Finite Group Databasebeta

← #26234049

Subgroups — #26234049

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689164C211yesmaximal, characteristic
2#20246269C211yesmaximal, characteristic
3#20246253C211yesmaximal, characteristic
4#20244129C211yesmaximal, characteristic, complemented
5#20244116C211yesmaximal, characteristic, complemented
6#20243978C211yesmaximal, characteristic, complemented
7#20243973C211yesmaximal, characteristic, complemented
8#20243964C211yesmaximal, characteristic, complemented
9#20243959C211yesmaximal, characteristic, complemented
10#20243482C211yesmaximal, characteristic
11#20243477C211yesmaximal, characteristic
12#20243422C211yesmaximal, characteristic
13#20243287C211yesmaximal, characteristic, complemented
14#20243171C211yesmaximal, characteristic
15C4×#107×C47C211yesmaximal, characteristic, complemented
16C22×#10625324C2211yescharacteristic
17C2×#13093982C2211yescharacteristic
18C2×#13093981—21no—
19C2×C4⋊C4×D47C2211yescharacteristic
20#15689164C418yescomplemented
21C2×#13093957C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×(C2×D47)⋊C8C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27#15689021C2211yescharacteristic
28#15689021C2211yescharacteristic
29#15689019C2211yescharacteristic
30#15689019C2211yescharacteristic
31#15689014—21no—
32#15689012—21no—
33#15689007—21no—
34#15689005—21no—
35#15688987—21no—
36#15688985—21no—
37#15688980C2211yescharacteristic
38#15688980C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4×(C2×D47)⋊C4C2211yescharacteristic
42#15688969—21no—
43#15688886C2211yescharacteristic
44#15688886C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688878C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#107×C47C2211yescharacteristic
55C2×C4×C8×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C4×D47—21no—
61C22×C4×D47C2311yescharacteristic
62C2×#10625324—22no—
63C2×#10625324—22no—
64C2×#10625324C2×C414yes—
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13093982—22no—
68#13093982C2×C414yes—
69#13093981—44no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C4⋊C4×D47—22no—
80C4⋊C4×D47C2×C414yes—
81C2×(C2×D47)⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4D411yescharacteristic
85C2×(C2×D47)⋊C4D411yescharacteristic
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47C2311yescharacteristic
90#13093957C2×C414yes—
91#13093956—22no—
92#13093956C2×C414yes—
93#13093955C2×C414yes—
94(C2×D47)⋊C8—22no—
95#13093934—22no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95D411yescharacteristic
105C47⋊#95D411yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C42×C47—21no—
113C2×C42×C47—21no—
114C2×C42×C47C2311yescharacteristic
115#107×C47—22no—
116#107×C47C2×C414yes—
117C4×C8×C47—22no—
118#95×C47—21no—
119#95×C47—21no—
120C23×D47C2×D411yescharacteristic
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2411yescharacteristic
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—21no—
131C22×Dic47—21no—
132C22×Dic47C2×D411yescharacteristic
133C22×Dic47C2×D411yescharacteristic
134C22×Dic47C2×D411yescharacteristic
135C2×C4×D47—21no—
136C2×C4×D47—21no—
137C2×C4×D47—21no—
138C2×C4×D47—21no—
139C2×C4×D47—24no—
140C2×C4×D47—42no—
141C2×C4×D47—42no—
142C2×C4×D47C22×C412yes—
143C2×C4×D47#4112yes—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47C22×C412yes—
148C2×C8×C47#4112yes—
149C2×C8×C47#4112yes—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47C22×C412yes—
157C4⋊C4×C47C22×C412yes—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—42no—
161C42×C47—42no—
162(C2×D47)⋊C4—24no—
163(C2×D47)⋊C4—22no—
164(C2×D47)⋊C4—41no—
165(C2×D47)⋊C4—42no—
166(C2×D47)⋊C4—81no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47C22×C412yes—
174C4⋊Dic47C22×C412yes—
175Dic47⋊C4—24no—
176Dic47⋊C4—42no—
177Dic47⋊C4—42no—
178Dic47⋊C4—81no—
179Dic47⋊C4—81no—
180C4×Dic47—21no—
181C4×Dic47—21no—
182C4×Dic47—24no—
183C4×Dic47—42no—
184C4×Dic47—42no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8C22×C412yes—
191#10625324—28no—
192C23×C47C22×D411yescharacteristic
193C22×D47—21no—
194C22×D47—21no—
195C22×D47—21no—
196C22×D47—21no—
197C22×D47—21no—
198C22×D47—21no—
199C22×D47C4×D412yes—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47AGL(1,ℤ/8)12yes—
218C2×C4×C47C23×C411yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C472−1+411yescharacteristic
223C2×C4×C472−1+411yescharacteristic
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47C4×D412yes—
245C4×D47—22no—
246C4×D47—22no—
247C4×D47—22no—
248C4×D47—22no—
249C4×D47—48no—
250C8×C47—41no—
251C8×C47#11714yescomplemented
252C47⋊C8—22no—
253C47⋊C8—41no—
254#22624—471nomaximal, Sylow
255C22×C47C2×C4×D411yescharacteristic
256C22×C47#54711yescharacteristic
257C22×C47#54711yescharacteristic
258C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C47#57511yescharacteristic
260C22×C47#57711yescharacteristic
261C22×C47#57811yescharacteristic
262C2×D47—22no—
263C2×D47—22no—
264C2×D47—22no—
265C2×D47—41no—
266C2×D47—41no—
267C2×D47—41no—
268C2×D47—41no—
269C4×C47—22no—
270C4×C47—41no—
271C4×C47—42no—
272C4×C47—42no—
273C4×C47—81no—
274C4×C47—81no—
275C4×C47C2×#11711yescharacteristic
276C4×C47#51911yescharacteristic
277C4×C47#55211yescharacteristic
278C4×C47#55211yescharacteristic
279Dic47—22no—
280Dic47—41no—
281Dic47—41no—
282Dic47—42no—
283Dic47—42no—
284Dic47—81no—
285Dic47—81no—
286C2×#428—471no—
287#2346—471no—
288#2344—471no—
289#1944—471no—
290#1941—471no—
291#1921—471no—
292#1920—471no—
293#1919—471no—
294#1918—471no—
295#1825—471no—
296#1824—471no—
297#1809—471no—
298#1776—471no—
299C4×#107D4711yescharacteristic, complemented
300#1753—471no—
301C2×C47#234411yescharacteristic
302C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C47#294711yescharacteristic
304C2×C47#334311yescharacteristic
305C2×C47#334811yescharacteristic
306C2×C47#335211yescharacteristic
307C2×C47#335411yescharacteristic
308D47—42no—
309C22×M16—471no—
310C2×#117—471no—
311C2×#117—941no—
312C22×C4⋊C4—471no—
313#428—478no—
314C2×#107—471no—
315C2×#107C2×D4711yescharacteristic
316C2×#107C2×D4711yescharacteristic
317C2×#106—471no—
318C2×#106—471no—
319C2×#106—471no—
320C2×C22⋊C8—471no—
321C2×#97—471no—
322C2×C4×C8C2×D4711yescharacteristic
323#389—471no—
324#389—471no—
325#389—471no—
326#389—471no—
327#388—941no—
328#388—941no—
329#387—941no—
330#387—941no—
331#382—941no—
332#381—941no—
333#380—471no—
334#380—471no—
335C4×C4⋊C4—471no—
336C4×C4⋊C4—471no—
337C4×C4⋊C4C2×D4711yescharacteristic
338C4×C4⋊C4C2×D4711yescharacteristic
339C4×C22⋊C4—471no—
340#376—941no—
341#340—471no—
342#340—471no—
343#340—471no—
344#340—471no—
345#340—471no—
346#340—471no—
347#340—471no—
348#340—471no—
349#340C2×D4711yescharacteristic
350#340C2×D4711yescharacteristic
351#336—471no—
352#336—471no—
353C47#2262411yescharacteristic, complemented, Sylow
354C23×C4—471no—
355C23×C4—941no—
356C2×M16—474no—
357C2×M16—942no—
358C2×M16—942no—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8C22×D4711yescharacteristic
362C22×C8C22×D4711yescharacteristic
363#117—474no—
364#117—942no—
365#117—1884no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—474no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—942no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C22⋊C4—471no—
382C2×C22⋊C4—471no—
383C2×C22⋊C4—941no—
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—471no—
389C2×C42—941no—
390C2×C42—941no—
391C2×C42—941no—
392C2×C42C22×D4711yescharacteristic
393#107—22no—
394#107—474no—
395#107C4×D4714yes—
396#106—474no—
397#106—474no—
398#106—942no—
399C22⋊C8—942no—
400#97—942no—
401C4×C8—22no—
402#95—21no—
403#95—21no—
404#95—471no—
405#95—471no—
406#95—941no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95—941no—
414C24—471no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—471no—
420C22×C4—472no—
421C22×C4—472no—
422C22×C4—471no—
423C22×C4—471no—
424C22×C4—944no—
425C22×C4—941no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—1882no—
434C22×C4—1882no—
435C22×C4D4×D4711yescharacteristic
436C22×C4D4×D4711yescharacteristic
437C22×C4C23×D4711yescharacteristic
438M16—948no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—472no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C2×C8C2×C4×D4712yes—
449C2×C8#1062535012yes—
450C2×C8#1062535012yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—472no—
458C4⋊C4—472no—
459C4⋊C4—944no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—3761no—
467C4⋊C4—3761no—
468C4⋊C4—1882no—
469C4⋊C4—1882no—
470C4⋊C4C2×C4×D4712yes—
471C4⋊C4C2×C4×D4712yes—
472C22⋊C4—944no—
473C22⋊C4—942no—
474C22⋊C4—3761no—
475C22⋊C4—1881no—
476C22⋊C4—1882no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—42no—
481C42—944no—
482C42—941no—
483C42—941no—
484C42—1882no—
485C42—1882no—
486C23—472no—
487C23—941no—
488C23—941no—
489C23—941no—
490C23—941no—
491C23—941no—
492C23—941no—
493C23C2×D4×D4711yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—472no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—941no—
526C2×C4—1888no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4#1309404812yes—
538C2×C4C22×C4×D4711yescharacteristic
539C2×C4#41×D4711yescharacteristic
540C2×C4C2×#1062535011yescharacteristic
541C2×C4C2×#1062535011yescharacteristic
542C2×C4#1309407111yescharacteristic
543C2×C4#1309407211yescharacteristic
544C8—41no—
545C8—942no—
546C8—1881no—
547C8#1309398114yescomplemented
548C22—942no—
549C22—942no—
550C22—942no—
551C22—1881no—
552C22—1881no—
553C22—1881no—
554C22—1881no—
555C22C4×D4×D4711yescharacteristic
556C22#1568977411yescharacteristic
557C22#1568977811yescharacteristic
558C22C2×#1309404811yescharacteristic
559C22#1568992011yescharacteristic
560C22#1568993011yescharacteristic
561C22#1568993611yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—942no—
569C4—3761no—
570C4—3761no—
571C4—1882no—
572C4—1882no—
573C4—1881no—
574C4—1881no—
575C4C2×#1309398111yescharacteristic
576C4#1568959911yescharacteristic
577C4#1568981211yescharacteristic
578C4#1568981411yescharacteristic
579C2—1882no—
580C2#2024625311yescharacteristic
581C2C4×#1309404811yescharacteristic
582C2#2025222611yescharacteristic
583C2#2025582411yescharacteristic
584C2#2025589911yescharacteristic
585C2#2025596411yescharacteristic
586C2#2025598511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#107×C47C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenterC4×#107D47
hypocenterC47#22624
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
4. #20244129C2semidirect
5. #20244116C2semidirect
6. #20243978C2semidirect
7. #20243973C2semidirect
8. #20243964C2semidirect
9. #20243959C2semidirect
13. #20243287C2semidirect
15. C4×#107×C47C2semidirect
20. #15689164C4semidirect
251. C8×C47#117semidirect
299. C4×#107D47semidirect
353. C47#22624semidirect
547. C8#13093981semidirect

Maximal subgroups

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

subgroupindex
C2×#156891642
#202462692
#202462532
#202441292
#202441162
#202439782
#202439732
#202439642
#202439592
#202434822
#202434772
#202434222
#202432872
#202431712
C4×#107×C472
#2262447