Finite Group Databasebeta

← #26217055

Subgroups — #26217055

Order 12032.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689606C211yesmaximal, characteristic
2C2×#15689165C211yesmaximal, characteristic, complemented
3C2×#15689161C211yesmaximal, characteristic, complemented
4#20245980C211yesmaximal, characteristic
5#20243589C212yesmaximal, complemented
6#20243589C212yesmaximal, complemented
7#20243583C212yesmaximal, complemented
8#20243578C212yesmaximal, complemented
9#20243108C211yesmaximal, characteristic, complemented
10C2×#15688886C211yesmaximal, characteristic, complemented
11C2×C47×#408C211yesmaximal, characteristic
12#15689606—24no—
13#15689606—24no—
14C2×Q8×Dic47—21no—
15C2×D4×Dic47—22no—
16C2×D4×Dic47—21no—
17C2×#13093983C2211yescharacteristic
18C2×#13093983C2212yes—
19C2×#13093983C2211yescharacteristic
20C22×C4⋊Dic47C2212yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23#15689165—24no—
24#15689161—24no—
25C2×#13093958C2211yescharacteristic
26C2×#13093958C2211yescharacteristic
27C2×#13093958C2212yes—
28C2×#13093956C2211yescharacteristic
29C2×#13093956C2211yescharacteristic
30C2×#13093956C2212yes—
31#15689030—21no—
32#15689026—21no—
33#15689003—22no—
34#15688983C2212yes—
35#15688983C2212yes—
36#15688886C2212yes—
37#15688886C2212yes—
38#15688886C2212yes—
39#15688886C2212yes—
40C22×#41×C47C2211yescharacteristic
41C22×M16×C47C2211yescharacteristic
42C23×C8×C47C2211yescharacteristic
43C47×#408C418yescomplemented
44C2×C22⋊C8×C47C2212yes—
45C2×C22⋊C8×C47C2212yes—
46C23×Dic47—22no—
47C23×Dic47—21no—
48C23×Dic47—21no—
49Q8×Dic47—28no—
50D4×Dic47—216no—
51D4×Dic47—28no—
52#13093983—22no—
53#13093983—24no—
54#13093983—24no—
55#13093983—22no—
56#13093983—22no—
57#13093983D414yes—
58C2×C4⋊Dic47—22no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—22no—
61C2×C4⋊Dic47—24no—
62C2×C4⋊Dic47—24no—
63C2×C4⋊Dic47—22no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47D414yes—
67C2×C4⋊Dic47C2312yes—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2312yes—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C22⋊Dic47—22no—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—22no—
77C2×C22⋊Dic47—22no—
78C2×C4×Dic47—22no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81#13093958—22no—
82#13093958—22no—
83#13093958—24no—
84#13093956—22no—
85#13093956—22no—
86#13093956—24no—
87C47⋊#95—22no—
88C47⋊#95—22no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47—22no—
92C2×#41×C47—22no—
93C2×#41×C47C2×C414yes—
94C2×#41×C47Q814yes—
95C22×Q8×C47D411yescharacteristic
96C22×D4×C47D412yes—
97C22×D4×C47D411yescharacteristic
98C23×C4×C47C2312yes—
99C23×C4×C47C2311yescharacteristic
100C2×M16×C47—22no—
101C2×M16×C47—22no—
102C2×M16×C47C2×C414yescomplemented
103C22×C8×C47—22no—
104C22×C8×C47—22no—
105C22×C8×C47C2×C414yescomplemented
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108C22×C8×C47C2312yes—
109C22⋊C8×C47C2×C418yescomplemented
110C22⋊C8×C47C2×C418yescomplemented
111C24×C47C2×D412yes—
112C24×C47C2×D411yescharacteristic
113#41×C47—28no—
114#41×C47—28no—
115#41×C47—28no—
116#41×C47—28no—
117C2×Q8×C47—22no—
118C2×Q8×C47—22no—
119C2×Q8×C47C4⋊C414yes—
120C2×D4×C47—24no—
121C2×D4×C47—24no—
122C2×D4×C47—22no—
123C2×D4×C47—22no—
124C2×D4×C47C4⋊C418yes—
125C2×D4×C47C4⋊C414yes—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47C22×C414yes—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×Q814yes—
137C22×C4×C47C2×Q812yes—
138C22×C4×C47#4112yes—
139C22×C4×C47#4112yes—
140C22×C4×C47C2411yescharacteristic
141C22×Dic47—21no—
142C22×Dic47—22no—
143C22×Dic47—21no—
144C22×Dic47—28no—
145C22×Dic47—24no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—22no—
149C22×Dic47—42no—
150C22×Dic47—81no—
151C22×Dic47—82no—
152C22×Dic47—81no—
153C22×Dic47—82no—
154C22×Dic47—81no—
155C22×Dic47—81no—
156M16×C47—28no—
157C2×C8×C47—28no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—22no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—22no—
164C2×C8×C47C22×C412yes—
165C2×C8×C47C22×C412yes—
166C2×C8×C47C22×C414yes—
167C22⋊Dic47—28no—
168C22⋊Dic47—24no—
169C22⋊Dic47—42no—
170C22⋊Dic47—82no—
171C22⋊Dic47—82no—
172C4⋊Dic47—28no—
173C4⋊Dic47—24no—
174C4⋊Dic47—22no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183C4⋊Dic47—42no—
184C4⋊Dic47C2×D412yes—
185C4⋊Dic47C2×D414yes—
186C4⋊Dic47C2×D412yes—
187C4×Dic47—28no—
188C4×Dic47—24no—
189C4×Dic47—42no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47C2×C4⋊C414yes—
201C23×C47C2×C4⋊C412yes—
202C23×C47#12012yes—
203C23×C47#12312yes—
204C23×C47C22×D411yescharacteristic
205Q8×C47—28no—
206D4×C47—216no—
207D4×C47—28no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47C2×C4⋊C414yes—
215C2×C4×C47C2×C4⋊C412yes—
216C2×C4×C47C4×D414yes—
217C2×C4×C47C4×D414yes—
218C2×C4×C47#12012yes—
219C2×C4×C47C22⋊Q814yes—
220C2×C4×C47C22⋊Q814yes—
221C2×C4×C47#12312yes—
222C2×C4×C47C23×C411yescharacteristic
223C2×C4×C47C22×D411yescharacteristic
224C2×C4×C47C22×D411yescharacteristic
225C2×C4×C47C22×Q811yescharacteristic
226C2×C4×C47C2×#4111yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×Dic47—22no—
229C2×Dic47—24no—
230C2×Dic47—22no—
231C2×Dic47—42no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—42no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—42no—
240C2×Dic47—41no—
241C2×Dic47—42no—
242C2×Dic47—41no—
243C2×Dic47—42no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—84no—
247C2×Dic47—88no—
248C2×Dic47—84no—
249C8×C47—22no—
250C8×C47—22no—
251C8×C47—24no—
252#20849—471nomaximal, Sylow
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—24no—
260C22×C47—24no—
261C22×C47—24no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—42no—
266C22×C47#38514yes—
267C22×C47C22×C4⋊C411yescharacteristic
268C22×C47C2×#12011yescharacteristic
269C22×C47C2×#12311yescharacteristic
270C22×C47#57611yescharacteristic
271C22×C47#57711yescharacteristic
272C22×C47#57711yescharacteristic
273C22×C47#57811yescharacteristic
274C4×C47—24no—
275C4×C47#38514yes—
276C4×C47C2×C4×D411yescharacteristic
277C4×C47C2×C4×D411yescharacteristic
278C4×C47C2×C22⋊Q811yescharacteristic
279C4×C47C2×C22⋊Q811yescharacteristic
280Dic47—44no—
281Dic47—42no—
282Dic47—42no—
283Dic47—82no—
284Dic47—81no—
285Dic47—81no—
286C2×#520—471no—
287C2×#428—471no—
288C2×#427—471no—
289C2×#408D4711yescharacteristic
290#2311—471no—
291#1854—472no—
292#1854—472no—
293#1853—472no—
294#1852—472no—
295#1736—471no—
296C2×#340—471no—
297C2×C47—24no—
298C2×C47—44no—
299C2×C47C2×#38511yescharacteristic
300C2×C47#291211yescharacteristic
301C2×C47#291311yescharacteristic
302C2×C47#301211yescharacteristic
303C2×C47#301311yescharacteristic
304C2×C47#309611yescharacteristic
305C2×C47#309711yescharacteristic
306C22×#41C2×D4711yescharacteristic
307C22×M16C2×D4711yescharacteristic
308C23×C8C2×D4711yescharacteristic
309#520—944no—
310#520—944no—
311C2×C4×Q8—941no—
312C2×C4×D4—942no—
313C2×C4×D4—941no—
314C2×#117—471no—
315C2×#117—471no—
316C2×#117—472no—
317C22×C4⋊C4—471no—
318C22×C4⋊C4—471no—
319C22×C4⋊C4—472no—
320#428—944no—
321#427—944no—
322C2×#107—471no—
323C2×#107—472no—
324C2×#107—471no—
325C2×#106—471no—
326C2×#106—472no—
327C2×#106—471no—
328#408Dic4718yescomplemented
329C2×C22⋊C8C2×D4712yes—
330C2×C22⋊C8C2×D4712yes—
331#391—941no—
332#390—941no—
333#386—942no—
334#380—472no—
335#380—472no—
336#340—472no—
337#340—472no—
338#340—472no—
339#340—472no—
340C47#2084911yescharacteristic, complemented, Sylow
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41Dic9414yes—
346C2×#41C2×Dic4714yes—
347C22×Q8D18811yescharacteristic
348C22×D4D18812yes—
349C22×D4D18811yescharacteristic
350C23×C4—942no—
351C23×C4—941no—
352C23×C4—941no—
353C23×C4C22×D4712yes—
354C23×C4C22×D4711yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C2×Dic4714yescomplemented
358C22×C8—22no—
359C22×C8—22no—
360C22×C8C2×Dic4714yescomplemented
361C22×C8C22×D4711yescharacteristic
362C22×C8C22×D4711yescharacteristic
363C22×C8C22×D4712yes—
364C4×Q8—948no—
365C4×D4—9416no—
366C4×D4—948no—
367#117—474no—
368#117—944no—
369#117—944no—
370#117—942no—
371#117—942no—
372#117—942no—
373C2×C4⋊C4—474no—
374C2×C4⋊C4—472no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—472no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—944no—
381C2×C4⋊C4—944no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—941no—
388C2×C22⋊C4—942no—
389C2×C22⋊C4—941no—
390C2×C22⋊C4—942no—
391C2×C22⋊C4—942no—
392C2×C22⋊C4—941no—
393C2×C42—942no—
394C2×C42—941no—
395C2×C42—941no—
396#107—942no—
397#107—942no—
398#107—944no—
399#106—942no—
400#106—942no—
401#106—944no—
402C22⋊C8C2×Dic4718yescomplemented
403C22⋊C8C2×Dic4718yescomplemented
404#95—942no—
405#95—942no—
406C24C2×D18812yes—
407C24C2×D18811yescharacteristic
408#41—28no—
409#41—28no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8C4⋊Dic4714yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4C4⋊Dic4718yes—
420C2×D4C4⋊Dic4714yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—948no—
426C22×C4—944no—
427C22×C4—941no—
428C22×C4—942no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—942no—
433C22×C4—3761no—
434C22×C4—3761no—
435C22×C4—3761no—
436C22×C4—3761no—
437C22×C4—3762no—
438C22×C4—3762no—
439C22×C4—1882no—
440C22×C4C2×Dic9414yes—
441C22×C4C2×Dic9412yes—
442C22×C4C2×D18812yes—
443C22×C4C2×D18811yescharacteristic
444C22×C4D4×D4712yes—
445C22×C4D4×D4712yes—
446C22×C4#1062535212yes—
447C22×C4#1062535212yes—
448C22×C4C22×Dic4714yes—
449C22×C4C22×Dic4712yes—
450C22×C4C23×D4711yescharacteristic
451M16—28no—
452C2×C8—28no—
453C2×C8—21no—
454C2×C8—22no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8C22×Dic4712yes—
460C2×C8C22×Dic4712yes—
461C2×C8C22×Dic4714yes—
462C4⋊C4—474no—
463C4⋊C4—472no—
464C4⋊C4—472no—
465C4⋊C4—948no—
466C4⋊C4—944no—
467C4⋊C4—941no—
468C4⋊C4—942no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—942no—
475C4⋊C4—942no—
476C4⋊C4—1882no—
477C22⋊C4—948no—
478C22⋊C4—944no—
479C22⋊C4—3762no—
480C22⋊C4—3762no—
481C22⋊C4—1882no—
482C42—948no—
483C42—944no—
484C42—1882no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C2×C4⋊Dic4714yes—
496C23C2×C4⋊Dic4712yes—
497C23#1309399412yes—
498C23#1309400812yes—
499C23C22×D18811yescharacteristic
500Q8—28no—
501D4—216no—
502D4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—942no—
510C2×C4—944no—
511C2×C4—942no—
512C2×C4—3764no—
513C2×C4—3764no—
514C2×C4—3768no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1882no—
518C2×C4—1881no—
519C2×C4—1882no—
520C2×C4—1881no—
521C2×C4—1882no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1882no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1882no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4C2×C4⋊Dic4714yes—
531C2×C4C2×C4⋊Dic4712yes—
532C2×C4D4×Dic4714yes—
533C2×C4D4×Dic4714yes—
534C2×C4#1309399412yes—
535C2×C4#1309400114yes—
536C2×C4#1309400114yes—
537C2×C4#1309400812yes—
538C2×C4C23×Dic4711yescharacteristic
539C2×C4C2×D4×D4711yescharacteristic
540C2×C4C2×D4×D4711yescharacteristic
541C2×C4C22×Dic9411yescharacteristic
542C2×C4C2×#1062535211yescharacteristic
543C2×C4C2×#1062535211yescharacteristic
544C8—22no—
545C8—22no—
546C8—24no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22#1568899814yes—
561C22C22×C4⋊Dic4711yescharacteristic
562C22C2×#1309399411yescharacteristic
563C22C2×#1309400811yescharacteristic
564C22#1568992811yescharacteristic
565C22#1568993511yescharacteristic
566C22#1568993511yescharacteristic
567C22#1568994011yescharacteristic
568C4—24no—
569C4—3761no—
570C4—3761no—
571C4—3762no—
572C4—1884no—
573C4—1882no—
574C4—1882no—
575C4#1568899814yes—
576C4C2×D4×Dic4711yescharacteristic
577C4C2×D4×Dic4711yescharacteristic
578C4C2×#1309400111yescharacteristic
579C4C2×#1309400111yescharacteristic
580C2—24no—
581C2—44no—
582C2C2×#1568899811yescharacteristic
583C2#2025198311yescharacteristic
584C2#2025198811yescharacteristic
585C2#2025269511yescharacteristic
586C2#2025270611yescharacteristic
587C2#2025334711yescharacteristic
588C2#2025335811yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#408C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#408D47
hypocenterC47#20849
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C2×#15689165C2semidirect
3. C2×#15689161C2semidirect
5. #20243589C2semidirect
6. #20243589C2semidirect
7. #20243583C2semidirect
8. #20243578C2semidirect
9. #20243108C2semidirect
10. C2×#15688886C2semidirect
43. C47×#408C4semidirect
102. C2×M16×C47C2×C4semidirect
105. C22×C8×C47C2×C4semidirect
109. C22⋊C8×C47C2×C4semidirect
110. C22⋊C8×C47C2×C4semidirect
328. #408Dic47semidirect
340. C47#20849semidirect
357. C2×M16C2×Dic47semidirect
360. C22×C8C2×Dic47semidirect
402. C22⋊C8C2×Dic47semidirect
403. C22⋊C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896062
C2×#156891652
C2×#156891612
#202459802
#202435892
#202435832
#202435782
#202431082
C2×#156888862
C2×C47×#4082
#2084947