Finite Group Databasebeta

← #13160249

Subgroups — #13160249

Order 1728.

1484 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C2×C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4#11722061C214yesmaximal
5#10592628—31nomaximal
6#10591220—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C22×C3⋊(C3⋊Dic3)C2212yes—
9C22×C3×C3⋊Dic3C2212yes—
10C2×C3×#634—22no—
11C2×C3×#634—22no—
12C2×C3×#634C414yescomplemented
13C22×C32×Dic3C2212yes—
14C4⋊(C3⋊(C3⋊Dic3))—22no—
15C32×C4×Dic3—22no—
16C22×#634D311yescharacteristic
17C22×C3×Dic6—34no—
18C2×C32×C42—31no—
19C2×C4⋊(C3⋊Dic3)—31no—
20C2×C4⋊(C3⋊Dic3)—34no—
21C2×C4⋊(C3⋊Dic3)—64no—
22C2×C3×C4×Dic3D314yes—
23#63658—34no—
24#63579—316no—
25C23×C33C2311yescharacteristic
26C2×C33×C4C2×C412yes—
27C2×C33×C4D412yes—
28C2×C33×C4D412yes—
29C2×C3⋊(C3⋊Dic3)—22no—
30C2×C3⋊(C3⋊Dic3)—24no—
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3C2×C414yes—
34C3×#634—28no—
35C2×C32×Dic3—22no—
36C2×C32×Dic3—22no—
37C2×C32×Dic3Q814yescomplemented
38#5229—91no—
39#4929—94no—
40C22×C32×C4—24no—
41C22×C32×C4—32no—
42C22×C32×C4C2×D314yes—
43C22×C32×C4C2×D311yescharacteristic
44C22×C3⋊Dic3—32no—
45C22×C3⋊Dic3—38no—
46C22×C3⋊Dic3—68no—
47C22×C3⋊Dic3C2×D312yes—
48C2×#634—22no—
49C2×#634—22no—
50C2×#634Dic314yescomplemented
51C22×C3×Dic3—38no—
52C22×C3×Dic3C2×D318yes—
53C2×C3×Dic6—316no—
54C2×C3×Dic6—68no—
55C2×C3×Dic6—68no—
56C32×C42—62no—
57C4⋊(C3⋊Dic3)—616no—
58C4⋊(C3⋊Dic3)—68no—
59C4⋊(C3⋊Dic3)—62no—
60C3×C4×Dic3—28no—
61C22×C33C22×C411yescharacteristic
62C22×C33C2×D411yescharacteristic
63C22×C33C2×D411yescharacteristic
64C22×C33C2×Q812yes—
65C22×C33#4112yes—
66C33×C4C22⋊C414yes—
67C3⋊(C3⋊Dic3)—42no—
68C3×C3⋊Dic3—24no—
69C32×Dic3—24no—
70C22×C3×Q8—91no—
71C22×Dic6—34no—
72C2×C3×C42—34no—
73C2×C4⋊Dic3—94no—
74C2×C4⋊Dic3—91no—
75C2×C4⋊Dic3—184no—
76C2×C4×Dic3C3⋊D311yescharacteristic
77C3⋊#95—916no—
78C3⋊#95—94no—
79C23×C32—24no—
80C23×C32C22×D314yes—
81C23×C32C22×D311yescharacteristic
82C2×C32×C4—28no—
83C2×C32×C4—28no—
84C2×C32×C4—28no—
85C2×C32×C4—34no—
86C2×C32×C4—62no—
87C2×C32×C4—62no—
88C2×C32×C4C4×D318yes—
89C2×C32×C4D1218yes—
90C2×C32×C4C2×Dic312yes—
91C2×C32×C4#6718yes—
92C2×C32×C4#6712yes—
93C2×C32×C4#6712yes—
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—616no—
97C2×C3⋊Dic3—68no—
98C2×C3⋊Dic3—64no—
99C2×C3⋊Dic3—62no—
100C2×C3⋊Dic3—616no—
101C2×C3⋊Dic3—1216no—
102C2×C3⋊Dic3C2×Dic314yes—
103#634—28no—
104C2×C3×Dic3—28no—
105C2×C3×Dic3—28no—
106C2×C3×Dic3—316no—
107C2×C3×Dic3—68no—
108C2×C3×Dic3—68no—
109C2×C3×Dic3Dic6116yescomplemented
110C3×Dic6—632no—
111#391—271noSylow
112C2×C33C2×C22⋊C411yescharacteristic
113C2×C33C4×Q812yes—
114C2×C33C22⋊Q812yes—
115C2×C33#12411yescharacteristic
116C2×C33C4⋊Q811yescharacteristic
117C2×C3×Q8—94no—
118C2×C3×Q8—182no—
119C2×C3×Q8—182no—
120C22×C3×C4—24no—
121C22×C3×C4—38no—
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⋊D312yes—
130C2×Dic6—316no—
131C2×Dic6—68no—
132C2×Dic6—68no—
133C3×C42—68no—
134C4⋊Dic3—1816no—
135C4⋊Dic3—188no—
136C4⋊Dic3—182no—
137C4×Dic3—22no—
138C22×C32—28no—
139C22×C32—28no—
140C22×C32—24no—
141C22×C32—24no—
142C22×C32—24no—
143C22×C32C2×Dic618yes—
144C22×C32C2×C4×D314yes—
145C22×C32C2×D1214yes—
146C22×C32#23518yes—
147C22×C32D3×Q812yes—
148C22×C32#23912yes—
149C22×C32C22×Dic311yescharacteristic
150C22×C32C2×#6714yes—
151C22×C32C2×#6711yescharacteristic
152C22×C32C2×#6711yescharacteristic
153C32×C4—216no—
154C32×C4—64no—
155C32×C4(C2×D3)⋊C4116yes—
156C32×C4#21714yes—
157C3⋊Dic3—24no—
158C3⋊Dic3—1216no—
159C3⋊Dic3—128no—
160C3⋊Dic3—122no—
161C3×Dic3—216no—
162C3×Dic3—616no—
163C22×Q8—91no—
164C2×C4⋊C4—271no—
165C2×C42—31no—
166#95—274no—
167C33#39111yescharacteristic, complemented, Sylow
168C23×C3—24no—
169C23×C3C2×D3214yes—
170C23×C3C22×C3⋊D311yescharacteristic
171C3×Q8—188no—
172C2×C3×C4—28no—
173C2×C3×C4—28no—
174C2×C3×C4—28no—
175C2×C3×C4—316no—
176C2×C3×C4—68no—
177C2×C3×C4—68no—
178C2×C3×C4—94no—
179C2×C3×C4—182no—
180C2×C3×C4—182no—
181C2×C3×C4D3×Dic318yes—
182C2×C3×C4#62518yes—
183C2×C3×C4#62618yes—
184C2×C3×C4C4×C3⋊D312yes—
185C2×C3×C4C4⋊(C3⋊D3)12yes—
186C2×C3×C4#63812yes—
187C2×Dic3—22no—
188C2×Dic3—22no—
189C2×Dic3—316no—
190C2×Dic3—68no—
191C2×Dic3—68no—
192C2×Dic3—1816no—
193C2×Dic3—1816no—
194C2×Dic3—188no—
195C2×Dic3—182no—
196C2×Dic3—184no—
197C2×Dic3—3616no—
198C2×Dic3#63414yescomplemented
199Dic6—632no—
200C2×C32—28no—
201C2×C32—28no—
202C2×C32—24no—
203C2×C32—24no—
204C2×C32—24no—
205C2×C32C4×Dic618yes—
206C2×C32#86914yes—
207C2×C32#87514yes—
208C2×C32#92418yes—
209C2×C32C2×(C2×D3)⋊C414yes—
210C2×C32#94411yescharacteristic
211C2×C32Q8×Dic312yes—
212C2×C32#94612yes—
213C2×C32#94711yescharacteristic
214C2×C32C2×#21711yescharacteristic
215C2×Q8—94no—
216C2×Q8—182no—
217C2×Q8—182no—
218C22×C4—32no—
219C22×C4—92no—
220C22×C4—272no—
221C22×C4D3×C3⋊D311yescharacteristic
222C4⋊C4—542no—
223C42—62no—
224C22×C3—28no—
225C22×C3—28no—
226C22×C3—24no—
227C22×C3—24no—
228C22×C3—24no—
229C22×C3D3×Dic618yes—
230C22×C3#379918yes—
231C22×C3C2×D3×Dic314yes—
232C22×C3C2×#62514yes—
233C22×C3C2×#62614yes—
234C22×C3C2×#63412yes—
235C22×C3C2×C4×C3⋊D311yescharacteristic
236C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
237C22×C3#382812yes—
238C22×C3C2×#63811yescharacteristic
239C3×C4—216no—
240C3×C4—616no—
241C3×C4—184no—
242C3×C4(C2×D3)⋊Dic3116yes—
243C3×C4#375214yes—
244Dic3—24no—
245Dic3—616no—
246Dic3—3616no—
247Dic3—362no—
248Dic3—368no—
249C32—24no—
250C32#492914yescomplemented
251C32#522911yescharacteristic, complemented
252C23C2×D3×C3⋊D311yescharacteristic
253Q8—188no—
254C2×C4—34no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—94no—
258C2×C4—182no—
259C2×C4—182no—
260C2×C4—544no—
261C2×C4—542no—
262C2×C4Dic3×C3⋊D312yes—
263C2×C4#629012yes—
264C2×C4#629212yes—
265C2×C3—24no—
266C2×C3—28no—
267C2×C3—28no—
268C2×C3—24no—
269C2×C3—24no—
270C2×C3Dic3×Dic618yes—
271C2×C3#6385618yes—
272C2×C3#6386414yes—
273C2×C3#6386614yes—
274C2×C3C2×(C2×D3)⋊Dic314yes—
275C2×C3C4×#63412yes—
276C2×C3#6407811yescharacteristic
277C2×C3#6408411yescharacteristic
278C2×C3#6413312yes—
279C2×C3C2×#375211yescharacteristic
280C22D3×#63412yes—
281C22#8902212yes—
282C22C2×Dic3×C3⋊D311yescharacteristic
283C22C2×#629011yescharacteristic
284C22C2×#629211yescharacteristic
285C4—64no—
286C4—184no—
287C4—1082no—
288C4#8880314yes—
289C3—24no—
290C3#1059122014yescomplemented
291C3#1059262811yescharacteristic, complemented
292C2Dic3×#63412yes—
293C2#1172342512yes—
294C2#1172344111yescharacteristic
295C2#1172344311yescharacteristic
296C2C2×#8880311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC22×C33×C4C22
socle, prime radical, cosocleC23×C33C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#391
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
12. C2×C3×#634C4semidirect
37. C2×C32×Dic3Q8semidirect
50. C2×#634Dic3semidirect
109. C2×C3×Dic3Dic6semidirect
167. C33#391semidirect
198. C2×Dic3#634semidirect
250. C32#4929semidirect
251. C32#5229semidirect
290. C3#10591220semidirect
291. C3#10592628semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C2×C4⋊(C3⋊(C3⋊Dic3))2
C2×C32×C4×Dic32
#117220612
#105926283
#105912203