Finite Group Databasebeta

← #13160443

Subgroups — #13160443

Order 1728.

1372 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Dic3C211yesmaximal, characteristic
2C2×Dic3×C3⋊Dic3C212yesmaximal
3#11722201C211yesmaximal, characteristic
4C3×#63658C211yesmaximal, characteristic
5#11722061C212yesmaximal
6#10592631—31nomaximal
7#10591239—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2212yes—
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2211yescharacteristic
13C32×C4×Dic3C414yescomplemented
14Dic3×C3⋊Dic3C418yescomplemented
15C2×C32×C42—31no—
16C2×C4×C3⋊Dic3—32no—
17C2×C3×C4×Dic3D314yes—
18C2×Dic32—38no—
19#63658—31no—
20#63658—32no—
21#63658—34no—
22#63658—64no—
23#63658D311yescharacteristic
24C3×C3⋊#95—34no—
25#63579—38no—
26C23×C33C2311yescharacteristic
27C2×C33×C4—22no—
28C2×C33×C4C2×C412yes—
29C2×C3⋊(C3⋊Dic3)—22no—
30C2×C3⋊(C3⋊Dic3)—22no—
31C2×C3⋊(C3⋊Dic3)C2×C414yes—
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3C2×C414yes—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3C2×C412yes—
38C2×C32×Dic3C2×C414yes—
39C2×C32×Dic3C2×C412yes—
40#4932—94no—
41#4648—91no—
42C22×C32×C4—24no—
43C22×C32×C4—31no—
44C22×C32×C4—31no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C2×D314yes—
47C22×C3⋊Dic3—32no—
48C22×C3⋊Dic3—38no—
49C22×C3⋊Dic3—68no—
50C22×C3⋊Dic3C2×D312yes—
51C22×C3×Dic3—38no—
52C22×C3×Dic3C2×D314yes—
53C22×C3×Dic3C2×D314yes—
54C32×C42—34no—
55C4×C3⋊Dic3—38no—
56C3×C4×Dic3Dic3116yescomplemented
57Dic32—332no—
58C22×C33C22×C412yes—
59C22×C33C22×C411yescharacteristic
60C22×C33#4111yescharacteristic
61C22×C33#4112yes—
62C22×C33#4111yescharacteristic
63C33×C4—22no—
64C3⋊(C3⋊Dic3)—24no—
65C3×C3⋊Dic3—24no—
66C32×Dic3—22no—
67C32×Dic3C4214yes—
68C2×C3×C42—34no—
69C2×C4×Dic3—98no—
70C2×C4×Dic3—92no—
71C2×C4×Dic3C3⋊D311yescharacteristic
72C3×#95—91no—
73C3⋊#95—34no—
74C3⋊#95—98no—
75C3⋊#95—94no—
76C3⋊#95—91no—
77C3⋊#95—92no—
78C3⋊#95—184no—
79C23×C32—24no—
80C23×C32C22×D314yes—
81C23×C32C22×D311yescharacteristic
82C2×C32×C4—22no—
83C2×C32×C4—28no—
84C2×C32×C4—28no—
85C2×C32×C4—34no—
86C2×C32×C4—32no—
87C2×C32×C4—32no—
88C2×C32×C4—48no—
89C2×C32×C4—61no—
90C2×C32×C4—61no—
91C2×C32×C4C4×D312yes—
92C2×C32×C4C2×Dic318yes—
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—34no—
96C2×C3⋊Dic3—316no—
97C2×C3⋊Dic3—616no—
98C2×C3⋊Dic3—62no—
99C2×C3⋊Dic3—62no—
100C2×C3⋊Dic3—68no—
101C2×C3⋊Dic3—68no—
102C2×C3⋊Dic3—128no—
103C2×C3⋊Dic3—128no—
104C2×C3⋊Dic3C4×D314yes—
105C2×C3×Dic3—24no—
106C2×C3×Dic3—24no—
107C2×C3×Dic3—316no—
108C2×C3×Dic3—68no—
109C2×C3×Dic3—68no—
110C2×C3×Dic3C4×D3116yes—
111C2×C3×Dic3C2×Dic318yes—
112C2×C3×Dic3C2×Dic318yes—
113#376—271noSylow
114C2×C33C2×C4211yescharacteristic
115C2×C33#11712yes—
116C2×C33#11712yes—
117C2×C33#11711yescharacteristic
118C2×C33#11711yescharacteristic
119C22×C3×C4—24no—
120C22×C3×C4—34no—
121C22×C3×C4—34no—
122C22×C3×C4—92no—
123C22×C3×C4D3214yes—
124C22×C3×C4C2×C3⋊D311yescharacteristic
125C22×Dic3—38no—
126C22×Dic3—98no—
127C22×Dic3—92no—
128C22×Dic3—188no—
129C22×Dic3C2×C3⋊D311yescharacteristic
130C22×Dic3C2×C3⋊D311yescharacteristic
131C3×C42—316no—
132C4×Dic3—932no—
133C4×Dic3—98no—
134C4×Dic3C3⋊Dic314yescomplemented
135C22×C32—24no—
136C22×C32—24no—
137C22×C32—28no—
138C22×C32—24no—
139C22×C32—28no—
140C22×C32C2×C4×D318yes—
141C22×C32C2×C4×D312yes—
142C22×C32C2×C4×D311yescharacteristic
143C22×C32#23514yes—
144C22×C32#23514yes—
145C22×C32#23518yes—
146C22×C32#23711yescharacteristic
147C22×C32#23712yes—
148C22×C32#23911yescharacteristic
149C22×C32C22×Dic314yes—
150C32×C4—22no—
151C32×C4—28no—
152C32×C4—34no—
153C32×C4—48no—
154C32×C4—62no—
155C3⋊Dic3—24no—
156C3⋊Dic3—64no—
157C3⋊Dic3—616no—
158C3⋊Dic3—1216no—
159C3×Dic3—28no—
160C3×Dic3—616no—
161C3×Dic3C4×Dic3116yes—
162C2×C42—31no—
163C2×C42—272no—
164#95—91no—
165#95—272no—
166#95—271no—
167C33#37611yescharacteristic, complemented, Sylow
168C23×C3—24no—
169C23×C3C2×D3214yes—
170C23×C3C22×C3⋊D311yescharacteristic
171C2×C3×C4—28no—
172C2×C3×C4—22no—
173C2×C3×C4—28no—
174C2×C3×C4—316no—
175C2×C3×C4—38no—
176C2×C3×C4—38no—
177C2×C3×C4—48no—
178C2×C3×C4—64no—
179C2×C3×C4—64no—
180C2×C3×C4—94no—
181C2×C3×C4—182no—
182C2×C3×C4—182no—
183C2×C3×C4D3×Dic318yes—
184C2×C3×C4C2×C3⋊Dic312yes—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—316no—
188C2×Dic3—68no—
189C2×Dic3—68no—
190C2×Dic3—916no—
191C2×Dic3—94no—
192C2×Dic3—1816no—
193C2×Dic3—188no—
194C2×Dic3—188no—
195C2×Dic3—182no—
196C2×Dic3—182no—
197C2×Dic3—368no—
198C2×Dic3—368no—
199C2×Dic3C4×C3⋊D314yes—
200C2×Dic3C2×C3⋊Dic312yes—
201C2×Dic3C2×C3⋊Dic312yes—
202C2×C32—28no—
203C2×C32—24no—
204C2×C32—24no—
205C2×C32—28no—
206C2×C32—24no—
207C2×C32C42×D311yescharacteristic
208C2×C32#87218yes—
209C2×C32#87218yes—
210C2×C32#87712yes—
211C2×C32#87711yescharacteristic
212C2×C32#89212yes—
213C2×C32#89211yescharacteristic
214C2×C32C2×C4×Dic314yes—
215C2×C32#92614yes—
216C2×C32#92614yes—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—92no—
220C22×C4—272no—
221C22×C4D3×C3⋊D311yescharacteristic
222C42—34no—
223C42—278no—
224C22×C3—24no—
225C22×C3—24no—
226C22×C3—28no—
227C22×C3—24no—
228C22×C3—28no—
229C22×C3#379514yes—
230C22×C3#379914yes—
231C22×C3C4×D3218yes—
232C22×C3C2×D3×Dic314yes—
233C22×C3#380418yes—
234C22×C3C2×C4×C3⋊D312yes—
235C22×C3#382811yescharacteristic
236C22×C3#382811yescharacteristic
237C22×C3#382812yes—
238C22×C3C22×C3⋊Dic311yescharacteristic
239C3×C4—22no—
240C3×C4—28no—
241C3×C4—316no—
242C3×C4—48no—
243C3×C4—68no—
244C3×C4—184no—
245Dic3—22no—
246Dic3—616no—
247Dic3—184no—
248Dic3—1816no—
249Dic3—3616no—
250Dic3C4×C3⋊Dic314yes—
251C32—24no—
252C32#464811yescharacteristic, complemented
253C32#493214yescomplemented
254C23C2×D3×C3⋊D311yescharacteristic
255C2×C4—22no—
256C2×C4—34no—
257C2×C4—32no—
258C2×C4—32no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—94no—
262C2×C4—182no—
263C2×C4—182no—
264C2×C4—274no—
265C2×C4—542no—
266C2×C4—542no—
267C2×C4D3×C3⋊Dic312yes—
268C2×C3—24no—
269C2×C3—24no—
270C2×C3—24no—
271C2×C3—28no—
272C2×C3—28no—
273C2×C3#6383618yes—
274C2×C3#6384114yes—
275C2×C3C4×D3×Dic314yes—
276C2×C3#6387718yes—
277C2×C3#6395514yes—
278C2×C3#6408112yes—
279C2×C3#6408112yes—
280C2×C3C2×C4×C3⋊Dic311yescharacteristic
281C2×C3#6413511yescharacteristic
282C2×C3#6413511yescharacteristic
283C22#8902111yescharacteristic
284C22#8902211yescharacteristic
285C22C4×D3×C3⋊D312yes—
286C22C2×D3×C3⋊Dic311yescharacteristic
287C22#8903112yes—
288C4—22no—
289C4—34no—
290C4—62no—
291C4—184no—
292C4—544no—
293C3—24no—
294C3#1059123914yescomplemented
295C3#1059263111yescharacteristic, complemented
296C2#1172342411yescharacteristic
297C2C4×D3×C3⋊Dic311yescharacteristic
298C2#1172346412yes—
299C2#1172346812yes—
300C2#1172359411yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC2×C33C2×C42
FittingC22×C33×C4C22
socle, prime radical, cosocleC23×C33C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#376
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
13. C32×C4×Dic3C4semidirect
14. Dic3×C3⋊Dic3C4semidirect
56. C3×C4×Dic3Dic3semidirect
134. C4×Dic3C3⋊Dic3semidirect
167. C33#376semidirect
252. C32#4648semidirect
253. C32#4932semidirect
294. C3#10591239semidirect
295. C3#10592631semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Dic32
C2×Dic3×C3⋊Dic32
#117222012
C3×#636582
#117220612
#105926313
#105912393