C2×C3⋊D312yes—
128C3×C4⋊C4—182no—
129C4⋊Dic3—1816no—
130C4⋊Dic3—188no—
131C4⋊Dic3—182no—
132Dic3⋊C4—68no—
133C22×C32—24no—
134C22×C32—24no—
135C22×C32—28no—
136C22×C32—28no—
137C22×C32—24no—
138C22×C32C2×Dic611yescharacteristic
139C22×C32C2×D1214yes—
140C22×C32#23518yes—
141C22×C32#23511yescharacteristic
142C22×C32#23714yes—
143C22×C32#23718yes—
144C22×C32#23712yes—
145C22×C32D3×Q814yes—
146C22×C32#23912yes—
147C22×C32C2×#6711yescharacteristic
148C32×C4—216no—
149C32×C4—28no—
150C32×C4—22no—
151C32×C4—122no—
152C3⋊Dic3—22no—
153C3⋊Dic3—41no—
154C3⋊Dic3—1216no—
155C3⋊Dic3—128no—
156C3⋊Dic3—122no—
157C3×Dic3—48no—
158C3×Dic3—68no—
159C3×Dic3—124no—
160C2×C4⋊C4—91no—
161C2×C4⋊C4—271no—
162#95—31no—
163#95—272no—
164#95—272no—
165C33#40011yescharacteristic, complemented, Sylow
166C23×C3—24no—
167C23×C3C2×D3214yes—
168C23×C3C22×C3⋊D311yescharacteristic
169C2×C3×C4—216no—
170C2×C3×C4—28no—
171C2×C3×C4—22no—
172C2×C3×C4—28no—
173C2×C3×C4—68no—
174C2×C3×C4—68no—
175C2×C3×C4—68no—
176C2×C3×C4—92no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4—181no—
180C2×C3×C4—182no—
181C2×C3×C4#62618yes—
182C2×C3×C4C4⋊(C3⋊D3)12yes—
183C2×Dic3—22no—
184C2×Dic3—22no—
185C2×Dic3—22no—
186C2×Dic3—38no—
187C2×Dic3—64no—
188C2×Dic3—64no—
189C2×Dic3—68no—
190C2×Dic3—64no—
191C2×Dic3—1816no—
192C2×Dic3—1816no—
193C2×Dic3—188no—
194C2×Dic3—182no—
195C2×Dic3—184no—
196C2×Dic3—3616no—
197C2×C32—28no—
198C2×C32—24no—
199C2×C32—28no—
200C2×C32—24no—
201C2×C32—24no—
202C2×C32#87514yes—
203C2×C32#87918yes—
204C2×C32#88614yes—
205C2×C32#88918yes—
206C2×C32#89012yes—
207C2×C32#89714yes—
208C2×C32#89812yes—
209C2×C32#92411yescharacteristic
210C2×C32#93511yescharacteristic
211C2×C32#94711yescharacteristic
212C22×C4—32no—
213C22×C4—91no—
214C22×C4—91no—
215C22×C4—272no—
216C22×C4D3×C3⋊D311yescharacteristic
217C4⋊C4—182no—
218C4⋊C4—542no—
219C22×C3—24no—
220C22×C3—24no—
221C22×C3—28no—
222C22×C3—28no—
223C22×C3—24no—
224C22×C3D3×Dic614yes—
225C22×C3#379514yes—
226C22×C3#379918yes—
227C22×C3#380518yes—
228C22×C3C2×#62614yes—
229C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
230C22×C3#382812yes—
231C22×C3#383011yescharacteristic
232C22×C3#383012yes—
233C22×C3Q8×C3⋊D311yescharacteristic
234C3×C4—216no—
235C3×C4—22no—
236C3×C4—28no—
237C3×C4—128no—
238C3×C4—182no—
239C3×C4—361no—
240Dic3—42no—
241Dic3—68no—
242Dic3—124no—
243Dic3—3616no—
244Dic3—362no—
245Dic3—368no—
246C32—24no—
247C32#465914yescomplemented
248C32#498611yescharacteristic, complemented
249C23C2×D3×C3⋊D311yescharacteristic
250C2×C4—22no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—92no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4—182no—
258C2×C4—181no—
259C2×C4—544no—
260C2×C4—542no—
261C2×C4#629212yes—
262C2×C3—24no—
263C2×C3—28no—
264C2×C3—28no—
265C2×C3—24no—
266C2×C3—24no—
267C2×C3#6384518yes—
268C2×C3#6385714yes—
269C2×C3#6386118yes—
270C2×C3#6386414yes—
271C2×C3#6396214yes—
272C2×C3#6408411yescharacteristic
273C2×C3#6408812yes—
274C2×C3#6409511yescharacteristic
275C2×C3#6409812yes—
276C2×C3#6410611yescharacteristic
277C22#8901411yescharacteristic
278C22C3⋊D3×Dic611yescharacteristic
279C22#8902212yes—
280C22#8902912yes—
281C22C2×#629211yescharacteristic
282C4—22no—
283C4—122no—
284C4—182no—
285C4—361no—
286C4—1082no—
287C3—24no—
288C3#1059122614yescomplemented
289C3#1059235811yescharacteristic, complemented
290C2#1172340912yes—
291C2#1172343012yes—
292C2#1172343111yescharacteristic
293C2#1172344111yescharacteristic
294C2#1172360911yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, prime radical, cosocleC23×C33C23
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#400
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
165. C33#400semidirect
247. C32#4659semidirect
248. C32#4986semidirect
288. C3#10591226semidirect
289. C3#10592358semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊(C3⋊Dic3))2
C2×C3×#37502
C32×C3⋊#952
#117220612
#105923583
#105912263