Finite Group Databasebeta

← #26228357

Subgroups — #26228357

Order 12032.

796 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689423C211yesmaximal, characteristic
2#20245943C211yesmaximal, characteristic, complemented
3#20245853C211yesmaximal, characteristic
4#20244343C211yesmaximal, characteristic, complemented
5#20244154C211yesmaximal, characteristic, complemented
6#20244120C211yesmaximal, characteristic, complemented
7#20244115C211yesmaximal, characteristic, complemented
8#20244068C211yesmaximal, characteristic
9#20244068C211yesmaximal, characteristic
10#20244018C211yesmaximal, characteristic
11#20244011C211yesmaximal, characteristic
12#20243983C211yesmaximal, characteristic, complemented
13#20243976C211yesmaximal, characteristic, complemented
14C4⋊C8×Dic47C211yesmaximal, characteristic, complemented
15C47×#1804C211yesmaximal, characteristic
16C2×#13094016C2211yescharacteristic
17C2×#13093983—21no—
18#15689423C418yescomplemented
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×C8×Dic47C2211yescharacteristic
27C2×C8×Dic47C2211yescharacteristic
28#15689026C2211yescharacteristic
29#15689026C2211yescharacteristic
30#15689022C2211yescharacteristic
31#15689022C2211yescharacteristic
32#15689019C2211yescharacteristic
33#15689019C2211yescharacteristic
34#15689016—21no—
35#15689016—21no—
36#15689005—21no—
37#15689005—21no—
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41#15688974—21no—
42#15688974—21no—
43C4×C22⋊Dic47—21no—
44C42×Dic47C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×C4⋊C8×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C47×#382C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62#13094016—22no—
63#13094016C2×C414yes—
64#13093983—44no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88#13093957—41no—
89#13093957C2×C414yes—
90#13093955—41no—
91#13093955C2×C414yes—
92C4⋊(C47⋊C8)C2×C414yes—
93#13093945C2×C414yes—
94#13093945C2×C414yes—
95C8×Dic47—41no—
96C8×Dic47—41no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C22×D4×C47C2311yescharacteristic
105C23×C4×C47—21no—
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108C4×D4×C47—22no—
109C4×D4×C47C2×C414yes—
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C22⋊C4×C47—21no—
114C2×C42×C47C2311yescharacteristic
115C4⋊C8×C47—22no—
116#102×C47—41no—
117#102×C47—41no—
118#95×C47—21no—
119#95×C47—21no—
120C24×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47C22×C412yes—
127C22×C4×C47—21no—
128C22×C4×C47—24no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2411yescharacteristic
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143C22×Dic47C2×D411yescharacteristic
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47C22×C412yes—
156C22⋊C4×C47—24no—
157C42×C47C22×C412yes—
158C42×C47#4112yes—
159C22⋊Dic47—42no—
160C22⋊Dic47—42no—
161C22⋊Dic47—42no—
162C22⋊Dic47—42no—
163C4⋊Dic47—21no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—42no—
170C4⋊Dic47—42no—
171C4⋊Dic47C22×C412yes—
172C4⋊Dic47C22×C412yes—
173Dic47⋊C4—24no—
174Dic47⋊C4—24no—
175Dic47⋊C4—42no—
176Dic47⋊C4—42no—
177C4×Dic47—21no—
178C4×Dic47—21no—
179C4×Dic47—21no—
180C4×Dic47—21no—
181C4×Dic47—22no—
182C4×Dic47—22no—
183C4×Dic47—22no—
184C4×Dic47—41no—
185C4×Dic47—42no—
186C4×Dic47—42no—
187C4×Dic47—42no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8C22×C412yes—
193C2×C47⋊C8C22×C412yes—
194C23×C47—22no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47C22×D411yescharacteristic
202D4×C47—22no—
203D4×C47—22no—
204D4×C47—41no—
205D4×C47—41no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—22no—
211C2×C4×C47—48no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47C4×D412yes—
221C2×C4×C47C4×D412yes—
222C2×C4×C47C23×C411yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C47H(5,𝔽2)11yescharacteristic
225C2×C4×C472−1+411yescharacteristic
226C2×C4×C472−1+411yescharacteristic
227C2×C4×C472−1+411yescharacteristic
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
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#13512yes—
250C2×Dic47#13512yes—
251C2×Dic47AGL(1,ℤ/8)12yes—
252C8×C47—41no—
253C8×C47—41no—
254C47⋊C8—22no—
255C47⋊C8—22no—
256#22114—471nomaximal, Sylow
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—81no—
261C22×C47—81no—
262C22×C47—81no—
263C22×C47—81no—
264C22×C47C2×C4×D411yescharacteristic
265C22×C47#53611yescharacteristic
266C22×C47#53711yescharacteristic
267C22×C47C2×#13511yescharacteristic
268C22×C47C2×#13511yescharacteristic
269C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C47#57511yescharacteristic
271C4×C47—22no—
272C4×C47—22no—
273C4×C47—42no—
274C4×C47—81no—
275C4×C47—81no—
276C4×C47#51911yescharacteristic
277C4×C47#52011yescharacteristic
278C4×C47#54111yescharacteristic
279C4×C47#54211yescharacteristic
280Dic47—22no—
281Dic47—22no—
282Dic47—22no—
283Dic47—42no—
284Dic47—42no—
285Dic47—41no—
286Dic47—81no—
287Dic47—81no—
288C2×#478—471no—
289C4×#125—471no—
290#2293—471no—
291#1989—471no—
292#1949—471no—
293#1942—471no—
294#1941—471no—
295#1934—471no—
296#1934—471no—
297#1926—471no—
298#1925—471no—
299#1921—471no—
300#1920—471no—
301#1804D4711yescharacteristic
302C4×C4⋊C8—471no—
303C2×C47—82no—
304C2×C47#232311yescharacteristic
305C2×C47C4×#13511yescharacteristic
306C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
307C2×C47#322811yescharacteristic
308C2×C47#323211yescharacteristic
309C2×C47#324711yescharacteristic
310C2×C47#325011yescharacteristic
311C2×#125—471no—
312C2×C4×D4C2×D4711yescharacteristic
313C2×#117—941no—
314#478—478no—
315C2×#107—471no—
316C2×#107—471no—
317C2×#106—471no—
318C2×#106—471no—
319C2×C4⋊C8—471no—
320C2×C4⋊C8C2×D4711yescharacteristic
321C2×#102—471no—
322C2×#102—471no—
323C2×#102C2×D4711yescharacteristic
324C2×#102C2×D4711yescharacteristic
325C2×C4×C8—471no—
326C2×C4×C8—471no—
327#390—471no—
328#390—471no—
329#389—471no—
330#389—471no—
331#389—471no—
332#389—471no—
333#388—941no—
334#388—941no—
335#387—941no—
336#387—941no—
337#382C2×D4711yescharacteristic
338C4×C4⋊C4—471no—
339C4×C4⋊C4—471no—
340C4×C4⋊C4—471no—
341C4×C4⋊C4—941no—
342C4×C4⋊C4—941no—
343C4×C22⋊C4—941no—
344C43—471no—
345#340—471no—
346#340—471no—
347#340—471no—
348#340—471no—
349#340—471no—
350#340—471no—
351#340C2×D4711yescharacteristic
352#340C2×D4711yescharacteristic
353#336—471no—
354#336—471no—
355C47#2211411yescharacteristic, complemented, Sylow
356C22×D4C22×D4711yescharacteristic
357C23×C4—21no—
358C22×C8—471no—
359C22×C8—471no—
360C22×C8C22×D4711yescharacteristic
361C22×C8C22×D4711yescharacteristic
362#125—474no—
363#125—942no—
364C4×D4—22no—
365C4×D4C4×D4714yes—
366#117—1884no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—941no—
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×C22⋊C4—21no—
381C2×C22⋊C4—941no—
382C2×C22⋊C4—941no—
383C2×C22⋊C4—941no—
384C2×C22⋊C4—941no—
385C2×C42—471no—
386C2×C42—471no—
387C2×C42—471no—
388C2×C42—471no—
389C2×C42—471no—
390C2×C42—471no—
391C2×C42—941no—
392C2×C42—941no—
393C2×C42—941no—
394C2×C42C22×D4711yescharacteristic
395#107—474no—
396#107—1881no—
397#106—474no—
398#106—1881no—
399C4⋊C8—22no—
400C4⋊C8—474no—
401#102—41no—
402#102—41no—
403#102—474no—
404#102—474no—
405C4×C8—1881no—
406C4×C8—1881no—
407#95—21no—
408#95—21no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D4712yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—471no—
430C22×C4—471no—
431C22×C4—471no—
432C22×C4—471no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4D4×D4711yescharacteristic
438C22×C4D4×D4711yescharacteristic
439C22×C4C23×D4711yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—472no—
447C2×C8—472no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
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—944no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—1882no—
468C4⋊C4—1882no—
469C4⋊C4—1882no—
470C4⋊C4—1882no—
471C4⋊C4C2×C4×D4712yes—
472C22⋊C4—24no—
473C22⋊C4—1882no—
474C22⋊C4—1882no—
475C22⋊C4—1882no—
476C22⋊C4—1882no—
477C42—942no—
478C42—942no—
479C42—942no—
480C42—941no—
481C42—941no—
482C42—941no—
483C42—941no—
484C42—1881no—
485C42—1882no—
486C42—1882no—
487C42—1882no—
488C42C2×C4×D4712yes—
489C42#1062535212yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D4711yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—472no—
517C2×C4—472no—
518C2×C4—472no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—941no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4(C22⋊D47)⋊C412yes—
541C2×C4(C22⋊D47)⋊C412yes—
542C2×C4C22×C4×D4711yescharacteristic
543C2×C4C2×#1062535211yescharacteristic
544C2×C4#1309406911yescharacteristic
545C2×C4#1309407111yescharacteristic
546C2×C4#1309407111yescharacteristic
547C2×C4#1309407211yescharacteristic
548C8—41no—
549C8—41no—
550C8—942no—
551C8—942no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×(C22⋊D47)⋊C411yescharacteristic
560C22#1568969611yescharacteristic
561C22#1568970211yescharacteristic
562C22#135×D4711yescharacteristic
563C22#135×D4711yescharacteristic
564C22AGL(1,ℤ/8)×D4711yescharacteristic
565C22#1568991911yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4—942no—
572C4—942no—
573C4—942no—
574C4—3761no—
575C4—3761no—
576C4—1882no—
577C4—1881no—
578C4—1882no—
579C4#1568960011yescharacteristic
580C4#1568960411yescharacteristic
581C4#1568973311yescharacteristic
582C4#1568974011yescharacteristic
583C2—82no—
584C2#2024606611yescharacteristic
585C2#2025215611yescharacteristic
586C2#2025219211yescharacteristic
587C2#2025452311yescharacteristic
588C2#2025455411yescharacteristic
589C2#2025471111yescharacteristic
590C2#2025473711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1804C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1804D47
hypocenterC47#22114
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. #20245943C2semidirect
4. #20244343C2semidirect
5. #20244154C2semidirect
6. #20244120C2semidirect
7. #20244115C2semidirect
12. #20243983C2semidirect
13. #20243976C2semidirect
14. C4⋊C8×Dic47C2semidirect
18. #15689423C4semidirect
355. C47#22114semidirect

Maximal subgroups

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

subgroupindex
C2×#156894232
#202459432
#202458532
#202443432
#202441542
#202441202
#202441152
#202440682
#202440182
#202440112
#202439832
#202439762
C4⋊C8×Dic472
C47×#18042
#2211447