Finite Group Databasebeta

← C4⋊#88813 #13161840

Subgroups — C4⋊#88813 #13161840

Order 1728.

1122 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C212yesmaximal
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C2×#88813C212yesmaximal
4#11722082C212yesmaximal
5#10591274—31nomaximal
6Dic3×C4⋊Dic3—32nomaximal
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C3×C3⋊Dic3C2214yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C418yescomplemented
12C3×C4×C3⋊Dic3C414yescomplemented
13#88813—24no—
14C2×C4⋊(C3⋊Dic3)D312yes—
15C2×C4×C3⋊Dic3D311yescharacteristic
16C2×C3×C4⋊Dic3—32no—
17C2×C3×C4⋊Dic3—32no—
18C2×C3×C4⋊Dic3—62no—
19C2×C3×C4×Dic3—32no—
20C2×C3×C4×Dic3—61no—
21C2×Dic32—34no—
22C2×Dic32—32no—
23#63579—34no—
24#63579—32no—
25C23×C33C2311yescharacteristic
26C2×C33×C4C2×C412yes—
27C2×C33×C4C2×C414yes—
28C2×C3×C3⋊Dic3—24no—
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3—24no—
31C2×C3×C3⋊Dic3C2×C412yes—
32C2×C3×C3⋊Dic3C2×C418yes—
33C2×C3×C3⋊Dic3C2×C412yes—
34C2×C3×C3⋊Dic3D412yes—
35C2×C3×C3⋊Dic3Q812yes—
36Dic3×C4⋊C4—92no—
37C4×C4⋊Dic3—91no—
38C22×C32×C4—21no—
39C22×C32×C4—22no—
40C22×C32×C4—41no—
41C22×C32×C4C2×D312yes—
42C22×C32×C4C2×D311yescharacteristic
43C22×C3⋊Dic3C2×D311yescharacteristic
44C22×C3⋊Dic3C2×D314yes—
45C22×C3⋊Dic3C2×D311yescharacteristic
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—34no—
49C22×C3×Dic3—34no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C22×C3×Dic3—64no—
53C4⋊(C3⋊Dic3)Dic318yescomplemented
54C4×C3⋊Dic3Dic314yescomplemented
55C3×C4⋊Dic3—38no—
56C3×C4⋊Dic3—38no—
57C3×C4⋊Dic3—68no—
58C3×C4×Dic3—38no—
59C3×C4×Dic3—64no—
60Dic32—68no—
61Dic32—64no—
62C22×C33C22×C411yescharacteristic
63C22×C33C22×C412yes—
64C22×C33C2×D411yescharacteristic
65C22×C33C2×Q811yescharacteristic
66C22×C33#4112yes—
67C33×C4C4214yescomplemented
68C3×C3⋊Dic3—22no—
69C3×C3⋊Dic3—28no—
70C3×C3⋊Dic3C4⋊C414yescomplemented
71C2×C3×C4⋊C4—92no—
72C2×C3×C42—91no—
73C2×C4⋊Dic3—32no—
74C2×C4⋊Dic3—32no—
75C2×C4⋊Dic3—62no—
76C2×C4×Dic3—32no—
77C2×C4×Dic3—61no—
78C2×C4×Dic3—92no—
79C2×C4×Dic3—94no—
80C3⋊#95—92no—
81C3⋊#95—94no—
82C23×C32—22no—
83C23×C32—21no—
84C23×C32—41no—
85C23×C32C22×D311yescharacteristic
86C23×C32C22×D312yes—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4—24no—
90C2×C32×C4—22no—
91C2×C32×C4—24no—
92C2×C32×C4—42no—
93C2×C32×C4—44no—
94C2×C32×C4C4×D314yes—
95C2×C32×C4C4×D314yes—
96C2×C32×C4C4×D314yes—
97C2×C32×C4C2×Dic312yes—
98C2×C32×C4C2×Dic314yes—
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—24no—
101C2×C3⋊Dic3—24no—
102C2×C3⋊Dic3Dic612yes—
103C2×C3⋊Dic3D1212yes—
104C2×C3⋊Dic3C2×Dic312yes—
105C2×C3⋊Dic3C2×Dic318yes—
106C2×C3⋊Dic3C2×Dic312yes—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—34no—
110C2×C3×Dic3—34no—
111C2×C3×Dic3—38no—
112C2×C3×Dic3—38no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—68no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—64no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—124no—
126C2×C3×Dic3—124no—
127C4×C4⋊C4—271noSylow
128C2×C33C2×C4211yescharacteristic
129C2×C33C2×C4⋊C411yescharacteristic
130C2×C33#11711yescharacteristic
131C2×C33C4×D412yes—
132C2×C33C4×Q812yes—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—41no—
136C22×C3×C4—91no—
137C22×C3×C4—91no—
138C22×C3×C4—94no—
139C22×C3×C4D3212yes—
140C22×C3×C4D3211yescharacteristic
141C22×Dic3—32no—
142C22×Dic3—32no—
143C22×Dic3—34no—
144C22×Dic3—34no—
145C22×Dic3—61no—
146C22×Dic3—61no—
147C22×Dic3—64no—
148C3×C4⋊C4—98no—
149C3×C42—94no—
150C4⋊Dic3—38no—
151C4⋊Dic3—38no—
152C4⋊Dic3—68no—
153C4×Dic3—38no—
154C4×Dic3—64no—
155C4×Dic3—188no—
156C4×Dic3—184no—
157C22×C32—22no—
158C22×C32—22no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—21no—
169C22×C32—41no—
170C22×C32—42no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—42no—
174C22×C32C2×Dic611yescharacteristic
175C22×C32C2×C4×D312yes—
176C22×C32C2×C4×D312yes—
177C22×C32C2×C4×D312yes—
178C22×C32C2×D1211yescharacteristic
179C22×C32#23512yes—
180C22×C32D3×D412yes—
181C22×C32#23712yes—
182C22×C32D3×Q812yes—
183C22×C32#23912yes—
184C22×C32C22×Dic311yescharacteristic
185C22×C32C22×Dic312yes—
186C32×C4—24no—
187C32×C4—28no—
188C32×C4—44no—
189C32×C4C4×Dic314yescomplemented
190C32×C4C4×Dic318yescomplemented
191C3⋊Dic3—22no—
192C3⋊Dic3—28no—
193C3⋊Dic3C4⋊Dic314yescomplemented
194C3×Dic3—38no—
195C3×Dic3—64no—
196C3×Dic3—64no—
197C3×Dic3—68no—
198C3×Dic3—64no—
199C3×Dic3—68no—
200C3×Dic3—128no—
201C2×C4⋊C4—92no—
202C2×C42—91no—
203C2×C42—272no—
204#95—272no—
205C33C4×C4⋊C411yescharacteristic, complemented, Sylow
206C23×C3—21no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3C2×D3212yes—
210C23×C3C2×D3211yescharacteristic
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—22no—
215C2×C3×C4—24no—
216C2×C3×C4—42no—
217C2×C3×C4—44no—
218C2×C3×C4—92no—
219C2×C3×C4—92no—
220C2×C3×C4—92no—
221C2×C3×C4—92no—
222C2×C3×C4—98no—
223C2×C3×C4—182no—
224C2×C3×C4—184no—
225C2×C3×C4—184no—
226C2×C3×C4D3×Dic314yes—
227C2×C3×C4D3×Dic314yes—
228C2×C3×C4D3×Dic314yes—
229C2×C3×C4#62412yes—
230C2×C3×C4#62414yes—
231C2×Dic3—34no—
232C2×Dic3—34no—
233C2×Dic3—34no—
234C2×Dic3—34no—
235C2×Dic3—38no—
236C2×Dic3—38no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—68no—
242C2×Dic3—64no—
243C2×Dic3—62no—
244C2×Dic3—64no—
245C2×Dic3—64no—
246C2×Dic3—62no—
247C2×Dic3—64no—
248C2×Dic3—64no—
249C2×Dic3—124no—
250C2×Dic3—124no—
251C2×C32—22no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—22no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—42no—
268C2×C32C4×Dic612yes—
269C2×C32C4×D1212yes—
270C2×C32#88712yes—
271C2×C32D3×C4⋊C412yes—
272C2×C32#89212yes—
273C2×C32#89312yes—
274C2×C32C2×C4×Dic311yescharacteristic
275C2×C32C2×C4×Dic312yes—
276C2×C32C2×C4⋊Dic311yescharacteristic
277C2×C32#92611yescharacteristic
278C2×C32D4×Dic312yes—
279C2×C32Q8×Dic312yes—
280C22×C4—91no—
281C22×C4—91no—
282C22×C4—94no—
283C22×C4#111311yescharacteristic
284C4⋊C4—98no—
285C42—94no—
286C42—544no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—21no—
293C22×C3—22no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—42no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—42no—
304C22×C3D3×Dic612yes—
305C22×C3#379512yes—
306C22×C3#379612yes—
307C22×C3#379711yescharacteristic
308C22×C3#379912yes—
309C22×C3D3×D1212yes—
310C22×C3#380211yescharacteristic
311C22×C3C2×D3×Dic312yes—
312C22×C3C2×D3×Dic312yes—
313C22×C3C2×D3×Dic312yes—
314C22×C3C2×#62412yes—
315C22×C3C2×#62411yescharacteristic
316C3×C4—24no—
317C3×C4—28no—
318C3×C4—44no—
319C3×C4—94no—
320C3×C4—182no—
321C3×C4—188no—
322C3×C4Dic3218yescomplemented
323C3×C4Dic3214yescomplemented
324Dic3—38no—
325Dic3—64no—
326Dic3—64no—
327Dic3—64no—
328Dic3—68no—
329Dic3—68no—
330Dic3—128no—
331C32—22no—
332C32—21no—
333C32—41no—
334C32C4×C4⋊Dic311yescharacteristic, complemented
335C32Dic3×C4⋊C412yescomplemented
336C23C2×#111311yescharacteristic
337C2×C4—92no—
338C2×C4—92no—
339C2×C4—92no—
340C2×C4—92no—
341C2×C4—98no—
342C2×C4—182no—
343C2×C4—184no—
344C2×C4—184no—
345C2×C4#629412yes—
346C2×C4#629414yes—
347C2×C3—21no—
348C2×C3—22no—
349C2×C3—22no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—22no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—21no—
358C2×C3—22no—
359C2×C3—41no—
360C2×C3—42no—
361C2×C3—41no—
362C2×C3—42no—
363C2×C3—41no—
364C2×C3#6384112yes—
365C2×C3Dic3×Dic612yes—
366C2×C3#6384312yes—
367C2×C3#6384412yes—
368C2×C3#6384911yescharacteristic
369C2×C3D3×C4⋊Dic312yes—
370C2×C3Dic3×D1212yes—
371C2×C3#6389412yes—
372C2×C3#6389812yes—
373C2×C3#6390011yescharacteristic
374C2×C3C2×Dic3212yes—
375C2×C3C2×Dic3211yescharacteristic
376C22#8904011yescharacteristic
377C22#8904112yes—
378C22C4⋊#111311yescharacteristic
379C22C2×#629411yescharacteristic
380C22C2×#629412yes—
381C4—94no—
382C4—182no—
383C4—188no—
384C4#8881314yescomplemented
385C3—21no—
386C3—22no—
387C3—41no—
388C3Dic3×C4⋊Dic312yescomplemented
389C3#1059127411yescharacteristic, complemented
390C2#1172366711yescharacteristic
391C2#1172366812yes—
392C2#1172368911yescharacteristic
393C2#1172369112yes—
394C2C2×#8881311yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutatorC2×C33C2×C42
FittingC22×C33×C4C22
socle, prime radical, cosocleC23×C33C23
hypercenterC22×C4#1113
hypocenterC33C4×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
11. C3×C4⋊(C3⋊Dic3)C4semidirect
12. C3×C4×C3⋊Dic3C4semidirect
53. C4⋊(C3⋊Dic3)Dic3semidirect
54. C4×C3⋊Dic3Dic3semidirect
67. C33×C4C42semidirect
70. C3×C3⋊Dic3C4⋊C4semidirect
189. C32×C4C4×Dic3semidirect
190. C32×C4C4×Dic3semidirect
193. C3⋊Dic3C4⋊Dic3semidirect
205. C33C4×C4⋊C4semidirect
322. C3×C4Dic32semidirect
323. C3×C4Dic32semidirect
334. C32C4×C4⋊Dic3semidirect
335. C32Dic3×C4⋊C4semidirect
384. C4#88813semidirect
388. C3Dic3×C4⋊Dic3semidirect
389. C3#10591274semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×C4×C3⋊Dic32
C2×#888132
#117220822
#105912743
Dic3×C4⋊Dic33