Finite Group Databasebeta

← #26217635

Subgroups — #26217635

Order 12032.

2234 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094050C211yesmaximal, characteristic
2C22×D4×Dic47C211yesmaximal, characteristic, complemented
3C2×#15689125C211yesmaximal, characteristic, complemented
4#20245876C211yesmaximal, characteristic, complemented
5#20243734C214yesmaximal, complemented
6#20243720C214yesmaximal, complemented
7#20243109C211yesmaximal, characteristic, complemented
8#20243081C211yesmaximal, characteristic, complemented
9C2×C47×#418C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C24×Dic47—21no—
12C2×#13094050—24no—
13C2×#13094050—24no—
14C2×#13094050C418yescomplemented
15C22×#10625335C2212yes—
16C22×#10625334C2212yes—
17C22×#10625329C2211yescharacteristic
18C2×D4×Dic47—24no—
19C2×D4×Dic47—22no—
20C2×D4×Dic47—24no—
21C2×D4×Dic47C2214yes—
22C22×C4⋊Dic47C2211yescharacteristic
23C22×C22⋊Dic47—21no—
24C22×C4×Dic47C2211yescharacteristic
25#15689125—24no—
26C2×#13093949C2212yescomplemented
27C2×#13093946C2212yescomplemented
28#15689028C2212yescomplemented
29#15689024C2212yescomplemented
30#15689000—21no—
31#15689000—21no—
32C4×(C2×D47)⋊C4—22no—
33#15688885C2214yescomplemented
34#15688877C2214yescomplemented
35C23×D4×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C47×#418—24no—
39C2×#102×C47C2214yes—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350C2×C414yescomplemented
46C22×Dic94C2311yescharacteristic
47C22×D188C2311yescharacteristic
48C22×C22⋊D47—21no—
49C23×Dic47—28no—
50C23×Dic47—21no—
51C23×Dic47—24no—
52C23×Dic47—48no—
53C23×Dic47D411yescharacteristic
54C23×Dic47D411yescharacteristic
55C22×C4×D47—21no—
56#13094050—232no—
57C2×#10625335—24no—
58C2×#10625335—24no—
59C2×#10625335C2×C418yescomplemented
60C2×#10625334—24no—
61C2×#10625334—24no—
62C2×#10625334C2×C418yescomplemented
63C2×#10625329—22no—
64C2×#10625329—22no—
65C2×#10625329C2×C414yescomplemented
66C22×C47⋊C8C2312yes—
67D4×Dic47—216no—
68D4×Dic47—28no—
69D4×Dic47—28no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—22no—
72C2×C4⋊Dic47D414yes—
73C2×C4⋊Dic47C2312yes—
74C2×C22⋊Dic47—28no—
75C2×C22⋊Dic47—24no—
76C2×(C2×D47)⋊C4—22no—
77C2×(C2×D47)⋊C4—22no—
78C2×C4×Dic47—22no—
79C2×C4×Dic47—22no—
80C2×C4×Dic47D412yes—
81C2×C4×Dic47C2312yes—
82#13093949—24no—
83#13093946—24no—
84C47⋊#95—22no—
85C47⋊#95—22no—
86C25×C47—21no—
87C22×D4×C47—21no—
88C22×D4×C47—22no—
89C22×D4×C47—22no—
90C22×D4×C47—22no—
91C22×D4×C47C2×C414yes—
92C22×D4×C47D414yes—
93C22×D4×C47D414yes—
94C22×D4×C47C2312yes—
95C23×C4×C47C2311yescharacteristic
96C2×M16×C47—22no—
97C2×M16×C47—22no—
98C2×M16×C47—22no—
99C22×C8×C47C2312yes—
100#117×C47—22no—
101#117×C47—22no—
102C2×C4⋊C4×C47C2312yes—
103C2×C22⋊C4×C47—21no—
104C2×C42×C47—21no—
105#102×C47—28no—
106C24×C47—24no—
107C24×C47—24no—
108C24×C47—24no—
109C24×C47—22no—
110C24×C47—44no—
111C24×C47—44no—
112C24×C47C2×D411yescharacteristic
113C23×D47—21no—
114C2×D4×C47—28no—
115C2×D4×C47—24no—
116C2×D4×C47—24no—
117C2×D4×C47—28no—
118C2×D4×C47—24no—
119C2×D4×C47—28no—
120C2×D4×C47—22no—
121C2×D4×C47—22no—
122C2×D4×C47—22no—
123C2×D4×C47C22⋊C4116yescomplemented
124C2×D4×C47C22×C414yes—
125C2×D4×C47C2×D414yes—
126C2×D4×C47C2×D414yes—
127C22×C4×C47—22no—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D412yes—
133C22×C4×C47#4112yes—
134C22×C4×C47#4112yes—
135C22×C4×C47C2411yescharacteristic
136C2×C22⋊D47—24no—
137C2×C22⋊D47—42no—
138C2×C22⋊D47—42no—
139C22×Dic47—22no—
140C22×Dic47—24no—
141C22×Dic47—216no—
142C22×Dic47—22no—
143C22×Dic47—24no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—42no—
147C22×Dic47—432no—
148C22×Dic47—48no—
149C22×Dic47—44no—
150C22×Dic47—48no—
151C22×Dic47C2×D412yes—
152C22×Dic47C2×D412yes—
153#10625350—28no—
154#10625350—28no—
155#10625350—44no—
156#10625350—44no—
157C2×D188—21no—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188—21no—
161C2×D188—21no—
162C2×D188C22×C412yescomplemented
163C2×C4×D47—24no—
164C2×C4×D47—42no—
165C2×C4×D47—42no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94C22×C412yescomplemented
172M16×C47—44no—
173C2×C8×C47—22no—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—22no—
178C22⋊C4×C47—42no—
179C42×C47—42no—
180C22⋊Dic47—216no—
181#10625335—216no—
182#10625334—216no—
183(C2×D47)⋊C4—44no—
184(C2×D47)⋊C4—44no—
185C4⋊Dic47—24no—
186C4⋊Dic47—22no—
187C4⋊Dic47C2×D414yes—
188C4×Dic47—24no—
189C4×Dic47—22no—
190C4×Dic47—22no—
191#10625329—28no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8C22×C414yescomplemented
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—24no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—216no—
201C23×C47—48no—
202C23×C47—416no—
203C23×C47—48no—
204C23×C47—416no—
205C23×C47—44no—
206C23×C47—44no—
207C23×C47C4×D412yes—
208C23×C47#12012yes—
209C23×C47#12112yes—
210C23×C47C22×D411yescharacteristic
211C22×D47—22no—
212C22×D47—41no—
213C22×D47—41no—
214C22×D47—42no—
215C22×D47—42no—
216D4×C47—24no—
217D4×C47—24no—
218D4×C47—24no—
219D4×C47—216no—
220D4×C47C2×C22⋊C418yes—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47C2×C22⋊C414yes—
225C2×C4×C47C4×D414yes—
226C2×C4×C47C4×D412yes—
227C2×C4×C47#12012yes—
228C2×C4×C47#12114yes—
229C2×C4×C47#12112yes—
230C2×C4×C47#12314yes—
231C2×C4×C47C23×C411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C22⋊D47—48no—
238C2×Dic47—28no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—24no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—432no—
247C2×Dic47—44no—
248C2×Dic47—48no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—44no—
252C2×Dic47AGL(1,ℤ/8)14yescomplemented
253D188—22no—
254D188—22no—
255D188—41no—
256D188—41no—
257C4×D47—48no—
258Dic94—22no—
259Dic94—22no—
260Dic94—41no—
261Dic94—41no—
262C8×C47—42no—
263C47⋊C8—24no—
264#20925—471nomaximal, Sylow
265C22×C47—24no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—24no—
269C22×C47—28no—
270C22×C47—432no—
271C22×C47—48no—
272C22×C47—44no—
273C22×C47—44no—
274C22×C47—48no—
275C22×C47#38614yes—
276C22×C47C2×C4×D411yescharacteristic
277C22×C47C2×#12011yescharacteristic
278C22×C47C2×#12111yescharacteristic
279C22×C47C2×AGL(1,ℤ/8)12yes—
280C22×C47#57512yes—
281C2×D47—44no—
282C2×D47—42no—
283C2×D47—81no—
284C2×D47—81no—
285C2×D47—82no—
286C4×C47—82no—
287C4×C47#38614yes—
288C4×C47C22×C22⋊C411yescharacteristic
289C4×C47C2×C4×D411yescharacteristic
290C4×C47C2×#12111yescharacteristic
291C4×C47C2×#12311yescharacteristic
292Dic47—24no—
293Dic47—44no—
294Dic47—42no—
295Dic47—42no—
296C22×AGL(1,ℤ/8)—471no—
297C22×C4×D4—471no—
298C2×#417—471no—
299C2×#418D4711yescharacteristic, complemented
300#2297—471no—
301#1879—474no—
302#1877—474no—
303#1736—471no—
304#1728—471no—
305C2×C47—24no—
306C2×C47—48no—
307C2×C47C2×#38611yescharacteristic
308C2×C47C4×AGL(1,ℤ/8)12yes—
309C2×C47#300311yescharacteristic
310C2×C47#300511yescharacteristic
311C2×C47#303612yes—
312D47—82no—
313C22×#41—471no—
314C23×D4C2×D4711yescharacteristic
315C24×C4—941no—
316C2×AGL(1,ℤ/8)—478no—
317C2×AGL(1,ℤ/8)—944no—
318C2×AGL(1,ℤ/8)—944no—
319C22×SD16—472no—
320C22×D8—472no—
321C22×M16—471no—
322C22×M16C2×D4711yescharacteristic
323C2×C4×D4—474no—
324C2×C4×D4—942no—
325C2×C4×D4—944no—
326C2×C4×D4—944no—
327C2×#117C2×D4711yescharacteristic
328C22×C4⋊C4—471no—
329C22×C22⋊C4—941no—
330C22×C42—471no—
331#418—24no—
332#417—944no—
333C2×#103—472no—
334C2×#102—472no—
335C2×#102C2×D4714yes—
336#391—472no—
337#390—472no—
338#386—941no—
339#386—941no—
340C4×C22⋊C4—942no—
341#340—474no—
342#336—474no—
343C47#2092511yescharacteristic, complemented, Sylow
344C25—21no—
345C2×#41—474no—
346C2×#41—942no—
347C2×#41—942no—
348C2×#41—942no—
349C2×#41—942no—
350C2×#41—942no—
351C22×Q8—471no—
352C22×D4—21no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—22no—
356C22×D4—471no—
357C22×D4—941no—
358C22×D4C4×D4714yes—
359C22×D4D18814yes—
360C22×D4C22⋊D4714yes—
361C22×D4C22×D4712yes—
362C23×C4—471no—
363C23×C4—471no—
364C23×C4—944no—
365C23×C4—948no—
366C23×C4—941no—
367C23×C4—941no—
368C23×C4—1888no—
369C23×C4C22×D4711yescharacteristic
370AGL(1,ℤ/8)—9432no—
371C2×SD16—478no—
372C2×SD16—944no—
373C2×SD16—944no—
374C2×D8—478no—
375C2×D8—944no—
376C2×D8—944no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—474no—
381C2×M16—942no—
382C2×M16—942no—
383C22×C8—472no—
384C22×C8C22×D4712yes—
385C4×D4—948no—
386C4×D4—9416no—
387C4×D4—948no—
388#117—22no—
389#117—22no—
390C2×C4⋊C4—474no—
391C2×C4⋊C4—472no—
392C2×C4⋊C4—942no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—948no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4—942no—
400C2×C42—21no—
401C2×C42—472no—
402C2×C42—472no—
403C2×C42—942no—
404C2×C42—942no—
405#103—944no—
406#102—28no—
407#102—944no—
408#95—942no—
409#95—942no—
410C24—24no—
411C24—24no—
412C24—24no—
413C24—22no—
414C24—44no—
415C24—44no—
416C24—941no—
417C24D4×D4711yescharacteristic
418#41—948no—
419#41—948no—
420#41—1884no—
421#41—1884no—
422C2×Q8—472no—
423C2×Q8—941no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8—941no—
427C2×Q8—941no—
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—28no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—472no—
438C2×D4—944no—
439C2×D4—941no—
440C2×D4—941no—
441C2×D4—941no—
442C2×D4—941no—
443C2×D4—941no—
444C2×D4—1882no—
445C2×D4—1882no—
446C2×D4(C2×D47)⋊C4116yescomplemented
447C2×D4C2×C4×D4714yes—
448C2×D4C2×D18814yes—
449C2×D4C2×C22⋊D4714yes—
450C22×C4—22no—
451C22×C4—472no—
452C22×C4—472no—
453C22×C4—944no—
454C22×C4—942no—
455C22×C4—9416no—
456C22×C4—942no—
457C22×C4—944no—
458C22×C4—941no—
459C22×C4—942no—
460C22×C4—944no—
461C22×C4—1888no—
462C22×C4—1882no—
463C22×C4—18832no—
464C22×C4—1888no—
465C22×C4—1884no—
466C22×C4—1882no—
467C22×C4—1882no—
468C22×C4C2×C4×D4712yes—
469C22×C4C2×D18812yes—
470C22×C4D4×D4711yescharacteristic
471C22×C4D4×D4712yes—
472C22×C4#1062535212yes—
473C22×C4#1062535212yes—
474C22×C4C2×C22⋊D4712yes—
475C22×C4C23×D4711yescharacteristic
476SD16—9416no—
477D8—9416no—
478M16—44no—
479M16—948no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—474no—
484C2×C8—942no—
485C2×C8—942no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—474no—
489C4⋊C4—944no—
490C4⋊C4—942no—
491C22⋊C4—42no—
492C22⋊C4—9416no—
493C22⋊C4—1884no—
494C22⋊C4—1884no—
495C42—42no—
496C42—944no—
497C42—942no—
498C42—942no—
499C23—22no—
500C23—22no—
501C23—216no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—48no—
506C23—416no—
507C23—416no—
508C23—48no—
509C23—44no—
510C23—44no—
511C23—942no—
512C23—1881no—
513C23—1881no—
514C23—1882no—
515C23—1882no—
516C23(C22⋊D47)⋊C412yes—
517C23#1309399412yes—
518C23#1309399612yes—
519C23C2×D4×D4711yescharacteristic
520Q8—942no—
521Q8—942no—
522Q8—1881no—
523Q8—1881no—
524D4—216no—
525D4—24no—
526D4—24no—
527D4—24no—
528D4—942no—
529D4—942no—
530D4—1888no—
531D4—1881no—
532D4—1881no—
533D4C2×(C2×D47)⋊C418yes—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—474no—
538C2×C4—948no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—944no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—18832no—
547C2×C4—1884no—
548C2×C4—1888no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1884no—
552C2×C4—1888no—
553C2×C4C2×(C2×D47)⋊C414yes—
554C2×C4(C22⋊D47)⋊C412yes—
555C2×C4(C22⋊D47)⋊C414yes—
556C2×C4#1309399412yes—
557C2×C4#1309399612yes—
558C2×C4#1309399614yes—
559C2×C4#1309400814yes—
560C2×C4C22×C4×D4711yescharacteristic
561C2×C4C2×D4×D4711yescharacteristic
562C2×C4C22×C22⋊D4711yescharacteristic
563C2×C4C22×D18811yescharacteristic
564C2×C4C2×#1062535211yescharacteristic
565C2×C4C2×#1062535211yescharacteristic
566C8—42no—
567C8—944no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—28no—
573C22—432no—
574C22—48no—
575C22—48no—
576C22—44no—
577C22—44no—
578C22—3762no—
579C22—3761no—
580C22—3761no—
581C22—1884no—
582C22—1882no—
583C22#1568900014yes—
584C22C2×(C22⋊D47)⋊C411yescharacteristic
585C22C2×#1309399411yescharacteristic
586C22C2×#1309399611yescharacteristic
587C22AGL(1,ℤ/8)×D4712yes—
588C22#1568991912yes—
589C4—82no—
590C4—944no—
591C4—1882no—
592C4—1884no—
593C4—1882no—
594C4#1568900014yes—
595C4C22×(C2×D47)⋊C411yescharacteristic
596C4C2×(C22⋊D47)⋊C411yescharacteristic
597C4C2×#1309399611yescharacteristic
598C4C2×#1309400811yescharacteristic
599C2—24no—
600C2—48no—
601C2—3762no—
602C2C2×#1568900011yescharacteristic
603C2#2025219212yes—
604C2#2025261711yescharacteristic
605C2#2025263211yescharacteristic
606C2#2025284912yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#418C2
socleC23×C47C22×D4
prime radicalC22×#13094050C2
cosocleC22×C4×C47C24
hypercenterC2×#418D47
hypocenterC47#20925
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C22×D4×Dic47C2semidirect
3. C2×#15689125C2semidirect
4. #20245876C2semidirect
5. #20243734C2semidirect
6. #20243720C2semidirect
7. #20243109C2semidirect
8. #20243081C2semidirect
9. C2×C47×#418C2semidirect
14. C2×#13094050C4semidirect
26. C2×#13093949C22semidirect
27. C2×#13093946C22semidirect
28. #15689028C22semidirect
29. #15689024C22semidirect
33. #15688885C22semidirect
34. #15688877C22semidirect
45. C2×#10625350C2×C4semidirect
59. C2×#10625335C2×C4semidirect
62. C2×#10625334C2×C4semidirect
65. C2×#10625329C2×C4semidirect
123. C2×D4×C47C22⋊C4semidirect
162. C2×D188C22×C4semidirect
171. C2×Dic94C22×C4semidirect
194. C2×C47⋊C8C22×C4semidirect
252. C2×Dic47AGL(1,ℤ/8)semidirect
299. C2×#418D47semidirect
343. C47#20925semidirect
446. C2×D4(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130940502
C22×D4×Dic472
C2×#156891252
#202458762
#202437342
#202437202
#202431092
#202430812
C2×C47×#4182
#2092547