Finite Group Databasebeta

← C22×#15688889 #26213705

Subgroups — C22×#15688889 #26213705

Order 12032.

4334 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Dic47C211yesmaximal, characteristic
2C23×C22⋊Dic47C211yesmaximal, characteristic
3C2×#15688889C2112yesmaximal, direct product factor
4C23×C22⋊C4×C47C211yesmaximal, characteristic
5C24×Dic47—21no—
6C24×Dic47C412yescomplemented
7C22×C22⋊Dic47—21no—
8C22×C22⋊Dic47—23no—
9C22×C22⋊Dic47—23no—
10C22×C22⋊Dic47—21no—
11C22×C22⋊Dic47—23no—
12C22×C22⋊Dic47—24no—
13C22×C22⋊Dic47C416yescomplemented
14C22×C22⋊Dic47C418yescomplemented
15C22×C22⋊Dic47C2216yes—
16C22×C22⋊Dic47C2216yes—
17#15688889C22116yesdirect product factor
18C26×C47C2211yescharacteristic
19C24×C4×C47—21no—
20C22×C22⋊C4×C47—23no—
21C22×C22⋊C4×C47—24no—
22C22×C22⋊C4×C47C418yescomplemented
23C22×C22⋊C4×C47C2216yes—
24C23×Dic47—26no—
25C23×Dic47—22no—
26C23×Dic47—28no—
27C23×Dic47—26no—
28C23×Dic47—48no—
29C23×Dic47—48no—
30C23×Dic47C2×C4112yescomplemented
31C2×C22⋊Dic47—26no—
32C2×C22⋊Dic47—26no—
33C2×C22⋊Dic47—212no—
34C2×C22⋊Dic47—224no—
35C2×C22⋊Dic47—26no—
36C2×C22⋊Dic47—26no—
37C2×C22⋊Dic47—26no—
38C2×C22⋊Dic47—26no—
39C2×C22⋊Dic47—26no—
40C2×C22⋊Dic47—212no—
41C2×C22⋊Dic47C2×C4112yescomplemented
42C2×C22⋊Dic47C2×C4124yescomplemented
43C2×C22⋊Dic47C2314yes—
44C2×C22⋊Dic47C2314yes—
45C25×C47—48no—
46C25×C47C2×C411yescharacteristic
47C25×C47C2×C414yes—
48C25×C47C2×C413yes—
49C25×C47C2×C414yes—
50C25×C47D411yescharacteristic
51C25×C47D411yescharacteristic
52C25×C47D413yes—
53C25×C47D413yes—
54C25×C47D414yes—
55C25×C47Q814yes—
56C25×C47C2313yes—
57C23×C4×C47—28no—
58C23×C4×C47—26no—
59C23×C4×C47—48no—
60C2×C22⋊C4×C47—224no—
61C2×C22⋊C4×C47—26no—
62C2×C22⋊C4×C47—212no—
63C2×C22⋊C4×C47C2×C4124yescomplemented
64C2×C22⋊C4×C47C2314yes—
65C24×C47—28no—
66C24×C47—28no—
67C24×C47—28no—
68C24×C47—416no—
69C24×C47—416no—
70C24×C47—412no—
71C24×C47—416no—
72C24×C47—412no—
73C24×C47—412no—
74C24×C47—416no—
75C24×C47—412no—
76C24×C47C4214yes—
77C24×C47C42112yes—
78C24×C47C22⋊C413yes—
79C24×C47C22⋊C413yes—
80C24×C47C22⋊C414yes—
81C24×C47C22⋊C413yes—
82C24×C47C22⋊C414yes—
83C24×C47C22⋊C4112yes—
84C24×C47C22⋊C416yes—
85C24×C47C22⋊C4112yes—
86C24×C47C22⋊C411yescharacteristic
87C24×C47C4⋊C414yes—
88C24×C47C4⋊C414yes—
89C24×C47C4⋊C414yes—
90C24×C47C4⋊C4112yes—
91C24×C47C4⋊C4112yes—
92C24×C47C4⋊C4112yes—
93C24×C47C22×C413yes—
94C24×C47C22×C416yes—
95C24×C47C22×C413yes—
96C24×C47C22×C416yes—
97C24×C47C2×D413yes—
98C24×C47C2×D413yes—
99C24×C47C2×D413yes—
100C24×C47C2×D413yes—
101C24×C47C2×D416yes—
102C24×C47C2×Q816yes—
103C24×C47C2411yescharacteristic
104C22×C4×C47—224no—
105C22×C4×C47—24no—
106C22×C4×C47—432no—
107C22×C4×C47—424no—
108C22×Dic47—212no—
109C22×Dic47—212no—
110C22×Dic47—224no—
111C22×Dic47—24no—
112C22×Dic47—432no—
113C22×Dic47—424no—
114C22×Dic47—424no—
115C22×Dic47—48no—
116C22×Dic47—424no—
117C22×Dic47C22×C418yescomplemented
118C22⋊C4×C47—224no—
119C22⋊C4×C47—24no—
120C22⋊C4×C47C22×C418yescomplemented
121C22⋊Dic47—212no—
122C22⋊Dic47—212no—
123C22⋊Dic47—224no—
124C22⋊Dic47—24no—
125C22⋊Dic47—24no—
126C22⋊Dic47—24no—
127C22⋊Dic47C22×C418yescomplemented
128C23×C47—216no—
129C23×C47—216no—
130C23×C47—216no—
131C23×C47—216no—
132C23×C47—216no—
133C23×C47—216no—
134C23×C47—212no—
135C23×C47—212no—
136C23×C47—212no—
137C23×C47—212no—
138C23×C47—212no—
139C23×C47—212no—
140C23×C47—424no—
141C23×C47—412no—
142C23×C47—424no—
143C23×C47—424no—
144C23×C47—424no—
145C23×C47—424no—
146C23×C47—424no—
147C23×C47—46no—
148C23×C47—412no—
149C23×C47—412no—
150C23×C47—46no—
151C23×C47—46no—
152C23×C47—412no—
153C23×C47—412no—
154C23×C47—42no—
155C23×C47#95112yes—
156C23×C47#95112yes—
157C23×C47#95112yes—
158C23×C47#95124yes—
159C23×C47#9514yes—
160C23×C47#9918yes—
161C23×C47C2×C4216yes—
162C23×C47C2×C4216yes—
163C23×C47C2×C22⋊C413yes—
164C23×C47C2×C22⋊C413yes—
165C23×C47C2×C22⋊C413yes—
166C23×C47C2×C22⋊C416yes—
167C23×C47C2×C22⋊C416yes—
168C23×C47C2×C22⋊C413yes—
169C23×C47C2×C22⋊C416yes—
170C23×C47C2×C22⋊C416yes—
171C23×C47C2×C4⋊C416yes—
172C23×C47C2×C4⋊C416yes—
173C23×C47C2×C4⋊C416yes—
174C23×C47C2×C4⋊C416yes—
175C23×C47C2×C4⋊C416yes—
176C23×C47C2×C4⋊C416yes—
177C23×C47C23×C411yescharacteristic
178C23×C47C23×C411yescharacteristic
179C23×C47C23×C411yescharacteristic
180C23×C47C22×D411yescharacteristic
181C23×C47C22×D411yescharacteristic
182C23×C47C22×D411yescharacteristic
183C23×C47C22×Q811yescharacteristic
184C2×C4×C47—28no—
185C2×C4×C47—48no—
186C2×C4×C47—448no—
187C2×Dic47—28no—
188C2×Dic47—28no—
189C2×Dic47—48no—
190C2×Dic47—448no—
191C2×Dic47—424no—
192C2×Dic47—424no—
193C2×Dic47—48no—
194C22×#342—471nomaximal, Sylow
195C22×C47—26no—
196C22×C47—22no—
197C22×C47—22no—
198C22×C47—22no—
199C22×C47—26no—
200C22×C47—26no—
201C22×C47—212no—
202C22×C47—212no—
203C22×C47—224no—
204C22×C47—224no—
205C22×C47—224no—
206C22×C47—212no—
207C22×C47—212no—
208C22×C47—212no—
209C22×C47—212no—
210C22×C47—424no—
211C22×C47—46no—
212C22×C47—46no—
213C22×C47—46no—
214C22×C47—46no—
215C22×C47—46no—
216C22×C47—46no—
217C22×C47—41no—
218C22×C47—41no—
219C22×C47—41no—
220C22×C47—41no—
221C22×C47#342116yescomplemented
222C22×C47C2×#9516yes—
223C22×C47C2×#9516yes—
224C22×C47C2×#9516yes—
225C22×C47C2×#9516yes—
226C22×C47C2×#99112yes—
227C22×C47C22×C4211yescharacteristic
228C22×C47C22×C22⋊C411yescharacteristic
229C22×C47C22×C22⋊C411yescharacteristic
230C22×C47C22×C22⋊C411yescharacteristic
231C22×C47C22×C4⋊C411yescharacteristic
232C22×C47C22×C4⋊C411yescharacteristic
233C22×C47C22×C4⋊C411yescharacteristic
234C4×C47—48no—
235Dic47—48no—
236Dic47—48no—
237C23×C22⋊C4—471no—
238C23×C22⋊C4—471no—
239C23×C22⋊C4D4711yescharacteristic
240C2×#342—4712no—
241C2×C47—21no—
242C2×C47—21no—
243C2×C47—21no—
244C2×C47—21no—
245C2×C47—21no—
246C2×C47—26no—
247C2×C47—26no—
248C2×C47—26no—
249C2×C47—21no—
250C2×C47C2×#342112yescomplemented
251C2×C47C22×#9511yescharacteristic
252C2×C47C22×#9912yes—
253C26C2×D4711yescharacteristic
254C24×C4—21no—
255C24×C4—472no—
256C24×C4—941no—
257C22×C22⋊C4—23no—
258C22×C22⋊C4—24no—
259C22×C22⋊C4—478no—
260C22×C22⋊C4—476no—
261C22×C22⋊C4—476no—
262C22×C22⋊C4—476no—
263C22×C22⋊C4—943no—
264C22×C22⋊C4—943no—
265C22×C22⋊C4—941no—
266C22×C22⋊C4—944no—
267C22×C22⋊C4—943no—
268C22×C22⋊C4—941no—
269C22×C22⋊C4Dic4718yescomplemented
270C22×C22⋊C4C2×D4716yes—
271#342—4716no—
272C47C22×#34211yescharacteristic, complemented, Sylow
273C25—48no—
274C25Dic9414yes—
275C25C4×D4711yescharacteristic
276C25C4×D4714yes—
277C25C4×D4713yes—
278C25D18811yescharacteristic
279C25D18813yes—
280C25C2×Dic4714yes—
281C25C22⋊D4711yescharacteristic
282C25C22⋊D4714yes—
283C25C22⋊D4713yes—
284C25C22×D4713yes—
285C23×C4—28no—
286C23×C4—26no—
287C23×C4—48no—
288C23×C4—4712no—
289C23×C4—948no—
290C23×C4—946no—
291C23×C4—942no—
292C23×C4—946no—
293C23×C4—1888no—
294C23×C4—1888no—
295C2×C22⋊C4—224no—
296C2×C22⋊C4—26no—
297C2×C22⋊C4—212no—
298C2×C22⋊C4—4724no—
299C2×C22⋊C4—474no—
300C2×C22⋊C4—4712no—
301C2×C22⋊C4—474no—
302C2×C22⋊C4—9424no—
303C2×C22⋊C4—946no—
304C2×C22⋊C4—946no—
305C2×C22⋊C4—9412no—
306C2×C22⋊C4—946no—
307C2×C22⋊C4—946no—
308C2×C22⋊C4—946no—
309C2×C22⋊C4—9412no—
310C2×C22⋊C4—946no—
311C2×C22⋊C4—946no—
312C2×C22⋊C4C2×Dic47124yescomplemented
313C2×C22⋊C4C22×D4714yes—
314C24—28no—
315C24—28no—
316C24—28no—
317C24—416no—
318C24—412no—
319C24—416no—
320C24—412no—
321C24—416no—
322C24—412no—
323C24—416no—
324C24—412no—
325C24C4×Dic4714yes—
326C24C4×Dic47112yes—
327C24Dic47⋊C414yes—
328C24Dic47⋊C414yes—
329C24Dic47⋊C4112yes—
330C24Dic47⋊C4112yes—
331C24C4⋊Dic4714yes—
332C24C4⋊Dic47112yes—
333C24(C2×D47)⋊C413yes—
334C24(C2×D47)⋊C413yes—
335C24(C2×D47)⋊C413yes—
336C24(C2×D47)⋊C414yes—
337C24(C2×D47)⋊C4112yes—
338C24(C2×D47)⋊C416yes—
339C24(C2×D47)⋊C411yescharacteristic
340C24C22⋊Dic4714yes—
341C24C22⋊Dic47112yes—
342C24C2×Dic9416yes—
343C24C2×C4×D4713yes—
344C24C2×C4×D4716yes—
345C24C2×C4×D4713yes—
346C24C2×D18813yes—
347C24C2×D18813yes—
348C24C22×Dic4716yes—
349C24C2×C22⋊D4713yes—
350C24C2×C22⋊D4713yes—
351C24C2×C22⋊D4716yes—
352C24C23×D4711yescharacteristic
353C22×C4—224no—
354C22×C4—24no—
355C22×C4—424no—
356C22×C4—432no—
357C22×C4—478no—
358C22×C4—9424no—
359C22×C4—9412no—
360C22×C4—9412no—
361C22×C4—944no—
362C22×C4—18824no—
363C22×C4—18832no—
364C22×C4—18824no—
365C22×C4—1888no—
366C22×C4—18824no—
367C22⋊C4—224no—
368C22⋊C4—24no—
369C22⋊C4—478no—
370C22⋊C4—9424no—
371C22⋊C4—9412no—
372C22⋊C4—9412no—
373C22⋊C4—944no—
374C22⋊C4—944no—
375C22⋊C4—944no—
376C22⋊C4C22×Dic4718yescomplemented
377C23—212no—
378C23—212no—
379C23—212no—
380C23—212no—
381C23—212no—
382C23—212no—
383C23—216no—
384C23—216no—
385C23—216no—
386C23—216no—
387C23—216no—
388C23—216no—
389C23—424no—
390C23—424no—
391C23—424no—
392C23—424no—
393C23—424no—
394C23—412no—
395C23—424no—
396C23—412no—
397C23—412no—
398C23—46no—
399C23—46no—
400C23—412no—
401C23—412no—
402C23—46no—
403C23—42no—
404C23C47⋊#95112yes—
405C23C47⋊#95112yes—
406C23C47⋊#95112yes—
407C23C47⋊#95124yes—
408C23C47⋊#9514yes—
409C23#1309393818yes—
410C23C2×C4×Dic4716yes—
411C23C2×C4×Dic4716yes—
412C23C2×(C2×D47)⋊C413yes—
413C23C2×(C2×D47)⋊C413yes—
414C23C2×(C2×D47)⋊C416yes—
415C23C2×(C2×D47)⋊C413yes—
416C23C2×(C2×D47)⋊C416yes—
417C23C2×(C2×D47)⋊C413yes—
418C23C2×C22⋊Dic4716yes—
419C23C2×C22⋊Dic4716yes—
420C23C2×Dic47⋊C416yes—
421C23C2×Dic47⋊C416yes—
422C23C2×Dic47⋊C416yes—
423C23C2×Dic47⋊C416yes—
424C23C2×C4⋊Dic4716yes—
425C23C2×C4⋊Dic4716yes—
426C23C22×C4×D4711yescharacteristic
427C23C22×C4×D4711yescharacteristic
428C23C23×Dic4711yescharacteristic
429C23C22×C22⋊D4711yescharacteristic
430C23C22×C22⋊D4711yescharacteristic
431C23C22×D18811yescharacteristic
432C23C22×Dic9411yescharacteristic
433C2×C4—28no—
434C2×C4—448no—
435C2×C4—48no—
436C2×C4—948no—
437C2×C4—948no—
438C2×C4—18848no—
439C2×C4—1888no—
440C2×C4—18824no—
441C2×C4—18824no—
442C2×C4—1888no—
443C22—212no—
444C22—212no—
445C22—224no—
446C22—224no—
447C22—224no—
448C22—212no—
449C22—212no—
450C22—212no—
451C22—212no—
452C22—22no—
453C22—26no—
454C22—22no—
455C22—22no—
456C22—26no—
457C22—26no—
458C22—424no—
459C22—46no—
460C22—46no—
461C22—46no—
462C22—46no—
463C22—46no—
464C22—46no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22#15688889116yesdirect product factor
470C22C2×C47⋊#9516yes—
471C22C2×C47⋊#9516yes—
472C22C2×C47⋊#9516yes—
473C22C2×C47⋊#9516yes—
474C22C2×#13093938112yes—
475C22C22×C4×Dic4711yescharacteristic
476C22C22×(C2×D47)⋊C411yescharacteristic
477C22C22×(C2×D47)⋊C411yescharacteristic
478C22C22×C22⋊Dic4711yescharacteristic
479C22C22×Dic47⋊C411yescharacteristic
480C22C22×Dic47⋊C411yescharacteristic
481C22C22×C4⋊Dic4711yescharacteristic
482C4—48no—
483C4—1888no—
484C4—1888no—
485C2—21no—
486C2—21no—
487C2—26no—
488C2—26no—
489C2—26no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2C2×#15688889112yesdirect product factor
495C2C22×C47⋊#9511yescharacteristic
496C2C22×#1309393812yes—
subgroupG/H
centerC24(C2×D47)⋊C4
commutatorC22×C47C22×C42
FrattiniC24C23×D47
FittingC23×C22⋊C4×C47C2
socleC24×C47C22⋊C4
prime radicalC26×C47C22
cosocleC24×C47C24
hypercenterC23×C22⋊C4D47
hypocenterC47C22×#342
LCSG ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C26 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#15688889C2direct (N × H)
6. C24×Dic47C4semidirect
13. C22×C22⋊Dic47C4semidirect
14. C22×C22⋊Dic47C4semidirect
17. #15688889C22direct (N × H)
22. C22×C22⋊C4×C47C4semidirect
30. C23×Dic47C2×C4semidirect
41. C2×C22⋊Dic47C2×C4semidirect
42. C2×C22⋊Dic47C2×C4semidirect
63. C2×C22⋊C4×C47C2×C4semidirect
117. C22×Dic47C22×C4semidirect
120. C22⋊C4×C47C22×C4semidirect
127. C22⋊Dic47C22×C4semidirect
221. C22×C47#342semidirect
250. C2×C47C2×#342semidirect
269. C22×C22⋊C4Dic47semidirect
272. C47C22×#342semidirect
312. C2×C22⋊C4C2×Dic47semidirect
376. C22⋊C4C22×Dic47semidirect
469. C22#15688889direct (N × H)
494. C2C2×#15688889direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Dic472
C2×#156888892
C23×C22⋊C4×C472
C22×#34247