Finite Group Databasebeta

← #26255022

Subgroups — #26255022

Order 12032.

1034 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689262C211yesmaximal, characteristic
2C2×#15689180C211yesmaximal, characteristic, complemented
3#20251719C211yesmaximal, characteristic
4#20244783C212yesmaximal, complemented
5#20244703C212yesmaximal, complemented
6#20244272C212yesmaximal, complemented
7#20244081C212yesmaximal, complemented
8#20243705C211yesmaximal, characteristic
9#20243576C211yesmaximal, characteristic, complemented
10#20243323C211yesmaximal, characteristic, complemented
11C2×C47×#391C211yesmaximal, characteristic, complemented
12C22×#10625336C2212yes—
13C2×#13094007—21no—
14C2×#13093998C2211yescharacteristic
15#15689262—24no—
16#15689180—24no—
17C2×#13093957C2212yes—
18C2×C4⋊(C47⋊C8)C2212yes—
19C2×#13093948C2212yes—
20C2×#13093948C2212yes—
21C2×#13093948C2212yes—
22C2×#13093944C2212yes—
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×#13093935C2212yes—
26#15689025C2211yescharacteristic
27#15689025C2211yescharacteristic
28#15689021C2211yescharacteristic
29#15689021C2211yescharacteristic
30#15689014—21no—
31#15689014—21no—
32#15689002—21no—
33#15688983C2211yescharacteristic
34#15688885C2212yes—
35#15688885C2212yes—
36C23×Q8×C47C2211yescharacteristic
37C22×C4⋊C4×C47C2211yescharacteristic
38C22×C42×C47C2211yescharacteristic
39C47×#391—21no—
40C47×#391—21no—
41C47×#391C2212yes—
42C47×#391C2212yes—
43C2×#95×C47—21no—
44C2×#95×C47—21no—
45C22×D188C2311yescharacteristic
46C22×C22⋊D47—21no—
47C2×#10625336—24no—
48C2×#10625336—24no—
49C2×#10625336—42no—
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52#13094007—44no—
53#13093998—22no—
54#13093998—22no—
55C2×C4⋊Dic47C2311yescharacteristic
56C2×C4⋊Dic47C2311yescharacteristic
57C2×C4⋊Dic47C2311yescharacteristic
58C2×Dic47⋊C4—21no—
59C2×Dic47⋊C4—21no—
60C2×C22⋊Dic47—21no—
61C2×(C2×D47)⋊C4—21no—
62C2×(C2×D47)⋊C4—21no—
63C2×(C2×D47)⋊C4—21no—
64C2×(C2×D47)⋊C4—21no—
65C2×(C2×D47)⋊C4—21no—
66#13093957—42no—
67C4⋊(C47⋊C8)—24no—
68#13093948—24no—
69#13093948—24no—
70#13093948—42no—
71#13093944—42no—
72C22⋊(C47⋊C8)—22no—
73C22⋊(C47⋊C8)D414yes—
74#13093935—24no—
75C47⋊#95—21no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C22×Q8×C47—24no—
79C22×Q8×C47—21no—
80C22×Q8×C47—22no—
81C22×Q8×C47—22no—
82C22×Q8×C47—42no—
83C22×Q8×C47C2312yes—
84C23×C4×C47—21no—
85C23×C4×C47—21no—
86C23×C4×C47D411yescharacteristic
87C23×C4×C47D411yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C2×C4⋊C4×C47—21no—
90C2×C4⋊C4×C47—42no—
91C2×C4⋊C4×C47C2312yes—
92C2×C42×C47—22no—
93C2×C42×C47—22no—
94C2×C42×C47D411yescharacteristic
95C2×C42×C47D411yescharacteristic
96C2×C42×C47C2311yescharacteristic
97C2×C42×C47C2311yescharacteristic
98#95×C47—22no—
99#95×C47—22no—
100#95×C47—22no—
101#95×C47—22no—
102C24×C47C2×D411yescharacteristic
103C23×D47—21no—
104C2×Q8×C47—24no—
105C2×Q8×C47—22no—
106C2×Q8×C47—22no—
107C2×Q8×C47—22no—
108C2×Q8×C47—22no—
109C2×Q8×C47—22no—
110C2×Q8×C47—48no—
111C2×Q8×C47—48no—
112C2×Q8×C47—42no—
113C2×Q8×C47—42no—
114C2×Q8×C47#4114yes—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—24no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—41no—
126C22×C4×C47—42no—
127C22×C4×C47—42no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47#4112yes—
138C22×C4×C47C2411yescharacteristic
139C2×C22⋊D47—42no—
140C2×C22⋊D47—42no—
141C2×C22⋊D47—42no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C2×D188—21no—
146C2×D188—21no—
147C2×D188—22no—
148C2×D188—22no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—42no—
152C42×C47—22no—
153C42×C47—22no—
154C42×C47—21no—
155C42×C47—21no—
156C42×C47—21no—
157C42×C47—21no—
158C22⋊Dic47—42no—
159#10625336—48no—
160(C2×D47)⋊C4—42no—
161(C2×D47)⋊C4—42no—
162(C2×D47)⋊C4—42no—
163(C2×D47)⋊C4—42no—
164(C2×D47)⋊C4—42no—
165C4⋊Dic47—21no—
166C4⋊Dic47—22no—
167C4⋊Dic47—21no—
168C4⋊Dic47—22no—
169Dic47⋊C4—42no—
170Dic47⋊C4—42no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8C2×D414yes—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47#12112yes—
182C23×C47AGL(1,ℤ/8)12yes—
183C23×C47#13712yes—
184C23×C47C22×D411yescharacteristic
185C22×D47—41no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47—41no—
189C22×D47—41no—
190C22×D47—41no—
191C22×D47—41no—
192Q8×C47—24no—
193Q8×C47—24no—
194Q8×C47—48no—
195Q8×C47—42no—
196Q8×C47—42no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—24no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—44no—
214C2×C4×C47—48no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—44no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—84no—
222C2×C4×C47#12112yes—
223C2×C4×C47#13712yes—
224C2×C4×C47C22×D411yescharacteristic
225C2×C4×C47C2×#4111yescharacteristic
226C2×C4×C47C2×#4112yes—
227C2×C4×C47H(5,𝔽2)12yes—
228C22⋊D47—84no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238D188—42no—
239D188—42no—
240C47⋊C8—24no—
241C47⋊C8—42no—
242#24509—471nomaximal, Sylow
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47#46814yes—
250C22×C47C2×#12111yescharacteristic
251C22×C47#52511yescharacteristic
252C22×C47#53511yescharacteristic
253C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C47C2×#13711yescharacteristic
255C22×C47C2×#13711yescharacteristic
256C22×C47#57511yescharacteristic
257C2×D47—81no—
258C2×D47—81no—
259C2×D47—81no—
260C2×D47—81no—
261C2×D47—81no—
262C2×D47—81no—
263C2×D47—81no—
264C4×C47—22no—
265C4×C47—22no—
266C4×C47—44no—
267C4×C47—41no—
268C4×C47—41no—
269C4×C47—41no—
270C4×C47—82no—
271C4×C47#54612yes—
272C4×C47#55112yes—
273Dic47—81no—
274Dic47—81no—
275Dic47—81no—
276C2×#448—471no—
277C2×#432—471no—
278#2854—471no—
279C2×#391D4711yescharacteristic, complemented
280#2068—472no—
281#2055—472no—
282#1973—472no—
283#1935—472no—
284#1874—471no—
285#1852—471no—
286#1787—471no—
287C2×C47—24no—
288C2×C47#241111yescharacteristic
289C2×C47C2×#46811yescharacteristic
290C2×C47#305611yescharacteristic
291C2×C47#311211yescharacteristic
292C2×C47#311711yescharacteristic
293C2×C47#321311yescharacteristic
294C2×C47#321811yescharacteristic
295D47—161no—
296C23×Q8C2×D4711yescharacteristic
297C22×SD16—472no—
298C2×#123—941no—
299C2×#121—471no—
300C22×C4⋊C4C2×D4711yescharacteristic
301C22×C42C2×D4711yescharacteristic
302#448—944no—
303#432—944no—
304C2×#107—472no—
305C2×C4⋊C8—472no—
306C2×#103—472no—
307C2×#103—472no—
308C2×#103—472no—
309C2×#102—472no—
310C2×C22⋊C8—471no—
311C2×C22⋊C8—471no—
312C2×#97—472no—
313#391—21no—
314#391—21no—
315#391C2×D4712yes—
316#391C2×D4712yes—
317#390—471no—
318#390—471no—
319#389—471no—
320#389—471no—
321#388—941no—
322#388—941no—
323#386—941no—
324#380—471no—
325C2×#95—21no—
326C2×#95—21no—
327#340—472no—
328#340—472no—
329C47#2450911yescharacteristic, complemented, Sylow
330C22×Q8—24no—
331C22×Q8—21no—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—42no—
335C22×Q8C22×D4712yes—
336C22×D4—471no—
337C22×D4—941no—
338C23×C4—21no—
339C23×C4—21no—
340C23×C4C22⋊D4711yescharacteristic
341C23×C4C22⋊D4711yescharacteristic
342C23×C4C22×D4711yescharacteristic
343C2×SD16—944no—
344C2×SD16—944no—
345C2×SD16—1882no—
346C22×C8—472no—
347C22×C8—472no—
348#123—1884no—
349#121—942no—
350#121—942no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—471no—
354C2×C4⋊C4—471no—
355C2×C4⋊C4—471no—
356C2×C4⋊C4—941no—
357C2×C4⋊C4—941no—
358C2×C4⋊C4C22×D4712yes—
359C2×C22⋊C4—941no—
360C2×C22⋊C4—941no—
361C2×C22⋊C4—941no—
362C2×C22⋊C4—941no—
363C2×C22⋊C4—941no—
364C2×C22⋊C4—941no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42C22⋊D4711yescharacteristic
368C2×C42C22⋊D4711yescharacteristic
369C2×C42C22×D4711yescharacteristic
370C2×C42C22×D4711yescharacteristic
371#107—1882no—
372C4⋊C8—944no—
373#103—944no—
374#103—944no—
375#103—1882no—
376#102—1882no—
377C22⋊C8—474no—
378C22⋊C8—942no—
379#97—944no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—941no—
385#95—941no—
386#95—941no—
387C24—941no—
388C24C2×C22⋊D4711yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8#1062535014yes—
400C2×D4—941no—
401C2×D4—941no—
402C2×D4—942no—
403C2×D4—942no—
404C2×D4—1882no—
405C2×D4—1882no—
406C2×D4—1882no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—941no—
424C22×C4—941no—
425C22×C4—941no—
426C22×C4D4×D4712yes—
427C22×C4#1062535212yes—
428C22×C4C2×C22⋊D4711yescharacteristic
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C2×C22⋊D4711yescharacteristic
432C22×C4C2×C22⋊D4711yescharacteristic
433C22×C4C23×D4711yescharacteristic
434SD16—1888no—
435C2×C8—474no—
436C2×C8—942no—
437C2×C8—942no—
438C2×C8—942no—
439C2×C8—942no—
440C2×C8—942no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—941no—
445C4⋊C4—941no—
446C4⋊C4—942no—
447C4⋊C4—942no—
448C4⋊C4—1882no—
449C4⋊C4—1882no—
450C22⋊C4—1882no—
451C22⋊C4—1882no—
452C22⋊C4—1882no—
453C22⋊C4—1882no—
454C22⋊C4—1882no—
455C22⋊C4—1882no—
456C42—22no—
457C42—22no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—1881no—
467C23—1881no—
468C23—1881no—
469C23—1881no—
470C23—1881no—
471C23—1881no—
472C23—1881no—
473C23#1309399612yes—
474C23#1309404912yes—
475C23#1309405512yes—
476C23C22×C22⋊D4711yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482D4—3764no—
483D4—1882no—
484D4—1882no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—84no—
510C2×C4—1881no—
511C2×C4—1881no—
512C2×C4—1881no—
513C2×C4—1881no—
514C2×C4—1881no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1881no—
518C2×C4—1881no—
519C2×C4#1309399612yes—
520C2×C4#1309405512yes—
521C2×C4C2×D4×D4711yescharacteristic
522C2×C4C2×#1062535211yescharacteristic
523C2×C4C2×#1062535012yes—
524C2×C4#1309406912yes—
525C8—944no—
526C8—1882no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—3761no—
534C22—3761no—
535C22—3761no—
536C22—3761no—
537C22—3761no—
538C22—3761no—
539C22—3761no—
540C22#1568937114yes—
541C22C2×#1309399611yescharacteristic
542C22#1568963411yescharacteristic
543C22#1568969011yescharacteristic
544C22C2×#1309404911yescharacteristic
545C22C2×#1309405511yescharacteristic
546C22C2×#1309405511yescharacteristic
547C22#1568992111yescharacteristic
548C4—22no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—44no—
554C4—82no—
555C4—3761no—
556C4—3761no—
557C4—3761no—
558C4#1568976312yes—
559C4#1568979912yes—
560C2—24no—
561C2—7521no—
562C2#2024677311yescharacteristic
563C2C2×#1568937111yescharacteristic
564C2#2025301411yescharacteristic
565C2#2025351011yescharacteristic
566C2#2025355911yescharacteristic
567C2#2025437511yescharacteristic
568C2#2025441511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#391C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenterC2×#391D47
hypocenterC47#24509
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C2×#15689180C2semidirect
4. #20244783C2semidirect
5. #20244703C2semidirect
6. #20244272C2semidirect
7. #20244081C2semidirect
9. #20243576C2semidirect
10. #20243323C2semidirect
11. C2×C47×#391C2semidirect
279. C2×#391D47semidirect
329. C47#24509semidirect

Maximal subgroups

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

subgroupindex
C2×#156892622
C2×#156891802
#202517192
#202447832
#202447032
#202442722
#202440812
#202437052
#202435762
#202433232
C2×C47×#3912
#2450947