Finite Group Databasebeta

← #26217634

Subgroups — #26217634

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×#13094049C211yesmaximal, characteristic
2C22×#13093986C211yesmaximal, characteristic, complemented
3C2×#15689129C211yesmaximal, characteristic, complemented
4#20245879C211yesmaximal, characteristic, complemented
5#20243733C214yesmaximal, complemented
6#20243719C214yesmaximal, complemented
7#20243107C211yesmaximal, characteristic, complemented
8#20243081C211yesmaximal, characteristic, complemented
9C2×C47×#417C211yesmaximal, characteristic, complemented
10C23×D188C2211yescharacteristic
11C23×C4×D47—21no—
12C2×#13094049—24no—
13C2×#13094049—24no—
14C2×#13094049C418yescomplemented
15C22×#10625336C2212yes—
16C22×#10625334C2212yes—
17C22×#10625329C2211yescharacteristic
18C2×#13093986—24no—
19C2×#13093986—22no—
20C2×#13093986—24no—
21C2×#13093986C2214yes—
22C2×#13093983C2211yescharacteristic
23C22×(C2×D47)⋊C4—21no—
24C22×C4×Dic47C2211yescharacteristic
25#15689129—24no—
26C2×#13093946C2214yes—
27#15689030C2212yescomplemented
28#15689026C2212yescomplemented
29#15689003—21no—
30#15689003—21no—
31C4×C22⋊Dic47—22no—
32#15688885C2214yescomplemented
33#15688877C2214yescomplemented
34C22×#41×C47C2211yescharacteristic
35C22×M16×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2211yescharacteristic
37C47×#417—24no—
38C2×#103×C47C2212yescomplemented
39C2×#102×C47C2212yescomplemented
40C24×D47—21no—
41C22×D188—21no—
42C22×D188—22no—
43C22×D188—22no—
44C22×D188—22no—
45C22×D188C2×C414yes—
46C22×D188D414yes—
47C22×D188D414yes—
48C22×D188C2312yes—
49C23×Dic47D411yescharacteristic
50C23×Dic47D411yescharacteristic
51C22×C4×D47—28no—
52C22×C4×D47—24no—
53C22×C4×D47—48no—
54#13094049—232no—
55C2×#10625336—24no—
56C2×#10625336—24no—
57C2×#10625336C2×C418yescomplemented
58C2×#10625334—24no—
59C2×#10625334—24no—
60C2×#10625334C2×C418yescomplemented
61C2×#10625329—22no—
62C2×#10625329—22no—
63C2×#10625329C2×C414yescomplemented
64C22×C47⋊C8C2312yes—
65#13093986—216no—
66#13093986—28no—
67#13093986—28no—
68#13093983—22no—
69#13093983—22no—
70C2×C4⋊Dic47C2312yes—
71C2×C22⋊Dic47—21no—
72C2×C22⋊Dic47—22no—
73C2×C22⋊Dic47—22no—
74C2×(C2×D47)⋊C4—28no—
75C2×(C2×D47)⋊C4—24no—
76C2×C4×Dic47—22no—
77C2×C4×Dic47—22no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47D412yes—
80C2×C4×Dic47C2312yes—
81#13093946—28no—
82C47⋊#95—22no—
83C47⋊#95—22no—
84C2×#41×C47—22no—
85C2×#41×C47—22no—
86C2×#41×C47—22no—
87C2×#41×C47—22no—
88C2×#41×C47—22no—
89C2×#41×C47C2×C414yescomplemented
90C22×Q8×C47C2311yescharacteristic
91C22×D4×C47—21no—
92C22×D4×C47C2311yescharacteristic
93C23×C4×C47—21no—
94C23×C4×C47—21no—
95C23×C4×C47C2311yescharacteristic
96C2×M16×C47—22no—
97C2×M16×C47—22no—
98C2×M16×C47—22no—
99C22×C8×C47C2312yes—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—22no—
102C2×C4⋊C4×C47D414yes—
103C2×C4⋊C4×C47C2312yes—
104#103×C47—24no—
105#102×C47—24no—
106C24×C47—21no—
107C24×C47C2×D411yescharacteristic
108C23×D47—24no—
109C23×D47—24no—
110C23×D47—24no—
111C23×D47—22no—
112C23×D47—44no—
113C23×D47—44no—
114#41×C47—28no—
115#41×C47—28no—
116#41×C47—44no—
117#41×C47—44no—
118C2×Q8×C47—21no—
119C2×Q8×C47—21no—
120C2×Q8×C47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47C22×C412yescomplemented
124C2×D4×C47—24no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—42no—
131C2×D4×C47—42no—
132C2×D4×C47C22×C412yescomplemented
133C22×C4×C47—24no—
134C22×C4×C47—22no—
135C22×C4×C47—21no—
136C22×C4×C47—24no—
137C22×C4×C47—44no—
138C22×C4×C47—42no—
139C22×C4×C47—42no—
140C22×C4×C47C22×C412yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D412yes—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D412yes—
145C22×C4×C47#4112yes—
146C22×C4×C47#4112yes—
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—22no—
149C22×Dic47—24no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—42no—
153C22×Dic47C2×D412yes—
154C22×Dic47C2×D412yes—
155C2×D188—28no—
156C2×D188—24no—
157C2×D188—24no—
158C2×D188—28no—
159C2×D188—24no—
160C2×D188—28no—
161C2×D188—22no—
162C2×D188—22no—
163C2×D188—22no—
164C2×D188C22⋊C4116yescomplemented
165C2×D188C22×C414yes—
166C2×D188C2×D414yes—
167C2×D188C2×D414yes—
168C2×C4×D47—216no—
169C2×C4×D47—432no—
170C2×C4×D47—48no—
171C2×C4×D47—48no—
172M16×C47—44no—
173C2×C8×C47—22no—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47C2×D414yes—
179C22⋊Dic47—44no—
180C22⋊Dic47—42no—
181C22⋊Dic47—44no—
182#10625336—216no—
183#10625334—216no—
184(C2×D47)⋊C4—216no—
185C4⋊Dic47—22no—
186C4⋊Dic47—22no—
187C4×Dic47—24no—
188C4×Dic47—22no—
189C4×Dic47—22no—
190C4×Dic47—42no—
191#10625329—28no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8C22×C414yescomplemented
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—42no—
201C23×C47—42no—
202C23×C47C4×D412yes—
203C23×C47#12012yes—
204C23×C47#12112yes—
205C23×C47C22×D411yescharacteristic
206C22×D47—24no—
207C22×D47—24no—
208C22×D47—24no—
209C22×D47—216no—
210C22×D47—48no—
211C22×D47—416no—
212C22×D47—48no—
213C22×D47—416no—
214C22×D47—44no—
215C22×D47—44no—
216Q8×C47—22no—
217Q8×C47—22no—
218Q8×C47—41no—
219Q8×C47—41no—
220D4×C47—22no—
221D4×C47—22no—
222D4×C47—48no—
223D4×C47—41no—
224D4×C47—41no—
225C2×C4×C47—22no—
226C2×C4×C47—24no—
227C2×C4×C47—48no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—44no—
231C2×C4×C47—48no—
232C2×C4×C47C2×C22⋊C414yes—
233C2×C4×C47C4×D412yes—
234C2×C4×C47C4×D414yes—
235C2×C4×C47#12012yes—
236C2×C4×C47#12114yes—
237C2×C4×C47#12112yes—
238C2×C4×C47#12314yes—
239C2×C4×C47C23×C411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×Dic47—28no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—44no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47AGL(1,ℤ/8)14yescomplemented
256D188—24no—
257D188—24no—
258D188—24no—
259D188—216no—
260D188C2×C22⋊C418yes—
261C4×D47—432no—
262C8×C47—42no—
263C47⋊C8—24no—
264#20925—471nomaximal, Sylow
265C22×C47—24no—
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—24no—
269C22×C47—44no—
270C22×C47—42no—
271C22×C47—81no—
272C22×C47—81no—
273C22×C47—82no—
274C22×C47#38614yes—
275C22×C47C2×C4×D411yescharacteristic
276C22×C47C2×#12011yescharacteristic
277C22×C47C2×#12111yescharacteristic
278C22×C47C2×AGL(1,ℤ/8)12yes—
279C22×C47#57512yes—
280C2×D47—28no—
281C2×D47—432no—
282C2×D47—48no—
283C2×D47—44no—
284C2×D47—44no—
285C2×D47—48no—
286C4×C47—44no—
287C4×C47—42no—
288C4×C47#38614yes—
289C4×C47C22×C22⋊C411yescharacteristic
290C4×C47C2×C4×D411yescharacteristic
291C4×C47C2×#12111yescharacteristic
292C4×C47C2×#12311yescharacteristic
293Dic47—24no—
294Dic47—42no—
295Dic47—82no—
296C22×AGL(1,ℤ/8)—471no—
297C22×C4×D4—471no—
298C2×#417D4711yescharacteristic, complemented
299C2×#418—471no—
300#2297—471no—
301#1879—474no—
302#1877—474no—
303#1736—471no—
304#1728—471no—
305C2×C47—24no—
306C2×C47—82no—
307C2×C47C2×#38611yescharacteristic
308C2×C47C4×AGL(1,ℤ/8)12yes—
309C2×C47#300311yescharacteristic
310C2×C47#300511yescharacteristic
311C2×C47#303612yes—
312D47—48no—
313C22×#41C2×D4711yescharacteristic
314C23×D4—471no—
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×#117—471no—
328C22×C4⋊C4C2×D4711yescharacteristic
329C22×C22⋊C4—941no—
330C22×C42—471no—
331#418—944no—
332#417—24no—
333C2×#103C2×D4712yescomplemented
334C2×#102—474no—
335C2×#102C2×D4712yescomplemented
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—941no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41C4×D4714yescomplemented
351C22×Q8C22×D4711yescharacteristic
352C22×D4—21no—
353C22×D4—474no—
354C22×D4—474no—
355C22×D4—474no—
356C22×D4—472no—
357C22×D4—942no—
358C22×D4—941no—
359C22×D4—942no—
360C22×D4—942no—
361C22×D4C22×D4711yescharacteristic
362C23×C4—21no—
363C23×C4—21no—
364C23×C4—471no—
365C23×C4—471no—
366C23×C4—944no—
367C23×C4—948no—
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—942no—
389#117—942no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—472no—
393C2×C4⋊C4C22⋊D4714yes—
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—944no—
396C2×C22⋊C4—948no—
397C2×C22⋊C4—942no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4—941no—
400C2×C42—472no—
401C2×C42—472no—
402C2×C42—942no—
403C2×C42—942no—
404C2×C42—941no—
405#103—24no—
406#102—24no—
407#102—948no—
408#95—942no—
409#95—942no—
410C24—21no—
411C24—944no—
412C24—944no—
413C24—944no—
414C24—942no—
415C24—1884no—
416C24—1884no—
417C24D4×D4711yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C2×C4×D4712yescomplemented
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—4716no—
437C2×D4—474no—
438C2×D4—474no—
439C2×D4—474no—
440C2×D4—944no—
441C2×D4—948no—
442C2×D4—944no—
443C2×D4—948no—
444C2×D4—944no—
445C2×D4—948no—
446C2×D4—942no—
447C2×D4—942no—
448C2×D4—942no—
449C2×D4C2×C4×D4712yescomplemented
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—472no—
458C22×C4—472no—
459C22×C4—9416no—
460C22×C4—944no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—1888no—
465C22×C4—18832no—
466C22×C4—1888no—
467C22×C4—1882no—
468C22×C4C2×C4×D4712yes—
469C22×C4#1062535012yes—
470C22×C4D4×D4712yes—
471C22×C4D4×D4712yes—
472C22×C4D4×D4711yescharacteristic
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—24no—
487C4⋊C4—22no—
488C4⋊C4—942no—
489C4⋊C4—942no—
490C4⋊C4C2×C22⋊D4714yes—
491C22⋊C4—9416no—
492C22⋊C4—1884no—
493C22⋊C4—1884no—
494C22⋊C4—1882no—
495C42—944no—
496C42—942no—
497C42—942no—
498C42—1882no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—9416no—
507C23—944no—
508C23—944no—
509C23—944no—
510C23—1888no—
511C23—18816no—
512C23—18816no—
513C23—1888no—
514C23—1884no—
515C23—1884no—
516C23(C22⋊D47)⋊C412yes—
517C23#1309399212yes—
518C23#1309399712yes—
519C23C2×D4×D4711yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524D4—22no—
525D4—22no—
526D4—48no—
527D4—41no—
528D4—41no—
529D4—478no—
530D4—9416no—
531D4—944no—
532D4—944no—
533D4—944no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—48no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—474no—
542C2×C4—948no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—18832no—
549C2×C4—1882no—
550C2×C4—1882no—
551C2×C4—1882no—
552C2×C4—1884no—
553C2×C4C22⋊C4×D4714yes—
554C2×C4(C22⋊D47)⋊C412yes—
555C2×C4C4×C22⋊D4714yes—
556C2×C4#1309399212yes—
557C2×C4#1309399514yes—
558C2×C4#1309399712yes—
559C2×C4#1309400614yes—
560C2×C4C22×C4×D4711yescharacteristic
561C2×C4C2×D4×D4711yescharacteristic
562C2×C4C2×D4×D4711yescharacteristic
563C2×C4C22×C22⋊D4711yescharacteristic
564C2×C4C2×#1062535211yescharacteristic
565C2×C4C2×#1062535011yescharacteristic
566C8—42no—
567C8—944no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—42no—
574C22—82no—
575C22—81no—
576C22—81no—
577C22—948no—
578C22—18832no—
579C22—1888no—
580C22—1888no—
581C22—1884no—
582C22—1884no—
583C22#1568899914yes—
584C22C2×(C22⋊D47)⋊C411yescharacteristic
585C22C2×#1309399211yescharacteristic
586C22C2×#1309399711yescharacteristic
587C22AGL(1,ℤ/8)×D4712yes—
588C22#1568992212yes—
589C4—42no—
590C4—44no—
591C4—944no—
592C4—3762no—
593C4—1882no—
594C4#1568899914yes—
595C4C2×C22⋊C4×D4711yescharacteristic
596C4C2×C4×C22⋊D4711yescharacteristic
597C4C2×#1309399511yescharacteristic
598C4C2×#1309400611yescharacteristic
599C2—24no—
600C2—82no—
601C2—1888no—
602C2C2×#1568899911yescharacteristic
603C2#2025220112yes—
604C2#2025261311yescharacteristic
605C2#2025262911yescharacteristic
606C2#2025285112yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#417C2
socleC23×C47C22×D4
prime radicalC22×#13094049C2
cosocleC22×C4×C47C24
hypercenterC2×#417D47
hypocenterC47#20925
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#13093986C2semidirect
3. C2×#15689129C2semidirect
4. #20245879C2semidirect
5. #20243733C2semidirect
6. #20243719C2semidirect
7. #20243107C2semidirect
8. #20243081C2semidirect
9. C2×C47×#417C2semidirect
14. C2×#13094049C4semidirect
27. #15689030C22semidirect
28. #15689026C22semidirect
32. #15688885C22semidirect
33. #15688877C22semidirect
38. C2×#103×C47C22semidirect
39. C2×#102×C47C22semidirect
57. C2×#10625336C2×C4semidirect
60. C2×#10625334C2×C4semidirect
63. C2×#10625329C2×C4semidirect
89. C2×#41×C47C2×C4semidirect
123. C2×Q8×C47C22×C4semidirect
132. C2×D4×C47C22×C4semidirect
164. C2×D188C22⋊C4semidirect
194. C2×C47⋊C8C22×C4semidirect
255. C2×Dic47AGL(1,ℤ/8)semidirect
298. C2×#417D47semidirect
333. C2×#103C2×D47semidirect
335. C2×#102C2×D47semidirect
343. C47#20925semidirect
350. C2×#41C4×D47semidirect
427. C2×Q8C2×C4×D47semidirect
449. C2×D4C2×C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940492
C22×#130939862
C2×#156891292
#202458792
#202437332
#202437192
#202431072
#202430812
C2×C47×#4172
#2092547