Finite Group Databasebeta

← #13160271

Subgroups — #13160271

Order 1728.

908 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2#11722200C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic
4#11722051C212yesmaximal, complemented
5C3⋊C8×C3⋊Dic3C212yesmaximal, complemented
6#10592417—31nomaximal
7#10591053—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C411yescharacteristic
10C22×C3×C3⋊Dic3C411yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
12C3×C4×C3⋊Dic3C411yescharacteristic
13C3×C4×C3⋊Dic3C411yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2212yes—
17C2×C4×C3⋊Dic3D311yescharacteristic
18C2×C3×C4×Dic3—34no—
19C32×C22⋊C8—31no—
20#63657—31no—
21#63657—34no—
22#63657—64no—
23#63641—32no—
24C8×C3⋊Dic3—32no—
25C3×#830D314yes—
26Dic3⋊(C3⋊C8)—38no—
27Dic3×C3⋊C8—38no—
28C23×C33C2×C411yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4C2×C411yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3C814yescomplemented
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C3⋊(C3⋊(C3⋊C8))—22no—
39C32×C3⋊C8—22no—
40#4925—91no—
41#4718—94no—
42C22×C32×C4—24no—
43C22×C32×C4C2×D314yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3Dic311yescharacteristic
46C22×C3⋊Dic3Dic311yescharacteristic
47C22×C3×Dic3—34no—
48C22×C3×Dic3—34no—
49C2×C32×C8—32no—
50C4×C3⋊Dic3Dic311yescharacteristic
51C4×C3⋊Dic3Dic311yescharacteristic
52C4×C3⋊Dic3C2×D311yescharacteristic
53C4×C3⋊Dic3C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)—32no—
55C2×C3⋊(C3⋊C8)—38no—
56C2×C3⋊(C3⋊C8)—68no—
57C3×C4×Dic3—34no—
58C3×C4×Dic3—34no—
59C3×C4×Dic3—34no—
60C3×C4×Dic3—34no—
61C2×C3×C3⋊C8C2×D318yes—
62C22×C33—21no—
63C22×C33—21no—
64C22×C33C2×C812yes—
65C22×C33C22×C411yescharacteristic
66C33×C4—21no—
67C33×C4#4112yes—
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3C2×C812yes—
70C3×C3⋊Dic3C2×C812yes—
71C3×C3⋊Dic3M1612yes—
72C2×C3×C42—91no—
73C2×C4×Dic3—34no—
74C3×C22⋊C8—34no—
75#830—94no—
76#830—91no—
77#830—184no—
78#830C3⋊D311yescharacteristic
79Dic3⋊C8—98no—
80C8×Dic3—98no—
81C4⋊(C3⋊C8)—92no—
82C4×C3⋊C8—92no—
83C23×C32—24no—
84C23×C32C4×D314yes—
85C23×C32C2×Dic311yescharacteristic
86C2×C32×C4—22no—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4—28no—
90C2×C32×C4—48no—
91C2×C32×C4C4×D314yes—
92C2×C32×C4C2×Dic311yescharacteristic
93C2×C32×C4C22×D314yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3C3⋊C814yescomplemented
97C2×C3⋊Dic3C2×Dic311yescharacteristic
98C2×C3⋊Dic3C2×Dic311yescharacteristic
99C2×C3⋊Dic3C2×Dic311yescharacteristic
100C2×C3⋊Dic3C2×Dic311yescharacteristic
101C2×C3×Dic3—316no—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—34no—
105C2×C3×Dic3—34no—
106C2×C3×Dic3—68no—
107C32×C8—62no—
108C3⋊(C3⋊C8)—62no—
109C3⋊(C3⋊C8)—68no—
110C3⋊(C3⋊C8)—128no—
111C3×C3⋊C8—28no—
112#431—271noSylow
113C2×C33—22no—
114C2×C33#11711yescharacteristic
115C2×C33C22×C811yescharacteristic
116C2×C33C2×M1611yescharacteristic
117C22×C3×C4—24no—
118C22×C3×C4—91no—
119C22×C3×C4—91no—
120C22×C3×C4D3214yes—
121C22×C3×C4C2×C3⋊D311yescharacteristic
122C22×Dic3—34no—
123C22×Dic3—34no—
124C2×C3×C8—38no—
125C3×C42—91no—
126C3×C42—91no—
127C3×C42—91no—
128C3×C42—91no—
129C4×Dic3—34no—
130C4×Dic3—34no—
131C4×Dic3—34no—
132C4×Dic3—34no—
133C2×C3⋊C8—98no—
134C2×C3⋊C8—92no—
135C2×C3⋊C8—188no—
136C2×C3⋊C8C2×C3⋊D312yes—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—28no—
140C22×C32—24no—
141C22×C32—24no—
142C22×C32—24no—
143C22×C32—44no—
144C22×C32—44no—
145C22×C32D3×C818yes—
146C22×C32C2×C3⋊C812yes—
147C22×C32C2×C4×D314yes—
148C22×C32C22×Dic311yescharacteristic
149C32×C4—21no—
150C32×C4—28no—
151C32×C4—24no—
152C32×C4—44no—
153C32×C4#23512yes—
154C32×C4#23718yes—
155C3⋊Dic3—21no—
156C3⋊Dic3C2×C3⋊C812yes—
157C3⋊Dic3C2×C3⋊C812yes—
158C3⋊Dic3#20812yes—
159C3×Dic3—38no—
160C3×Dic3—38no—
161C3×Dic3—38no—
162C3×Dic3—64no—
163C2×C42—91no—
164C4⋊C8—272no—
165C22⋊C8—31no—
166C22⋊C8—271no—
167C4×C8—272no—
168C33#43111yescharacteristic, complemented, Sylow
169C23×C3—24no—
170C23×C3D3×Dic314yes—
171C23×C3C4×C3⋊D311yescharacteristic
172C2×C3×C4—28no—
173C2×C3×C4—22no—
174C2×C3×C4—24no—
175C2×C3×C4—24no—
176C2×C3×C4—48no—
177C2×C3×C4—94no—
178C2×C3×C4—91no—
179C2×C3×C4—91no—
180C2×C3×C4—91no—
181C2×C3×C4—91no—
182C2×C3×C4—182no—
183C2×C3×C4D3×Dic314yes—
184C2×C3×C4C4×C3⋊D311yescharacteristic
185C2×C3×C4C2×D3214yes—
186C2×C3×C4C22×C3⋊D311yescharacteristic
187C2×Dic3—316no—
188C2×Dic3—34no—
189C2×Dic3—34no—
190C2×Dic3—34no—
191C2×Dic3—34no—
192C2×Dic3—68no—
193C3×C8—68no—
194C3⋊C8—22no—
195C3⋊C8—188no—
196C3⋊C8—182no—
197C3⋊C8—368no—
198C2×C32—22no—
199C2×C32—24no—
200C2×C32—24no—
201C2×C32—24no—
202C2×C32—28no—
203C2×C32—48no—
204C2×C32#87714yes—
205C2×C32C2×D3×C814yes—
206C2×C32D3×M1614yes—
207C2×C32C22×C3⋊C811yescharacteristic
208C2×C32C2×#20811yescharacteristic
209C2×C32#92611yescharacteristic
210C22×C4—91no—
211C22×C4—91no—
212C22×C4D3×C3⋊D311yescharacteristic
213C2×C8—32no—
214C2×C8—272no—
215C42—91no—
216C42—91no—
217C42—91no—
218C42—91no—
219C22×C3—24no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—24no—
223C22×C3—24no—
224C22×C3—28no—
225C22×C3—44no—
226C22×C3—44no—
227C22×C3D3×C3⋊C818yes—
228C22×C3C8×C3⋊D312yes—
229C22×C3C2×D3×Dic314yes—
230C22×C3C2×C4×C3⋊D311yescharacteristic
231C3×C4—21no—
232C3×C4—24no—
233C3×C4—28no—
234C3×C4—44no—
235C3×C4—92no—
236C3×C4—92no—
237C3×C4—92no—
238C3×C4—181no—
239C3×C4#380418yes—
240C3×C4#383012yes—
241Dic3—38no—
242Dic3—38no—
243Dic3—38no—
244Dic3—64no—
245C32—24no—
246C32#471814yescomplemented
247C32#492511yescharacteristic, complemented
248C23Dic3×C3⋊D311yescharacteristic
249C2×C4—22no—
250C2×C4—91no—
251C2×C4—91no—
252C2×C4—91no—
253C2×C4—94no—
254C2×C4—91no—
255C2×C4—182no—
256C2×C4Dic3×C3⋊D311yescharacteristic
257C2×C4C2×D3×C3⋊D311yescharacteristic
258C8—62no—
259C8—542no—
260C2×C3—28no—
261C2×C3—22no—
262C2×C3—24no—
263C2×C3—24no—
264C2×C3—24no—
265C2×C3—48no—
266C2×C3C2×D3×C3⋊C814yes—
267C2×C3D3×#20814yes—
268C2×C3#6395514yes—
269C2×C3#6408611yescharacteristic
270C2×C3C2×C8×C3⋊D311yescharacteristic
271C2×C3M16×C3⋊D311yescharacteristic
272C22—21no—
273C22—21no—
274C22C3⋊D3×C3⋊C812yes—
275C22C2×Dic3×C3⋊D311yescharacteristic
276C4—21no—
277C4—92no—
278C4—92no—
279C4—92no—
280C4—181no—
281C4#8902812yes—
282C3—24no—
283C3#1059105314yescomplemented
284C3#1059241711yescharacteristic, complemented
285C2—22no—
286C2C2×C3⋊D3×C3⋊C811yescharacteristic
287C2C3⋊D3×#20811yescharacteristic
288C2#1172359511yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutatorC2×C33C22×C8
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#431
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722051C2semidirect
5. C3⋊C8×C3⋊Dic3C2semidirect
33. C2×C3×C3⋊Dic3C8semidirect
96. C2×C3⋊Dic3C3⋊C8semidirect
168. C33#431semidirect
246. C32#4718semidirect
247. C32#4925semidirect
283. C3#10591053semidirect
284. C3#10592417semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
#117222002
C32×#8302
#117220512
C3⋊C8×C3⋊Dic32
#105924173
#105910533