Finite Group Databasebeta

← C2×#20254763 #27089136

Subgroups — C2×#20254763 #27089136

Order 12032.

1898 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic, complemented
2C2×#15689655C211yesmaximal, characteristic
3#20254763C2116yesmaximal, direct product factor
4C2×#15689433C212yesmaximal, complemented
5C2×#15689415C211yesmaximal, characteristic, complemented
6C2×#15689382C212yesmaximal, complemented
7C2×#15689368C212yesmaximal
8C2×#15689344C212yesmaximal, complemented
9C2×#15689334C211yesmaximal, characteristic, complemented
10C2×#15689328C211yesmaximal, characteristic, complemented
11C2×#15689145C211yesmaximal, characteristic, complemented
12C2×C47×#476C211yesmaximal, characteristic, complemented
13C22×Dic188C2212yes—
14C22×#10625335C2212yes—
15C22×#10625324C2212yes—
16#15689678—24no—
17#15689678C2218yescomplemented
18C2×#13094024C2211yescharacteristic
19#15689655—24no—
20#15689655C2218yes—
21C2×#13094017—21no—
22C2×#13094015—21no—
23C2×#13094015C2211yescharacteristic
24C2×#13094011C2211yescharacteristic
25C2×#13094006—21no—
26C2×#13094004C2212yes—
27C2×#13094004C2211yescharacteristic
28C2×#13094000—21no—
29C2×#13093995C2211yescharacteristic
30C2×Q8×Dic47—21no—
31C2×C4×Dic94C2211yescharacteristic
32C2×#13093989C2212yes—
33C2×C4×C22⋊D47—21no—
34C2×#13093983—21no—
35C2×#13093982—21no—
36C2×C42×D47C2211yescharacteristic
37#15689433C22116yescomplemented
38#15689415C2218yescomplemented
39#15689382C22116yescomplemented
40#15689368C22116yes—
41#15689344C22116yescomplemented
42#15689334C2218yescomplemented
43#15689328C2218yescomplemented
44C2×#13093957C2212yes—
45#15689145C2218yescomplemented
46C2×C4⋊(C47⋊C8)C2211yescharacteristic
47C2×Dic47⋊C8C2212yes—
48C2×#13093949C2212yes—
49C2×#13093947C2212yes—
50C2×#13093947C2212yes—
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×C4×D4×C47C2211yescharacteristic
53C47×#476C2218yescomplemented
54C2×#106×C47C2212yes—
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#102×C47C2212yes—
57C22×Dic94C2312yes—
58C22×Dic94C2311yescharacteristic
59C22×C22⋊D47—21no—
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C2×Dic188—44no—
64C2×Dic188C2318yes—
65C2×#10625335—44no—
66C2×#10625335C2318yes—
67C2×#10625324—44no—
68C2×#10625324C2318yes—
69C22×C47⋊C8C2312yes—
70#13094024—22no—
71#13094024C2314yes—
72#13094017—28no—
73#13094015—28no—
74#13094015—22no—
75#13094015C2314yes—
76#13094011—22no—
77#13094011C2314yes—
78#13094006—28no—
79#13094004—22no—
80#13094004—24no—
81#13094004C2318yes—
82#13094004C2314yes—
83#13094000—28no—
84#13093995—22no—
85#13093995C2314yes—
86Q8×Dic47—28no—
87C4×Dic94—22no—
88C4×Dic94C2314yes—
89#13093989—24no—
90#13093989C2318yes—
91C4×C22⋊D47—28no—
92#13093983—28no—
93#13093982—28no—
94C2×C4⋊Dic47—21no—
95C2×C4⋊Dic47—21no—
96C2×C4⋊Dic47—21no—
97C2×C4⋊Dic47—21no—
98C2×C4⋊Dic47—22no—
99C2×C4⋊Dic47C2311yescharacteristic
100C2×Dic47⋊C4—21no—
101C2×Dic47⋊C4—21no—
102C2×Dic47⋊C4—22no—
103C2×C22⋊Dic47—21no—
104C2×C22⋊Dic47—21no—
105C2×C22⋊Dic47—21no—
106C2×(C2×D47)⋊C4—22no—
107C2×(C2×D47)⋊C4—21no—
108C2×C4×Dic47—21no—
109C2×C4×Dic47—22no—
110C2×C4×Dic47D411yescharacteristic
111C2×C4×Dic47D411yescharacteristic
112C2×C4×Dic47C2311yescharacteristic
113C42×D47D414yes—
114C42×D47C2314yes—
115#13093957C2318yes—
116C4⋊(C47⋊C8)C2314yes—
117Dic47⋊C8C2318yes—
118#13093949C2318yes—
119#13093947C2318yes—
120#13093947C2318yes—
121C22×Q8×C47—21no—
122C22×D4×C47C2311yescharacteristic
123C23×C4×C47—21no—
124C22×C8×C47C2312yes—
125C4⋊Q8×C47—22no—
126C4⋊Q8×C47C2314yes—
127C4×D4×C47—22no—
128C4×D4×C47C2314yes—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47C2311yescharacteristic
131C2×C4⋊C4×C47C2312yes—
132C2×C22⋊C4×C47—21no—
133C2×C42×C47C2311yescharacteristic
134#106×C47C2318yes—
135C4⋊C8×C47C2314yes—
136#102×C47C2318yes—
137C24×C47—21no—
138C23×D47C2×D411yescharacteristic
139C2×Q8×C47—24no—
140C2×Q8×C47—44no—
141C2×D4×C47—21no—
142C2×D4×C47—22no—
143C2×D4×C47—22no—
144C2×D4×C47C2412yes—
145C22×C4×C47—21no—
146C22×C4×C47—22no—
147C22×C4×C47—24no—
148C22×C4×C47—44no—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2411yescharacteristic
152C2×C22⋊D47—24no—
153C2×C22⋊D47—44no—
154C22×Dic47—21no—
155C22×Dic47—22no—
156C22×Dic47—21no—
157C22×Dic47C2×D411yescharacteristic
158C22×Dic47C2×D411yescharacteristic
159C22×Dic47C2×D411yescharacteristic
160C2×C4×D47—24no—
161C2×C4×D47—22no—
162C2×C4×D47—22no—
163C2×C4×D47—42no—
164C2×C4×D47C2×D412yes—
165C2×C4×D47C2×D414yes—
166C2×C4×D47C2×D414yes—
167C2×C4×D47C2412yes—
168C2×Dic94—24no—
169C2×Dic94—22no—
170C2×Dic94—24no—
171C2×Dic94—22no—
172C2×Dic94—22no—
173C2×Dic94—21no—
174C2×Dic94C2412yes—
175C2×Dic94C2414yes—
176C2×C8×C47—24no—
177C2×C8×C47C2414yes—
178C4⋊C4×C47—24no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47C2412yes—
182C4⋊C4×C47C2414yes—
183C22⋊C4×C47—24no—
184C42×C47C2×D412yes—
185C42×C47C2412yes—
186C22⋊Dic47—24no—
187C22⋊Dic47—24no—
188C22⋊Dic47—24no—
189#10625335—48no—
190(C2×D47)⋊C4—28no—
191(C2×D47)⋊C4—24no—
192C4⋊Dic47—24no—
193C4⋊Dic47—24no—
194C4⋊Dic47—21no—
195C4⋊Dic47—24no—
196C4⋊Dic47—24no—
197C4⋊Dic47—28no—
198C4⋊Dic47C2412yes—
199Dic47⋊C4—24no—
200Dic47⋊C4—24no—
201Dic47⋊C4—28no—
202C4×Dic47—24no—
203C4×Dic47—28no—
204C4×Dic47C2×D412yes—
205C4×Dic47C2×D414yes—
206C4×Dic47C2×D414yes—
207C4×Dic47C2412yes—
208C2×C47⋊C8—24no—
209C2×C47⋊C8C2414yes—
210Dic188—48no—
211#10625324—48no—
212C23×C47—22no—
213C23×C47—42no—
214C23×C47—42no—
215C23×C47—42no—
216C23×C47C22×D411yescharacteristic
217C22×D47—22no—
218C22×D47—22no—
219C22×D47—22no—
220C22×D47C22×D412yes—
221Q8×C47—48no—
222D4×C47—22no—
223D4×C47—22no—
224D4×C47—42no—
225C2×C4×C47—22no—
226C2×C4×C47—24no—
227C2×C4×C47—22no—
228C2×C4×C47—42no—
229C2×C4×C47—44no—
230C2×C4×C47—48no—
231C2×C4×C47#13714yes—
232C2×C4×C47C22×D412yes—
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×D412yes—
235C2×C4×C47H(5,𝔽2)12yes—
236C2×C4×C472−1+412yes—
237C2×C4×C47C2511yescharacteristic
238C22⋊D47—48no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—24no—
244C2×Dic47—22no—
245C2×Dic47—44no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47C22×D412yes—
249C2×Dic47C22×D412yes—
250C2×Dic47C22×D412yes—
251C4×D47—24no—
252C4×D47—28no—
253C4×D47—22no—
254C4×D47—22no—
255C4×D47—44no—
256Dic94—22no—
257Dic94—24no—
258Dic94—22no—
259Dic94—24no—
260Dic94—44no—
261Dic94—42no—
262C8×C47—24no—
263C47⋊C8—24no—
264C2×#3252—471nomaximal, Sylow
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—84no—
269C22×C47#53712yes—
270C22×C47C2×#13712yes—
271C22×C47#57512yes—
272C22×C47C23×D411yescharacteristic
273C2×D47—22no—
274C2×D47—22no—
275C2×D47—22no—
276C2×D47—44no—
277C4×C47—22no—
278C4×C47—42no—
279C4×C47—44no—
280C4×C47#53712yes—
281C4×C47C2×#13714yes—
282C4×C47C2×H(5,𝔽2)11yescharacteristic
283C4×C47C2×2−1+411yescharacteristic
284Dic47—22no—
285Dic47—22no—
286Dic47—22no—
287Dic47—42no—
288Dic47—44no—
289Dic47—42no—
290C2×#533—471no—
291C2×#529—471no—
292#3252—4716no—
293C2×#479—472no—
294C2×#477—471no—
295C2×#476D4711yescharacteristic, complemented
296C2×#470—472no—
297C2×#468—472no—
298C2×#464—472no—
299C2×#462—471no—
300C2×#461—471no—
301C2×#423—471no—
302C2×C47—82no—
303C2×C47#325214yescomplemented
304C2×C47C2×#53711yescharacteristic
305C2×C47C22×#13711yescharacteristic
306C2×C47C2×#57511yescharacteristic
307D47—42no—
308C22×Q16—472no—
309C22×SD16—472no—
310C22×M16—472no—
311#533—478no—
312#533—944no—
313C2×C4⋊Q8—471no—
314C2×C4⋊Q8C2×D4711yescharacteristic
315#529—478no—
316#529—944no—
317C2×#126—941no—
318C2×#125—471no—
319C2×#125—941no—
320C2×#124—471no—
321C2×#123—941no—
322C2×C22⋊Q8—471no—
323C2×C22⋊Q8—472no—
324C2×C22⋊Q8—941no—
325C2×#121—471no—
326C2×C4×Q8—472no—
327C2×C4×Q8—471no—
328C2×C4×Q8—941no—
329C2×C4×D4—941no—
330C2×C4×D4C2×D4711yescharacteristic
331C2×#117—941no—
332C2×#117—941no—
333C22×C42—471no—
334#479—4716no—
335#477—478no—
336#476C2×D4718yescomplemented
337#470—4716no—
338#468—4716no—
339#464—4716no—
340#462—478no—
341#461—478no—
342C2×#107—472no—
343C2×#106C2×D4712yes—
344#423—478no—
345C2×C4⋊C8—472no—
346C2×C4⋊C8—471no—
347C2×C4⋊C8C2×D4711yescharacteristic
348C2×#103—472no—
349C2×#103—472no—
350C2×#103—472no—
351C2×#102C2×D4712yes—
352C47C2×#325211yescharacteristic, complemented, Sylow
353C22×Q8—21no—
354C22×Q8—471no—
355C22×Q8—472no—
356C22×D4—941no—
357C22×D4C22×D4711yescharacteristic
358C23×C4—21no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—471no—
362C2×Q16—478no—
363C2×Q16—1884no—
364C2×SD16—478no—
365C2×SD16—1884no—
366C2×M16—478no—
367C2×M16—1884no—
368C22×C8—472no—
369C22×C8C22×D4712yes—
370C4⋊Q8—22no—
371C4⋊Q8—474no—
372C4⋊Q8—942no—
373C4⋊Q8C22×D4714yes—
374#126—948no—
375#125—474no—
376#125—942no—
377#125—948no—
378#124—474no—
379#124—942no—
380#123—948no—
381C22⋊Q8—474no—
382C22⋊Q8—478no—
383C22⋊Q8—942no—
384C22⋊Q8—944no—
385C22⋊Q8—948no—
386#121—474no—
387#121—942no—
388C4×Q8—478no—
389C4×Q8—474no—
390C4×Q8—944no—
391C4×Q8—948no—
392C4×Q8—942no—
393C4×D4—22no—
394C4×D4—948no—
395C4×D4C22×D4714yes—
396#117—948no—
397#117—948no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4—942no—
404C2×C4⋊C4—942no—
405C2×C4⋊C4—941no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4C22×D4711yescharacteristic
409C2×C4⋊C4C22×D4712yes—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—942no—
412C2×C22⋊C4—941no—
413C2×C22⋊C4—941no—
414C2×C22⋊C4—941no—
415C2×C22⋊C4—941no—
416C2×C42—471no—
417C2×C42—471no—
418C2×C42—471no—
419C2×C42—474no—
420C2×C42—474no—
421C2×C42—942no—
422C2×C42—941no—
423C2×C42C22×D4711yescharacteristic
424#107—478no—
425#106C22×D4718yes—
426C4⋊C8—478no—
427C4⋊C8—474no—
428C4⋊C8C22×D4714yes—
429#103—478no—
430#103—478no—
431#103—478no—
432#102C22×D4718yes—
433C24—21no—
434C24—471no—
435C2×Q8—24no—
436C2×Q8—44no—
437C2×Q8—472no—
438C2×Q8—474no—
439C2×Q8—941no—
440C2×Q8—942no—
441C2×Q8—944no—
442C2×Q8—942no—
443C2×Q8—942no—
444C2×Q8—944no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—944no—
449C2×D4—1884no—
450C2×D4C23×D4712yes—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—471no—
458C22×C4—474no—
459C22×C4—472no—
460C22×C4—472no—
461C22×C4—474no—
462C22×C4—944no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—942no—
468C22×C4—1882no—
469C22×C4D4×D4711yescharacteristic
470C22×C4D4×D4711yescharacteristic
471C22×C4C23×D4711yescharacteristic
472Q16—1888no—
473SD16—1888no—
474M16—1888no—
475C2×C8—24no—
476C2×C8—474no—
477C2×C8—944no—
478C2×C8C23×D4714yes—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—472no—
483C4⋊C4—944no—
484C4⋊C4—948no—
485C4⋊C4—944no—
486C4⋊C4—944no—
487C4⋊C4—948no—
488C4⋊C4—944no—
489C4⋊C4—941no—
490C4⋊C4—944no—
491C4⋊C4—944no—
492C4⋊C4C23×D4714yes—
493C4⋊C4C23×D4712yes—
494C22⋊C4—24no—
495C22⋊C4—948no—
496C22⋊C4—944no—
497C22⋊C4—944no—
498C22⋊C4—944no—
499C22⋊C4—944no—
500C42—474no—
501C42—472no—
502C42—474no—
503C42—472no—
504C42—948no—
505C42—944no—
506C42D4×D4712yes—
507C42C23×D4712yes—
508C23—22no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—472no—
513C23—942no—
514C23—942no—
515C23—942no—
516C23C2×D4×D4711yescharacteristic
517Q8—48no—
518Q8—944no—
519Q8—942no—
520Q8—942no—
521Q8—944no—
522Q8—1882no—
523Q8—1884no—
524D4—22no—
525D4—22no—
526D4—42no—
527D4—1888no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—472no—
535C2×C4—472no—
536C2×C4—472no—
537C2×C4—942no—
538C2×C4—942no—
539C2×C4—942no—
540C2×C4—948no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—944no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—944no—
547C2×C4—1884no—
548C2×C4—1882no—
549C2×C4—1882no—
550C2×C4—1884no—
551C2×C4#1309405314yes—
552C2×C4C2×D4×D4712yes—
553C2×C4C2×D4×D4712yes—
554C2×C4C2×D4×D4711yescharacteristic
555C2×C4#1309407012yes—
556C2×C4#1309407212yes—
557C2×C4C24×D4711yescharacteristic
558C8—24no—
559C8—944no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—84no—
564C22—942no—
565C22—942no—
566C22—942no—
567C22—1884no—
568C22#1568970312yes—
569C22C2×#1309405312yes—
570C22#1568991912yes—
571C22C22×D4×D4711yescharacteristic
572C4—22no—
573C4—42no—
574C4—44no—
575C4—942no—
576C4—942no—
577C4—942no—
578C4—1882no—
579C4—1882no—
580C4—1884no—
581C4#1568970312yes—
582C4C2×#1309405314yes—
583C4C2×#1309407011yescharacteristic
584C4C2×#1309407211yescharacteristic
585C2—82no—
586C2—1882no—
587C2#2025476314yesdirect product factor
588C2C2×#1568970311yescharacteristic
589C2C22×#1309405311yescharacteristic
590C2C2×#1568991911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#476C2
socleC23×C47C22×D4
prime radicalC2×#13093995C22
hypercenterC2×#476D47
hypocenterC47C2×#3252
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#15689678C2semidirect
3. #20254763C2direct (N × H)
4. C2×#15689433C2semidirect
5. C2×#15689415C2semidirect
6. C2×#15689382C2semidirect
8. C2×#15689344C2semidirect
9. C2×#15689334C2semidirect
10. C2×#15689328C2semidirect
11. C2×#15689145C2semidirect
12. C2×C47×#476C2semidirect
17. #15689678C22semidirect
37. #15689433C22semidirect
38. #15689415C22semidirect
39. #15689382C22semidirect
41. #15689344C22semidirect
42. #15689334C22semidirect
43. #15689328C22semidirect
45. #15689145C22semidirect
53. C47×#476C22semidirect
295. C2×#476D47semidirect
303. C2×C47#3252semidirect
336. #476C2×D47semidirect
352. C47C2×#3252semidirect
587. C2#20254763direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896782
C2×#156896552
#202547632
C2×#156894332
C2×#156894152
C2×#156893822
C2×#156893682
C2×#156893442
C2×#156893342
C2×#156893282
C2×#156891452
C2×C47×#4762
C2×#325247