Finite Group Databasebeta

← #26319882

Subgroups — #26319882

Order 12032.

828 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251719C211yesmaximal, characteristic
2#20244824C211yesmaximal, characteristic, complemented
3#20244588C211yesmaximal, characteristic, complemented
4#20244570C211yesmaximal, characteristic
5#20244389C211yesmaximal, characteristic
6#20244245C211yesmaximal, characteristic, complemented
7#20244118C211yesmaximal, characteristic, complemented
8#20244070C211yesmaximal, characteristic, complemented
9#20244067C211yesmaximal, characteristic, complemented
10#20244008C211yesmaximal, characteristic, complemented
11#20243973C211yesmaximal, characteristic, complemented
12#20243743C211yesmaximal, characteristic, complemented
13#20243642C211yesmaximal, characteristic, complemented
14C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8×C47C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C2×#13094007—21no—
18C2×#13093998C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×C22⋊(C47⋊C8)C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31C2×C4×C47⋊C8C2211yescharacteristic
32#15689025C2211yescharacteristic
33#15689025C2211yescharacteristic
34#15689021C2211yescharacteristic
35#15689021C2211yescharacteristic
36#15689014—21no—
37#15689014—21no—
38#15689002—21no—
39#15688983C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×C22⋊Q8×C47C2211yescharacteristic
49C2×C4×Q8×C47C2211yescharacteristic
50C22×C42×C47C2211yescharacteristic
51C47×#391C2211yescharacteristic
52C47×#389C2211yescharacteristic
53C47×#387—21no—
54C47×#385—21no—
55C47×#380C2211yescharacteristic
56C4×C4⋊C4×C47—21no—
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C22⋊C4×C47—21no—
59C22×D188C2311yescharacteristic
60C22×C22⋊D47—21no—
61C2×#10625336—22no—
62C2×#10625336—22no—
63C2×#10625336—41no—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68#13094007—44no—
69#13093998—41no—
70#13093998—41no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82#13093957—22no—
83#13093957—41no—
84#13093955—22no—
85#13093948—22no—
86#13093948—22no—
87#13093948—41no—
88#13093944—22no—
89#13093944—22no—
90#13093944—41no—
91C22⋊(C47⋊C8)—22no—
92C22⋊(C47⋊C8)—22no—
93C4×C47⋊C8—41no—
94C4×C47⋊C8—41no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C22×Q8×C47C2311yescharacteristic
99C23×C4×C47D411yescharacteristic
100C23×C4×C47D411yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C22⋊Q8×C47—22no—
103C22⋊Q8×C47—22no—
104C4×Q8×C47—41no—
105C4×Q8×C47—41no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47—21no—
116C2×C42×C47—41no—
117C2×C42×C47—41no—
118C2×C42×C47D411yescharacteristic
119C2×C42×C47D411yescharacteristic
120C2×C42×C47C2311yescharacteristic
121C2×C42×C47C2311yescharacteristic
122#95×C47—21no—
123#95×C47—21no—
124#95×C47—21no—
125#95×C47—21no—
126#95×C47—21no—
127C24×C47C2×D411yescharacteristic
128C23×D47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47—21no—
133C2×Q8×C47—21no—
134C2×Q8×C47#4112yes—
135C22×C4×C47—22no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—41no—
150C22×C4×C47—41no—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2411yescharacteristic
157C2×C22⋊D47—42no—
158C2×C22⋊D47—42no—
159C2×C22⋊D47—42no—
160C22×Dic47—21no—
161C22×Dic47—21no—
162C22×Dic47—21no—
163C2×D188—21no—
164C2×D188—21no—
165C2×D188—21no—
166C2×D188—21no—
167C2×D188—21no—
168C2×D188—21no—
169C4⋊C4×C47—24no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—42no—
176C4⋊C4×C47—42no—
177C4⋊C4×C47#4112yes—
178C22⋊C4×C47—24no—
179C22⋊C4×C47—42no—
180C42×C47—21no—
181C42×C47—21no—
182C42×C47—21no—
183C42×C47—21no—
184C42×C47—41no—
185C42×C47—41no—
186C42×C47—42no—
187C42×C47—42no—
188C22⋊Dic47—42no—
189#10625336—44no—
190(C2×D47)⋊C4—42no—
191(C2×D47)⋊C4—42no—
192(C2×D47)⋊C4—42no—
193(C2×D47)⋊C4—42no—
194(C2×D47)⋊C4—42no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201Dic47⋊C4—42no—
202Dic47⋊C4—42no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8#4112yes—
214C2×C47⋊C8#4112yes—
215C23×C47—21no—
216C23×C47—21no—
217C23×C47—21no—
218C23×C47—21no—
219C23×C47—21no—
220C23×C47—21no—
221C23×C47#13512yes—
222C23×C47C22×D411yescharacteristic
223C22×D47—41no—
224C22×D47—41no—
225C22×D47—41no—
226C22×D47—41no—
227C22×D47—41no—
228C22×D47—41no—
229C22×D47—41no—
230Q8×C47—22no—
231Q8×C47—22no—
232Q8×C47—41no—
233Q8×C47—41no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—44no—
249C2×C4×C47—44no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—41no—
262C2×C4×C47—41no—
263C2×C4×C47—41no—
264C2×C4×C47#13512yes—
265C2×C4×C47AGL(1,ℤ/8)12yes—
266C2×C4×C47#13712yes—
267C2×C4×C47C2×#4111yescharacteristic
268C2×C4×C47C2×#4111yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×C4×C47H(5,𝔽2)11yescharacteristic
272C2×C4×C47H(5,𝔽2)11yescharacteristic
273C22⋊D47—84no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283D188—41no—
284D188—41no—
285D188—41no—
286D188—41no—
287C47⋊C8—22no—
288C47⋊C8—22no—
289C47⋊C8—41no—
290C47⋊C8—41no—
291#29467—471nomaximal, Sylow
292C22×C47—22no—
293C22×C47—22no—
294C22×C47—22no—
295C22×C47—41no—
296C22×C47—41no—
297C22×C47—41no—
298C22×C47—41no—
299C22×C47#52511yescharacteristic
300C22×C47#53211yescharacteristic
301C22×C47#53411yescharacteristic
302C22×C47C2×#13511yescharacteristic
303C22×C47C2×#13511yescharacteristic
304C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C47C2×#13711yescharacteristic
306C2×D47—81no—
307C2×D47—81no—
308C2×D47—81no—
309C2×D47—81no—
310C2×D47—81no—
311C2×D47—81no—
312C2×D47—81no—
313C4×C47—22no—
314C4×C47—22no—
315C4×C47—22no—
316C4×C47—42no—
317C4×C47—41no—
318C4×C47—42no—
319C4×C47—41no—
320C4×C47—81no—
321C4×C47—81no—
322C4×C47#55111yescharacteristic
323C4×C47#55311yescharacteristic
324C4×C47#55311yescharacteristic
325C4×C47#55911yescharacteristic
326Dic47—81no—
327Dic47—81no—
328Dic47—81no—
329#2854—471no—
330C4×C22⋊Q8D4711yescharacteristic, complemented
331#2077—471no—
332#2036—471no—
333#2033—471no—
334#2000—471no—
335#1968—471no—
336#1942—471no—
337#1934—471no—
338#1933—471no—
339#1925—471no—
340#1920—471no—
341#1880—471no—
342#1863—471no—
343C4×C22⋊C8—471no—
344C2×C47—42no—
345C2×C47#260211yescharacteristic
346C2×C47#310611yescharacteristic
347C2×C47#310811yescharacteristic
348C2×C47#316811yescharacteristic
349C2×C47#318211yescharacteristic
350C2×C47#318811yescharacteristic
351C2×C47#319011yescharacteristic
352D47—161no—
353C22×SD16—471no—
354C2×#123—941no—
355C2×C22⋊Q8C2×D4711yescharacteristic
356C2×#121—471no—
357C2×C4×Q8C2×D4711yescharacteristic
358C22×C42C2×D4711yescharacteristic
359C2×#107—471no—
360C2×#107—471no—
361C2×#106—471no—
362C2×#103—471no—
363C2×#103—471no—
364C2×#103—471no—
365C2×#102—471no—
366C2×#102—471no—
367C2×#102—471no—
368C2×C22⋊C8—471no—
369C2×C22⋊C8—471no—
370C2×C4×C8—471no—
371C2×C4×C8—471no—
372#391C2×D4711yescharacteristic
373#390—471no—
374#390—471no—
375#389—471no—
376#389—471no—
377#389C2×D4711yescharacteristic
378#388—941no—
379#388—941no—
380#387—21no—
381#386—941no—
382#385—21no—
383#380—471no—
384#380C2×D4711yescharacteristic
385C4×C4⋊C4—21no—
386C4×C4⋊C4C2×D4711yescharacteristic
387C4×C22⋊C4—21no—
388#340—471no—
389#340—471no—
390#340—471no—
391#340—471no—
392#340—471no—
393#340—471no—
394#336—471no—
395#336—471no—
396C47#2946711yescharacteristic, complemented, Sylow
397C22×Q8C22×D4711yescharacteristic
398C22×D4—471no—
399C22×D4—941no—
400C23×C4C22⋊D4711yescharacteristic
401C23×C4C22⋊D4711yescharacteristic
402C23×C4C22×D4711yescharacteristic
403C2×SD16—942no—
404C2×SD16—942no—
405C2×SD16—1881no—
406C22×C8—471no—
407C22×C8—471no—
408C22×C8—471no—
409C22×C8—471no—
410#123—1884no—
411C22⋊Q8—22no—
412C22⋊Q8—22no—
413#121—1881no—
414#121—1881no—
415C4×Q8—41no—
416C4×Q8—41no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—471no—
422C2×C4⋊C4—471no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4C22×D4711yescharacteristic
426C2×C4⋊C4C22×D4711yescharacteristic
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—941no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C22⋊C4—941no—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—41no—
439C2×C42—41no—
440C2×C42C22⋊D4711yescharacteristic
441C2×C42C22⋊D4711yescharacteristic
442C2×C42C22×D4711yescharacteristic
443C2×C42C22×D4711yescharacteristic
444#107—942no—
445#107—1881no—
446#106—942no—
447#103—942no—
448#103—942no—
449#103—1881no—
450#102—942no—
451#102—942no—
452#102—1881no—
453C22⋊C8—942no—
454C22⋊C8—942no—
455C4×C8—1881no—
456C4×C8—1881no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—941no—
463#95—941no—
464#95—941no—
465C24—941no—
466C24C2×C22⋊D4711yescharacteristic
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8#1062535012yes—
473C2×D4—941no—
474C2×D4—941no—
475C2×D4—941no—
476C2×D4—941no—
477C2×D4—941no—
478C2×D4—941no—
479C2×D4—1882no—
480C2×D4—1882no—
481C2×D4—1882no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—941no—
501C22×C4C2×C22⋊D4711yescharacteristic
502C22×C4C2×C22⋊D4711yescharacteristic
503C22×C4C2×C22⋊D4711yescharacteristic
504C22×C4C2×C22⋊D4711yescharacteristic
505C22×C4C2×C22⋊D4711yescharacteristic
506C22×C4C23×D4711yescharacteristic
507SD16—1884no—
508C2×C8—472no—
509C2×C8—472no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8—941no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—941no—
529C4⋊C4—941no—
530C4⋊C4—941no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4—941no—
534C4⋊C4—1882no—
535C4⋊C4—1882no—
536C4⋊C4#1062535012yes—
537C22⋊C4—24no—
538C22⋊C4—42no—
539C22⋊C4—1882no—
540C22⋊C4—1882no—
541C22⋊C4—1882no—
542C22⋊C4—1882no—
543C22⋊C4—1882no—
544C22⋊C4—1882no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—41no—
550C42—41no—
551C42—42no—
552C42—42no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—1881no—
560C23—1881no—
561C23—1881no—
562C23—1881no—
563C23—1881no—
564C23—1881no—
565C23—1881no—
566C23#1309404412yes—
567C23C22×C22⋊D4711yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572D4—3764no—
573D4—1881no—
574D4—1881no—
575D4—1881no—
576D4—1881no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4—1881no—
615C2×C4—1881no—
616C2×C4#1309404412yes—
617C2×C4#1309404912yes—
618C2×C4#1309405512yes—
619C2×C4#41×D4711yescharacteristic
620C2×C4#41×D4711yescharacteristic
621C2×C4C2×#1062535011yescharacteristic
622C2×C4C2×#1062535011yescharacteristic
623C2×C4#1309406911yescharacteristic
624C2×C4#1309406911yescharacteristic
625C8—942no—
626C8—942no—
627C8—1881no—
628C8—1881no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—3761no—
637C22—3761no—
638C22—3761no—
639C22—3761no—
640C22—3761no—
641C22—3761no—
642C22—3761no—
643C22#1568963411yescharacteristic
644C22#1568966911yescharacteristic
645C22#1568968211yescharacteristic
646C22C2×#1309404411yescharacteristic
647C22C2×#1309404411yescharacteristic
648C22C2×#1309404911yescharacteristic
649C22C2×#1309405511yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—81no—
658C4—81no—
659C4—3761no—
660C4—3761no—
661C4—3761no—
662C4#1568980211yescharacteristic
663C4#1568982211yescharacteristic
664C4#1568982311yescharacteristic
665C4#1568985611yescharacteristic
666C2—42no—
667C2—7521no—
668C2#2024900011yescharacteristic
669C2#2025344811yescharacteristic
670C2#2025346811yescharacteristic
671C2#2025395211yescharacteristic
672C2#2025410711yescharacteristic
673C2#2025414911yescharacteristic
674C2#2025416911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC4×C22⋊Q8×C47C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenterC4×C22⋊Q8D47
hypocenterC47#29467
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20244824C2semidirect
3. #20244588C2semidirect
6. #20244245C2semidirect
7. #20244118C2semidirect
8. #20244070C2semidirect
9. #20244067C2semidirect
10. #20244008C2semidirect
11. #20243973C2semidirect
12. #20243743C2semidirect
13. #20243642C2semidirect
14. C4×C22⋊(C47⋊C8)C2semidirect
15. C4×C22⋊Q8×C47C2semidirect
330. C4×C22⋊Q8D47semidirect
396. C47#29467semidirect

Maximal subgroups

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

subgroupindex
#202517192
#202448242
#202445882
#202445702
#202443892
#202442452
#202441182
#202440702
#202440672
#202440082
#202439732
#202437432
#202436422
C4×C22⋊(C47⋊C8)2
C4×C22⋊Q8×C472
#2946747