Finite Group Databasebeta

← #26234051

Subgroups — #26234051

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×#15689165C211yesmaximal, characteristic
2#20246271C211yesmaximal, characteristic
3#20246255C211yesmaximal, characteristic
4#20244131C211yesmaximal, characteristic, complemented
5#20244116C211yesmaximal, characteristic, complemented
6#20243981C211yesmaximal, characteristic, complemented
7#20243974C211yesmaximal, characteristic, complemented
8#20243967C211yesmaximal, characteristic, complemented
9#20243960C211yesmaximal, characteristic, complemented
10#20243484C211yesmaximal, characteristic
11#20243479C211yesmaximal, characteristic
12#20243421C211yesmaximal, characteristic
13#20243285C211yesmaximal, characteristic, complemented
14C4×#13093958C211yesmaximal, characteristic, complemented
15C47×#1753C211yesmaximal, characteristic
16C2×#13093983C2211yescharacteristic
17C2×#13093980—21no—
18C22×C4⋊Dic47C2211yescharacteristic
19#15689165C418yescomplemented
20C2×#13093958C2211yescharacteristic
21C2×#13093958C2211yescharacteristic
22C2×#13093958C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×#13093956C2211yescharacteristic
25C2×#13093956C2211yescharacteristic
26#15689021C2211yescharacteristic
27#15689021C2211yescharacteristic
28#15689021C2211yescharacteristic
29#15689021C2211yescharacteristic
30#15689011—21no—
31#15689011—21no—
32#15689008—21no—
33#15689008—21no—
34#15688987—21no—
35#15688984—21no—
36#15688983C2211yescharacteristic
37#15688983C2211yescharacteristic
38C4×C4⋊Dic47C2211yescharacteristic
39C4×C4⋊Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41C4×C4⋊Dic47C2211yescharacteristic
42#15688968—21no—
43#15688886C2211yescharacteristic
44#15688886C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688886C2211yescharacteristic
52#15688886C2211yescharacteristic
53C22×M16×C47C2211yescharacteristic
54C2×C22⋊C8×C47C2211yescharacteristic
55C2×#97×C47C2211yescharacteristic
56C2×C4×C8×C47C2211yescharacteristic
57C4×C22⋊C4×C47C2211yescharacteristic
58C47×#336C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C23×Dic47—21no—
61#13093983—22no—
62#13093983C2×C414yes—
63#13093980—44no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47C2×C414yes—
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87#13093958—22no—
88#13093958C2×C414yes—
89#13093958C2×C414yes—
90#13093956—22no—
91#13093956C2×C414yes—
92#13093956C2×C414yes—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C23×C4×C47C2311yescharacteristic
104C2×M16×C47—22no—
105C2×M16×C47—22no—
106C2×M16×C47C2×C414yes—
107C22×C8×C47C2311yescharacteristic
108C22×C8×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C2×C22⋊C4×C47D411yescharacteristic
112C2×C22⋊C4×C47D411yescharacteristic
113C2×C42×C47C2311yescharacteristic
114C2×C42×C47C2311yescharacteristic
115C22⋊C8×C47—22no—
116#97×C47—22no—
117C4×C8×C47—22no—
118#95×C47D411yescharacteristic
119#95×C47D411yescharacteristic
120C24×C47C2×D411yescharacteristic
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47C22×C412yes—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47#4112yes—
132C22×C4×C47C2411yescharacteristic
133C22×Dic47—24no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—42no—
143C22×Dic47—42no—
144M16×C47—28no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47C22×C412yes—
154C2×C8×C47C22×C412yes—
155C2×C8×C47#4112yes—
156C2×C8×C47#4112yes—
157C22⋊C4×C47—22no—
158C22⋊C4×C47—41no—
159C42×C47—21no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—21no—
163C22⋊Dic47—24no—
164C22⋊Dic47—42no—
165C22⋊Dic47—81no—
166C4⋊Dic47—24no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47C22×C412yes—
180C4⋊Dic47C22×C412yes—
181C4⋊Dic47C22×C412yes—
182C4⋊Dic47C22×C412yes—
183Dic47⋊C4—42no—
184Dic47⋊C4—42no—
185Dic47⋊C4—81no—
186Dic47⋊C4—81no—
187C4×Dic47—24no—
188C4×Dic47—42no—
189C4×Dic47—42no—
190C4×Dic47—42no—
191C4×Dic47—42no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47C4×D412yes—
199C23×C47C22×D411yescharacteristic
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47C4×D412yes—
218C2×C4×C47AGL(1,ℤ/8)12yes—
219C2×C4×C47C23×C411yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×C4×C472−1+411yescharacteristic
224C2×C4×C472−1+411yescharacteristic
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—48no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
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×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C8×C47—22no—
251C8×C47—41no—
252C8×C47—41no—
253C8×C47#11714yescomplemented
254#22624—471nomaximal, Sylow
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47—41no—
262C22×C47C2×C4×D411yescharacteristic
263C22×C47#54711yescharacteristic
264C22×C47#54711yescharacteristic
265C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C47#57511yescharacteristic
267C22×C47#57711yescharacteristic
268C22×C47#57811yescharacteristic
269C4×C47—22no—
270C4×C47—22no—
271C4×C47—41no—
272C4×C47—41no—
273C4×C47—41no—
274C4×C47C2×#11711yescharacteristic
275C4×C47#51911yescharacteristic
276C4×C47#55211yescharacteristic
277C4×C47#55211yescharacteristic
278Dic47—42no—
279Dic47—42no—
280Dic47—42no—
281Dic47—42no—
282Dic47—81no—
283Dic47—81no—
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×#107—471no—
300#1753D4711yescharacteristic
301C2×C47—42no—
302C2×C47#234411yescharacteristic
303C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C47#294711yescharacteristic
305C2×C47#334311yescharacteristic
306C2×C47#334811yescharacteristic
307C2×C47#335211yescharacteristic
308C2×C47#335411yescharacteristic
309C22×M16C2×D4711yescharacteristic
310C2×#117—471no—
311C2×#117—941no—
312C22×C4⋊C4—471no—
313#428—478no—
314C2×#107—471no—
315C2×#107—471no—
316C2×#107—471no—
317C2×#106—471no—
318C2×#106—471no—
319C2×#106—471no—
320C2×C22⋊C8C2×D4711yescharacteristic
321C2×#97C2×D4711yescharacteristic
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⋊C4—471no—
338C4×C4⋊C4—471no—
339C4×C22⋊C4C2×D4711yescharacteristic
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#340—471no—
350#340—471no—
351#336C2×D4711yescharacteristic
352#336C2×D4711yescharacteristic
353C47#2262411yescharacteristic, complemented, Sylow
354C23×C4—941no—
355C23×C4C22×D4711yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C4×D4714yes—
359C22×C8C22×D4711yescharacteristic
360C22×C8C22×D4711yescharacteristic
361C22×C8C22×D4711yescharacteristic
362C22×C8C22×D4711yescharacteristic
363#117—474no—
364#117—942no—
365#117—1884no—
366C2×C4⋊C4—474no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C22⋊C4—941no—
382C2×C22⋊C4—941no—
383C2×C22⋊C4—941no—
384C2×C22⋊C4D18811yescharacteristic
385C2×C22⋊C4D18811yescharacteristic
386C2×C42—941no—
387C2×C42—941no—
388C2×C42—941no—
389C2×C42—941no—
390C2×C42—941no—
391C2×C42C22×D4711yescharacteristic
392C2×C42C22×D4711yescharacteristic
393#107—474no—
394#107—474no—
395#107—942no—
396#106—474no—
397#106—474no—
398#106—942no—
399C22⋊C8—22no—
400#97—22no—
401C4×C8—22no—
402#95—941no—
403#95—941no—
404#95—941no—
405#95—941no—
406#95—941no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95D18811yescharacteristic
413#95D18811yescharacteristic
414C24C2×D18811yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—944no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4—941no—
423C22×C4—941no—
424C22×C4—941no—
425C22×C4—941no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—1882no—
429C22×C4—1882no—
430C22×C4C2×C4×D4712yes—
431C22×C4C2×D18811yescharacteristic
432C22×C4C2×D18811yescharacteristic
433C22×C4C2×D18811yescharacteristic
434C22×C4C2×D18811yescharacteristic
435C22×C4C2×D18811yescharacteristic
436C22×C4#1062535012yes—
437C22×C4C23×D4711yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8C2×C4×D4712yes—
448C2×C8C2×C4×D4712yes—
449C2×C8#1062535212yes—
450C2×C8#1062535212yes—
451C4⋊C4—472no—
452C4⋊C4—472no—
453C4⋊C4—472no—
454C4⋊C4—472no—
455C4⋊C4—944no—
456C4⋊C4—941no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—941no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—3761no—
469C4⋊C4—3761no—
470C4⋊C4—1882no—
471C4⋊C4—1882no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—944no—
475C22⋊C4—3761no—
476C22⋊C4—1882no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—944no—
482C42—1882no—
483C42—1882no—
484C42—1882no—
485C42—1882no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×D18812yes—
493C23C22×D18811yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—942no—
512C2×C4—942no—
513C2×C4—942no—
514C2×C4—942no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1881no—
518C2×C4—1881no—
519C2×C4—1881no—
520C2×C4—1881no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1888no—
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×C4C4×D18812yes—
537C2×C4#1309405212yes—
538C2×C4C22×C4×D4711yescharacteristic
539C2×C4C2×#1062535211yescharacteristic
540C2×C4C2×#1062535211yescharacteristic
541C2×C4C2×#1062535011yescharacteristic
542C2×C4#1309407111yescharacteristic
543C2×C4#1309407111yescharacteristic
544C8—22no—
545C8—41no—
546C8—41no—
547C8#1309398014yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×D18811yescharacteristic
556C22#1568977711yescharacteristic
557C22#1568977711yescharacteristic
558C22C2×#1309405211yescharacteristic
559C22#1568992311yescharacteristic
560C22#1568993511yescharacteristic
561C22#1568994011yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—3761no—
568C4—3761no—
569C4—3761no—
570C4—3761no—
571C4—1882no—
572C4—1882no—
573C4—1882no—
574C4—1882no—
575C4C2×#1309398011yescharacteristic
576C4#1568960011yescharacteristic
577C4#1568981311yescharacteristic
578C4#1568981311yescharacteristic
579C2—42no—
580C2#2024625511yescharacteristic
581C2C4×#1309405211yescharacteristic
582C2#2025223311yescharacteristic
583C2#2025583011yescharacteristic
584C2#2025590511yescharacteristic
585C2#2025596211yescharacteristic
586C2#2025598411yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1753C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1753D47
hypocenterC47#22624
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #20244131C2semidirect
5. #20244116C2semidirect
6. #20243981C2semidirect
7. #20243974C2semidirect
8. #20243967C2semidirect
9. #20243960C2semidirect
13. #20243285C2semidirect
14. C4×#13093958C2semidirect
19. #15689165C4semidirect
253. C8×C47#117semidirect
353. C47#22624semidirect
547. C8#13093980semidirect

Maximal subgroups

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

subgroupindex
C2×#156891652
#202462712
#202462552
#202441312
#202441162
#202439812
#202439742
#202439672
#202439602
#202434842
#202434792
#202434212
#202432852
C4×#130939582
C47×#17532
#2262447