Finite Group Databasebeta

← #26187683

Subgroups — #26187683

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253400C211yesmaximal, characteristic
2#20252517C211yesmaximal, characteristic, complemented
3#20242288C211yesmaximal, characteristic, complemented
4#20242271C211yesmaximal, characteristic, complemented
5#20242266C211yesmaximal, characteristic, complemented
6#20242264C211yesmaximal, characteristic, complemented
7#20242236C211yesmaximal, characteristic, complemented
8#20242217C211yesmaximal, characteristic, complemented
9#20242200C211yesmaximal, characteristic, complemented
10C8×#10625337C211yesmaximal, characteristic, complemented
11#20242162C211yesmaximal, characteristic, complemented
12#20241853C211yesmaximal, characteristic, complemented
13#20241773C211yesmaximal, characteristic, complemented
14#20241688C211yesmaximal, characteristic, complemented
15C47×#2994C211yesmaximal, characteristic, complemented
16C4×#10625350C2211yescharacteristic
17#15689461C411yescharacteristic
18#15689444C411yescharacteristic
19#15689427C411yescharacteristic
20#15689420C411yescharacteristic
21#15689342C411yescharacteristic
22#15689335C411yescharacteristic
23#15689283C411yescharacteristic
24#15689266C411yescharacteristic
25C8×Dic94C2211yescharacteristic
26C8×Dic94C2211yescharacteristic
27C4×#10625337C2211yescharacteristic
28C4×#10625336C2211yescharacteristic
29C4×#10625335C2211yescharacteristic
30C4×#10625334C2211yescharacteristic
31#15689202C2211yescharacteristic
32#15689202C2211yescharacteristic
33#15689198—21no—
34#15689198—21no—
35#15689176C2211yescharacteristic
36#15689176C2211yescharacteristic
37#15689175—21no—
38#15689147C2211yescharacteristic
39#15689145—21no—
40#15688875C2211yescharacteristic
41#15688875C2211yescharacteristic
42#15688875C2211yescharacteristic
43#15688875C2211yescharacteristic
44Q8⋊(C47⋊C8)C2211yescharacteristic
45Q8⋊(C47⋊C8)C2211yescharacteristic
46#15688853C2211yescharacteristic
47#15688853C2211yescharacteristic
48D4⋊(C47⋊C8)C2211yescharacteristic
49D4⋊(C47⋊C8)C2211yescharacteristic
50#15688850C2211yescharacteristic
51#15688850C2211yescharacteristic
52#15688845C2211yescharacteristic
53#15688845C2211yescharacteristic
54#15688844C2211yescharacteristic
55C8×C47⋊C8C2211yescharacteristic
56C47×#529C2211yescharacteristic
57C47×#446C2211yescharacteristic
58C8×Q8×C47C2211yescharacteristic
59C47×#436C2211yescharacteristic
60C47×#435—21no—
61C8×D4×C47—21no—
62C8×D4×C47C2211yescharacteristic
63C47×#433—21no—
64C47×#431C2211yescharacteristic
65C47×#424—21no—
66C2×C4⋊C8×C47C2211yescharacteristic
67C2×#10625350C2×C411yescharacteristic
68C2×#10625337C2×C411yescharacteristic
69C2×#10625335C2×C411yescharacteristic
70C2×#10625336C2×C411yescharacteristic
71C2×#10625334C2×C411yescharacteristic
72C2×#10625329—21no—
73C2×#10625324—21no—
74C2×#10625324—21no—
75C4×Dic94C2×C411yescharacteristic
76C4×Dic94C2311yescharacteristic
77C4×D188C2×C411yescharacteristic
78C4×D188C2311yescharacteristic
79C4×C22⋊D47—21no—
80C4×C22⋊D47—21no—
81#13093983C2×C411yescharacteristic
82#13093981—21no—
83C42×D47—21no—
84#13093957C2×C411yescharacteristic
85#13093957C2×C411yescharacteristic
86#13093955C2×C411yescharacteristic
87#13093955C2×C411yescharacteristic
88C4⋊(C47⋊C8)—21no—
89C4⋊(C47⋊C8)—21no—
90C4⋊(C47⋊C8)C2×C411yescharacteristic
91C4⋊(C47⋊C8)C2×C411yescharacteristic
92C4⋊(C47⋊C8)C2311yescharacteristic
93C4⋊(C47⋊C8)C2311yescharacteristic
94Dic47⋊C8—21no—
95Dic47⋊C8—21no—
96#13093948C2×C411yescharacteristic
97#13093948C2×C411yescharacteristic
98#13093947C2×C411yescharacteristic
99#13093947C2×C411yescharacteristic
100#13093945C2×C411yescharacteristic
101#13093945C2×C411yescharacteristic
102#13093944C2×C411yescharacteristic
103#13093944C2×C411yescharacteristic
104C22⋊(C47⋊C8)—21no—
105C22⋊(C47⋊C8)C2×C411yescharacteristic
106C22⋊(C47⋊C8)C2×C411yescharacteristic
107(C2×D47)⋊C8—21no—
108(C2×D47)⋊C8—21no—
109C4×C47⋊C8C2311yescharacteristic
110C4×C47⋊C8C2311yescharacteristic
111C4×C47⋊C8C2311yescharacteristic
112C4×C47⋊C8C2311yescharacteristic
113C8×Dic47—21no—
114C8×Dic47—21no—
115C2×M16×C47—21no—
116C22×C8×C47—21no—
117C22×C8×C47—21no—
118#126×C47—21no—
119#125×C47C2×C411yescharacteristic
120#124×C47C2×C411yescharacteristic
121#123×C47—21no—
122C22⋊Q8×C47C2×C411yescharacteristic
123#121×C47C2×C411yescharacteristic
124C4×Q8×C47C2311yescharacteristic
125C4×D4×C47—21no—
126C4×D4×C47C2311yescharacteristic
127#117×C47—21no—
128C2×C42×C47C2311yescharacteristic
129C4⋊C8×C47—21no—
130C4⋊C8×C47—21no—
131C4⋊C8×C47—21no—
132C4⋊C8×C47D411yescharacteristic
133C4⋊C8×C47D411yescharacteristic
134C4⋊C8×C47C2311yescharacteristic
135C4⋊C8×C47C2311yescharacteristic
136C22⋊C8×C47—21no—
137C22⋊C8×C47—21no—
138C22⋊C8×C47D411yescharacteristic
139C22⋊C8×C47D411yescharacteristic
140#97×C47—21no—
141C4×C8×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47C22×C411yescharacteristic
145C2×D4×C47—21no—
146C2×D4×C47C22×C411yescharacteristic
147C22×C4×C47—21no—
148C22×C4×C47C22×C411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47#4111yescharacteristic
151C2×C22⋊D47—21no—
152#10625350—22no—
153#10625350—41no—
154C2×D188C22×C411yescharacteristic
155C2×C4×D47—21no—
156C2×C4×D47—21no—
157C2×C4×D47—21no—
158C2×Dic94C22×C411yescharacteristic
159M16×C47—81no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—81no—
163C2×C8×C47—81no—
164C2×C8×C47C2×D411yescharacteristic
165C2×C8×C47C2×D411yescharacteristic
166C2×C8×C47C2×D411yescharacteristic
167C2×C8×C47C2×D411yescharacteristic
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47C22×C411yescharacteristic
173C4⋊C4×C47C22×C411yescharacteristic
174C22⋊C4×C47—21no—
175C22⋊C4×C47—21no—
176C22⋊C4×C47—21no—
177C22⋊C4×C47—21no—
178C22⋊C4×C47—21no—
179C42×C47—21no—
180C42×C47C22×C411yescharacteristic
181C42×C47C2×D411yescharacteristic
182C42×C47#4111yescharacteristic
183C42×C47C2411yescharacteristic
184C22⋊Dic47—21no—
185#10625337—22no—
186#10625336—41no—
187#10625335—22no—
188#10625334—41no—
189(C2×D47)⋊C4—21no—
190(C2×D47)⋊C4—21no—
191C4⋊Dic47C22×C411yescharacteristic
192C4⋊Dic47C22×C411yescharacteristic
193Dic47⋊C4—21no—
194Dic47⋊C4—21no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—21no—
198#10625329—81no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8C22×C411yescharacteristic
204C2×C47⋊C8C22×C411yescharacteristic
205C2×C47⋊C8C22×C411yescharacteristic
206C2×C47⋊C8C22×C411yescharacteristic
207#10625324—81no—
208#10625324—81no—
209C23×C47—21no—
210C23×C47C4×D411yescharacteristic
211C22×D47—21no—
212Q8×C47—21no—
213Q8×C47—21no—
214D4×C47—21no—
215D4×C47—21no—
216D4×C47—42no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47C4×D411yescharacteristic
228C2×C4×C47C4×D411yescharacteristic
229C2×C4×C47C4×D411yescharacteristic
230C2×C4×C47C23×C411yescharacteristic
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C22⋊D47—42no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237D188—21no—
238D188—21no—
239C4×D47—42no—
240C4×D47—42no—
241C4×D47—41no—
242C4×D47—41no—
243Dic94—21no—
244Dic94#13112yes—
245C8×C47—21no—
246C8×C47—21no—
247C8×C47—21no—
248C8×C47—21no—
249C8×C47—41no—
250C8×C47—41no—
251C47⋊C8—21no—
252C47⋊C8—21no—
253C47⋊C8—21no—
254C47⋊C8—21no—
255C47⋊C8—41no—
256C47⋊C8—41no—
257C47⋊C8—41no—
258C47⋊C8—41no—
259#18033—471nomaximal, Sylow
260C22×C47—21no—
261C22×C47—21no—
262C22×C47—21no—
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47C2×C4×D411yescharacteristic
267C2×D47—41no—
268C2×D47—41no—
269C2×D47—41no—
270C4×C47—21no—
271C4×C47—21no—
272C4×C47—21no—
273C4×C47—21no—
274C4×C47—41no—
275C4×C47—41no—
276C4×C47—41no—
277C4×C47C2×#13111yescharacteristic
278C4×C47#56811yescharacteristic
279C4×C47#57611yescharacteristic
280C4×C47#57811yescharacteristic
281Dic47—22no—
282Dic47—41no—
283Dic47—41no—
284#3102—471no—
285#2994D4711yescharacteristic, complemented
286#2991—471no—
287#1592—471no—
288#1589—471no—
289#1588—471no—
290#1588—471no—
291#1584—471no—
292#1581—471no—
293SD16⋊C8—471no—
294C8×Q16—471no—
295#1571—471no—
296#1495—471no—
297#1477—471no—
298#1454—471no—
299C2×C47—41no—
300C2×C47—81no—
301C2×C47#293311yescharacteristic
302C2×C47#295111yescharacteristic
303C2×C47#295511yescharacteristic
304D47—81no—
305#529C2×D4711yescharacteristic
306C4×#41—471no—
307#485—471no—
308#482—471no—
309#479—471no—
310#478—471no—
311#464—471no—
312#463—471no—
313#452—471no—
314#449—471no—
315#446C2×D4711yescharacteristic
316C8×Q8—471no—
317C8×Q8—471no—
318C8×Q8C2×D4711yescharacteristic
319C4×Q16—471no—
320C4×SD16—471no—
321C4×SD16—471no—
322C4×D8—471no—
323#436—471no—
324#436—471no—
325#436—941no—
326#436—941no—
327#436C2×D4711yescharacteristic
328#435—21no—
329C8×D4—21no—
330C8×D4C2×D4711yescharacteristic
331#433—21no—
332#431—471no—
333#431—471no—
334#431—941no—
335#431C2×D4711yescharacteristic
336#424—21no—
337#423—471no—
338#423—941no—
339C2×C4⋊C8C2×D4711yescharacteristic
340#335—471no—
341#335—471no—
342#335—471no—
343#335—471no—
344Q8⋊C8—471no—
345Q8⋊C8—471no—
346Q8⋊C8—471no—
347Q8⋊C8—471no—
348D4⋊C8—471no—
349D4⋊C8—471no—
350D4⋊C8—471no—
351D4⋊C8—471no—
352#322—471no—
353#322—471no—
354#322—471no—
355C82—471no—
356C47#1803311yescharacteristic, complemented, Sylow
357C2×#41—471no—
358C2×Q16—471no—
359C2×SD16—471no—
360C2×SD16—471no—
361C2×D8—471no—
362C2×M16—21no—
363C2×M16—941no—
364C2×M16—941no—
365C2×M16—941no—
366C22×C8—21no—
367C22×C8—21no—
368#126—21no—
369#125C4×D4711yescharacteristic
370#124C4×D4711yescharacteristic
371#123—21no—
372C22⋊Q8C4×D4711yescharacteristic
373#121C4×D4711yescharacteristic
374C4×Q8—471no—
375C4×Q8—471no—
376C4×Q8C22×D4711yescharacteristic
377C4×D4—21no—
378C4×D4—471no—
379C4×D4—471no—
380C4×D4—941no—
381C4×D4—941no—
382C4×D4C22×D4711yescharacteristic
383#117—21no—
384#117—471no—
385#117—941no—
386C2×C42—941no—
387C2×C42C22×D4711yescharacteristic
388#107—471no—
389#107—471no—
390#106—471no—
391#106—471no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—471no—
396C4⋊C8—471no—
397C4⋊C8—471no—
398C4⋊C8—471no—
399C4⋊C8—941no—
400C4⋊C8—941no—
401C4⋊C8—941no—
402C4⋊C8—941no—
403C4⋊C8C22⋊D4711yescharacteristic
404C4⋊C8C22⋊D4711yescharacteristic
405C4⋊C8C22×D4711yescharacteristic
406C4⋊C8C22×D4711yescharacteristic
407#103—471no—
408#103—471no—
409#103—471no—
410#103—471no—
411#102—471no—
412#102—471no—
413#102—471no—
414#102—471no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—471no—
418C22⋊C8—471no—
419C22⋊C8—941no—
420C22⋊C8—941no—
421C22⋊C8—941no—
422C22⋊C8C22⋊D4711yescharacteristic
423C22⋊C8C22⋊D4711yescharacteristic
424#97—21no—
425C4×C8—21no—
426C4×C8—471no—
427C4×C8—471no—
428C4×C8—471no—
429C4×C8—471no—
430C4×C8—941no—
431C4×C8—941no—
432C4×C8C22×D4711yescharacteristic
433C4×C8C22×D4711yescharacteristic
434#41—942no—
435#41—1881no—
436C2×Q8—471no—
437C2×Q8C2×C4×D4711yescharacteristic
438C2×D4—21no—
439C2×D4—471no—
440C2×D4—941no—
441C2×D4C2×C4×D4711yescharacteristic
442C22×C4—21no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4C2×C4×D4711yescharacteristic
447C22×C4#1062535011yescharacteristic
448C22×C4C2×C22⋊D4711yescharacteristic
449Q16—942no—
450SD16—942no—
451SD16—1881no—
452D8—1881no—
453M16—81no—
454M16—3761no—
455M16—3761no—
456M16—3761no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—81no—
460C2×C8—81no—
461C2×C8—471no—
462C2×C8—471no—
463C2×C8—471no—
464C2×C8—471no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8C2×C22⋊D4711yescharacteristic
470C2×C8C2×C22⋊D4711yescharacteristic
471C2×C8C2×C22⋊D4711yescharacteristic
472C2×C8C2×C22⋊D4711yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—471no—
478C4⋊C4—471no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4C2×C4×D4711yescharacteristic
482C4⋊C4C2×C4×D4711yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—941no—
489C22⋊C4—941no—
490C22⋊C4—941no—
491C42—21no—
492C42—941no—
493C42—941no—
494C42—941no—
495C42C2×C4×D4711yescharacteristic
496C42#1062535011yescharacteristic
497C42C2×C22⋊D4711yescharacteristic
498C42C23×D4711yescharacteristic
499C23—21no—
500C23—941no—
501C23C4×C22⋊D4711yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—472no—
505Q8—941no—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—941no—
510D4—941no—
511D4—1882no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4C4×C22⋊D4711yescharacteristic
530C2×C4C4×C22⋊D4711yescharacteristic
531C2×C4C4×C22⋊D4711yescharacteristic
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C22×C22⋊D4711yescharacteristic
534C2×C4C2×#1062535011yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—941no—
542C8—941no—
543C8—941no—
544C8—941no—
545C8—1881no—
546C8—1881no—
547C8—1881no—
548C8—1881no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—1881no—
556C22—1881no—
557C22—1881no—
558C22C2×C4×C22⋊D4711yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—942no—
567C4—1881no—
568C4—1881no—
569C4#131×D4711yescharacteristic
570C4#1568988811yescharacteristic
571C4#1568992611yescharacteristic
572C4#1568993811yescharacteristic
573C2—41no—
574C2—81no—
575C2—3761no—
576C2#2025211011yescharacteristic
577C2#2025226111yescharacteristic
578C2#2025228911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#2994C2
socleC22×C47C2×C4×D4
prime radical#15689266C4
cosocleC42×C47C24
hypercenter#2994D47
hypocenterC47#18033
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2994

Complements

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

NH ≅ G/Ntype
2. #20252517C2semidirect
3. #20242288C2semidirect
4. #20242271C2semidirect
5. #20242266C2semidirect
6. #20242264C2semidirect
7. #20242236C2semidirect
8. #20242217C2semidirect
9. #20242200C2semidirect
10. C8×#10625337C2semidirect
11. #20242162C2semidirect
12. #20241853C2semidirect
13. #20241773C2semidirect
14. #20241688C2semidirect
15. C47×#2994C2semidirect
285. #2994D47semidirect
356. C47#18033semidirect

Maximal subgroups

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

subgroupindex
#202534002
#202525172
#202422882
#202422712
#202422662
#202422642
#202422362
#202422172
#202422002
C8×#106253372
#202421622
#202418532
#202417732
#202416882
C47×#29942
#1803347