Finite Group Databasebeta

← #26250111

Subgroups — #26250111

Order 12032.

1088 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689456C211yesmaximal, characteristic, complemented
2C2×#15689166C211yesmaximal, characteristic, complemented
3#20247696C211yesmaximal, characteristic
4#20244723C212yesmaximal, complemented
5#20244581C212yesmaximal, complemented
6#20243822C211yesmaximal, characteristic, complemented
7#20243686C211yesmaximal, characteristic
8#20243642C212yesmaximal, complemented
9#20243576C212yesmaximal, complemented
10#20243483C211yesmaximal, characteristic, complemented
11C2×C47×#552C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13C2×#13094005—21no—
14C22×C4⋊Dic47C2211yescharacteristic
15#15689456—24no—
16#15689166—24no—
17C2×#13093957C2212yes—
18C2×#13093957C2212yes—
19C2×#13093955C2212yes—
20C2×#13093948C2212yes—
21C2×#13093948C2212yes—
22C2×C22⋊(C47⋊C8)C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2212yes—
24#15689053—21no—
25#15689048—21no—
26#15689021C2212yes—
27#15688984—21no—
28#15688983C2211yescharacteristic
29#15688983C2212yes—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688885C2212yes—
33#15688878C2212yes—
34C47×#552—24no—
35C47×#552—24no—
36C2×#126×C47—21no—
37C2×#125×C47—21no—
38C2×#125×C47C2211yescharacteristic
39C2×#123×C47—21no—
40C2×C22⋊Q8×C47C2211yescharacteristic
41C2×C22⋊Q8×C47C2211yescharacteristic
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4×D4×C47C2211yescharacteristic
44C2×#117×C47—21no—
45C2×#117×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C23×Dic47—21no—
48C2×#10625329—22no—
49C2×#10625329—24no—
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52#13094005—44no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47—22no—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47C2312yes—
57C2×C4⋊Dic47C2312yes—
58C2×Dic47⋊C4—22no—
59C2×C22⋊Dic47—21no—
60C2×C22⋊Dic47—22no—
61#13093957—24no—
62#13093957—24no—
63#13093955—24no—
64#13093948—24no—
65#13093948—24no—
66C22⋊(C47⋊C8)—22no—
67C22⋊(C47⋊C8)—24no—
68C47⋊#95—22no—
69C47⋊#95—21no—
70C47⋊#95—22no—
71C22×Q8×C47C2311yescharacteristic
72C22×D4×C47D411yescharacteristic
73C23×C4×C47—21no—
74C23×C4×C47C2311yescharacteristic
75C23×C4×C47C2311yescharacteristic
76#126×C47—28no—
77#125×C47—28no—
78#125×C47—22no—
79#125×C47—22no—
80#123×C47—28no—
81C22⋊Q8×C47—22no—
82C22⋊Q8×C47—22no—
83C22⋊Q8×C47—22no—
84C22⋊Q8×C47D414yes—
85C4×Q8×C47—22no—
86C4×Q8×C47—22no—
87C4×D4×C47—22no—
88C4×D4×C47—22no—
89#117×C47—28no—
90#117×C47—22no—
91#117×C47D414yes—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—22no—
99C2×C4⋊C4×C47—22no—
100C2×C4⋊C4×C47D411yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47—21no—
107C2×C22⋊C4×C47D411yescharacteristic
108C2×C22⋊C4×C47D411yescharacteristic
109C2×C42×C47—21no—
110C2×C42×C47—21no—
111C2×C42×C47C2311yescharacteristic
112C24×C47C2×D411yescharacteristic
113C24×C47C2×D411yescharacteristic
114C2×Q8×C47—21no—
115C2×Q8×C47—22no—
116C2×Q8×C47—22no—
117C2×Q8×C47C2×D412yes—
118C2×D4×C47—22no—
119C2×D4×C47—22no—
120C2×D4×C47—41no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—24no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2411yescharacteristic
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—42no—
144C22×Dic47—42no—
145C22×Dic47—42no—
146C4⋊C4×C47—24no—
147C4⋊C4×C47—24no—
148C4⋊C4×C47—24no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—24no—
151C4⋊C4×C47—24no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47C2×D412yes—
157C4⋊C4×C47C2×D412yes—
158C4⋊C4×C47C2×D412yes—
159C22⋊C4×C47—24no—
160C22⋊C4×C47—24no—
161C22⋊C4×C47—24no—
162C22⋊C4×C47—22no—
163C22⋊C4×C47—22no—
164C42×C47—24no—
165C42×C47—24no—
166C42×C47—21no—
167C42×C47—21no—
168C22⋊Dic47—42no—
169C22⋊Dic47—44no—
170C4⋊Dic47—22no—
171C4⋊Dic47—22no—
172C4⋊Dic47—22no—
173C4⋊Dic47—22no—
174C4⋊Dic47—42no—
175Dic47⋊C4—44no—
176#10625329—28no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8#4114yes—
183C23×C47—21no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—21no—
187C23×C47—21no—
188C23×C47—21no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47#12012yes—
192C23×C47AGL(1,ℤ/8)12yes—
193C23×C47C22×D411yescharacteristic
194Q8×C47—24no—
195Q8×C47—41no—
196Q8×C47—41no—
197D4×C47—44no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—24no—
202C2×C4×C47—21no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—22no—
213C2×C4×C47—48no—
214C2×C4×C47—42no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—42no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—42no—
223C2×C4×C47#12012yes—
224C2×C4×C47C22×D411yescharacteristic
225C2×C4×C47C22×D411yescharacteristic
226C2×C4×C47C2×#4112yes—
227C2×C4×C472−1+412yes—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—84no—
235C47⋊C8—24no—
236C47⋊C8—42no—
237#24110—471nomaximal, Sylow
238C22×C47—24no—
239C22×C47—22no—
240C22×C47—22no—
241C22×C47—22no—
242C22×C47—22no—
243C22×C47—42no—
244C22×C47—42no—
245C22×C47—42no—
246C22×C47—41no—
247C22×C47—41no—
248C22×C47C2×#12011yescharacteristic
249C22×C47#54712yes—
250C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C47#57511yescharacteristic
252C22×C47#57812yes—
253C4×C47—22no—
254C4×C47—41no—
255C4×C47—42no—
256C4×C47—41no—
257C4×C47—42no—
258C4×C47—42no—
259C4×C47—41no—
260C4×C47—42no—
261C4×C47#52511yescharacteristic
262C4×C47#53611yescharacteristic
263C4×C47#54712yes—
264Dic47—82no—
265Dic47—82no—
266C2×#552D4711yescharacteristic
267C2×#484—471no—
268C2×#428—471no—
269#2486—471no—
270#2059—472no—
271#2035—472no—
272#1894—471no—
273#1870—471no—
274#1863—472no—
275#1852—472no—
276#1825—471no—
277C2×C47—42no—
278C2×C47—42no—
279C2×C47#240611yescharacteristic
280C2×C47#300611yescharacteristic
281C2×C47#301111yescharacteristic
282C2×C47#334312yes—
283C2×C47#335212yes—
284C22×M16—471no—
285#552—24no—
286#552—24no—
287C2×#126—21no—
288C2×#125—21no—
289C2×#125C2×D4711yescharacteristic
290C2×#123—21no—
291C2×#123—941no—
292C2×C22⋊Q8C2×D4711yescharacteristic
293C2×C22⋊Q8C2×D4711yescharacteristic
294C2×C4×Q8C2×D4711yescharacteristic
295C2×C4×D4C2×D4711yescharacteristic
296C2×#117—21no—
297C2×#117C2×D4711yescharacteristic
298C22×C4⋊C4—471no—
299C22×C4⋊C4C2×D4711yescharacteristic
300#484—944no—
301#428—944no—
302C2×#107—472no—
303C2×#107—472no—
304C2×#106—472no—
305C2×#103—472no—
306C2×#103—472no—
307C2×C22⋊C8—471no—
308C2×C22⋊C8—472no—
309#398—941no—
310#397—941no—
311#389—472no—
312#381—941no—
313#380—471no—
314#380—472no—
315#340—472no—
316#340—472no—
317#340—472no—
318#336—472no—
319C47#2411011yescharacteristic, complemented, Sylow
320C22×Q8C22×D4711yescharacteristic
321C22×D4C22⋊D4711yescharacteristic
322C23×C4—21no—
323C23×C4—941no—
324C23×C4C22×D4711yescharacteristic
325C23×C4C22×D4711yescharacteristic
326C2×M16—944no—
327C2×M16—942no—
328C22×C8—472no—
329C22×C8—472no—
330#126—28no—
331#125—28no—
332#125—22no—
333#125—22no—
334#123—28no—
335#123—1884no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8C22⋊D4714yes—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×D4—22no—
343C4×D4—22no—
344#117—28no—
345#117—22no—
346#117C22⋊D4714yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—472no—
356C2×C4⋊C4—472no—
357C2×C4⋊C4—942no—
358C2×C4⋊C4—942no—
359C2×C4⋊C4—941no—
360C2×C4⋊C4—942no—
361C2×C4⋊C4C22⋊D4711yescharacteristic
362C2×C4⋊C4C22×D4711yescharacteristic
363C2×C4⋊C4C22×D4711yescharacteristic
364C2×C4⋊C4C22×D4711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—941no—
369C2×C22⋊C4—942no—
370C2×C22⋊C4C22⋊D4711yescharacteristic
371C2×C22⋊C4C22⋊D4711yescharacteristic
372C2×C42—21no—
373C2×C42—21no—
374C2×C42C22×D4711yescharacteristic
375#107—944no—
376#107—944no—
377#106—944no—
378#103—944no—
379#103—944no—
380C22⋊C8—944no—
381C22⋊C8—942no—
382#95—942no—
383#95—941no—
384#95—942no—
385C24C2×C22⋊D4711yescharacteristic
386C24C2×C22⋊D4711yescharacteristic
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8C2×C22⋊D4712yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—942no—
408C22×C4—942no—
409C22×C4—1882no—
410C22×C4—1882no—
411C22×C4—1882no—
412C22×C4C2×C22⋊D4711yescharacteristic
413C22×C4C2×C22⋊D4711yescharacteristic
414C22×C4C2×C22⋊D4711yescharacteristic
415C22×C4C2×C22⋊D4711yescharacteristic
416C22×C4C2×C22⋊D4712yes—
417C22×C4C2×C22⋊D4712yes—
418C22×C4C23×D4711yescharacteristic
419M16—948no—
420C2×C8—474no—
421C2×C8—942no—
422C2×C8—942no—
423C2×C8—942no—
424C2×C8—942no—
425C2×C8—942no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—942no—
437C4⋊C4—942no—
438C4⋊C4—942no—
439C4⋊C4—942no—
440C4⋊C4—1884no—
441C4⋊C4—1882no—
442C4⋊C4C2×C22⋊D4712yes—
443C4⋊C4C2×C22⋊D4712yes—
444C4⋊C4C2×C22⋊D4712yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—1882no—
451C22⋊C4—1884no—
452C42—24no—
453C42—24no—
454C42—21no—
455C42—21no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—22no—
464C23#1309399312yes—
465C23#1309404912yes—
466C23C22×C22⋊D4711yescharacteristic
467Q8—24no—
468Q8—41no—
469Q8—41no—
470D4—44no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—3764no—
497C2×C4—1882no—
498C2×C4—1882no—
499C2×C4—1882no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4#1309399312yes—
504C2×C4C22×C22⋊D4711yescharacteristic
505C2×C4C22×C22⋊D4711yescharacteristic
506C2×C4#41×D4712yes—
507C2×C4#1309407112yes—
508C8—944no—
509C8—1882no—
510C22—22no—
511C22—24no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—42no—
517C22—41no—
518C22—42no—
519C22—42no—
520C22C2×#1309399311yescharacteristic
521C22#1568977212yes—
522C22C2×#1309404911yescharacteristic
523C22#1568992111yescharacteristic
524C22#1568993812yes—
525C4—22no—
526C4—41no—
527C4—42no—
528C4—41no—
529C4—42no—
530C4—41no—
531C4—42no—
532C4—42no—
533C4—3762no—
534C4—3762no—
535C4#1568963011yescharacteristic
536C4#1568969411yescharacteristic
537C4#1568977212yes—
538C2—42no—
539C2—42no—
540C2#2024671911yescharacteristic
541C2#2025264011yescharacteristic
542C2#2025268311yescharacteristic
543C2#2025582812yes—
544C2#2025595712yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#552C2
socleC23×C47C22×D4
prime radicalC22×D4×C47D4
hypercenterC2×#552D47
hypocenterC47#24110
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#15689456C2semidirect
2. C2×#15689166C2semidirect
4. #20244723C2semidirect
5. #20244581C2semidirect
6. #20243822C2semidirect
8. #20243642C2semidirect
9. #20243576C2semidirect
10. #20243483C2semidirect
319. C47#24110semidirect

Maximal subgroups

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

subgroupindex
C2×#156894562
C2×#156891662
#202476962
#202447232
#202445812
#202438222
#202436862
#202436422
#202435762
#202434832
C2×C47×#5522
#2411047