Finite Group Databasebeta

← #26217530

Subgroups — #26217530

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×#15689697C211yesmaximal, characteristic
2C2×#15689604C211yesmaximal, characteristic
3C2×#15689132C211yesmaximal, characteristic, complemented
4C2×#15689089C211yesmaximal, characteristic
5#20243697C214yesmaximal, complemented
6#20243692C214yesmaximal, complemented
7#20243107C211yesmaximal, characteristic, complemented
8#20243105C211yesmaximal, characteristic, complemented
9C22×#103×C47C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12#15689697—24no—
13#15689697C418yescomplemented
14C2×#13094022—21no—
15C2×#13094013—21no—
16C2×#13094007—22no—
17C2×#13094000C2214yes—
18#15689604—24no—
19#15689604—24no—
20C2×#13093989—21no—
21C2×#13093986—21no—
22C2×C4×C22⋊D47—22no—
23C2×#13093983C2211yescharacteristic
24C2×#13093983C2211yescharacteristic
25C2×#13093982C2212yes—
26C2×C4⋊C4×D47C2212yes—
27#15689132—24no—
28C2×#13093948C2212yes—
29C2×#13093948C2212yes—
30#15689089—24no—
31C2×(C2×D47)⋊C8C2214yes—
32#15688886C2212yes—
33#15688886C2212yes—
34#15688885C2214yes—
35C23×Q8×C47C2211yescharacteristic
36C23×C8×C47C2211yescharacteristic
37C22×C4⋊C4×C47C2211yescharacteristic
38C2×#103×C47—24no—
39C2×#103×C47C2214yes—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350C2×C414yes—
45C2×#10625350D414yes—
46C22×Dic94D411yescharacteristic
47C22×D188D411yescharacteristic
48C22×C22⋊D47D412yes—
49C22×C4×D47—22no—
50C22×C4×D47C2312yes—
51C2×#10625329—22no—
52C2×#10625329—22no—
53C2×#10625329—22no—
54C22×C47⋊C8C2312yes—
55#13094022—28no—
56#13094013—28no—
57#13094007—216no—
58#13094000—28no—
59#13094000C2×C4116yescomplemented
60#13093989—28no—
61#13093986—28no—
62C4×C22⋊D47—216no—
63#13093983—22no—
64#13093983—22no—
65#13093983C2×C414yescomplemented
66#13093983D414yes—
67#13093982—24no—
68#13093982—24no—
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—22no—
73C2×Dic47⋊C4—24no—
74C4⋊C4×D47—24no—
75C4⋊C4×D47—24no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×(C2×D47)⋊C4—22no—
79C2×(C2×D47)⋊C4—24no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—22no—
83#13093948—24no—
84#13093948—24no—
85(C2×D47)⋊C8—28no—
86C22×Q8×C47—21no—
87C22×Q8×C47—22no—
88C22×Q8×C47—22no—
89C22×Q8×C47—22no—
90C22×Q8×C47—22no—
91C22×Q8×C47—22no—
92C22×Q8×C47C2×C414yescomplemented
93C22×Q8×C47C2312yes—
94C23×C4×C47—21no—
95C23×C4×C47—21no—
96C23×C4×C47C2311yescharacteristic
97C22×C8×C47—22no—
98C22×C8×C47—22no—
99C22×C8×C47—22no—
100C22×C8×C47C2312yes—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—22no—
103C2×C4⋊C4×C47D414yes—
104C2×C4⋊C4×C47C2312yes—
105#103×C47—28no—
106C24×C47C2×D411yescharacteristic
107C23×D47C2×D412yes—
108C2×Q8×C47—24no—
109C2×Q8×C47—22no—
110C2×Q8×C47—28no—
111C2×Q8×C47—28no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47—22no—
116C2×Q8×C47—42no—
117C2×Q8×C47—44no—
118C2×Q8×C47—44no—
119C2×Q8×C47—42no—
120C2×Q8×C47C22×C414yes—
121C22×C4×C47—24no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—24no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47—82no—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47#4112yes—
134C22×C4×C47#4112yes—
135C22×C4×C47C2411yescharacteristic
136C2×C22⋊D47—24no—
137C2×C22⋊D47—24no—
138C2×C22⋊D47C22⋊C418yes—
139C22×Dic47—21no—
140C22×Dic47—22no—
141C22×Dic47—21no—
142C22×Dic47C2×D412yes—
143#10625350—28no—
144#10625350—28no—
145#10625350—28no—
146#10625350—28no—
147C2×D188—22no—
148C2×D188—22no—
149C2×D188C22⋊C414yes—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—28no—
155C2×C4×D47—84no—
156C2×C4×D47C22×C414yes—
157C2×C4×D47C2×D414yes—
158C2×Dic94—22no—
159C2×Dic94—22no—
160C2×Dic94C22⋊C414yes—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C2×C8×C47—22no—
164C2×C8×C47—44no—
165C4⋊C4×C47—24no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47C2×D414yes—
168C22⋊Dic47—24no—
169C22⋊Dic47—24no—
170(C2×D47)⋊C4—28no—
171(C2×D47)⋊C4—216no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—22no—
175C4⋊Dic47C22×C414yes—
176C4⋊Dic47C2×D412yes—
177C4⋊Dic47C2×D412yes—
178Dic47⋊C4—28no—
179Dic47⋊C4—216no—
180C4×Dic47—28no—
181C4×Dic47—24no—
182C4×Dic47—24no—
183#10625329—44no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47C2×C22⋊C412yes—
190C23×C47#12012yes—
191C23×C47#12312yes—
192C23×C47C22×D411yescharacteristic
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—24no—
196C22×D47—24no—
197C22×D47C2×C22⋊C414yes—
198Q8×C47—24no—
199Q8×C47—24no—
200Q8×C47—42no—
201Q8×C47—42no—
202Q8×C47—48no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—44no—
208C2×C4×C47—48no—
209C2×C4×C47—84no—
210C2×C4×C47C2×C22⋊C412yes—
211C2×C4×C47C4×D414yes—
212C2×C4×C47C4×D414yes—
213C2×C4×C47#12012yes—
214C2×C4×C47#12114yes—
215C2×C4×C47#12114yes—
216C2×C4×C47#12312yes—
217C2×C4×C47C23×C411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C22⋊D47—216no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—24no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47C2×C22⋊C414yes—
234D188—28no—
235C4×D47—24no—
236C4×D47—24no—
237C4×D47—24no—
238C4×D47—24no—
239C4×D47—88no—
240Dic94—28no—
241C8×C47—42no—
242C47⋊C8—42no—
243#20913—471nomaximal, Sylow
244C22×C47—24no—
245C22×C47—24no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47#38614yes—
249C22×C47C22×C22⋊C411yescharacteristic
250C22×C47C2×#12011yescharacteristic
251C22×C47C2×#12311yescharacteristic
252C22×C47#57712yes—
253C22×C47#57812yes—
254C2×D47—28no—
255C2×D47—24no—
256C2×D47—44no—
257C2×D47—42no—
258C2×D47—42no—
259C4×C47—44no—
260C4×C47—44no—
261C4×C47#38614yes—
262C4×C47C2×C4×D411yescharacteristic
263C4×C47C2×C4×D411yescharacteristic
264C4×C47C2×#12111yescharacteristic
265C4×C47C2×#12111yescharacteristic
266Dic47—24no—
267Dic47—44no—
268Dic47—42no—
269Dic47—42no—
270C2×#536—471no—
271C2×#520—471no—
272C2×#419—471no—
273C22×#103D4711yescharacteristic, complemented
274C2×#408—471no—
275#1873—474no—
276#1872—474no—
277#1736—471no—
278#1735—471no—
279C2×C47—24no—
280C2×C47C2×#38611yescharacteristic
281C2×C47#289812yes—
282C2×C47#300911yescharacteristic
283C2×C47#301111yescharacteristic
284C2×C47#309712yes—
285D47—44no—
286C22×#41—471no—
287C23×Q8C2×D4711yescharacteristic
288C22×M16—471no—
289C23×C8C2×D4711yescharacteristic
290#536—478no—
291#536—944no—
292C2×C4⋊Q8—941no—
293C2×#124—941no—
294C2×#123—942no—
295C2×C22⋊Q8—474no—
296#520—944no—
297#520—944no—
298C2×C4×Q8—941no—
299C2×C4×D4—941no—
300C2×C4×D4—942no—
301C2×#117—471no—
302C2×#117—471no—
303C2×#117—472no—
304C22×C4⋊C4—472no—
305C22×C4⋊C4C2×D4711yescharacteristic
306#419—944no—
307C2×#103—24no—
308C2×#103—472no—
309C2×#103—472no—
310C2×#103C2×D4714yes—
311#408—944no—
312C2×C22⋊C8—474no—
313#340—472no—
314#340—472no—
315#340—474no—
316C47#2091311yescharacteristic, complemented, Sylow
317C2×#41—474no—
318C2×#41—474no—
319C2×#41—942no—
320C2×#41—942no—
321C2×#41—942no—
322C2×#41—942no—
323C22×Q8—21no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—471no—
330C22×Q8C4×D4714yescomplemented
331C22×Q8C22×D4712yes—
332C22×D4—472no—
333C22×D4—471no—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—472no—
337C23×C4—942no—
338C23×C4C22×D4711yescharacteristic
339C2×M16—942no—
340C2×M16—942no—
341C2×M16—942no—
342C22×C8—22no—
343C22×C8—22no—
344C22×C8—22no—
345C22×C8—472no—
346C22×C8C22×D4712yes—
347C4⋊Q8—948no—
348#124—948no—
349#123—9416no—
350C22⋊Q8—4716no—
351C22⋊Q8—948no—
352C4×Q8—948no—
353C4×D4—948no—
354C4×D4—9416no—
355#117—474no—
356#117—474no—
357#117—944no—
358#117—942no—
359#117—942no—
360#117—944no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—472no—
366C2×C4⋊C4—944no—
367C2×C4⋊C4—942no—
368C2×C4⋊C4—944no—
369C2×C4⋊C4—944no—
370C2×C4⋊C4C22⋊D4714yes—
371C2×C4⋊C4C22×D4712yes—
372C2×C22⋊C4—942no—
373C2×C22⋊C4—944no—
374C2×C22⋊C4—941no—
375C2×C22⋊C4—941no—
376C2×C42—942no—
377C2×C42—941no—
378C2×C42—941no—
379#103—28no—
380#103—944no—
381#103—944no—
382C22⋊C8—948no—
383C24—472no—
384C24D4×D4711yescharacteristic
385#41—948no—
386#41—948no—
387#41—948no—
388#41—948no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—474no—
402C2×Q8—942no—
403C2×Q8—942no—
404C2×Q8C2×C4×D4714yes—
405C2×D4—478no—
406C2×D4—474no—
407C2×D4—944no—
408C2×D4—944no—
409C2×D4—942no—
410C2×D4—942no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—82no—
418C22×C4—472no—
419C22×C4—474no—
420C22×C4—474no—
421C22×C4—948no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4—941no—
427C22×C4—942no—
428C22×C4—941no—
429C22×C4—3764no—
430C22×C4C2×C4×D4712yes—
431C22×C4#1062535012yes—
432C22×C4D4×D4711yescharacteristic
433C22×C4D4×D4712yes—
434C22×C4D4×D4712yes—
435C22×C4#1062535212yes—
436C22×C4C2×C22⋊D4712yes—
437C22×C4C23×D4711yescharacteristic
438M16—1884no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—44no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—942no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—474no—
449C4⋊C4—472no—
450C4⋊C4—472no—
451C4⋊C4—9416no—
452C4⋊C4—948no—
453C4⋊C4—942no—
454C4⋊C4—941no—
455C4⋊C4—941no—
456C4⋊C4C2×C22⋊D4714yes—
457C22⋊C4—9416no—
458C22⋊C4—948no—
459C22⋊C4—944no—
460C22⋊C4—944no—
461C42—948no—
462C42—944no—
463C42—944no—
464C23—22no—
465C23—22no—
466C23—474no—
467C23—942no—
468C23—942no—
469C23—944no—
470C23—944no—
471C23C22⋊C4×D4712yes—
472C23#1309399212yes—
473C23#1309400612yes—
474C23C2×D4×D4711yescharacteristic
475Q8—24no—
476Q8—24no—
477Q8—48no—
478Q8—42no—
479Q8—42no—
480Q8—948no—
481D4—9416no—
482D4—948no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—84no—
490C2×C4—474no—
491C2×C4—942no—
492C2×C4—942no—
493C2×C4—944no—
494C2×C4—944no—
495C2×C4—944no—
496C2×C4—944no—
497C2×C4—944no—
498C2×C4—942no—
499C2×C4—942no—
500C2×C4—3768no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4C22⋊C4×D4712yes—
506C2×C4(C22⋊D47)⋊C414yes—
507C2×C4C4×C22⋊D4714yes—
508C2×C4#1309399212yes—
509C2×C4#1309399514yes—
510C2×C4#1309399714yes—
511C2×C4#1309400612yes—
512C2×C4C22×C4×D4711yescharacteristic
513C2×C4C2×D4×D4711yescharacteristic
514C2×C4C2×D4×D4711yescharacteristic
515C2×C4C22×C22⋊D4711yescharacteristic
516C2×C4C2×#1062535211yescharacteristic
517C2×C4C2×#1062535011yescharacteristic
518C8—42no—
519C8—1882no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—22no—
524C22—948no—
525C22—944no—
526C22—1882no—
527C22—1884no—
528C22—1882no—
529C22#1568899914yes—
530C22C2×C22⋊C4×D4711yescharacteristic
531C22C2×#1309399211yescharacteristic
532C22C2×#1309400611yescharacteristic
533C22#1568992912yes—
534C22#1568993612yes—
535C4—44no—
536C4—44no—
537C4—944no—
538C4—1882no—
539C4—1884no—
540C4—1882no—
541C4#1568899914yes—
542C4C2×(C22⋊D47)⋊C411yescharacteristic
543C4C2×C4×C22⋊D4711yescharacteristic
544C4C2×#1309399511yescharacteristic
545C4C2×#1309399711yescharacteristic
546C2—24no—
547C2—1884no—
548C2C2×#1568899911yescharacteristic
549C2#2025191612yes—
550C2#2025266111yescharacteristic
551C2#2025267911yescharacteristic
552C2#2025335312yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#103×C47C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC22×#103D47
hypocenterC47#20913
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C2×#15689132C2semidirect
5. #20243697C2semidirect
6. #20243692C2semidirect
7. #20243107C2semidirect
8. #20243105C2semidirect
9. C22×#103×C47C2semidirect
13. #15689697C4semidirect
59. #13094000C2×C4semidirect
65. #13093983C2×C4semidirect
92. C22×Q8×C47C2×C4semidirect
273. C22×#103D47semidirect
316. C47#20913semidirect
330. C22×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896972
C2×#156896042
C2×#156891322
C2×#156890892
#202436972
#202436922
#202431072
#202431052
C22×#103×C472
#2091347