Finite Group Databasebeta

← C2×#20254764 #27089137

Subgroups — C2×#20254764 #27089137

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×#15689660C211yesmaximal, characteristic
3#20254764C2116yesmaximal, direct product factor
4C2×#15689429C212yesmaximal, complemented
5C2×#15689410C211yesmaximal, characteristic, complemented
6C2×#15689380C212yesmaximal, complemented
7C2×#15689370C212yesmaximal
8C2×#15689348C212yesmaximal, complemented
9C2×#15689333C211yesmaximal, characteristic, complemented
10C2×#15689329C211yesmaximal, characteristic, complemented
11C2×#15689145C211yesmaximal, characteristic, complemented
12C2×C47×#477C211yesmaximal, characteristic, complemented
13C22×#10625337C2212yes—
14C22×#10625325C2212yes—
15C22×#10625324C2212yes—
16#15689678—24no—
17#15689678C2218yescomplemented
18C2×#13094024C2211yescharacteristic
19#15689660—24no—
20#15689660C2218yes—
21C2×#13094019—21no—
22C2×#13094015—21no—
23C2×#13094015C2211yescharacteristic
24C2×#13094013C2211yescharacteristic
25C2×#13094009—21no—
26C2×#13094004C2212yes—
27C2×#13094000—21no—
28C2×#13094000C2211yescharacteristic
29C2×#13093999C2211yescharacteristic
30C2×Q8×Dic47—21no—
31C2×#13093989C2212yes—
32C2×C4×D188—21no—
33C2×C4×D188C2211yescharacteristic
34C2×#13093982—21no—
35C2×#13093981—21no—
36C2×C42×D47C2211yescharacteristic
37#15689429C22116yescomplemented
38#15689410C2218yescomplemented
39#15689380C22116yescomplemented
40#15689370C22116yes—
41#15689348C22116yescomplemented
42#15689333C2218yescomplemented
43#15689329C2218yescomplemented
44C2×#13093955C2212yes—
45#15689145C2218yescomplemented
46C2×C4⋊(C47⋊C8)C2211yescharacteristic
47C2×Dic47⋊C8C2212yes—
48C2×#13093949C2212yes—
49C2×#13093947C2212yes—
50C2×#13093944C2212yes—
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C47×#477C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#103×C47C2212yes—
57C22×Dic94C2312yes—
58C22×D188—21no—
59C22×D188C2311yescharacteristic
60C22×C4×D47—21no—
61C22×C4×D47D411yescharacteristic
62C22×C4×D47D411yescharacteristic
63C22×C4×D47C2311yescharacteristic
64C2×#10625337—44no—
65C2×#10625337C2318yes—
66C2×#10625325—44no—
67C2×#10625325C2318yes—
68C2×#10625324—44no—
69C2×#10625324C2318yes—
70C22×C47⋊C8C2312yes—
71#13094024—22no—
72#13094024C2314yes—
73#13094019—28no—
74#13094015—28no—
75#13094015—22no—
76#13094015C2314yes—
77#13094013—22no—
78#13094013C2314yes—
79#13094009—28no—
80#13094004—24no—
81#13094004C2318yes—
82#13094000—28no—
83#13094000—22no—
84#13094000C2314yes—
85#13093999—22no—
86#13093999C2314yes—
87Q8×Dic47—28no—
88#13093989—24no—
89#13093989C2318yes—
90C4×D188—28no—
91C4×D188—22no—
92C4×D188C2314yes—
93#13093982—28no—
94#13093981—28no—
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×(C2×D47)⋊C4—22no—
104C2×(C2×D47)⋊C4—21no—
105C2×(C2×D47)⋊C4—21no—
106C2×(C2×D47)⋊C4—21no—
107C2×(C2×D47)⋊C4—21no—
108C2×(C2×D47)⋊C4—21no—
109C2×C4×Dic47—22no—
110C2×C4×Dic47D411yescharacteristic
111C2×C4×Dic47D411yescharacteristic
112C2×C4×Dic47C2311yescharacteristic
113C42×D47D414yes—
114C42×D47C2314yes—
115#13093955C2318yes—
116C4⋊(C47⋊C8)C2314yes—
117Dic47⋊C8C2318yes—
118#13093949C2318yes—
119#13093947C2318yes—
120#13093944C2318yes—
121C22×Q8×C47—21no—
122C22×Q8×C47C2311yescharacteristic
123C22×C8×C47C2312yes—
124C4⋊Q8×C47—22no—
125C4⋊Q8×C47C2314yes—
126C4×Q8×C47—22no—
127C4×Q8×C47C2314yes—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47C2311yescharacteristic
131C2×C4⋊C4×C47C2312yes—
132C2×C42×C47—21no—
133C2×C42×C47C2311yescharacteristic
134#107×C47C2318yes—
135C4⋊C8×C47C2314yes—
136#103×C47C2318yes—
137C23×D47—21no—
138C23×D47C2×D411yescharacteristic
139C2×Q8×C47—21no—
140C2×Q8×C47—24no—
141C2×Q8×C47—22no—
142C2×Q8×C47—22no—
143C2×Q8×C47—44no—
144C2×Q8×C47C2412yes—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—22no—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2411yescharacteristic
151C22×Dic47—22no—
152C22×Dic47—21no—
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
155C22×Dic47C2×D411yescharacteristic
156C2×D188—22no—
157C2×D188—22no—
158C2×D188—24no—
159C2×D188—21no—
160C2×D188—44no—
161C2×D188C2412yes—
162C2×C4×D47—24no—
163C2×C4×D47—22no—
164C2×C4×D47—22no—
165C2×C4×D47—24no—
166C2×C4×D47—42no—
167C2×C4×D47—44no—
168C2×C4×D47C2×D412yes—
169C2×C4×D47C2×D414yes—
170C2×C4×D47C2×D414yes—
171C2×C4×D47C2412yes—
172C2×Dic94—24no—
173C2×Dic94—24no—
174C2×Dic94—22no—
175C2×Dic94C2414yes—
176C2×C8×C47—24no—
177C2×C8×C47C2414yes—
178C4⋊C4×C47—24no—
179C4⋊C4×C47—24no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—22no—
182C4⋊C4×C47C2414yes—
183C4⋊C4×C47C2412yes—
184C42×C47—24no—
185C42×C47C2×D412yes—
186C42×C47C2412yes—
187#10625337—48no—
188(C2×D47)⋊C4—28no—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—24no—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193(C2×D47)⋊C4—24no—
194C4⋊Dic47—24no—
195C4⋊Dic47—24no—
196C4⋊Dic47—21no—
197C4⋊Dic47—24no—
198C4⋊Dic47—28no—
199C4⋊Dic47C2412yes—
200Dic47⋊C4—24no—
201Dic47⋊C4—24no—
202Dic47⋊C4—28no—
203C4×Dic47—28no—
204C4×Dic47C2×D412yes—
205C4×Dic47C2×D414yes—
206C4×Dic47C2×D414yes—
207C4×Dic47C2412yes—
208C2×C47⋊C8—24no—
209C2×C47⋊C8C2414yes—
210#10625325—48no—
211#10625324—48no—
212C23×C47C22×D411yescharacteristic
213C22×D47—22no—
214C22×D47—22no—
215C22×D47—22no—
216C22×D47—22no—
217C22×D47—42no—
218C22×D47—42no—
219C22×D47—42no—
220C22×D47C22×D412yes—
221Q8×C47—22no—
222Q8×C47—22no—
223Q8×C47—48no—
224Q8×C47—42no—
225C2×C4×C47—22no—
226C2×C4×C47—24no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—44no—
232C2×C4×C47#13714yes—
233C2×C4×C47C22×D412yes—
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×D412yes—
236C2×C4×C47H(5,𝔽2)12yes—
237C2×C4×C472−1+412yes—
238C2×C4×C47C2511yescharacteristic
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—24no—
243C2×Dic47—22no—
244C2×Dic47—44no—
245C2×Dic47—42no—
246C2×Dic47C22×D412yes—
247C2×Dic47C22×D412yes—
248C2×Dic47C22×D412yes—
249D188—22no—
250D188—22no—
251D188—48no—
252D188—42no—
253C4×D47—24no—
254C4×D47—28no—
255C4×D47—22no—
256C4×D47—22no—
257C4×D47—44no—
258C4×D47—48no—
259Dic94—24no—
260Dic94—24no—
261Dic94—44no—
262C8×C47—24no—
263C47⋊C8—24no—
264C2×#3252—471nomaximal, Sylow
265C22×C47#53712yes—
266C22×C47C2×#13712yes—
267C22×C47#57512yes—
268C22×C47C23×D411yescharacteristic
269C2×D47—22no—
270C2×D47—22no—
271C2×D47—22no—
272C2×D47—44no—
273C2×D47—42no—
274C2×D47—42no—
275C2×D47—42no—
276C2×D47—84no—
277C4×C47—22no—
278C4×C47—42no—
279C4×C47—42no—
280C4×C47—44no—
281C4×C47#53712yes—
282C4×C47C2×#13714yes—
283C4×C47C2×H(5,𝔽2)11yescharacteristic
284C4×C47C2×2−1+411yescharacteristic
285Dic47—22no—
286Dic47—22no—
287Dic47—22no—
288Dic47—44no—
289Dic47—42no—
290C2×#533—471no—
291C2×#529—471no—
292#3252—4716no—
293C2×#479—472no—
294C2×#477D4711yescharacteristic, complemented
295C2×#476—471no—
296C2×#470—472no—
297C2×#468—472no—
298C2×#464—472no—
299C2×#462—471no—
300C2×#461—471no—
301C2×#423—471no—
302C2×C47#325214yescomplemented
303C2×C47C2×#53711yescharacteristic
304C2×C47C22×#13711yescharacteristic
305C2×C47C2×#57511yescharacteristic
306D47—42no—
307D47—82no—
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—941no—
328C2×C4×Q8C2×D4711yescharacteristic
329C2×C4×D4—471no—
330C2×C4×D4—941no—
331C2×#117—941no—
332C2×#117—941no—
333C22×C42—471no—
334#479—4716no—
335#477C2×D4718yescomplemented
336#476—478no—
337#470—4716no—
338#468—4716no—
339#464—4716no—
340#462—478no—
341#461—478no—
342C2×#107C2×D4712yes—
343C2×#106—472no—
344#423—478no—
345C2×C4⋊C8—472no—
346C2×C4⋊C8—471no—
347C2×C4⋊C8C2×D4711yescharacteristic
348C2×#103—472no—
349C2×#103—472no—
350C2×#103C2×D4712yes—
351C2×#102—472no—
352C47C2×#325211yescharacteristic, complemented, Sylow
353C22×Q8—21no—
354C22×Q8—472no—
355C22×Q8C22×D4711yescharacteristic
356C22×D4—471no—
357C22×D4—941no—
358C23×C4—471no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—941no—
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—944no—
384C22⋊Q8—942no—
385C22⋊Q8—948no—
386#121—474no—
387#121—942no—
388C4×Q8—22no—
389C4×Q8—478no—
390C4×Q8—944no—
391C4×Q8—948no—
392C4×Q8C22×D4714yes—
393C4×D4—474no—
394C4×D4—948no—
395C4×D4—942no—
396#117—948no—
397#117—948no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—471no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4—941no—
404C2×C4⋊C4—942no—
405C2×C4⋊C4—942no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4C22×D4712yes—
409C2×C4⋊C4C22×D4711yescharacteristic
410C2×C22⋊C4—941no—
411C2×C22⋊C4—941no—
412C2×C22⋊C4—941no—
413C2×C22⋊C4—942no—
414C2×C22⋊C4—941no—
415C2×C22⋊C4—941no—
416C2×C42—21no—
417C2×C42—471no—
418C2×C42—471no—
419C2×C42—471no—
420C2×C42—474no—
421C2×C42—474no—
422C2×C42—942no—
423C2×C42C22×D4711yescharacteristic
424#107C22×D4718yes—
425#106—478no—
426C4⋊C8—478no—
427C4⋊C8—474no—
428C4⋊C8C22×D4714yes—
429#103—478no—
430#103—478no—
431#103C22×D4718yes—
432#102—478no—
433C24—471no—
434C24—941no—
435C2×Q8—21no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—474no—
441C2×Q8—942no—
442C2×Q8—944no—
443C2×Q8—944no—
444C2×Q8C23×D4712yes—
445C2×D4—472no—
446C2×D4—944no—
447C2×D4—941no—
448C2×D4—942no—
449C2×D4—942no—
450C2×D4—1884no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—474no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—474no—
461C22×C4—944no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—941no—
466C22×C4—944no—
467C22×C4—1882no—
468C22×C4—1884no—
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—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—472no—
484C4⋊C4—944no—
485C4⋊C4—944no—
486C4⋊C4—948no—
487C4⋊C4—944no—
488C4⋊C4—948no—
489C4⋊C4—941no—
490C4⋊C4—944no—
491C4⋊C4—944no—
492C4⋊C4C23×D4714yes—
493C4⋊C4C23×D4712yes—
494C22⋊C4—944no—
495C22⋊C4—944no—
496C22⋊C4—944no—
497C22⋊C4—948no—
498C22⋊C4—944no—
499C22⋊C4—944no—
500C42—24no—
501C42—474no—
502C42—472no—
503C42—474no—
504C42—472no—
505C42—948no—
506C42D4×D4712yes—
507C42C23×D4712yes—
508C23—472no—
509C23—942no—
510C23—942no—
511C23—942no—
512C23—942no—
513C23—1882no—
514C23—1882no—
515C23—1882no—
516C23C2×D4×D4711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—48no—
520Q8—42no—
521Q8—944no—
522Q8—944no—
523Q8—1884no—
524D4—942no—
525D4—942no—
526D4—1888no—
527D4—1882no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—472no—
536C2×C4—472no—
537C2×C4—472no—
538C2×C4—942no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—948no—
542C2×C4—942no—
543C2×C4—942no—
544C2×C4—944no—
545C2×C4—944no—
546C2×C4—942no—
547C2×C4—1884no—
548C2×C4—1882no—
549C2×C4—1884no—
550C2×C4—1888no—
551C2×C4#1309405614yes—
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—942no—
561C22—942no—
562C22—942no—
563C22—3764no—
564C22—1882no—
565C22—1882no—
566C22—1882no—
567C22—1884no—
568C22#1568970312yes—
569C22C2×#1309405612yes—
570C22#1568991912yes—
571C22C22×D4×D4711yescharacteristic
572C4—22no—
573C4—42no—
574C4—44no—
575C4—42no—
576C4—942no—
577C4—942no—
578C4—942no—
579C4—1882no—
580C4—1884no—
581C4#1568970312yes—
582C4C2×#1309405614yes—
583C4C2×#1309407011yescharacteristic
584C4C2×#1309407211yescharacteristic
585C2—3762no—
586C2—1882no—
587C2#2025476414yesdirect product factor
588C2C2×#1568970311yescharacteristic
589C2C22×#1309405611yescharacteristic
590C2C2×#1568991911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#477C2
socleC23×C47C22×D4
prime radicalC2×#13093999C22
hypercenterC2×#477D47
hypocenterC47C2×#3252
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×#15689678C2semidirect
3. #20254764C2direct (N × H)
4. C2×#15689429C2semidirect
5. C2×#15689410C2semidirect
6. C2×#15689380C2semidirect
8. C2×#15689348C2semidirect
9. C2×#15689333C2semidirect
10. C2×#15689329C2semidirect
11. C2×#15689145C2semidirect
12. C2×C47×#477C2semidirect
17. #15689678C22semidirect
37. #15689429C22semidirect
38. #15689410C22semidirect
39. #15689380C22semidirect
41. #15689348C22semidirect
42. #15689333C22semidirect
43. #15689329C22semidirect
45. #15689145C22semidirect
53. C47×#477C22semidirect
294. C2×#477D47semidirect
302. C2×C47#3252semidirect
335. #477C2×D47semidirect
352. C47C2×#3252semidirect
587. C2#20254764direct (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×#156896602
#202547642
C2×#156894292
C2×#156894102
C2×#156893802
C2×#156893702
C2×#156893482
C2×#156893332
C2×#156893292
C2×#156891452
C2×C47×#4772
C2×#325247