Finite Group Databasebeta

← C2×#20254563 #27088352

Subgroups — C2×#20254563 #27088352

Order 12032.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689680C211yesmaximal, characteristic, complemented
2#20254563C2116yesmaximal, direct product factor
3C2×#15689522C212yesmaximal, complemented
4C2×#15689477C212yesmaximal, complemented
5C2×#15689461C212yesmaximal, complemented
6C2×#15689444C212yesmaximal
7C2×#15689418C211yesmaximal, characteristic, complemented
8C2×#15689408C211yesmaximal, characteristic, complemented
9C2×#15689413C211yesmaximal, characteristic, complemented
10C2×#15689403C211yesmaximal, characteristic, complemented
11C2×#15689176C211yesmaximal, characteristic, complemented
12C2×C47×#529C211yesmaximal, characteristic
13#15689680—24no—
14#15689680C2218yescomplemented
15C2×#13094024C2211yescharacteristic
16C2×#13094024C2211yescharacteristic
17C2×#13094016C2212yes—
18C2×#13094003—21no—
19C2×#13094003—21no—
20C2×C4×Dic94—21no—
21C2×C4×Dic94—21no—
22C2×#13093983C2212yes—
23#15689522C22116yescomplemented
24#15689477C22116yescomplemented
25#15689461C22116yescomplemented
26#15689444C22116yes—
27#15689418C2218yescomplemented
28#15689413C2218yescomplemented
29#15689408C2218yescomplemented
30#15689403C2218yescomplemented
31#15689176C2218yescomplemented
32C2×#13093957C2212yes—
33C2×#13093957C2212yes—
34C2×#13093955C2212yes—
35C2×#13093955C2212yes—
36C2×C4⋊(C47⋊C8)C2211yescharacteristic
37C2×C4⋊(C47⋊C8)C2211yescharacteristic
38C2×#13093948C2212yes—
39C2×#13093948C2212yes—
40C2×#13093945C2212yes—
41C2×#13093945C2212yes—
42C2×C22⋊(C47⋊C8)C2212yes—
43C2×C4×C47⋊C8C2212yes—
44C47×#529—24no—
45C47×#529C2218yes—
46C2×#126×C47—21no—
47C2×#125×C47C2211yescharacteristic
48C2×#124×C47C2211yescharacteristic
49C2×#123×C47—21no—
50C2×C22⋊Q8×C47C2211yescharacteristic
51C2×#121×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×D4×C47—21no—
54C2×C4×D4×C47C2211yescharacteristic
55C2×#117×C47—21no—
56C22×C42×C47C2211yescharacteristic
57C22×Dic94—21no—
58C22×Dic94—21no—
59C22×C47⋊C8C2312yes—
60C22×C47⋊C8C2312yes—
61#13094024—22no—
62#13094024—22no—
63#13094024C2314yes—
64#13094024C2314yes—
65#13094016—24no—
66#13094016C2318yes—
67#13094003—28no—
68#13094003—28no—
69C4×Dic94—28no—
70C4×Dic94—28no—
71#13093983—24no—
72#13093983C2318yes—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47C2312yes—
76C2×C4⋊Dic47C2312yes—
77C2×Dic47⋊C4—22no—
78C2×Dic47⋊C4—22no—
79C2×C22⋊Dic47—22no—
80C2×C4×Dic47—22no—
81#13093957C2318yes—
82#13093957C2318yes—
83#13093955C2318yes—
84#13093955C2318yes—
85C4⋊(C47⋊C8)C2314yes—
86C4⋊(C47⋊C8)C2314yes—
87#13093948C2318yes—
88#13093948C2318yes—
89#13093945C2318yes—
90#13093945C2318yes—
91C22⋊(C47⋊C8)C2318yes—
92C4×C47⋊C8C2318yes—
93C22×Q8×C47C2311yescharacteristic
94C22×D4×C47—21no—
95C22×D4×C47C2311yescharacteristic
96C23×C4×C47—21no—
97C23×C4×C47D411yescharacteristic
98C23×C4×C47D411yescharacteristic
99C23×C4×C47C2311yescharacteristic
100#126×C47—28no—
101#125×C47—22no—
102#125×C47C2314yes—
103#124×C47—22no—
104#124×C47C2314yes—
105#123×C47—28no—
106C22⋊Q8×C47—22no—
107C22⋊Q8×C47C2314yes—
108#121×C47—22no—
109#121×C47C2314yes—
110C4×Q8×C47—22no—
111C4×Q8×C47C2314yes—
112C4×D4×C47—28no—
113C4×D4×C47—22no—
114C4×D4×C47C2314yes—
115#117×C47—28no—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47—21no—
119C2×C4⋊C4×C47—21no—
120C2×C4⋊C4×C47C2311yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C22⋊C4×C47—21no—
123C2×C22⋊C4×C47—21no—
124C2×C22⋊C4×C47—21no—
125C2×C22⋊C4×C47—21no—
126C2×C22⋊C4×C47—21no—
127C2×C42×C47—21no—
128C2×C42×C47D411yescharacteristic
129C2×C42×C47D411yescharacteristic
130C2×C42×C47D414yes—
131C2×C42×C47C2311yescharacteristic
132C2×C42×C47C2311yescharacteristic
133C2×C42×C47C2314yes—
134C24×C47—21no—
135C24×C47C2×D411yescharacteristic
136C2×Q8×C47—21no—
137C2×Q8×C47—22no—
138C2×Q8×C47—22no—
139C2×Q8×C47C2412yes—
140C2×D4×C47—24no—
141C2×D4×C47—21no—
142C2×D4×C47—22no—
143C2×D4×C47—22no—
144C2×D4×C47—44no—
145C2×D4×C47C2412yes—
146C22×C4×C47—24no—
147C22×C4×C47—22no—
148C22×C4×C47—22no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—24no—
153C22×C4×C47—42no—
154C22×C4×C47—44no—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D412yes—
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D414yes—
161C22×C4×C47C2×D414yes—
162C22×C4×C47C2×D411yescharacteristic
163C22×C4×C47C2412yes—
164C22×C4×C47C2411yescharacteristic
165C22×Dic47—22no—
166C22×Dic47—22no—
167C2×Dic94—24no—
168C2×Dic94—24no—
169C2×Dic94—44no—
170C2×Dic94—44no—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—24no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47C2412yes—
178C4⋊C4×C47C2412yes—
179C22⋊C4×C47—24no—
180C22⋊C4×C47—24no—
181C22⋊C4×C47—24no—
182C22⋊C4×C47—24no—
183C22⋊C4×C47—24no—
184C42×C47—24no—
185C42×C47C2×D414yes—
186C42×C47C2×D412yes—
187C42×C47C2×D412yes—
188C42×C47C2×D414yes—
189C42×C47C2412yes—
190C42×C47C2412yes—
191C22⋊Dic47—28no—
192C4⋊Dic47—24no—
193C4⋊Dic47—24no—
194C4⋊Dic47—22no—
195C4⋊Dic47—22no—
196C4⋊Dic47C2414yes—
197C4⋊Dic47C2414yes—
198Dic47⋊C4—28no—
199Dic47⋊C4—28no—
200C4×Dic47—28no—
201C2×C47⋊C8—24no—
202C2×C47⋊C8—24no—
203C2×C47⋊C8C2414yes—
204C2×C47⋊C8C2414yes—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47—22no—
209C23×C47—42no—
210C23×C47—42no—
211C23×C47—42no—
212C23×C47C22×D412yes—
213C23×C47C22×D411yescharacteristic
214Q8×C47—22no—
215Q8×C47—22no—
216Q8×C47—42no—
217D4×C47—22no—
218D4×C47—22no—
219D4×C47—48no—
220D4×C47—42no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—28no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—44no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—48no—
237C2×C4×C47#13514yes—
238C2×C4×C47C22×D412yes—
239C2×C4×C47C22×D412yes—
240C2×C4×C47C22×D412yes—
241C2×C4×C47C22×D412yes—
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C22×D412yes—
244C2×C4×C472−1+412yes—
245C2×C4×C472−1+412yes—
246C2×C4×C47C2511yescharacteristic
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—44no—
250C2×Dic47—44no—
251Dic94—48no—
252Dic94—48no—
253C47⋊C8—24no—
254C47⋊C8—24no—
255C2×#3233—471nomaximal, Sylow
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—44no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47—84no—
264C22×C47#53612yes—
265C22×C47C2×#13512yes—
266C22×C47#57512yes—
267C22×C47C23×D411yescharacteristic
268C4×C47—22no—
269C4×C47—22no—
270C4×C47—22no—
271C4×C47—22no—
272C4×C47—42no—
273C4×C47—42no—
274C4×C47—42no—
275C4×C47#53612yes—
276C4×C47C2×#13514yes—
277C4×C47C2×2−1+411yescharacteristic
278C4×C47C2×2−1+411yescharacteristic
279Dic47—44no—
280Dic47—44no—
281C2×#533—471no—
282C2×#529D4711yescharacteristic
283#3233—4716no—
284C2×#499—472no—
285C2×#488—472no—
286C2×#485—472no—
287C2×#482—472no—
288C2×#477—471no—
289C2×#475—471no—
290C2×#476—471no—
291C2×#474—471no—
292C2×#431—471no—
293C2×C47—42no—
294C2×C47—82no—
295C2×C47#323314yescomplemented
296C2×C47C2×#53611yescharacteristic
297C2×C47C22×#13511yescharacteristic
298C2×C47C2×#57511yescharacteristic
299#533—478no—
300#533—944no—
301C2×C4⋊Q8—471no—
302C2×C4⋊Q8—471no—
303#529—24no—
304#529C2×D4718yes—
305C2×#126—21no—
306C2×#125—472no—
307C2×#125C2×D4711yescharacteristic
308C2×#124C2×D4711yescharacteristic
309C2×#123—21no—
310C2×C22⋊Q8—941no—
311C2×C22⋊Q8—941no—
312C2×C22⋊Q8C2×D4711yescharacteristic
313C2×#121C2×D4711yescharacteristic
314C2×C4×Q8—941no—
315C2×C4×Q8—941no—
316C2×C4×Q8C2×D4711yescharacteristic
317C2×C4×D4—21no—
318C2×C4×D4C2×D4711yescharacteristic
319C2×#117—21no—
320C2×#117—472no—
321C22×C42C2×D4711yescharacteristic
322#499—4716no—
323#488—4716no—
324#485—4716no—
325#482—4716no—
326#477—478no—
327#476—478no—
328#475—478no—
329#474—478no—
330#431—478no—
331C2×#107—472no—
332C2×#107—472no—
333C2×#106—472no—
334C2×#106—472no—
335C2×C4⋊C8—471no—
336C2×C4⋊C8—471no—
337C2×#103—472no—
338C2×#103—472no—
339C2×#102—472no—
340C2×#102—472no—
341C2×C22⋊C8—472no—
342C2×C4×C8—472no—
343C47C2×#323311yescharacteristic, complemented, Sylow
344C22×Q8—941no—
345C22×Q8—941no—
346C22×Q8C22×D4711yescharacteristic
347C22×D4—21no—
348C22×D4C22×D4711yescharacteristic
349C23×C4—21no—
350C23×C4C22⋊D4711yescharacteristic
351C23×C4C22⋊D4711yescharacteristic
352C23×C4C22×D4711yescharacteristic
353C22×C8—472no—
354C22×C8—472no—
355C4⋊Q8—474no—
356C4⋊Q8—474no—
357C4⋊Q8—942no—
358C4⋊Q8—942no—
359#126—28no—
360#125—22no—
361#125—478no—
362#125—944no—
363#125C22×D4714yes—
364#124—22no—
365#124C22×D4714yes—
366#123—28no—
367C22⋊Q8—22no—
368C22⋊Q8—948no—
369C22⋊Q8—948no—
370C22⋊Q8C22×D4714yes—
371#121—22no—
372#121C22×D4714yes—
373C4×Q8—22no—
374C4×Q8—948no—
375C4×Q8—948no—
376C4×Q8C22×D4714yes—
377C4×D4—28no—
378C4×D4—22no—
379C4×D4C22×D4714yes—
380#117—28no—
381#117—478no—
382#117—944no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—472no—
388C2×C4⋊C4—472no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4C22×D4711yescharacteristic
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—942no—
401C2×C42—21no—
402C2×C42—942no—
403C2×C42C22⋊D4711yescharacteristic
404C2×C42C22⋊D4711yescharacteristic
405C2×C42C22⋊D4714yes—
406C2×C42C22×D4711yescharacteristic
407C2×C42C22×D4711yescharacteristic
408C2×C42C22×D4714yes—
409#107—478no—
410#107—478no—
411#106—478no—
412#106—478no—
413C4⋊C8—474no—
414C4⋊C8—474no—
415#103—478no—
416#103—478no—
417#102—478no—
418#102—478no—
419C22⋊C8—478no—
420C4×C8—478no—
421C24—21no—
422C24C2×C22⋊D4711yescharacteristic
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—944no—
427C2×Q8—944no—
428C2×Q8—1884no—
429C2×Q8—1884no—
430C2×Q8C23×D4712yes—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—44no—
436C2×D4C23×D4712yes—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4C2×C22⋊D4712yes—
449C22×C4C2×C22⋊D4714yes—
450C22×C4C2×C22⋊D4711yescharacteristic
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C2×C22⋊D4714yes—
455C22×C4C2×C22⋊D4711yescharacteristic
456C22×C4C23×D4712yes—
457C22×C4C23×D4711yescharacteristic
458C2×C8—474no—
459C2×C8—474no—
460C2×C8—944no—
461C2×C8—944no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—474no—
469C4⋊C4—474no—
470C4⋊C4—948no—
471C4⋊C4—948no—
472C4⋊C4—942no—
473C4⋊C4—942no—
474C4⋊C4—944no—
475C4⋊C4—944no—
476C4⋊C4C23×D4712yes—
477C4⋊C4C23×D4712yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—948no—
484C42—24no—
485C42—948no—
486C42C2×C22⋊D4714yes—
487C42C2×C22⋊D4712yes—
488C42C2×C22⋊D4714yes—
489C42C2×C22⋊D4712yes—
490C42C23×D4712yes—
491C42C23×D4712yes—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23C22×C22⋊D4712yes—
500C23C22×C22⋊D4711yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—42no—
504Q8—1888no—
505Q8—1888no—
506D4—22no—
507D4—22no—
508D4—48no—
509D4—42no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—944no—
527C2×C4—944no—
528C2×C4—1884no—
529C2×C4—1884no—
530C2×C4#1309404414yes—
531C2×C4C22×C22⋊D4712yes—
532C2×C4C22×C22⋊D4712yes—
533C2×C4C22×C22⋊D4712yes—
534C2×C4C22×C22⋊D4712yes—
535C2×C4C22×C22⋊D4712yes—
536C2×C4C22×C22⋊D4711yescharacteristic
537C2×C4#1309407112yes—
538C2×C4#1309407112yes—
539C2×C4C24×D4711yescharacteristic
540C8—944no—
541C8—944no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—84no—
550C22#1568969412yes—
551C22C2×#1309404412yes—
552C22#1568992112yes—
553C22C23×C22⋊D4711yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—1884no—
562C4—1884no—
563C4#1568969412yes—
564C4C2×#1309404414yes—
565C4C2×#1309407111yescharacteristic
566C4C2×#1309407111yescharacteristic
567C2—42no—
568C2—82no—
569C2#2025456314yesdirect product factor
570C2C2×#1568969411yescharacteristic
571C2C22×#1309404411yescharacteristic
572C2C2×#1568992111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#529C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
hypercenterC2×#529D47
hypocenterC47C2×#3233
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#15689680C2semidirect
2. #20254563C2direct (N × H)
3. C2×#15689522C2semidirect
4. C2×#15689477C2semidirect
5. C2×#15689461C2semidirect
7. C2×#15689418C2semidirect
8. C2×#15689408C2semidirect
9. C2×#15689413C2semidirect
10. C2×#15689403C2semidirect
11. C2×#15689176C2semidirect
14. #15689680C22semidirect
23. #15689522C22semidirect
24. #15689477C22semidirect
25. #15689461C22semidirect
27. #15689418C22semidirect
28. #15689413C22semidirect
29. #15689408C22semidirect
30. #15689403C22semidirect
31. #15689176C22semidirect
295. C2×C47#3233semidirect
343. C47C2×#3233semidirect
569. C2#20254563direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896802
#202545632
C2×#156895222
C2×#156894772
C2×#156894612
C2×#156894442
C2×#156894182
C2×#156894082
C2×#156894132
C2×#156894032
C2×#156891762
C2×C47×#5292
C2×#323347