Finite Group Databasebeta

← #26225362

Subgroups — #26225362

Order 12032.

1558 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689634C211yesmaximal, characteristic
2C2×#15689087C211yesmaximal, characteristic, complemented
3#20243824C212yesmaximal, complemented
4#20243819C212yesmaximal, complemented
5#20243705C212yesmaximal, complemented
6#20243695C212yesmaximal, complemented
7#20243480C212yesmaximal, complemented
8#20243326C212yesmaximal, complemented
9C2×C47×#380C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329C2212yes—
12#15689634—24no—
13#15689634C418yescomplemented
14C2×#13094007—21no—
15C2×#13094003C2211yescharacteristic
16C2×#13094003C2211yescharacteristic
17C2×#13093998C2211yescharacteristic
18C2×#13093998C2211yescharacteristic
19C2×#13093993—21no—
20C2×C4×C22⋊D47—21no—
21C2×C4×C22⋊D47—21no—
22C2×#13093983C2211yescharacteristic
23C2×#13093947C2214yescomplemented
24C2×#13093944C2214yescomplemented
25#15689087—24no—
26C2×C22⋊(C47⋊C8)C2212yes—
27C2×C22⋊(C47⋊C8)C2212yes—
28#15688885C2214yescomplemented
29#15688885C2214yescomplemented
30C24×C4×C47C2211yescharacteristic
31C22×C4⋊C4×C47C2212yes—
32C22×C22⋊C4×C47—21no—
33C47×#380—21no—
34C47×#380—21no—
35C47×#380C2212yes—
36C47×#380C2212yes—
37C2×#95×C47—21no—
38C2×#10625350—22no—
39C2×#10625350—44no—
40C2×#10625350C2×C414yescomplemented
41C22×Dic94C2311yescharacteristic
42C22×D188C2311yescharacteristic
43C22×C22⋊D47—21no—
44C22×C22⋊D47—21no—
45C22×C22⋊D47—21no—
46C22×C4×D47—21no—
47C2×#10625329—24no—
48C2×#10625329—44no—
49C22×C47⋊C8C2314yes—
50#13094007—28no—
51#13094003—22no—
52#13094003—22no—
53#13094003C2×C414yes—
54#13094003C2×C414yescomplemented
55#13093998—22no—
56#13093998—22no—
57#13093998C2×C414yes—
58#13093998C2×C414yescomplemented
59#13093993—28no—
60C4×C22⋊D47—28no—
61C4×C22⋊D47—28no—
62#13093983—22no—
63#13093983C2×C414yescomplemented
64C2×C4⋊Dic47C2311yescharacteristic
65C2×C4⋊Dic47C2311yescharacteristic
66C2×Dic47⋊C4—21no—
67C2×Dic47⋊C4—21no—
68C2×C22⋊Dic47—21no—
69C2×C22⋊Dic47—21no—
70C2×C22⋊Dic47—21no—
71C2×(C2×D47)⋊C4—21no—
72C2×(C2×D47)⋊C4—21no—
73C2×C4×Dic47—21no—
74#13093947—44no—
75#13093944—44no—
76C22⋊(C47⋊C8)—24no—
77C22⋊(C47⋊C8)—24no—
78C25×C47D411yescharacteristic
79C23×C4×C47—22no—
80C23×C4×C47—44no—
81C23×C4×C47C2×C414yes—
82C23×C4×C47D411yescharacteristic
83C23×C4×C47D411yescharacteristic
84C23×C4×C47D411yescharacteristic
85C23×C4×C47D414yes—
86C23×C4×C47C2311yescharacteristic
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C2×C4⋊C4×C47—22no—
90C2×C4⋊C4×C47—44no—
91C2×C4⋊C4×C47C2314yes—
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47—22no—
94C2×C22⋊C4×C47—44no—
95#95×C47—22no—
96#95×C47—22no—
97C24×C47—22no—
98C24×C47—22no—
99C24×C47—42no—
100C24×C47—42no—
101C24×C47C22⋊C414yes—
102C24×C47C2×D411yescharacteristic
103C24×C47C2×D411yescharacteristic
104C24×C47C2×D411yescharacteristic
105C23×D47—21no—
106C22×C4×C47—24no—
107C22×C4×C47—24no—
108C22×C4×C47—24no—
109C22×C4×C47—24no—
110C22×C4×C47—48no—
111C22×C4×C47—48no—
112C22×C4×C47—42no—
113C22×C4×C47—42no—
114C22×C4×C47—42no—
115C22×C4×C47—44no—
116C22×C4×C47—44no—
117C22×C4×C47—44no—
118C22×C4×C47C22⋊C414yes—
119C22×C4×C47C22⋊C414yes—
120C22×C4×C47C22⋊C414yes—
121C22×C4×C47C22×C412yes—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C22×C412yes—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2411yescharacteristic
131C2×C22⋊D47—24no—
132C2×C22⋊D47—24no—
133C2×C22⋊D47—24no—
134C2×C22⋊D47—44no—
135C2×C22⋊D47—44no—
136C2×C22⋊D47—42no—
137C2×C22⋊D47—42no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141#10625350—48no—
142#10625350—48no—
143C2×D188—24no—
144C2×D188—21no—
145C2×D188C22×C412yes—
146C2×C4×D47—24no—
147C2×C4×D47—44no—
148C2×Dic94—24no—
149C2×Dic94—21no—
150C2×Dic94C22×C412yes—
151C4⋊C4×C47—24no—
152C4⋊C4×C47—24no—
153C4⋊C4×C47—44no—
154C22⋊C4×C47—44no—
155C22⋊C4×C47—44no—
156C22⋊Dic47—24no—
157C22⋊Dic47—24no—
158C22⋊Dic47—24no—
159(C2×D47)⋊C4—24no—
160(C2×D47)⋊C4—24no—
161C4⋊Dic47—21no—
162C4⋊Dic47—21no—
163C4⋊Dic47C22×C412yes—
164C4⋊Dic47C22×C412yes—
165Dic47⋊C4—24no—
166Dic47⋊C4—24no—
167C4×Dic47—24no—
168#10625329—48no—
169C2×C47⋊C8—24no—
170C2×C47⋊C8—24no—
171C2×C47⋊C8—24no—
172C23×C47—22no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—22no—
176C23×C47—21no—
177C23×C47—21no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—28no—
183C23×C47—44no—
184C23×C47—44no—
185C23×C47—48no—
186C23×C47—48no—
187C23×C47C2×C22⋊C412yes—
188C23×C47C2×C22⋊C412yes—
189C23×C47C2×C22⋊C412yes—
190C23×C47AGL(1,ℤ/8)12yes—
191C23×C47#13712yes—
192C23×C47C22×D411yescharacteristic
193C22×D47—22no—
194C22×D47—42no—
195C22×D47—41no—
196C22×D47—41no—
197C22×D47—42no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—44no—
208C2×C4×C47—44no—
209C2×C4×C47—44no—
210C2×C4×C47—44no—
211C2×C4×C47—44no—
212C2×C4×C47—44no—
213C2×C4×C47—88no—
214C2×C4×C47C2×C22⋊C412yes—
215C2×C4×C47C2×C22⋊C412yes—
216C2×C4×C47C2×C22⋊C412yes—
217C2×C4×C47C23×C411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47H(5,𝔽2)14yes—
220C22⋊D47—48no—
221C22⋊D47—48no—
222C22⋊D47—48no—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—42no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—42no—
230D188—24no—
231D188—42no—
232C4×D47—48no—
233Dic94—24no—
234Dic94—42no—
235C47⋊C8—44no—
236#21769—471nomaximal, Sylow
237C22×C47—22no—
238C22×C47—22no—
239C22×C47—24no—
240C22×C47—24no—
241C22×C47—22no—
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—416no—
248C22×C47—42no—
249C22×C47—42no—
250C22×C47—42no—
251C22×C47—42no—
252C22×C47#41714yes—
253C22×C47C22×C22⋊C411yescharacteristic
254C22×C47#53412yes—
255C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C47C2×#13711yescharacteristic
257C22×C47#57512yes—
258C2×D47—42no—
259C2×D47—42no—
260C2×D47—42no—
261C2×D47—82no—
262C2×D47—82no—
263C4×C47—22no—
264C4×C47—22no—
265C4×C47—22no—
266C4×C47—84no—
267C4×C47#51812yes—
268C4×C47#53412yes—
269Dic47—42no—
270Dic47—42no—
271Dic47—42no—
272C2×#525—471no—
273C2×#407—471no—
274C2×#380D4711yescharacteristic, complemented
275#1895—472no—
276#1894—472no—
277#1874—472no—
278#1872—472no—
279#1825—472no—
280#1788—472no—
281C2×C47—24no—
282C2×C47—42no—
283C2×C47—42no—
284C2×C47#230811yescharacteristic
285C2×C47C2×#41711yescharacteristic
286C2×C47#289611yescharacteristic
287C2×C47#319412yes—
288C2×C47#319612yes—
289D47—82no—
290C22×#41—471no—
291C24×C4C2×D4711yescharacteristic
292C22×M16—472no—
293#525—478no—
294#525—944no—
295C2×#123—941no—
296C2×C22⋊Q8—471no—
297C2×C22⋊Q8—471no—
298C2×#121—471no—
299C2×#121—471no—
300C2×#120—941no—
301C2×C4×D4—941no—
302C2×C4×D4—941no—
303C2×#117—471no—
304C22×C4⋊C4C2×D4712yes—
305C22×C22⋊C4—21no—
306C2×#103—474no—
307C2×#102—474no—
308#407—944no—
309C2×C22⋊C8—472no—
310C2×C22⋊C8—472no—
311#380—21no—
312#380—21no—
313#380C2×D4712yes—
314#380C2×D4712yes—
315C2×#95—21no—
316#340—474no—
317#340—474no—
318C47#2176911yescharacteristic, complemented, Sylow
319C25C22⋊D4711yescharacteristic
320C2×#41—474no—
321C2×#41—942no—
322C2×#41—1884no—
323C22×Q8—471no—
324C22×D4—471no—
325C22×D4—941no—
326C22×D4—941no—
327C22×D4—941no—
328C23×C4—22no—
329C23×C4—44no—
330C23×C4—941no—
331C23×C4C4×D4714yes—
332C23×C4D18814yes—
333C23×C4C22⋊D4711yescharacteristic
334C23×C4C22⋊D4711yescharacteristic
335C23×C4C22⋊D4711yescharacteristic
336C23×C4C22×D4711yescharacteristic
337C23×C4C22×D4711yescharacteristic
338C23×C4C22×D4711yescharacteristic
339C2×M16—944no—
340C2×M16—1884no—
341C22×C8—474no—
342#123—948no—
343C22⋊Q8—474no—
344C22⋊Q8—474no—
345C22⋊Q8—942no—
346C22⋊Q8—942no—
347#121—474no—
348#121—474no—
349#121—942no—
350#121—942no—
351#120—948no—
352C4×D4—948no—
353C4×D4—948no—
354#117—474no—
355#117—942no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4—471no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4—941no—
362C2×C4⋊C4C22×D4714yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—44no—
366C2×C22⋊C4—941no—
367C2×C22⋊C4—941no—
368C2×C22⋊C4—941no—
369C2×C22⋊C4—941no—
370C2×C22⋊C4—941no—
371C2×C42—941no—
372#103—1884no—
373#102—1884no—
374C22⋊C8—944no—
375C22⋊C8—944no—
376#95—22no—
377#95—22no—
378C24—22no—
379C24—22no—
380C24—42no—
381C24—42no—
382C24—941no—
383C24(C2×D47)⋊C414yes—
384C24C2×C22⋊D4711yescharacteristic
385C24C2×C22⋊D4711yescharacteristic
386C24C2×C22⋊D4711yescharacteristic
387#41—1888no—
388#41—1888no—
389C2×Q8—472no—
390C2×Q8—941no—
391C2×Q8—944no—
392C2×D4—472no—
393C2×D4—944no—
394C2×D4—944no—
395C2×D4—944no—
396C2×D4—941no—
397C2×D4—944no—
398C2×D4—1884no—
399C2×D4—1884no—
400C2×D4—1882no—
401C2×D4—1882no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—941no—
415C22×C4—941no—
416C22×C4—941no—
417C22×C4—944no—
418C22×C4—1884no—
419C22×C4(C2×D47)⋊C414yes—
420C22×C4(C2×D47)⋊C414yes—
421C22×C4(C2×D47)⋊C414yes—
422C22×C4C2×C4×D4712yes—
423C22×C4C2×C4×D4712yes—
424C22×C4C2×C4×D4712yes—
425C22×C4C2×D18812yes—
426C22×C4C2×D18812yes—
427C22×C4C2×D18812yes—
428C22×C4C2×C22⋊D4711yescharacteristic
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C23×D4711yescharacteristic
432M16—1888no—
433C2×C8—944no—
434C2×C8—944no—
435C2×C8—944no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—44no—
439C4⋊C4—472no—
440C4⋊C4—472no—
441C4⋊C4—944no—
442C4⋊C4—944no—
443C4⋊C4—941no—
444C4⋊C4—941no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—944no—
448C22⋊C4—944no—
449C22⋊C4—944no—
450C22⋊C4—944no—
451C22⋊C4—944no—
452C42—944no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—44no—
465C23—44no—
466C23—48no—
467C23—48no—
468C23—942no—
469C23—1881no—
470C23—1881no—
471C23—1882no—
472C23—1882no—
473C23C2×(C2×D47)⋊C412yes—
474C23C2×(C2×D47)⋊C412yes—
475C23C2×(C2×D47)⋊C412yes—
476C23#1309405012yes—
477C23#1309405512yes—
478C23C22×C22⋊D4711yescharacteristic
479Q8—944no—
480Q8—1882no—
481D4—944no—
482D4—1888no—
483D4—1888no—
484D4—1888no—
485D4—1882no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—88no—
502C2×C4—942no—
503C2×C4—942no—
504C2×C4—942no—
505C2×C4—1888no—
506C2×C4—1881no—
507C2×C4—1881no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4C2×(C2×D47)⋊C412yes—
511C2×C4C2×(C2×D47)⋊C412yes—
512C2×C4C2×(C2×D47)⋊C412yes—
513C2×C4C22×C4×D4711yescharacteristic
514C2×C4C22×D18811yescharacteristic
515C2×C4#1309406914yes—
516C8—1884no—
517C22—22no—
518C22—24no—
519C22—22no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—416no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—3762no—
533C22—3762no—
534C22—1882no—
535C22—1882no—
536C22—1882no—
537C22#1568912214yes—
538C22C22×(C2×D47)⋊C411yescharacteristic
539C22#1568968212yes—
540C22C2×#1309405011yescharacteristic
541C22C2×#1309405511yescharacteristic
542C22#1568992112yes—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—84no—
547C4—1882no—
548C4—1882no—
549C4—1882no—
550C4#1568959612yes—
551C4#1568968512yes—
552C2—24no—
553C2—42no—
554C2—42no—
555C2—3762no—
556C2#2024596311yescharacteristic
557C2C2×#1568912211yescharacteristic
558C2#2025190511yescharacteristic
559C2#2025420212yes—
560C2#2025421812yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#380C2
socleC23×C47C22×D4
prime radicalC2×#15689634C2
cosocleC22×C4×C47C24
hypercenterC2×#380D47
hypocenterC47#21769
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#15689087C2semidirect
3. #20243824C2semidirect
4. #20243819C2semidirect
5. #20243705C2semidirect
6. #20243695C2semidirect
7. #20243480C2semidirect
8. #20243326C2semidirect
9. C2×C47×#380C2semidirect
13. #15689634C4semidirect
23. C2×#13093947C22semidirect
24. C2×#13093944C22semidirect
28. #15688885C22semidirect
29. #15688885C22semidirect
40. C2×#10625350C2×C4semidirect
54. #13094003C2×C4semidirect
58. #13093998C2×C4semidirect
63. #13093983C2×C4semidirect
274. C2×#380D47semidirect
318. C47#21769semidirect

Maximal subgroups

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

subgroupindex
C2×#156896342
C2×#156890872
#202438242
#202438192
#202437052
#202436952
#202434802
#202433262
C2×C47×#3802
#2176947