Finite Group Databasebeta

← C22×#89046 #13181691

Subgroups — C22×#89046 #13181691

Order 1728.

4402 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C212yesmaximal, complemented
2C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C22×#6296C211yesmaximal, characteristic, complemented
4C22×#6295C211yesmaximal, characteristic, complemented
5C2×#89046C2124yesmaximal, direct product factor
6C22×#6294C212yesmaximal, complemented
7C22×#3805—31nomaximal
8C22×#3804—32nomaximal
9C24×C33C2211yescharacteristic
10C23×C3×C3⋊D3C2212yes—
11C2×C3×#638C22124yescomplemented
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C3×C3⋊Dic3C2212yes—
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C22112yescomplemented
16C2×#6296C22112yescomplemented
17C2×#6295C22112yescomplemented
18#89046C22164yesdirect product factor
19C2×#6294C22124yescomplemented
20C22×#638D312yescomplemented
21C23×C3⋊Dic3D311yescharacteristic, complemented
22C22×C3×#67—32no—
23C22×C3×#67—32no—
24C22×C3×#67—62no—
25C23×C3×Dic3—32no—
26C23×C3×Dic3—61no—
27C22×#627—32no—
28C22×#627—31no—
29C22×#626—32no—
30C22×#625—31no—
31C22×#624—32no—
32C2×#3805—324no—
33C2×#3804—348no—
34C22×D3×Dic3—32no—
35C22×D3×Dic3—32no—
36C23×C33—24no—
37C23×C33C2316yes—
38C23×C33C2311yescharacteristic
39C22×C3×C3⋊D3—28no—
40C22×C3×C3⋊D3C23112yes—
41C3×#638C23132yescomplemented
42C2×C3×C3⋊Dic3C2316yes—
43C2×C3×C3⋊Dic3C23112yes—
44C2×C3×C3⋊Dic3C2316yes—
45C2×C3×C3⋊Dic3C23116yescomplemented
46#6296C23116yescomplemented
47#6295C23116yescomplemented
48#6294C23132yescomplemented
49C22×#237—92no—
50C22×#235—91no—
51C24×C32—21no—
52C24×C32—22no—
53C24×C32—41no—
54C24×C32C2×D312yes—
55C24×C32C2×D311yescharacteristic
56C23×C3⋊D3C2×D312yes—
57C23×C3×D3—32no—
58C23×C3×D3—32no—
59C23×C3×D3—62no—
60C2×#638C2×D3124yescomplemented
61C22×C3⋊Dic3C2×D311yescharacteristic
62C22×C3⋊Dic3C2×D312yes—
63C22×C3⋊Dic3C2×D311yescharacteristic
64C22×C3⋊Dic3C2×D3112yescomplemented
65C2×C3×#67—324no—
66C2×C3×#67—324no—
67C2×C3×#67—624no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—324no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—61no—
74C22×C3×Dic3—61no—
75C22×C3×Dic3—612no—
76C22×C3×Dic3—62no—
77C2×#627—324no—
78C2×#627—312no—
79C2×#626—324no—
80C2×#625—312no—
81C2×#624—324no—
82#3805—364no—
83#3804—3128no—
84C2×D3×Dic3—324no—
85C2×D3×Dic3—324no—
86C22×C33—212no—
87C22×C33#4114yescomplemented
88C22×C33C2414yes—
89C22×C33C2413yes—
90C2×C3×C3⋊D3—224no—
91C2×C3×C3⋊D3C2418yes—
92C3×C3⋊Dic3C2414yes—
93C3×C3⋊Dic3C2418yes—
94C3×C3⋊Dic3C2414yes—
95C22×C3×D4—92no—
96C23×C3×C4—91no—
97C22×#67—32no—
98C22×#67—32no—
99C22×#67—62no—
100C22×#67—92no—
101C23×Dic3—32no—
102C23×Dic3—61no—
103C23×Dic3—92no—
104C2×#237—948no—
105C2×#235—924no—
106C22×D12—91no—
107C22×C4×D3—92no—
108C22×C4×D3—92no—
109C22×Dic6—91no—
110C22×Dic6—92no—
111C23×C32—24no—
112C23×C32—28no—
113C23×C32—28no—
114C23×C32—22no—
115C23×C32—26no—
116C23×C32—21no—
117C23×C32—212no—
118C23×C32—46no—
119C23×C32—41no—
120C23×C32—48no—
121C23×C32—48no—
122C23×C32C22×D3112yes—
123C23×C32C22×D316yes—
124C23×C32C22×D311yescharacteristic
125C23×C32C22×D312yes—
126C22×C3⋊D3—28no—
127C22×C3⋊D3C22×D3112yes—
128C22×C3×D3—312no—
129C22×C3×D3—312no—
130C22×C3×D3—612no—
131C22×C3×D3—68no—
132C22×C3×D3—68no—
133C22×C3×D3—128no—
134#638C22×D3132yescomplemented
135C2×C3⋊Dic3C22×D316yes—
136C2×C3⋊Dic3C22×D3112yes—
137C2×C3⋊Dic3C22×D316yes—
138C2×C3⋊Dic3C22×D3116yescomplemented
139C3×#67—332no—
140C3×#67—332no—
141C3×#67—632no—
142C2×C3×Dic3—312no—
143C2×C3×Dic3—312no—
144C2×C3×Dic3—312no—
145C2×C3×Dic3—312no—
146C2×C3×Dic3—332no—
147C2×C3×Dic3—66no—
148C2×C3×Dic3—66no—
149C2×C3×Dic3—612no—
150C2×C3×Dic3—616no—
151#627—332no—
152#627—316no—
153#626—332no—
154#625—316no—
155#624—332no—
156D3×Dic3—332no—
157D3×Dic3—332no—
158C22×#41—271noSylow
159C2×C33—24no—
160C2×C33C2×#4116yescomplemented
161C2×C33C2511yescharacteristic
162C3×C3⋊D3—28no—
163C24×C3—21no—
164C24×C3—22no—
165C24×C3—41no—
166C24×C3—92no—
167C24×C3D3212yes—
168C24×C3D3211yescharacteristic
169C23×D3—32no—
170C23×D3—32no—
171C23×D3—62no—
172C2×C3×D4—924no—
173C22×C3×C4—92no—
174C22×C3×C4—91no—
175C22×C3×C4—91no—
176C22×C3×C4—912no—
177C2×#67—324no—
178C2×#67—324no—
179C2×#67—624no—
180C2×#67—924no—
181C22×Dic3—32no—
182C22×Dic3—32no—
183C22×Dic3—32no—
184C22×Dic3—32no—
185C22×Dic3—324no—
186C22×Dic3—61no—
187C22×Dic3—61no—
188C22×Dic3—612no—
189C22×Dic3—62no—
190C22×Dic3—924no—
191#237—9128no—
192#235—964no—
193C2×D12—912no—
194C2×C4×D3—924no—
195C2×C4×D3—924no—
196C2×Dic6—912no—
197C2×Dic6—924no—
198C22×C32—212no—
199C22×C32—224no—
200C22×C32—224no—
201C22×C32—24no—
202C22×C32—23no—
203C22×C32—26no—
204C22×C32—28no—
205C22×C32—28no—
206C22×C32—24no—
207C22×C32—44no—
208C22×C32—43no—
209C22×C32—44no—
210C22×C32—424no—
211C22×C32—424no—
212C22×C32#23514yescomplemented
213C22×C32#23718yescomplemented
214C22×C32C23×D318yes—
215C22×C32C23×D314yes—
216C22×C32C23×D313yes—
217C22×C32C23×D316yes—
218C2×C3⋊D3—224no—
219C2×C3⋊D3C23×D318yes—
220C2×C3×D3—38no—
221C2×C3×D3—38no—
222C2×C3×D3—68no—
223C2×C3×D3—624no—
224C2×C3×D3—624no—
225C2×C3×D3—1224no—
226C3⋊Dic3C23×D314yes—
227C3⋊Dic3C23×D318yes—
228C3⋊Dic3C23×D314yes—
229C3×Dic3—38no—
230C3×Dic3—38no—
231C3×Dic3—38no—
232C3×Dic3—38no—
233C3×Dic3—64no—
234C3×Dic3—64no—
235C3×Dic3—68no—
236C2×#41—2724no—
237C22×Q8—271no—
238C22×D4—92no—
239C22×D4—271no—
240C23×C4—91no—
241C23×C4—272no—
242C33C22×#4111yescharacteristic, complemented, Sylow
243C23×C3—28no—
244C23×C3—28no—
245C23×C3—24no—
246C23×C3—26no—
247C23×C3—212no—
248C23×C3—21no—
249C23×C3—22no—
250C23×C3—48no—
251C23×C3—48no—
252C23×C3—46no—
253C23×C3—41no—
254C23×C3—912no—
255C23×C3—188no—
256C23×C3C2×D32112yes—
257C23×C3C2×D3216yes—
258C23×C3C2×D3212yes—
259C23×C3C2×D3211yescharacteristic
260C22×D3—312no—
261C22×D3—312no—
262C22×D3—612no—
263C22×D3—68no—
264C22×D3—68no—
265C22×D3—128no—
266C3×D4—932no—
267C2×C3×C4—912no—
268C2×C3×C4—96no—
269C2×C3×C4—96no—
270C2×C3×C4—916no—
271#67—332no—
272#67—332no—
273#67—632no—
274#67—932no—
275C2×Dic3—312no—
276C2×Dic3—312no—
277C2×Dic3—312no—
278C2×Dic3—312no—
279C2×Dic3—332no—
280C2×Dic3—66no—
281C2×Dic3—66no—
282C2×Dic3—612no—
283C2×Dic3—616no—
284C2×Dic3—932no—
285D12—916no—
286C4×D3—932no—
287C4×D3—932no—
288Dic6—916no—
289Dic6—932no—
290C2×C32—24no—
291C2×C32—28no—
292C2×C32—28no—
293C2×C32—212no—
294C2×C32—26no—
295C2×C32—21no—
296C2×C32—22no—
297C2×C32—46no—
298C2×C32—41no—
299C2×C32—48no—
300C2×C32—48no—
301C2×C32C2×#23516yescomplemented
302C2×C32C2×#237112yescomplemented
303C2×C32C24×D311yescharacteristic
304C2×C32C24×D312yes—
305C3⋊D3—28no—
306C3×D3—68no—
307C3×D3—68no—
308C3×D3—128no—
309C24—92no—
310C24#111311yescharacteristic
311#41—2764no—
312C2×Q8—2712no—
313C2×D4—924no—
314C2×D4—2712no—
315C22×C4—912no—
316C22×C4—91no—
317C22×C4—91no—
318C22×C4—92no—
319C22×C4—2724no—
320C22×C3—224no—
321C22×C3—224no—
322C22×C3—212no—
323C22×C3—24no—
324C22×C3—23no—
325C22×C3—26no—
326C22×C3—28no—
327C22×C3—28no—
328C22×C3—24no—
329C22×C3—424no—
330C22×C3—424no—
331C22×C3—44no—
332C22×C3—43no—
333C22×C3—44no—
334C22×C3—98no—
335C22×C3—1824no—
336C22×C3#380418yescomplemented
337C22×C3#380514yescomplemented
338C22×C3C22×D3218yes—
339C22×C3C22×D3214yes—
340C22×C3C22×D3216yes—
341C22×C3C22×D3213yes—
342C2×D3—38no—
343C2×D3—38no—
344C2×D3—68no—
345C2×D3—624no—
346C2×D3—624no—
347C2×D3—1224no—
348C3×C4—98no—
349C3×C4—94no—
350C3×C4—94no—
351Dic3—38no—
352Dic3—38no—
353Dic3—38no—
354Dic3—38no—
355Dic3—64no—
356Dic3—64no—
357Dic3—68no—
358C32—22no—
359C32—21no—
360C32—41no—
361C32C22×#23511yescharacteristic, complemented
362C32C22×#23712yescomplemented
363C23—24no—
364C23—912no—
365C23—188no—
366C23C2×#111316yes—
367C23C2×#111311yescharacteristic
368Q8—2716no—
369D4—932no—
370D4—2716no—
371C2×C4—96no—
372C2×C4—96no—
373C2×C4—916no—
374C2×C4—912no—
375C2×C4—2732no—
376C2×C3—28no—
377C2×C3—28no—
378C2×C3—24no—
379C2×C3—212no—
380C2×C3—26no—
381C2×C3—21no—
382C2×C3—22no—
383C2×C3—48no—
384C2×C3—48no—
385C2×C3—46no—
386C2×C3—41no—
387C2×C3—188no—
388C2×C3C2×#3804112yescomplemented
389C2×C3C2×#380516yescomplemented
390C2×C3C23×D3212yes—
391C2×C3C23×D3211yescharacteristic
392D3—68no—
393D3—68no—
394D3—128no—
395C22—212no—
396C22—98no—
397C22—1824no—
398C22#8904614yesdirect product factor
399C22C22×#111314yes—
400C22C22×#111313yes—
401C4—94no—
402C4—94no—
403C4—98no—
404C3—22no—
405C3—21no—
406C3—41no—
407C3C22×#380412yescomplemented
408C3C22×#380511yescharacteristic, complemented
409C2—24no—
410C2—188no—
411C2C2×#8904616yesdirect product factor
412C2C23×#111311yescharacteristic
subgroupG/H
centerC23C2×#1113
commutator, cosocleC2×C33C25
FrattiniC2C23×#1113
FittingC24×C33C22
socleC23×C33C23
hypercenterC24#1113
hypocenterC33C22×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C3×#638C2semidirect
2. C23×C3×C3⋊Dic3C2semidirect
3. C22×#6296C2semidirect
4. C22×#6295C2semidirect
5. C2×#89046C2direct (N × H)
6. C22×#6294C2semidirect
11. C2×C3×#638C22semidirect
15. C22×C3×C3⋊Dic3C22semidirect
16. C2×#6296C22semidirect
17. C2×#6295C22semidirect
18. #89046C22direct (N × H)
19. C2×#6294C22semidirect
20. C22×#638D3semidirect
21. C23×C3⋊Dic3D3semidirect
41. C3×#638C23semidirect
45. C2×C3×C3⋊Dic3C23semidirect
46. #6296C23semidirect
47. #6295C23semidirect
48. #6294C23semidirect
60. C2×#638C2×D3semidirect
64. C22×C3⋊Dic3C2×D3semidirect
87. C22×C33#41semidirect
134. #638C22×D3semidirect
138. C2×C3⋊Dic3C22×D3semidirect
160. C2×C33C2×#41semidirect
212. C22×C32#235semidirect
213. C22×C32#237semidirect
242. C33C22×#41semidirect
301. C2×C32C2×#235semidirect
302. C2×C32C2×#237semidirect
336. C22×C3#3804semidirect
337. C22×C3#3805semidirect
361. C32C22×#235semidirect
362. C32C22×#237semidirect
388. C2×C3C2×#3804semidirect
389. C2×C3C2×#3805semidirect
398. C22#89046direct (N × H)
407. C3C22×#3804semidirect
408. C3C22×#3805semidirect
411. C2C2×#89046direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C23×C3×C3⋊Dic32
C22×#62962
C22×#62952
C2×#890462
C22×#62942
C22×#38053
C22×#38043