Finite Group Databasebeta

← C2×#20256251 #27096704

Subgroups — C2×#20256251 #27096704

Order 12032.

1918 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689833C211yesmaximal, characteristic, complemented
2#20256251C2116yesmaximal, direct product factor
3C2×#15689526C211yesmaximal, characteristic, complemented
4C2×#15689522C212yesmaximal, complemented
5C2×#15689423C212yesmaximal, complemented
6C2×#15689420C212yesmaximal, complemented
7C2×#15689413C211yesmaximal, characteristic, complemented
8C2×#15689410C211yesmaximal, characteristic, complemented
9C2×Q8×C47⋊C8C211yesmaximal, characteristic, complemented
10C2×C4×#10625334C211yesmaximal, characteristic
11C2×C4×#10625334C212yesmaximal
12C2×C47×#554C211yesmaximal, characteristic, complemented
13C22×#10625334C2211yescharacteristic
14#15689833—24no—
15#15689833C2218yescomplemented
16C2×#13094024C2211yescharacteristic
17C2×#13094016C2212yes—
18C2×#13094002—22no—
19C2×#13094000—21no—
20C2×C4×D188C2211yescharacteristic
21C2×C4×D188C2212yes—
22C2×C4⋊C4×D47—21no—
23#15689526C2218yescomplemented
24#15689522C22116yescomplemented
25#15689423C22116yescomplemented
26#15689420C22116yescomplemented
27#15689413C2218yescomplemented
28#15689410C2218yescomplemented
29Q8×C47⋊C8C2218yescomplemented
30C4×#10625334C2218yes—
31C4×#10625334C22116yes—
32C2×#13093957C2211yescharacteristic
33C2×#13093957C2212yes—
34C2×#13093957C2212yes—
35C2×#13093955C2212yes—
36C2×#13093955C2212yes—
37C2×C4⋊(C47⋊C8)C2211yescharacteristic
38C2×C4⋊(C47⋊C8)C2212yes—
39C2×#13093945C2212yes—
40C2×#13093945C2211yescharacteristic
41C2×#13093944C2212yes—
42C2×#13093944C2211yescharacteristic
43C2×C4×C47⋊C8C2212yes—
44C2×C4×C47⋊C8C2211yescharacteristic
45C47×#554—24no—
46C47×#554C2218yescomplemented
47C2×C4⋊Q8×C47C2211yescharacteristic
48C2×#125×C47C2212yes—
49C2×C22⋊Q8×C47—21no—
50C2×C22⋊Q8×C47—22no—
51C2×C4×Q8×C47C2211yescharacteristic
52C2×C4×D4×C47C2211yescharacteristic
53C2×C4×D4×C47C2212yes—
54C22×C4⋊C4×C47—21no—
55C22×Dic94—21no—
56C22×D188C2311yescharacteristic
57C22×C4×D47—21no—
58C22×C4×D47—22no—
59C2×#10625334Q818yescomplemented
60C2×#10625334C2314yes—
61C22×C47⋊C8C2312yes—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094024—22no—
65#13094024C2314yes—
66#13094016—24no—
67#13094016C2318yes—
68#13094002—216no—
69#13094000—28no—
70C4×D188—22no—
71C4×D188—24no—
72C4×D188C2314yes—
73C4×D188C2318yes—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2312yes—
77C2×Dic47⋊C4—22no—
78C2×Dic47⋊C4—22no—
79C4⋊C4×D47—28no—
80C2×(C2×D47)⋊C4—22no—
81C2×(C2×D47)⋊C4—21no—
82#13093957C2314yes—
83#13093957C2318yes—
84#13093957C2318yes—
85#13093955C2318yes—
86#13093955C2318yes—
87C4⋊(C47⋊C8)C2314yes—
88C4⋊(C47⋊C8)C2318yes—
89#13093945C2318yes—
90#13093945C2314yes—
91#13093944C2318yes—
92#13093944C2314yes—
93C4×C47⋊C8C2318yes—
94C4×C47⋊C8C2314yes—
95C22×Q8×C47—21no—
96C22×Q8×C47D411yescharacteristic
97C22×D4×C47C2311yescharacteristic
98C23×C4×C47—21no—
99C23×C4×C47—22no—
100C4⋊Q8×C47—22no—
101C4⋊Q8×C47C2314yes—
102#125×C47—24no—
103#125×C47C2318yes—
104C22⋊Q8×C47—28no—
105C22⋊Q8×C47—216no—
106C4×Q8×C47D414yes—
107C4×Q8×C47C2314yes—
108C4×D4×C47—24no—
109C4×D4×C47—22no—
110C4×D4×C47C2314yes—
111C4×D4×C47C2318yes—
112C2×C4⋊C4×C47—22no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47—22no—
115C2×C4⋊C4×C47—28no—
116C2×C4⋊C4×C47D411yescharacteristic
117C2×C4⋊C4×C47D412yes—
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2312yes—
120C2×C22⋊C4×C47—21no—
121C2×C22⋊C4×C47—22no—
122C2×C42×C47C2312yes—
123C2×C42×C47C2311yescharacteristic
124C24×C47—21no—
125C23×D47—21no—
126C2×Q8×C47—24no—
127C2×Q8×C47—24no—
128C2×Q8×C47—82no—
129C2×Q8×C47C2×D414yes—
130C2×D4×C47—21no—
131C2×D4×C47—22no—
132C2×D4×C47C2×Q814yes—
133C2×D4×C47C2412yes—
134C22×C4×C47—21no—
135C22×C4×C47—22no—
136C22×C4×C47—24no—
137C22×C4×C47—28no—
138C22×C4×C47—44no—
139C22×C4×C47—84no—
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D412yes—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2411yescharacteristic
145C22×Dic47—21no—
146C22×Dic47—22no—
147C2×D188—21no—
148C2×D188—22no—
149C2×D188C2×Q814yes—
150C2×D188C2412yes—
151C2×C4×D47—24no—
152C2×C4×D47—28no—
153C2×C4×D47—44no—
154C2×C4×D47—84no—
155C2×Dic94—24no—
156C2×Dic94—82no—
157C4⋊C4×C47—28no—
158C4⋊C4×C47—24no—
159C4⋊C4×C47—28no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47C2×D414yes—
163C4⋊C4×C47C2×D418yes—
164C4⋊C4×C47C2412yes—
165C4⋊C4×C47C2414yes—
166C22⋊C4×C47—24no—
167C22⋊C4×C47—28no—
168C42×C47C2×D414yes—
169C42×C47C2×D412yes—
170C42×C47C2414yes—
171C42×C47C2412yes—
172#10625334C2×Q8116yescomplemented
173(C2×D47)⋊C4—28no—
174(C2×D47)⋊C4—24no—
175C4⋊Dic47—24no—
176C4⋊Dic47—21no—
177C4⋊Dic47—22no—
178C4⋊Dic47C2412yes—
179C4⋊Dic47C2414yes—
180Dic47⋊C4—28no—
181Dic47⋊C4—28no—
182C2×C47⋊C8—24no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8C2×Q814yes—
185C2×C47⋊C8C2414yes—
186C2×C47⋊C8C2412yes—
187C2×C47⋊C8C2412yes—
188C23×C47—24no—
189C23×C47—22no—
190C23×C47—42no—
191C23×C47—42no—
192C23×C47C22×D411yescharacteristic
193C22×D47—24no—
194C22×D47—22no—
195C22×D47—42no—
196C22×D47—42no—
197Q8×C47—28no—
198Q8×C47—84no—
199D4×C47—24no—
200D4×C47—22no—
201D4×C47C22×Q814yes—
202C2×C4×C47—24no—
203C2×C4×C47—22no—
204C2×C4×C47—24no—
205C2×C4×C47—22no—
206C2×C4×C47—24no—
207C2×C4×C47—42no—
208C2×C4×C47—44no—
209C2×C4×C47—48no—
210C2×C4×C47—88no—
211C2×C4×C47#13514yes—
212C2×C4×C47C22×D414yes—
213C2×C4×C47C22×D414yes—
214C2×C4×C47C22×D412yes—
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C22×D412yes—
217C2×C4×C47C22×Q812yes—
218C2×C4×C472−1+412yes—
219C2×C4×C47C2511yescharacteristic
220C2×Dic47—22no—
221C2×Dic47—24no—
222C2×Dic47—42no—
223C2×Dic47—44no—
224D188—24no—
225D188—22no—
226D188C22×Q814yes—
227C4×D47—48no—
228C4×D47—88no—
229Dic94—84no—
230C47⋊C8—24no—
231C47⋊C8—22no—
232C47⋊C8C22×Q814yes—
233C2×#3380—471nomaximal, Sylow
234C22×C47—24no—
235C22×C47—48no—
236C22×C47—42no—
237C22×C47—42no—
238C22×C47D4×Q812yes—
239C22×C47C2×#13512yes—
240C22×C47#57612yes—
241C22×C47C23×D411yescharacteristic
242C2×D47—24no—
243C2×D47—48no—
244C2×D47—42no—
245C2×D47—42no—
246C4×C47—24no—
247C4×C47—22no—
248C4×C47—24no—
249C4×C47—42no—
250C4×C47—44no—
251C4×C47D4×Q812yes—
252C4×C47C2×#13514yes—
253C4×C47C23×Q811yescharacteristic
254C4×C47C2×2−1+411yescharacteristic
255Dic47—42no—
256Dic47—44no—
257C2×#554—471no—
258C2×#554D4711yescharacteristic, complemented
259#3380—4716no—
260C2×#500—471no—
261C2×#499—472no—
262C2×#478—472no—
263C2×#478—472no—
264C2×#476—471no—
265C2×#476—471no—
266C2×C8×Q8—471no—
267C2×C4×D8—472no—
268C2×C4×D8—471no—
269C2×C47—44no—
270C2×C47#338014yescomplemented
271C2×C47C2×D4×Q811yescharacteristic
272C2×C47C22×#13511yescharacteristic
273C2×C47C2×#57611yescharacteristic
274D47—44no—
275C22×D8—471no—
276#554—24no—
277#554—478no—
278#554—944no—
279#554C2×D4718yescomplemented
280C2×C4⋊Q8—471no—
281C2×C4⋊Q8C2×D4711yescharacteristic
282C2×#125—472no—
283C2×#125C2×D4712yes—
284C2×C22⋊Q8—21no—
285C2×C22⋊Q8—22no—
286C2×C22⋊Q8—941no—
287C2×C22⋊Q8—942no—
288C2×C4×Q8C2×D4711yescharacteristic
289C2×C4×D4—472no—
290C2×C4×D4—471no—
291C2×C4×D4C2×D4712yes—
292C2×C4×D4C2×D4711yescharacteristic
293C22×C4⋊C4—21no—
294C22×C4⋊C4—941no—
295#500—478no—
296#499—4716no—
297#478—4716no—
298#478—4716no—
299#476—478no—
300#476—478no—
301C8×Q8—478no—
302C4×D8—4716no—
303C4×D8—478no—
304C2×#107—472no—
305C2×#107—471no—
306C2×#107—472no—
307C2×#106—472no—
308C2×#106—472no—
309C2×C4⋊C8—472no—
310C2×C4⋊C8—471no—
311C2×#102—471no—
312C2×#102—472no—
313C2×#102—471no—
314C2×#102—472no—
315C2×C4×C8—472no—
316C2×C4×C8—471no—
317C47C2×#338011yescharacteristic, complemented, Sylow
318C22×Q8—21no—
319C22×Q8—941no—
320C22×Q8C22⋊D4711yescharacteristic
321C22×D4—471no—
322C22×D4C22×D4711yescharacteristic
323C23×C4—21no—
324C23×C4—22no—
325C23×C4—942no—
326C23×C4—941no—
327C2×D8—478no—
328C2×D8—474no—
329C22×C8—471no—
330C22×C8—472no—
331C22×C8—471no—
332C4⋊Q8—22no—
333C4⋊Q8—474no—
334C4⋊Q8—942no—
335C4⋊Q8C22×D4714yes—
336#125—24no—
337#125—478no—
338#125—944no—
339#125C22×D4718yes—
340C22⋊Q8—28no—
341C22⋊Q8—216no—
342C22⋊Q8—948no—
343C22⋊Q8—9416no—
344C4×Q8C22⋊D4714yes—
345C4×Q8C22×D4714yes—
346C4×D4—24no—
347C4×D4—22no—
348C4×D4—474no—
349C4×D4—478no—
350C4×D4—944no—
351C4×D4—942no—
352C4×D4C22×D4714yes—
353C4×D4C22×D4718yes—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—28no—
358C2×C4⋊C4—471no—
359C2×C4⋊C4—472no—
360C2×C4⋊C4—942no—
361C2×C4⋊C4—941no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—948no—
364C2×C4⋊C4C22⋊D4711yescharacteristic
365C2×C4⋊C4C22⋊D4712yes—
366C2×C4⋊C4C22×D4711yescharacteristic
367C2×C4⋊C4C22×D4712yes—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—941no—
371C2×C22⋊C4—942no—
372C2×C42C22×D4712yes—
373C2×C42C22×D4711yescharacteristic
374#107—474no—
375#107—478no—
376#107—478no—
377#106—478no—
378#106—478no—
379C4⋊C8—474no—
380C4⋊C8—478no—
381#102—478no—
382#102—474no—
383#102—478no—
384#102—474no—
385C4×C8—478no—
386C4×C8—474no—
387C24—21no—
388C24—941no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—82no—
392C2×Q8—944no—
393C2×Q8—3762no—
394C2×Q8C2×C22⋊D4714yes—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—474no—
398C2×D4—472no—
399C2×D4—941no—
400C2×D4—942no—
401C2×D4Q8×D4714yes—
402C2×D4C23×D4712yes—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—44no—
408C22×C4—84no—
409C22×C4—941no—
410C22×C4—942no—
411C22×C4—948no—
412C22×C4—944no—
413C22×C4—3764no—
414C22×C4—1884no—
415C22×C4C2×C22⋊D4712yes—
416C22×C4C2×C22⋊D4711yescharacteristic
417C22×C4C2×C22⋊D4712yes—
418C22×C4C2×C22⋊D4711yescharacteristic
419C22×C4C23×D4711yescharacteristic
420D8—4716no—
421C2×C8—474no—
422C2×C8—472no—
423C2×C8—474no—
424C2×C8—472no—
425C2×C8—944no—
426C2×C8—942no—
427C4⋊C4—28no—
428C4⋊C4—24no—
429C4⋊C4—28no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—472no—
433C4⋊C4—474no—
434C4⋊C4—948no—
435C4⋊C4—948no—
436C4⋊C4—941no—
437C4⋊C4—942no—
438C4⋊C4—944no—
439C4⋊C4C2×C22⋊D4714yes—
440C4⋊C4C2×C22⋊D4718yes—
441C4⋊C4C23×D4712yes—
442C4⋊C4C23×D4714yes—
443C22⋊C4—24no—
444C22⋊C4—28no—
445C22⋊C4—944no—
446C22⋊C4—948no—
447C42C2×C22⋊D4714yes—
448C42C2×C22⋊D4712yes—
449C42C23×D4714yes—
450C42C23×D4712yes—
451C23—24no—
452C23—22no—
453C23—42no—
454C23—42no—
455C23—944no—
456C23—942no—
457C23—1882no—
458C23—1882no—
459C23C22×C22⋊D4711yescharacteristic
460Q8—28no—
461Q8—84no—
462Q8—3764no—
463D4—24no—
464D4—22no—
465D4—474no—
466D4—944no—
467D4—942no—
468D4C2×Q8×D4714yes—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—48no—
477C2×C4—88no—
478C2×C4—942no—
479C2×C4—944no—
480C2×C4—3768no—
481C2×C4—1882no—
482C2×C4—1884no—
483C2×C4—1888no—
484C2×C4#1309404414yes—
485C2×C4C22×C22⋊D4714yes—
486C2×C4C22×C22⋊D4712yes—
487C2×C4C22×C22⋊D4714yes—
488C2×C4C22×C22⋊D4711yescharacteristic
489C2×C4C22×C22⋊D4712yes—
490C2×C4C2×Q8×D4712yes—
491C2×C4#1309407212yes—
492C2×C4C24×D4711yescharacteristic
493C8—474no—
494C8—944no—
495C8—942no—
496C22—24no—
497C22—42no—
498C22—42no—
499C22—48no—
500C22—944no—
501C22—1882no—
502C22—1882no—
503C22—1888no—
504C22Q8×C22⋊D4712yes—
505C22C2×#1309404412yes—
506C22#1568992612yes—
507C22C23×C22⋊D4711yescharacteristic
508C4—24no—
509C4—24no—
510C4—22no—
511C4—42no—
512C4—44no—
513C4—1882no—
514C4—1884no—
515C4Q8×C22⋊D4712yes—
516C4C2×#1309404414yes—
517C4C22×Q8×D4711yescharacteristic
518C4C2×#1309407211yescharacteristic
519C2—44no—
520C2—1884no—
521C2#2025625114yesdirect product factor
522C2C2×Q8×C22⋊D4711yescharacteristic
523C2C22×#1309404411yescharacteristic
524C2C2×#1568992611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#554C2
socleC23×C47C22×D4
prime radicalC22×#10625334C22
hypercenterC2×#554D47
hypocenterC47C2×#3380
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#15689833C2semidirect
2. #20256251C2direct (N × H)
3. C2×#15689526C2semidirect
4. C2×#15689522C2semidirect
5. C2×#15689423C2semidirect
6. C2×#15689420C2semidirect
7. C2×#15689413C2semidirect
8. C2×#15689410C2semidirect
9. C2×Q8×C47⋊C8C2semidirect
12. C2×C47×#554C2semidirect
15. #15689833C22semidirect
23. #15689526C22semidirect
24. #15689522C22semidirect
25. #15689423C22semidirect
26. #15689420C22semidirect
27. #15689413C22semidirect
28. #15689410C22semidirect
29. Q8×C47⋊C8C22semidirect
46. C47×#554C22semidirect
59. C2×#10625334Q8semidirect
172. #10625334C2×Q8semidirect
258. C2×#554D47semidirect
270. C2×C47#3380semidirect
279. #554C2×D47semidirect
317. C47C2×#3380semidirect
521. C2#20256251direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156898332
#202562512
C2×#156895262
C2×#156895222
C2×#156894232
C2×#156894202
C2×#156894132
C2×#156894102
C2×Q8×C47⋊C82
C2×C4×#106253342
C2×C47×#5542
C2×#338047