Finite Group Databasebeta

← #26217536

Subgroups — #26217536

Order 12032.

1778 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689606C211yesmaximal, characteristic
2C2×#15689132C211yesmaximal, characteristic, complemented
3C22×#13093948C211yesmaximal, characteristic, complemented
4C2×#15689090C211yesmaximal, characteristic
5#20243701C214yesmaximal, complemented
6#20243696C214yesmaximal, complemented
7#20243109C211yesmaximal, characteristic, complemented
8#20243104C211yesmaximal, characteristic, complemented
9C2×C47×#536C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12#15689606—24no—
13#15689606—24no—
14C2×Q8×Dic47—21no—
15C2×D4×Dic47—22no—
16C2×D4×Dic47—21no—
17C2×#13093983C2212yes—
18C2×#13093983C2211yescharacteristic
19C22×C4⋊Dic47C2212yes—
20C22×C4⋊Dic47C2211yescharacteristic
21#15689132—24no—
22C2×#13093948—24no—
23C2×#13093948C2214yes—
24C2×#13093948C2212yes—
25C2×#13093948C2212yes—
26#15689090—24no—
27C2×C22⋊(C47⋊C8)C2214yes—
28#15688885C2212yes—
29#15688885C2212yes—
30#15688885C2214yes—
31C22×#41×C47C2211yescharacteristic
32C23×Q8×C47C2211yescharacteristic
33C47×#536—24no—
34C47×#536C418yescomplemented
35C2×C4⋊Q8×C47—21no—
36C2×#124×C47—21no—
37C2×#123×C47—22no—
38C2×C22⋊Q8×C47C2214yes—
39C2×#117×C47C2211yescharacteristic
40C23×Dic47—22no—
41C23×Dic47—21no—
42C2×#10625329—22no—
43C2×#10625329—22no—
44C2×#10625329—22no—
45C22×C47⋊C8—22no—
46C22×C47⋊C8—22no—
47C22×C47⋊C8—22no—
48C22×C47⋊C8C2312yes—
49C22×C47⋊C8C2312yes—
50Q8×Dic47—28no—
51D4×Dic47—216no—
52D4×Dic47—28no—
53#13093983—22no—
54#13093983—24no—
55#13093983—24no—
56#13093983D414yes—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47—22no—
60C2×C4⋊Dic47—24no—
61C2×C4⋊Dic47—24no—
62C2×C4⋊Dic47D414yes—
63C2×C4⋊Dic47C2311yescharacteristic
64C2×C4⋊Dic47C2312yes—
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C22⋊Dic47—22no—
67C2×C22⋊Dic47—21no—
68C2×C4×Dic47—22no—
69C2×C4×Dic47—21no—
70#13093948—28no—
71#13093948—24no—
72#13093948—24no—
73C22⋊(C47⋊C8)—28no—
74C2×#41×C47—22no—
75C2×#41×C47—22no—
76C2×#41×C47—22no—
77C2×#41×C47—22no—
78C2×#41×C47C2×C414yes—
79C2×#41×C47D414yes—
80C22×Q8×C47—21no—
81C22×Q8×C47—22no—
82C22×Q8×C47—22no—
83C22×Q8×C47—22no—
84C22×Q8×C47—22no—
85C22×Q8×C47—22no—
86C22×Q8×C47C2×C414yescomplemented
87C22×Q8×C47D411yescharacteristic
88C22×Q8×C47C2312yes—
89C22×D4×C47D412yes—
90C22×D4×C47D411yescharacteristic
91C23×C4×C47—21no—
92C23×C4×C47C2312yes—
93C23×C4×C47C2311yescharacteristic
94C4⋊Q8×C47—28no—
95#124×C47—28no—
96#123×C47—216no—
97C22⋊Q8×C47—28no—
98C22⋊Q8×C47C2×C4116yescomplemented
99#117×C47—22no—
100#117×C47C2×C414yescomplemented
101C2×C4⋊C4×C47—24no—
102C2×C4⋊C4×C47C2312yes—
103C2×C22⋊C4×C47—21no—
104C2×C22⋊C4×C47—24no—
105C2×C42×C47—21no—
106C24×C47C2×D412yes—
107C24×C47C2×D411yescharacteristic
108#41×C47—28no—
109#41×C47—28no—
110#41×C47—28no—
111#41×C47—28no—
112C2×Q8×C47—24no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47—22no—
116C2×Q8×C47—28no—
117C2×Q8×C47—28no—
118C2×Q8×C47—22no—
119C2×Q8×C47—22no—
120C2×Q8×C47—22no—
121C2×Q8×C47—22no—
122C2×Q8×C47—42no—
123C2×Q8×C47—44no—
124C2×Q8×C47—44no—
125C2×Q8×C47—42no—
126C2×Q8×C47C22⋊C414yes—
127C2×Q8×C47C22×C414yes—
128C2×D4×C47—24no—
129C2×D4×C47—24no—
130C2×D4×C47—22no—
131C2×D4×C47—22no—
132C2×D4×C47C22⋊C418yes—
133C2×D4×C47C22⋊C414yes—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—24no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—42no—
142C22×C4×C47—42no—
143C22×C4×C47C22×C414yes—
144C22×C4×C47C22×C412yes—
145C22×C4×C47C2×D412yes—
146C22×C4×C47C2×D414yes—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D412yes—
151C22×C4×C47#4112yes—
152C22×C4×C47#4112yes—
153C22×C4×C47C2411yescharacteristic
154C22×Dic47—21no—
155C22×Dic47—22no—
156C22×Dic47—21no—
157C22×Dic47—28no—
158C22×Dic47—24no—
159C22×Dic47—82no—
160C22×Dic47—84no—
161C4⋊C4×C47—216no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47C22×C414yes—
164C22⋊C4×C47—24no—
165C22⋊C4×C47—216no—
166C42×C47—24no—
167C22⋊Dic47—28no—
168C22⋊Dic47—24no—
169C4⋊Dic47—28no—
170C4⋊Dic47—24no—
171C4⋊Dic47—22no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47C2×D412yes—
175C4⋊Dic47C2×D414yes—
176C4⋊Dic47C2×D412yes—
177C4×Dic47—28no—
178C4×Dic47—24no—
179#10625329—44no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—44no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—24no—
192C23×C47—24no—
193C23×C47C2×C22⋊C414yes—
194C23×C47C2×C22⋊C412yes—
195C23×C47#12012yes—
196C23×C47#12312yes—
197C23×C47C22×D411yescharacteristic
198Q8×C47—28no—
199Q8×C47—24no—
200Q8×C47—24no—
201Q8×C47—42no—
202Q8×C47—42no—
203Q8×C47—48no—
204D4×C47—216no—
205D4×C47—28no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—48no—
219C2×C4×C47C2×C22⋊C414yes—
220C2×C4×C47C2×C22⋊C412yes—
221C2×C4×C47C4×D414yes—
222C2×C4×C47C4×D414yes—
223C2×C4×C47#12012yes—
224C2×C4×C47#12114yes—
225C2×C4×C47#12114yes—
226C2×C4×C47#12312yes—
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×Dic47—22no—
234C2×Dic47—24no—
235C2×Dic47—22no—
236C2×Dic47—44no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—84no—
240C2×Dic47—88no—
241C47⋊C8—42no—
242C47⋊C8—42no—
243#20913—471nomaximal, Sylow
244C22×C47—24no—
245C22×C47—24no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—28no—
249C22×C47—24no—
250C22×C47—44no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47#38614yes—
254C22×C47C22×C22⋊C411yescharacteristic
255C22×C47C2×#12011yescharacteristic
256C22×C47C2×#12311yescharacteristic
257C22×C47#57712yes—
258C22×C47#57812yes—
259C4×C47—24no—
260C4×C47—44no—
261C4×C47—44no—
262C4×C47#38614yes—
263C4×C47C2×C4×D411yescharacteristic
264C4×C47C2×C4×D411yescharacteristic
265C4×C47C2×#12111yescharacteristic
266C4×C47C2×#12111yescharacteristic
267Dic47—44no—
268Dic47—42no—
269Dic47—42no—
270C2×#536D4711yescharacteristic
271C2×#520—471no—
272C2×#419—471no—
273C22×#103—471no—
274C2×#408—471no—
275#1873—474no—
276#1872—474no—
277#1736—471no—
278#1735—471no—
279C2×C47—24no—
280C2×C47—44no—
281C2×C47C2×#38611yescharacteristic
282C2×C47#289812yes—
283C2×C47#300911yescharacteristic
284C2×C47#301111yescharacteristic
285C2×C47#309712yes—
286C22×#41C2×D4711yescharacteristic
287C23×Q8C2×D4711yescharacteristic
288C22×M16—471no—
289C23×C8—471no—
290#536—24no—
291#536Dic4718yescomplemented
292C2×C4⋊Q8—21no—
293C2×#124—21no—
294C2×#123—22no—
295C2×C22⋊Q8C2×D4714yes—
296#520—944no—
297#520—944no—
298C2×C4×Q8—941no—
299C2×C4×D4—942no—
300C2×C4×D4—941no—
301C2×#117—471no—
302C2×#117—472no—
303C2×#117C2×D4711yescharacteristic
304C22×C4⋊C4—471no—
305C22×C4⋊C4—472no—
306#419—944no—
307C2×#103—474no—
308C2×#103—472no—
309C2×#103—472no—
310C2×#103—944no—
311#408—944no—
312C2×C22⋊C8—474no—
313#340—472no—
314#340—474no—
315#340—472no—
316C47#2091311yescharacteristic, complemented, Sylow
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41C2×Dic4714yes—
322C2×#41C22⋊D4714yes—
323C22×Q8—21no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8C2×Dic4714yescomplemented
330C22×Q8C22⋊D4711yescharacteristic
331C22×Q8C22×D4712yes—
332C22×D4C22⋊D4712yes—
333C22×D4C22⋊D4711yescharacteristic
334C23×C4—21no—
335C23×C4—942no—
336C23×C4—941no—
337C23×C4C22×D4712yes—
338C23×C4C22×D4711yescharacteristic
339C2×M16—942no—
340C2×M16—942no—
341C2×M16—942no—
342C22×C8—472no—
343C22×C8—472no—
344C22×C8—942no—
345C22×C8—942no—
346C22×C8—942no—
347C4⋊Q8—28no—
348#124—28no—
349#123—216no—
350C22⋊Q8—28no—
351C22⋊Q8C2×Dic47116yescomplemented
352C4×Q8—948no—
353C4×D4—9416no—
354C4×D4—948no—
355#117—22no—
356#117—474no—
357#117—944no—
358#117—944no—
359#117—942no—
360#117C2×Dic4714yescomplemented
361C2×C4⋊C4—24no—
362C2×C4⋊C4—474no—
363C2×C4⋊C4—472no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—944no—
367C2×C4⋊C4—944no—
368C2×C4⋊C4—942no—
369C2×C4⋊C4—942no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4C22×D4712yes—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—942no—
375C2×C22⋊C4—941no—
376C2×C42—21no—
377C2×C42—942no—
378C2×C42—941no—
379#103—948no—
380#103—944no—
381#103—944no—
382C22⋊C8—948no—
383C24C2×C22⋊D4712yes—
384C24C2×C22⋊D4711yescharacteristic
385#41—28no—
386#41—28no—
387#41—28no—
388#41—28no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8C22⋊Dic4714yes—
404C2×Q8C22×Dic4714yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4C22⋊Dic4718yes—
410C2×D4C22⋊Dic4714yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—948no—
421C22×C4—944no—
422C22×C4—941no—
423C22×C4—942no—
424C22×C4—941no—
425C22×C4—3762no—
426C22×C4—3764no—
427C22×C4D4×D4712yes—
428C22×C4D4×D4712yes—
429C22×C4#1062535212yes—
430C22×C4#1062535212yes—
431C22×C4C22×Dic4714yes—
432C22×C4C22×Dic4712yes—
433C22×C4C2×C22⋊D4712yes—
434C22×C4C2×C22⋊D4714yes—
435C22×C4C2×C22⋊D4711yescharacteristic
436C22×C4C2×C22⋊D4712yes—
437C22×C4C23×D4711yescharacteristic
438M16—1884no—
439C2×C8—942no—
440C2×C8—942no—
441C2×C8—942no—
442C2×C8—942no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—1884no—
446C4⋊C4—216no—
447C4⋊C4—22no—
448C4⋊C4—474no—
449C4⋊C4—472no—
450C4⋊C4—472no—
451C4⋊C4—948no—
452C4⋊C4—944no—
453C4⋊C4—941no—
454C4⋊C4—942no—
455C4⋊C4—941no—
456C4⋊C4C22×Dic4714yes—
457C22⋊C4—24no—
458C22⋊C4—216no—
459C22⋊C4—948no—
460C22⋊C4—944no—
461C42—24no—
462C42—948no—
463C42—944no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23C2×C22⋊Dic4714yes—
471C23C2×C22⋊Dic4712yes—
472C23#1309399212yes—
473C23#1309400512yes—
474C23C22×C22⋊D4711yescharacteristic
475Q8—28no—
476Q8—24no—
477Q8—24no—
478Q8—48no—
479Q8—42no—
480Q8—42no—
481D4—216no—
482D4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—942no—
497C2×C4—944no—
498C2×C4—942no—
499C2×C4—3764no—
500C2×C4—3768no—
501C2×C4—1884no—
502C2×C4—1882no—
503C2×C4—1882no—
504C2×C4C2×C22⋊Dic4714yes—
505C2×C4C2×C22⋊Dic4712yes—
506C2×C4D4×Dic4714yes—
507C2×C4D4×Dic4714yes—
508C2×C4#1309399212yes—
509C2×C4#1309399614yes—
510C2×C4#1309399614yes—
511C2×C4#1309400512yes—
512C2×C4C23×Dic4711yescharacteristic
513C2×C4C2×D4×D4711yescharacteristic
514C2×C4C2×D4×D4711yescharacteristic
515C2×C4C22×C22⋊D4711yescharacteristic
516C2×C4C2×#1062535211yescharacteristic
517C2×C4C2×#1062535211yescharacteristic
518C8—1882no—
519C8—1882no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—22no—
524C22—28no—
525C22—24no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22#1568900314yes—
530C22C22×C22⋊Dic4711yescharacteristic
531C22C2×#1309399211yescharacteristic
532C22C2×#1309400511yescharacteristic
533C22#1568993112yes—
534C22#1568993812yes—
535C4—24no—
536C4—44no—
537C4—44no—
538C4—1882no—
539C4—1884no—
540C4—1882no—
541C4#1568900314yes—
542C4C2×D4×Dic4711yescharacteristic
543C4C2×D4×Dic4711yescharacteristic
544C4C2×#1309399611yescharacteristic
545C4C2×#1309399611yescharacteristic
546C2—24no—
547C2—44no—
548C2C2×#1568900311yescharacteristic
549C2#2025192112yes—
550C2#2025266311yescharacteristic
551C2#2025268111yescharacteristic
552C2#2025335112yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#536C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#536D47
hypocenterC47#20913
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C2×#15689132C2semidirect
3. C22×#13093948C2semidirect
5. #20243701C2semidirect
6. #20243696C2semidirect
7. #20243109C2semidirect
8. #20243104C2semidirect
34. C47×#536C4semidirect
86. C22×Q8×C47C2×C4semidirect
98. C22⋊Q8×C47C2×C4semidirect
100. #117×C47C2×C4semidirect
291. #536Dic47semidirect
316. C47#20913semidirect
329. C22×Q8C2×Dic47semidirect
351. C22⋊Q8C2×Dic47semidirect
360. #117C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896062
C2×#156891322
C22×#130939482
C2×#156890902
#202437012
#202436962
#202431092
#202431042
C2×C47×#5362
#2091347