Finite Group Databasebeta

← C2×#20253993 #27086054

Subgroups — C2×#20253993 #27086054

Order 12032.

1986 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic, complemented
2C2×#15689658C211yesmaximal, characteristic
3#20253993C2116yesmaximal, direct product factor
4C2×#15689533C211yesmaximal, characteristic, complemented
5C2×#15689515C211yesmaximal, characteristic, complemented
6C2×#15689483C211yesmaximal, characteristic, complemented
7C2×#15689380C212yesmaximal, complemented
8C2×#15689372C212yesmaximal
9C2×#15689215C212yesmaximal, complemented
10C2×#15689208C212yesmaximal, complemented
11C2×#15689075C211yesmaximal, characteristic, complemented
12C2×C47×#491C211yesmaximal, characteristic, complemented
13C22×#10625337C2212yes—
14C22×#10625336C2212yes—
15C22×#10625324C2212yes—
16#15689678—24no—
17#15689678C2218yescomplemented
18C2×#13094024C2211yescharacteristic
19#15689658—24no—
20#15689658C2218yes—
21C2×#13094019—21no—
22C2×#13094015—21no—
23C2×#13094012C2211yescharacteristic
24C2×#13094009—21no—
25C2×#13094004—21no—
26C2×#13094004C2211yescharacteristic
27C2×#13094000C2212yes—
28C2×#13093999C2211yescharacteristic
29C2×Q8×Dic47C2212yes—
30C2×#13093989—21no—
31C2×#13093989C2211yescharacteristic
32C2×#13093986—21no—
33C2×#13093986C2211yescharacteristic
34C2×#13093982—21no—
35C2×#13093982—21no—
36C2×C42×D47C2211yescharacteristic
37#15689533C2218yescomplemented
38#15689515C2218yescomplemented
39#15689483C2218yescomplemented
40#15689380C22116yescomplemented
41#15689372C22116yes—
42#15689215C22116yescomplemented
43#15689208C22116yescomplemented
44C2×#13093957C2212yes—
45C2×#13093955C2212yes—
46C2×#13093949C2212yes—
47C2×#13093946C2212yes—
48#15689075C2218yescomplemented
49C2×#13093935C2211yescharacteristic
50C2×C8×Dic47C2212yes—
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×#125×C47C2211yescharacteristic
53C47×#491C2218yescomplemented
54C2×#103×C47C2212yes—
55C2×#103×C47C2212yes—
56C2×#97×C47C2211yescharacteristic
57C22×Dic94—21no—
58C22×Dic94C2311yescharacteristic
59C22×D188—21no—
60C22×D188C2311yescharacteristic
61C22×C4×D47—21no—
62C22×C4×D47D411yescharacteristic
63C22×C4×D47D411yescharacteristic
64C22×C4×D47C2311yescharacteristic
65C2×#10625337—28no—
66C2×#10625337C2318yes—
67C2×#10625336—28no—
68C2×#10625336C2318yes—
69C2×#10625324—28no—
70C2×#10625324C2318yes—
71C22×C47⋊C8C2312yes—
72#13094024—22no—
73#13094024C2314yes—
74#13094019—28no—
75#13094015—28no—
76#13094012—22no—
77#13094012C2314yes—
78#13094009—28no—
79#13094004—28no—
80#13094004—22no—
81#13094004C2314yes—
82#13094000—24no—
83#13094000C2318yes—
84#13093999—22no—
85#13093999C2314yes—
86Q8×Dic47—24no—
87Q8×Dic47C2318yes—
88#13093989—28no—
89#13093989—22no—
90#13093989C2314yes—
91#13093986—28no—
92#13093986—22no—
93#13093986C2314yes—
94#13093982—28no—
95#13093982—28no—
96C2×C4⋊Dic47—21no—
97C2×C4⋊Dic47—21no—
98C2×C4⋊Dic47—22no—
99C2×C4⋊Dic47C2312yes—
100C2×Dic47⋊C4—22no—
101C2×Dic47⋊C4—21no—
102C2×(C2×D47)⋊C4—22no—
103C2×(C2×D47)⋊C4—21no—
104C2×(C2×D47)⋊C4—21no—
105C2×(C2×D47)⋊C4—21no—
106C2×(C2×D47)⋊C4—21no—
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—
116#13093955C2318yes—
117#13093949C2318yes—
118#13093946C2318yes—
119#13093935C2314yes—
120C8×Dic47C2318yes—
121C22×Q8×C47C2312yes—
122C22×C8×C47C2312yes—
123C4⋊Q8×C47—22no—
124C4⋊Q8×C47C2314yes—
125#125×C47—22no—
126#125×C47C2314yes—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47C2311yescharacteristic
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C42×C47C2311yescharacteristic
134#103×C47C2318yes—
135#103×C47C2318yes—
136#97×C47C2314yes—
137C23×D47—21no—
138C23×D47C2×D411yescharacteristic
139C2×Q8×C47—22no—
140C2×Q8×C47—24no—
141C2×Q8×C47—24no—
142C2×Q8×C47C2414yes—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—22no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C22×Dic47—21no—
150C22×Dic47—22no—
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C2×D188—24no—
155C2×D188—24no—
156C2×D188—21no—
157C2×D188—44no—
158C2×D188C2412yes—
159C2×C4×D47—24no—
160C2×C4×D47—24no—
161C2×C4×D47—24no—
162C2×C4×D47—42no—
163C2×C4×D47—44no—
164C2×C4×D47C2×D412yes—
165C2×C4×D47C2×D414yes—
166C2×C4×D47C2×D414yes—
167C2×C4×D47C2412yes—
168C2×Dic94—24no—
169C2×Dic94—21no—
170C2×Dic94—24no—
171C2×Dic94—44no—
172C2×Dic94C2412yes—
173C2×C8×C47—24no—
174C2×C8×C47C2414yes—
175C4⋊C4×C47—24no—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—24no—
178C4⋊C4×C47—24no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47C2412yes—
182C4⋊C4×C47C2412yes—
183C42×C47C2×D412yes—
184C42×C47C2412yes—
185#10625337—216no—
186#10625336—216no—
187(C2×D47)⋊C4—28no—
188(C2×D47)⋊C4—24no—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—24no—
191(C2×D47)⋊C4—24no—
192(C2×D47)⋊C4—24no—
193C4⋊Dic47—24no—
194C4⋊Dic47—22no—
195C4⋊Dic47—24no—
196C4⋊Dic47—28no—
197C4⋊Dic47C2414yes—
198Dic47⋊C4—28no—
199Dic47⋊C4—24no—
200C4×Dic47—24no—
201C4×Dic47—28no—
202C4×Dic47C2×D412yes—
203C4×Dic47C2×D414yes—
204C4×Dic47C2×D414yes—
205C4×Dic47C2412yes—
206C2×C47⋊C8#4118yes—
207C2×C47⋊C8C2414yes—
208#10625324—216no—
209C23×C47C22×D411yescharacteristic
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—22no—
214C22×D47—42no—
215C22×D47—42no—
216C22×D47—42no—
217C22×D47C22×D412yes—
218Q8×C47—24no—
219Q8×C47—24no—
220Q8×C47—44no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—24no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—44no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47C22×D412yes—
230C2×C4×C47C22×D412yes—
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C47C2×#4114yes—
233C2×C4×C47C2511yescharacteristic
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47—22no—
238C2×Dic47—42no—
239C2×Dic47—44no—
240C2×Dic47#13714yes—
241C2×Dic47C22×D412yes—
242C2×Dic47C22×D412yes—
243C2×Dic47C22×D412yes—
244D188—24no—
245D188—48no—
246D188—42no—
247C4×D47—24no—
248C4×D47—24no—
249C4×D47—28no—
250C4×D47—44no—
251C4×D47—48no—
252Dic94—24no—
253Dic94—48no—
254Dic94—42no—
255C8×C47—24no—
256C47⋊C8C2×#4118yes—
257C2×#3172—471nomaximal, Sylow
258C22×C47#53212yes—
259C22×C47C2×#13712yes—
260C22×C47#57512yes—
261C22×C47C23×D411yescharacteristic
262C2×D47—22no—
263C2×D47—22no—
264C2×D47—22no—
265C2×D47—44no—
266C2×D47—42no—
267C2×D47—42no—
268C2×D47—42no—
269C2×D47—84no—
270C4×C47—22no—
271C4×C47—22no—
272C4×C47—44no—
273C4×C47—42no—
274C4×C47—42no—
275C4×C47#53212yes—
276C4×C47C22×#4112yes—
277Dic47—22no—
278Dic47—22no—
279Dic47—42no—
280Dic47—44no—
281Dic47C2×#13714yes—
282C2×#533—471no—
283C2×#529—471no—
284#3172—4716no—
285C2×#501—471no—
286C2×#497—471no—
287C2×#491D4711yescharacteristic, complemented
288C2×#489—471no—
289C2×#470—472no—
290C2×#468—472no—
291C2×C4×Q16—472no—
292C2×C4×SD16—472no—
293C2×C4×M16—471no—
294C2×C47#317214yescomplemented
295C2×C47C2×#53211yescharacteristic
296C2×C47C22×#13711yescharacteristic
297C2×C47C2×#57511yescharacteristic
298D47—42no—
299D47—82no—
300C22×Q16—472no—
301C22×SD16—472no—
302C22×M16—472no—
303#533—478no—
304#533—944no—
305C2×C4⋊Q8—471no—
306C2×C4⋊Q8C2×D4711yescharacteristic
307#529—478no—
308#529—944no—
309C2×#126—941no—
310C2×#125—941no—
311C2×#125C2×D4711yescharacteristic
312C2×#124—471no—
313C2×#123—941no—
314C2×C22⋊Q8—472no—
315C2×C22⋊Q8—471no—
316C2×C22⋊Q8—941no—
317C2×#121—471no—
318C2×C4×Q8—472no—
319C2×C4×Q8—471no—
320C2×C4×Q8—941no—
321C2×C4×D4—471no—
322C2×C4×D4—941no—
323C2×#117—941no—
324C2×#117—941no—
325C22×C42—471no—
326#501—478no—
327#497—478no—
328#491C2×D4718yescomplemented
329#489—478no—
330#470—4716no—
331#468—4716no—
332C4×Q16—4716no—
333C4×SD16—4716no—
334C2×#107—472no—
335C2×#106—472no—
336C2×#103—472no—
337C2×#103C2×D4712yes—
338C2×#103C2×D4712yes—
339C2×#102—472no—
340C4×M16—478no—
341C2×#97—471no—
342C2×#97C2×D4711yescharacteristic
343C2×C4×C8—472no—
344C47C2×#317211yescharacteristic, complemented, Sylow
345C22×Q8—471no—
346C22×Q8—941no—
347C22×Q8C22×D4712yes—
348C22×D4—471no—
349C22×D4—941no—
350C23×C4—471no—
351C23×C4—471no—
352C23×C4—471no—
353C23×C4—941no—
354C2×Q16—478no—
355C2×Q16—948no—
356C2×SD16—478no—
357C2×SD16—948no—
358C2×M16—478no—
359C2×M16—948no—
360C22×C8—472no—
361C22×C8C22×D4712yes—
362C4⋊Q8—22no—
363C4⋊Q8—474no—
364C4⋊Q8—942no—
365C4⋊Q8C22×D4714yes—
366#126—948no—
367#125—22no—
368#125—948no—
369#125C22×D4714yes—
370#124—474no—
371#124—942no—
372#123—948no—
373C22⋊Q8—478no—
374C22⋊Q8—474no—
375C22⋊Q8—942no—
376C22⋊Q8—948no—
377C22⋊Q8—944no—
378#121—474no—
379#121—942no—
380C4×Q8—478no—
381C4×Q8—474no—
382C4×Q8—942no—
383C4×Q8—948no—
384C4×Q8—944no—
385C4×D4—474no—
386C4×D4—942no—
387C4×D4—948no—
388#117—948no—
389#117—948no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—472no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—942no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—942no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4C22×D4711yescharacteristic
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C22⋊C4—941no—
403C2×C22⋊C4—941no—
404C2×C22⋊C4—942no—
405C2×C22⋊C4—941no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4—941no—
408C2×C42—471no—
409C2×C42—471no—
410C2×C42—471no—
411C2×C42—474no—
412C2×C42—474no—
413C2×C42—941no—
414C2×C42—942no—
415C2×C42C22×D4711yescharacteristic
416#107—478no—
417#106—478no—
418#103—478no—
419#103C22×D4718yes—
420#103C22×D4718yes—
421#102—478no—
422#97—474no—
423#97C22×D4714yes—
424C4×C8—478no—
425C24—471no—
426C24—941no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—472no—
431C2×Q8—941no—
432C2×Q8—944no—
433C2×Q8—944no—
434C2×Q8—1884no—
435C2×Q8C23×D4714yes—
436C2×D4—472no—
437C2×D4—944no—
438C2×D4—941no—
439C2×D4—944no—
440C2×D4—1884no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—471no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—474no—
448C22×C4—472no—
449C22×C4—472no—
450C22×C4—474no—
451C22×C4—944no—
452C22×C4—944no—
453C22×C4—944no—
454C22×C4—941no—
455C22×C4—942no—
456C22×C4—1882no—
457C22×C4—1884no—
458C22×C4D4×D4711yescharacteristic
459C22×C4D4×D4711yescharacteristic
460C22×C4C23×D4711yescharacteristic
461Q16—9416no—
462SD16—9416no—
463M16—9416no—
464C2×C8—24no—
465C2×C8—478no—
466C2×C8—474no—
467C2×C8C23×D4714yes—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—474no—
475C4⋊C4—944no—
476C4⋊C4—948no—
477C4⋊C4—944no—
478C4⋊C4—948no—
479C4⋊C4—942no—
480C4⋊C4—944no—
481C4⋊C4C23×D4712yes—
482C4⋊C4C23×D4712yes—
483C22⋊C4—944no—
484C22⋊C4—944no—
485C22⋊C4—948no—
486C22⋊C4—944no—
487C22⋊C4—944no—
488C22⋊C4—944no—
489C42—474no—
490C42—472no—
491C42—474no—
492C42—472no—
493C42—944no—
494C42—948no—
495C42D4×D4712yes—
496C42C23×D4712yes—
497C23—472no—
498C23—942no—
499C23—942no—
500C23—942no—
501C23—942no—
502C23—1882no—
503C23—1882no—
504C23—1882no—
505C23C2×D4×D4711yescharacteristic
506Q8—24no—
507Q8—24no—
508Q8—44no—
509Q8—944no—
510Q8—1888no—
511Q8—1882no—
512D4—944no—
513D4—1888no—
514D4—1882no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—474no—
524C2×C4—472no—
525C2×C4—472no—
526C2×C4—472no—
527C2×C4—948no—
528C2×C4—942no—
529C2×C4—942no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—942no—
533C2×C4—944no—
534C2×C4—1884no—
535C2×C4—1888no—
536C2×C4—1884no—
537C2×C4—1882no—
538C2×C4C2×D4×D4712yes—
539C2×C4C2×D4×D4712yes—
540C2×C4C2×D4×D4711yescharacteristic
541C2×C4#41×D4714yes—
542C2×C4C24×D4711yescharacteristic
543C8—24no—
544C8—478no—
545C22—942no—
546C22—942no—
547C22—942no—
548C22—3764no—
549C22—1882no—
550C22—1882no—
551C22—1882no—
552C22—1884no—
553C22#1568967212yes—
554C22#137×D4712yes—
555C22#1568992212yes—
556C22C22×D4×D4711yescharacteristic
557C4—22no—
558C4—22no—
559C4—42no—
560C4—42no—
561C4—44no—
562C4—474no—
563C4—942no—
564C4—942no—
565C4—1884no—
566C4—1882no—
567C4#1568967212yes—
568C4C2×#41×D4712yes—
569C2—3762no—
570C2—1882no—
571C2#2025399314yesdirect product factor
572C2C2×#1568967211yescharacteristic
573C2C2×#137×D4711yescharacteristic
574C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#491C2
socleC23×C47C22×D4
prime radicalC2×#13093999C22
hypercenterC2×#491D47
hypocenterC47C2×#3172
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
1. C2×#15689678C2semidirect
3. #20253993C2direct (N × H)
4. C2×#15689533C2semidirect
5. C2×#15689515C2semidirect
6. C2×#15689483C2semidirect
7. C2×#15689380C2semidirect
9. C2×#15689215C2semidirect
10. C2×#15689208C2semidirect
11. C2×#15689075C2semidirect
12. C2×C47×#491C2semidirect
17. #15689678C22semidirect
37. #15689533C22semidirect
38. #15689515C22semidirect
39. #15689483C22semidirect
40. #15689380C22semidirect
42. #15689215C22semidirect
43. #15689208C22semidirect
48. #15689075C22semidirect
53. C47×#491C22semidirect
287. C2×#491D47semidirect
294. C2×C47#3172semidirect
328. #491C2×D47semidirect
344. C47C2×#3172semidirect
571. C2#20253993direct (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×#156896582
#202539932
C2×#156895332
C2×#156895152
C2×#156894832
C2×#156893802
C2×#156893722
C2×#156892152
C2×#156892082
C2×#156890752
C2×C47×#4912
C2×#317247