Finite Group Databasebeta

← #26215866

Subgroups — #26215866

Order 12032.

1438 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094030C211yesmaximal, characteristic
2C2×#15689158C211yesmaximal, characteristic, complemented
3C2×#106×D47C211yesmaximal, characteristic, complemented
4#20245977C212yesmaximal
5#20243420C212yesmaximal, complemented
6#20243419C212yesmaximal, complemented
7#20243413C211yesmaximal, characteristic, complemented
8#20243410C211yesmaximal, characteristic, complemented
9#20243410C211yesmaximal, characteristic, complemented
10#20243106C212yesmaximal, complemented
11C47×#1807C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C2×#13094030—24no—
14C2×#13094030—24no—
15C2×#13094030C418yescomplemented
16C22×#10625329C2211yescharacteristic
17C22×#10625324C2211yescharacteristic
18C22×#10625324C2211yescharacteristic
19C22×C8×D47C2211yescharacteristic
20C22×C8×D47C2211yescharacteristic
21C2×#13093982C2212yes—
22C2×C4⋊C4×D47C2212yes—
23#15689158—24no—
24C2×#13093957C2212yes—
25C2×#13093956C2211yescharacteristic
26C2×#13093956C2211yescharacteristic
27C2×#13093955C2212yes—
28#106×D47—24no—
29#15689027—22no—
30#15689023—22no—
31#15689001—22no—
32#15688999—22no—
33#15688983C2212yes—
34#15688980C2212yes—
35#15688980C2212yes—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39C23×C8×C47C2211yescharacteristic
40C2×#106×C47C2211yescharacteristic
41C2×#106×C47C2211yescharacteristic
42C47×#380C2212yes—
43C47×#340C2212yes—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350C2×C414yes—
49C2×#10625350Q814yes—
50C22×Dic94D411yescharacteristic
51C22×D188D411yescharacteristic
52C22×C22⋊D47D411yescharacteristic
53C22×C22⋊D47D411yescharacteristic
54C22×C4×D47—22no—
55C22×C4×D47C2311yescharacteristic
56C22×C4×D47C2311yescharacteristic
57#13094030—232no—
58C2×#10625329—22no—
59C2×#10625329—22no—
60C2×#10625329C2×C414yescomplemented
61C2×#10625324—22no—
62C2×#10625324—22no—
63C2×#10625324—22no—
64C2×#10625324—22no—
65C2×#10625324C2×C414yes—
66C2×#10625324C2×C414yescomplemented
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C2×C8×D47—22no—
70C2×C8×D47—22no—
71C2×C8×D47C2×C414yes—
72C2×C8×D47C2×C414yescomplemented
73C2×C8×D47D414yes—
74C2×C8×D47D414yes—
75#13093982—24no—
76#13093982—24no—
77C2×C4⋊Dic47C2312yes—
78C2×C4⋊Dic47C2312yes—
79C2×Dic47⋊C4—22no—
80C4⋊C4×D47—24no—
81C4⋊C4×D47—24no—
82C2×C22⋊Dic47—22no—
83C2×(C2×D47)⋊C4—22no—
84C2×(C2×D47)⋊C4—22no—
85C2×(C2×D47)⋊C4—22no—
86C2×C4×Dic47—22no—
87#13093957—24no—
88#13093956—22no—
89#13093956—22no—
90#13093955—24no—
91C47⋊#95—22no—
92C47⋊#95—22no—
93C47⋊#95—22no—
94C23×C4×C47C2311yescharacteristic
95C22×C8×C47—22no—
96C22×C8×C47—22no—
97C22×C8×C47C2×C414yescomplemented
98C22×C8×C47C2311yescharacteristic
99C22×C8×C47C2311yescharacteristic
100C2×C4⋊C4×C47C2312yes—
101C2×C4⋊C4×C47C2312yes—
102C2×C22⋊C4×C47—22no—
103#106×C47—22no—
104#106×C47—22no—
105#95×C47—22no—
106C24×C47C2×D411yescharacteristic
107C23×D47C2×D411yescharacteristic
108C23×D47C2×D411yescharacteristic
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47C22×C412yes—
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×Q812yes—
118C22×C4×C47C2411yescharacteristic
119C2×C22⋊D47—22no—
120C2×C22⋊D47—22no—
121C2×C22⋊D47—22no—
122C2×C22⋊D47—22no—
123C2×C22⋊D47C4⋊C414yes—
124C2×C22⋊D47C4⋊C414yes—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47C2×D411yescharacteristic
128C22×Dic47C2×D411yescharacteristic
129#10625350—28no—
130#10625350—28no—
131#10625350—28no—
132#10625350—28no—
133C2×D188—22no—
134C2×D188—22no—
135C2×D188C4⋊C414yes—
136C2×C4×D47—21no—
137C2×C4×D47—21no—
138C2×C4×D47—21no—
139C2×C4×D47—21no—
140C2×C4×D47—44no—
141C2×C4×D47—82no—
142C2×C4×D47—82no—
143C2×C4×D47C22×C412yes—
144C2×C4×D47C22×C412yes—
145C2×C4×D47C2×D412yes—
146C2×C4×D47C2×D412yes—
147C2×C4×D47C2×Q812yes—
148C2×C4×D47C2×Q812yes—
149C2×C4×D47#4112yes—
150C2×C4×D47#4112yes—
151C2×Dic94—22no—
152C2×Dic94—22no—
153C2×Dic94C4⋊C414yes—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—28no—
157C2×C8×C47C22×C412yes—
158C2×C8×C47C22×C412yes—
159C2×C8×C47C2×D412yes—
160C2×C8×C47C2×D412yes—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C22⋊C4×C47—82no—
166C22⋊Dic47—82no—
167(C2×D47)⋊C4—44no—
168(C2×D47)⋊C4—82no—
169(C2×D47)⋊C4—82no—
170C4⋊Dic47—22no—
171C4⋊Dic47—22no—
172C4⋊Dic47—22no—
173C4⋊Dic47—22no—
174Dic47⋊C4—44no—
175C4×Dic47—44no—
176#10625329—28no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8C22×C412yes—
180C2×C47⋊C8C22×C412yes—
181C2×C47⋊C8C2×D412yes—
182C2×C47⋊C8C2×D412yes—
183#10625324—28no—
184#10625324—28no—
185C8×D47—28no—
186C8×D47C22⋊C4116yescomplemented
187C23×C47—21no—
188C23×C47—21no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47C2×C4⋊C412yes—
192C23×C47C22×D411yescharacteristic
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—21no—
196C22×D47—21no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47C2×C4⋊C412yes—
200C22×D47C2×C4⋊C412yes—
201C22×D47#12112yes—
202C22×D47#12112yes—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47C2×C4⋊C412yes—
216C2×C4×C47C23×C411yescharacteristic
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×Q811yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C22⋊D47—28no—
223C22⋊D47—28no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47C2×C4⋊C412yes—
233C2×Dic47C2×C4⋊C412yes—
234C2×Dic47#12112yes—
235C2×Dic47#12112yes—
236D188—28no—
237C4×D47—22no—
238C4×D47—22no—
239C4×D47—22no—
240C4×D47—22no—
241C4×D47—88no—
242C4×D47C2×C22⋊C414yes—
243C4×D47#11714yes—
244C4×D47C22⋊Q814yes—
245C4×D47C22⋊Q814yes—
246Dic94—28no—
247C8×C47—22no—
248C8×C47C2×C22⋊C414yes—
249C47⋊C8—22no—
250C47⋊C8C2×C22⋊C414yes—
251#20662—471nomaximal, Sylow
252C22×C47—22no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47C22×C4⋊C411yescharacteristic
258C22×C47C2×#12111yescharacteristic
259C22×C47C2×#12111yescharacteristic
260C22×C47#57712yes—
261C22×C47#57712yes—
262C2×D47—22no—
263C2×D47—22no—
264C2×D47—24no—
265C2×D47—24no—
266C2×D47—22no—
267C2×D47—22no—
268C2×D47—22no—
269C2×D47—42no—
270C2×D47—42no—
271C2×D47#38014yes—
272C4×C47—22no—
273C4×C47—82no—
274C4×C47—82no—
275C4×C47C22×C22⋊C411yescharacteristic
276C4×C47C2×#11711yescharacteristic
277C4×C47C2×C22⋊Q811yescharacteristic
278C4×C47C2×C22⋊Q811yescharacteristic
279Dic47—22no—
280Dic47—82no—
281Dic47—82no—
282Dic47#38014yes—
283C22×#131—471no—
284C2×#427—471no—
285C22×#106—471no—
286#2311—472no—
287#1809—472no—
288#1809—472no—
289#1807—471no—
290#1807—471no—
291#1807—471no—
292#1807D4711yescharacteristic, complemented
293#1736—472no—
294C2×C47—42no—
295C2×C47C2×#38011yescharacteristic
296C2×C47#291211yescharacteristic
297C2×C47#291211yescharacteristic
298C2×C47#305712yes—
299C2×C47#306012yes—
300D47—24no—
301D47—42no—
302C22×#41—471no—
303C2×#131—478no—
304C2×#131—944no—
305C2×#131—944no—
306C22×M16—471no—
307C22×M16—471no—
308C22×M16—471no—
309C23×C8—471no—
310C23×C8—471no—
311C23×C8C2×D4711yescharacteristic
312C2×#117—472no—
313C22×C4⋊C4—472no—
314#427—944no—
315C2×#107—472no—
316C2×#106—471no—
317C2×#106—471no—
318C2×#106—472no—
319C2×#106—944no—
320C2×#106C2×D4711yescharacteristic
321C2×#106C2×D4711yescharacteristic
322#391—942no—
323#390—942no—
324#386—942no—
325#386—942no—
326#380—472no—
327#380—472no—
328#380—472no—
329#380C2×D4712yes—
330#340—472no—
331#340—472no—
332#340—472no—
333#340C2×D4712yes—
334C47#2066211yescharacteristic, complemented, Sylow
335C2×#41—474no—
336C2×#41—474no—
337C2×#41—942no—
338C2×#41—942no—
339C2×#41—942no—
340C2×#41—942no—
341C22×Q8—471no—
342C22×D4—471no—
343C22×D4—471no—
344C22×D4—471no—
345C23×C4—471no—
346C23×C4—471no—
347C23×C4—942no—
348C23×C4C22×D4711yescharacteristic
349#131—9432no—
350C2×M16—474no—
351C2×M16—474no—
352C2×M16—474no—
353C2×M16—942no—
354C2×M16—942no—
355C2×M16—942no—
356C2×M16—942no—
357C2×M16—942no—
358C2×M16—942no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—471no—
362C22×C8—474no—
363C22×C8—474no—
364C22×C8—474no—
365C22×C8—471no—
366C22×C8—474no—
367C22×C8—942no—
368C22×C8—942no—
369C22×C8C4×D4714yescomplemented
370C22×C8C22×D4711yescharacteristic
371C22×C8C22×D4711yescharacteristic
372#117—944no—
373#117—944no—
374C2×C4⋊C4—472no—
375C2×C4⋊C4—472no—
376C2×C4⋊C4—944no—
377C2×C4⋊C4—944no—
378C2×C4⋊C4—942no—
379C2×C4⋊C4C22×D4712yes—
380C2×C4⋊C4C22×D4712yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4—942no—
384C2×C22⋊C4—942no—
385C2×C22⋊C4—942no—
386C2×C42—942no—
387#107—944no—
388#106—22no—
389#106—22no—
390#106—944no—
391#106—942no—
392#106—942no—
393#95—22no—
394#95—942no—
395#95—942no—
396#95—942no—
397C24—471no—
398C24—471no—
399C24D4×D4711yescharacteristic
400#41—948no—
401#41—948no—
402#41—948no—
403#41—948no—
404C2×Q8—474no—
405C2×Q8—942no—
406C2×Q8—942no—
407C2×D4—474no—
408C2×D4—474no—
409C2×D4—474no—
410C2×D4—942no—
411C2×D4—942no—
412C2×D4—942no—
413C2×D4—942no—
414C2×D4—942no—
415C2×D4—942no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—471no—
423C22×C4—472no—
424C22×C4—472no—
425C22×C4—472no—
426C22×C4—472no—
427C22×C4—472no—
428C22×C4—472no—
429C22×C4—471no—
430C22×C4—472no—
431C22×C4—472no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—3762no—
439C22×C4—3762no—
440C22×C4—1884no—
441C22×C4C2×C4×D4712yes—
442C22×C4D4×D4711yescharacteristic
443C22×C4Q8×D4712yes—
444C22×C4C23×D4711yescharacteristic
445M16—948no—
446M16—948no—
447M16—948no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—28no—
451C2×C8—472no—
452C2×C8—472no—
453C2×C8—472no—
454C2×C8—4716no—
455C2×C8—472no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—948no—
459C2×C8C2×C4×D4712yes—
460C2×C8C2×C4×D4712yes—
461C2×C8D4×D4712yes—
462C2×C8D4×D4712yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—942no—
468C4⋊C4—942no—
469C4⋊C4—942no—
470C4⋊C4—942no—
471C4⋊C4—1884no—
472C22⋊C4—82no—
473C22⋊C4—3762no—
474C22⋊C4—3762no—
475C22⋊C4—3762no—
476C22⋊C4—1884no—
477C42—1884no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—472no—
483C23—472no—
484C23—472no—
485C23—472no—
486C23—942no—
487C23—942no—
488C23—941no—
489C23—941no—
490C23—942no—
491C23—942no—
492C23C4⋊C4×D4712yes—
493C23C2×D4×D4711yescharacteristic
494Q8—948no—
495D4—948no—
496D4—948no—
497D4—948no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—472no—
511C2×C4—472no—
512C2×C4—472no—
513C2×C4—474no—
514C2×C4—474no—
515C2×C4—474no—
516C2×C4—474no—
517C2×C4—472no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—942no—
524C2×C4—3768no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4C4⋊C4×D4712yes—
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4C2×Q8×D4711yescharacteristic
536C2×C4#41×D4711yescharacteristic
537C2×C4#41×D4711yescharacteristic
538C8—22no—
539C8—474no—
540C8—942no—
541C8C22⋊C4×D4714yes—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—474no—
548C22—942no—
549C22—942no—
550C22—944no—
551C22—944no—
552C22—942no—
553C22—942no—
554C22—942no—
555C22—1882no—
556C22—1882no—
557C22C2×C4⋊C4×D4711yescharacteristic
558C22#121×D4711yescharacteristic
559C22#121×D4711yescharacteristic
560C22#1568992912yes—
561C22#1568993012yes—
562C4—22no—
563C4—82no—
564C4—82no—
565C4—474no—
566C4—942no—
567C4—3762no—
568C4—3762no—
569C4C2×C22⋊C4×D4711yescharacteristic
570C4#117×D4711yescharacteristic
571C4C22⋊Q8×D4711yescharacteristic
572C4C22⋊Q8×D4711yescharacteristic
573C2—42no—
574C2—944no—
575C2—1882no—
576C2#380×D4711yescharacteristic
577C2#2025197911yescharacteristic
578C2#2025198011yescharacteristic
579C2#2025302212yes—
580C2#2025304512yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1807C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenter#1807D47
hypocenterC47#20662
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
2. C2×#15689158C2semidirect
3. C2×#106×D47C2semidirect
5. #20243420C2semidirect
6. #20243419C2semidirect
7. #20243413C2semidirect
8. #20243410C2semidirect
9. #20243410C2semidirect
10. #20243106C2semidirect
11. C47×#1807C2semidirect
15. C2×#13094030C4semidirect
60. C2×#10625329C2×C4semidirect
66. C2×#10625324C2×C4semidirect
72. C2×C8×D47C2×C4semidirect
97. C22×C8×C47C2×C4semidirect
186. C8×D47C22⋊C4semidirect
292. #1807D47semidirect
334. C47#20662semidirect
369. C22×C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940302
C2×#156891582
C2×#106×D472
#202459772
#202434202
#202434192
#202434132
#202434102
#202431062
C47×#18072
#2066247