Finite Group Databasebeta

← C2×#11724511 #13181510

Subgroups — C2×#11724511 #13181510

Order 1728.

1176 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×#88993C212yesmaximal, complemented
3#11724511C218yesmaximal, direct product factor
4C2×#88989C212yesmaximal, complemented
5C2×#88988C212yesmaximal, complemented
6C2×#64283—31nomaximal
7C23×C3×C3⋊D3C411yescharacteristic
8C22×C33×C4C411yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C3×C4×C3⋊D3C414yes—
11C2×C3×C4×C3⋊D3C411yescharacteristic
12C2×C3×C4×C3⋊D3C411yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2214yes—
16#88993C2218yescomplemented
17C2×C3⋊#622C2214yescomplemented
18#88989C2218yescomplemented
19#88988C2218yescomplemented
20C22×C4×C3⋊D3D311yescharacteristic
21C22×C3×C4×D3—62no—
22C2×#3792—32no—
23#64283—38no—
24C2×#3788—32no—
25C2×#3787—32no—
26C23×C33C2×C411yescharacteristic
27C22×C3×C3⋊D3—24no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C22×C3×C3⋊D3C2×C414yes—
30C22×C3×C3⋊D3C2×C411yescharacteristic
31C2×C33×C4C2×C414yes—
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C412yes—
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C2×C3×C3⋊Dic3C2312yes—
38C3×C4×C3⋊D3C2×C412yes—
39C3×C4×C3⋊D3C2×C414yes—
40C3×C4×C3⋊D3C2×C412yes—
41C3×C4×C3⋊D3C2×C414yes—
42C3×C4×C3⋊D3C2312yes—
43C3×C4×C3⋊D3C2312yes—
44C3⋊#622C2318yescomplemented
45C22×#208—91nomaximal
46C23×C3⋊D3Dic311yescharacteristic
47C23×C3×D3—62no—
48C22×C32×C4—22no—
49C22×C32×C4—42no—
50C22×C32×C4Dic311yescharacteristic
51C22×C3⋊Dic3C2×D311yescharacteristic
52C2×C4×C3⋊D3Dic314yes—
53C2×C4×C3⋊D3Dic311yescharacteristic
54C2×C4×C3⋊D3Dic311yescharacteristic
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C4×C3⋊D3C2×D314yes—
58C22×C3×Dic3—62no—
59C2×C3×C4×D3—62no—
60C2×C3×C4×D3—68no—
61C2×C3×C4×D3—62no—
62C2×C3×C4×D3—62no—
63C2×C3×C4×D3—62no—
64C2×C3×C4×D3—68no—
65#3792—38no—
66C2×#622—34no—
67#3788—38no—
68#3787—38no—
69C22×C33—22no—
70C22×C33C22×C412yes—
71C22×C33C22×C411yescharacteristic
72C2×C3×C3⋊D3—22no—
73C2×C3×C3⋊D3—28no—
74C2×C3×C3⋊D3M1614yescomplemented
75C2×C3×C3⋊D3C22×C412yes—
76C2×C3×C3⋊D3C22×C412yes—
77C33×C4C22×C412yes—
78C33×C4C22×C412yes—
79C3×C3⋊Dic3C22×C411yescharacteristic
80C3×C3⋊Dic3C22×C412yes—
81C3×C3⋊Dic3C2411yescharacteristic
82C23×C3×C4—91no—
83C22×C4×D3—62no—
84C2×#208—92no—
85C2×#208—98no—
86C2×#208—92no—
87C22×C3⋊C8—92no—
88C23×C32—22no—
89C23×C32—42no—
90C23×C32C2×Dic311yescharacteristic
91C22×C3⋊D3—24no—
92C22×C3⋊D3C2×Dic311yescharacteristic
93C22×C3⋊D3C2×Dic314yes—
94C22×C3⋊D3C2×Dic311yescharacteristic
95C22×C3×D3—62no—
96C22×C3×D3—68no—
97C22×C3×D3—62no—
98C22×C3×D3—616no—
99C2×C32×C4—28no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—48no—
103C2×C32×C4—42no—
104C2×C32×C4—42no—
105C2×C32×C4C2×Dic314yes—
106C2×C32×C4C2×Dic311yescharacteristic
107C2×C32×C4C2×Dic311yescharacteristic
108C2×C3⋊Dic3C2×Dic312yes—
109C2×C3⋊Dic3C2×Dic311yescharacteristic
110C2×C3⋊Dic3C22×D311yescharacteristic
111C2×C3⋊Dic3C22×D312yes—
112C4×C3⋊D3C2×Dic312yes—
113C4×C3⋊D3C2×Dic314yes—
114C4×C3⋊D3C2×Dic312yes—
115C4×C3⋊D3C2×Dic314yes—
116C4×C3⋊D3C22×D312yes—
117C4×C3⋊D3C22×D312yes—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—64no—
122C3×C4×D3—64no—
123C3×C4×D3—68no—
124C3×C4×D3—64no—
125C3×C4×D3—68no—
126C3×C4×D3—64no—
127C3×C4×D3—64no—
128#622—38no—
129C22×M16—271noSylow
130C2×C33—22no—
131C2×C33C2×M1612yescomplemented
132C2×C33C23×C411yescharacteristic
133C3×C3⋊D3—22no—
134C3×C3⋊D3C2×M1614yescomplemented
135C24×C3—91no—
136C23×D3—62no—
137C22×C3×C4—22no—
138C22×C3×C4—42no—
139C22×C3×C4—91no—
140C22×C3×C4—91no—
141C22×C3×C4—91no—
142C22×C3×C4—94no—
143C22×C3×C4—91no—
144C22×C3×C4—94no—
145C22×C3×C4—91no—
146C22×C3×C4#15711yescharacteristic
147C22×Dic3—62no—
148C2×C4×D3—62no—
149C2×C4×D3—68no—
150C2×C4×D3—62no—
151C2×C4×D3—62no—
152C2×C4×D3—68no—
153C2×C4×D3—62no—
154#208—98no—
155#208—98no—
156C2×C3⋊C8—98no—
157C2×C3⋊C8—94no—
158C22×C32—22no—
159C22×C32—28no—
160C22×C32—24no—
161C22×C32—22no—
162C22×C32—48no—
163C22×C32—44no—
164C22×C32—42no—
165C22×C32C22×Dic312yes—
166C22×C32C22×Dic311yescharacteristic
167C2×C3⋊D3—22no—
168C2×C3⋊D3—28no—
169C2×C3⋊D3#20814yescomplemented
170C2×C3⋊D3C22×Dic312yes—
171C2×C3⋊D3C22×Dic312yes—
172C2×C3×D3—68no—
173C2×C3×D3—64no—
174C2×C3×D3—64no—
175C2×C3×D3—632no—
176C2×C3×D3—68no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—44no—
180C32×C4—44no—
181C32×C4C22×Dic312yes—
182C32×C4C22×Dic312yes—
183C3⋊Dic3C22×Dic311yescharacteristic
184C3⋊Dic3C22×Dic312yes—
185C3⋊Dic3C23×D311yescharacteristic
186C3×Dic3—62no—
187C3×Dic3—64no—
188C3×Dic3—62no—
189C23×C4—91no—
190C2×M16—272no—
191C2×M16—278no—
192C2×M16—272no—
193C22×C8—272no—
194C33C22×M1611yescharacteristic, complemented, Sylow
195C23×C3—22no—
196C23×C3—42no—
197C23×C3—91no—
198C23×C3—94no—
199C23×C3—91no—
200C23×C3—184no—
201C23×C3C2×#15711yescharacteristic
202C22×D3—62no—
203C22×D3—68no—
204C22×D3—62no—
205C22×D3—616no—
206C2×C3×C4—22no—
207C2×C3×C4—28no—
208C2×C3×C4—22no—
209C2×C3×C4—42no—
210C2×C3×C4—48no—
211C2×C3×C4—42no—
212C2×C3×C4—91no—
213C2×C3×C4—92no—
214C2×C3×C4—92no—
215C2×C3×C4—91no—
216C2×C3×C4—92no—
217C2×C3×C4—92no—
218C2×C3×C4—92no—
219C2×C3×C4—94no—
220C2×C3×C4—92no—
221C2×C3×C4—94no—
222C2×C3×C4C2×#15714yes—
223C2×C3×C4C2×#15711yescharacteristic
224C2×C3×C4C2×#15711yescharacteristic
225C2×Dic3—64no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—64no—
229C4×D3—68no—
230C4×D3—64no—
231C4×D3—64no—
232C4×D3—68no—
233C4×D3—64no—
234C4×D3—64no—
235C3⋊C8—98no—
236C2×C32—22no—
237C2×C32—28no—
238C2×C32—24no—
239C2×C32—22no—
240C2×C32—48no—
241C2×C32—44no—
242C2×C32—42no—
243C2×C32C2×#20812yescomplemented
244C2×C32C23×Dic311yescharacteristic
245C3⋊D3—22no—
246C3⋊D3C2×#20814yescomplemented
247C3×D3—68no—
248C3×D3—68no—
249C24—91no—
250C22×C4—91no—
251C22×C4—91no—
252C22×C4—94no—
253C22×C4—91no—
254C22×C4—94no—
255C22×C4—91no—
256C22×C4—91no—
257C22×C4C3⋊#15711yescharacteristic
258M16—278no—
259M16—278no—
260C2×C8—278no—
261C2×C8—274no—
262C22×C3—28no—
263C22×C3—22no—
264C22×C3—24no—
265C22×C3—22no—
266C22×C3—48no—
267C22×C3—44no—
268C22×C3—42no—
269C22×C3—94no—
270C22×C3—92no—
271C22×C3—92no—
272C22×C3—188no—
273C22×C3—182no—
274C22×C3C22×#15712yes—
275C22×C3C22×#15711yescharacteristic
276C2×D3—64no—
277C2×D3—64no—
278C2×D3—68no—
279C2×D3—632no—
280C2×D3—68no—
281C3×C4—24no—
282C3×C4—24no—
283C3×C4—44no—
284C3×C4—44no—
285C3×C4—91no—
286C3×C4—91no—
287C3×C4—92no—
288C3×C4C22×#15712yes—
289C3×C4C22×#15712yes—
290Dic3—62no—
291Dic3—64no—
292Dic3—62no—
293C32—22no—
294C32—42no—
295C32C22×#20811yescharacteristic, complemented
296C23—91no—
297C23—94no—
298C23—91no—
299C23—184no—
300C23C2×C3⋊#15711yescharacteristic
301C2×C4—91no—
302C2×C4—92no—
303C2×C4—92no—
304C2×C4—91no—
305C2×C4—92no—
306C2×C4—92no—
307C2×C4—94no—
308C2×C4—92no—
309C2×C4—92no—
310C2×C4—94no—
311C2×C4C2×C3⋊#15711yescharacteristic
312C2×C4C2×C3⋊#15714yes—
313C2×C4C2×C3⋊#15711yescharacteristic
314C8—278no—
315C2×C3—28no—
316C2×C3—22no—
317C2×C3—22no—
318C2×C3—24no—
319C2×C3—48no—
320C2×C3—42no—
321C2×C3—44no—
322C2×C3—94no—
323C2×C3—182no—
324C2×C3#6428312yescomplemented
325C2×C3C23×#15711yescharacteristic
326D3—68no—
327D3—68no—
328C22—22no—
329C22—92no—
330C22—94no—
331C22—92no—
332C22—188no—
333C22—182no—
334C22C22×C3⋊#15712yes—
335C22C22×C3⋊#15711yescharacteristic
336C4—91no—
337C4—91no—
338C4—92no—
339C4C22×C3⋊#15712yes—
340C4C22×C3⋊#15712yes—
341C3—22no—
342C3—42no—
343C3C2×#6428311yescharacteristic, complemented
344C2—22no—
345C2—94no—
346C2—182no—
347C2#1172451112yesdirect product factor
348C2C23×C3⋊#15711yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊#157
commutatorC2×C33C23×C4
FrattiniC2C23×C3⋊#157
FittingC22×C33×C4C4
socleC22×C33C22×C4
prime radicalC23×C3×C3⋊D3C4
cosocleC3×C3⋊Dic3C24
hypercenterC22×C4C3⋊#157
hypocenterC33C22×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. #11724511C2direct (N × H)
4. C2×#88989C2semidirect
5. C2×#88988C2semidirect
16. #88993C22semidirect
17. C2×C3⋊#622C22semidirect
18. #88989C22semidirect
19. #88988C22semidirect
44. C3⋊#622C23semidirect
74. C2×C3×C3⋊D3M16semidirect
131. C2×C33C2×M16semidirect
134. C3×C3⋊D3C2×M16semidirect
169. C2×C3⋊D3#208semidirect
194. C33C22×M16semidirect
243. C2×C32C2×#208semidirect
246. C3⋊D3C2×#208semidirect
295. C32C22×#208semidirect
324. C2×C3#64283semidirect
343. C3C2×#64283semidirect
347. C2#11724511direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×#889932
#117245112
C2×#889892
C2×#889882
C2×#642833
C22×#2089