Finite Group Databasebeta

← #26225424

Subgroups — #26225424

Order 12032.

1362 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256717C211yesmaximal, characteristic
2#20251833C211yesmaximal, characteristic, complemented
3#20243836C212yesmaximal, complemented
4#20243831C212yesmaximal, complemented
5#20243688C214yesmaximal, complemented
6#20243488C212yesmaximal, complemented
7#20243333C212yesmaximal, complemented
8C47×#2297C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C22×#10625329C2212yescomplemented
11#15689701C414yescomplemented
12#15689697—21no—
13#15689694C412yescomplemented
14#15689690—21no—
15#15689687C412yescomplemented
16#15689682—21no—
17#15689682—21no—
18#15689669—22no—
19#15689634C2212yes—
20#15689634C2212yes—
21#15689630—22no—
22C2×#13093983C2211yescharacteristic
23C2×#13093952C2214yes—
24C2×#13093952C2214yes—
25#15689090—22no—
26#15689090—22no—
27#15689087C2212yes—
28#15689087C2212yescomplemented
29#15688883C2214yescomplemented
30#15688883C2214yescomplemented
31C22×#41×C47C2211yescharacteristic
32C2×#117×C47C2211yescharacteristic, complemented
33C22×C42×C47C2211yescharacteristic, complemented
34C47×#391—21no—
35C47×#390—21no—
36C47×#386—21no—
37C47×#386—21no—
38C4×C22⋊C4×C47C2212yes—
39C4×C22⋊C4×C47C2212yescomplemented
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350—42no—
43C2×#10625350—42no—
44C2×#10625350C2×C412yescomplemented
45C2×#10625350C2312yes—
46C22×Dic94C2×C411yescharacteristic, complemented
47C22×D188C2×C411yescharacteristic, complemented
48C22×C22⋊D47—21no—
49C22×C4×D47—21no—
50C2×#10625329—22no—
51C2×#10625329—42no—
52C2×#10625329—42no—
53C2×#10625329C2314yes—
54C22×C47⋊C8—22no—
55#13094022—22no—
56#13094020—22no—
57#13094013—22no—
58#13094011—22no—
59#13094007—22no—
60#13094007—22no—
61#13094005—22no—
62#13094005—22no—
63#13094003C2×C414yes—
64#13094003C2×C414yescomplemented
65#13094000—24no—
66#13093998C2×C414yes—
67#13093998C2×C414yescomplemented
68#13093996—24no—
69#13093996—24no—
70#13093995—24no—
71#13093993—22no—
72#13093993—22no—
73#13093992—22no—
74#13093992—22no—
75C4×C22⋊D47—24no—
76C4×C22⋊D47—24no—
77#13093983—21no—
78#13093983—21no—
79#13093983C2×C412yescomplemented
80#13093983C2312yes—
81C2×C4⋊Dic47C2×C412yescomplemented
82C2×C22⋊Dic47—21no—
83C2×C4×Dic47—21no—
84#13093952—44no—
85#13093952—44no—
86C22⋊(C47⋊C8)—22no—
87C22⋊(C47⋊C8)—22no—
88C22⋊(C47⋊C8)—24no—
89C2×#41×C47—42no—
90C2×#41×C47—42no—
91C2×#41×C47C2×C412yes—
92C2×#41×C47C2×C412yes—
93C2×#41×C47D412yes—
94C2×#41×C47D412yes—
95C22×Q8×C47D411yescharacteristic
96C22×D4×C47D411yescharacteristic
97C22×D4×C47D411yescharacteristic
98C22×D4×C47D411yescharacteristic
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2311yescharacteristic
103#117×C47—42no—
104C2×C4⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47—22no—
107C2×C22⋊C4×C47—22no—
108C2×C42×C47—21no—
109C2×C42×C47—42no—
110C2×C42×C47C2312yes—
111C2×C42×C47C2312yes—
112#95×C47—22no—
113#95×C47—22no—
114C24×C47C2×D411yescharacteristic
115C24×C47C2×D411yescharacteristic
116C24×C47C2×D411yescharacteristic
117C23×D47—21no—
118#41×C47—44no—
119#41×C47—44no—
120#41×C47—42no—
121#41×C47—42no—
122#41×C47—42no—
123#41×C47—42no—
124C2×Q8×C47—41no—
125C2×Q8×C47—41no—
126C2×Q8×C47C22⋊C412yes—
127C2×Q8×C47C22⋊C412yes—
128C2×D4×C47—41no—
129C2×D4×C47—41no—
130C2×D4×C47—41no—
131C2×D4×C47—41no—
132C2×D4×C47—41no—
133C2×D4×C47—41no—
134C2×D4×C47C22⋊C412yes—
135C2×D4×C47C22⋊C412yes—
136C2×D4×C47C22⋊C412yes—
137C2×D4×C47C22⋊C412yes—
138C2×D4×C47C22⋊C412yes—
139C2×D4×C47C22⋊C412yes—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—44no—
149C22×C4×C47—42no—
150C22×C4×C47C22×C411yescharacteristic
151C22×C4×C47C22×C411yescharacteristic
152C22×C4×C47C22×C412yes—
153C22×C4×C47C22×C412yes—
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D412yes—
160C22×C4×C47C2×D412yes—
161C22×C4×C47C2411yescharacteristic
162C2×C22⋊D47—22no—
163C2×C22⋊D47—22no—
164C2×C22⋊D47—24no—
165C2×C22⋊D47—24no—
166C2×C22⋊D47—42no—
167C2×C22⋊D47—42no—
168C22×Dic47—21no—
169C22×Dic47—22no—
170#10625350—24no—
171#10625350—24no—
172#10625350—44no—
173#10625350—42no—
174#10625350—42no—
175#10625350—44no—
176C2×D188—21no—
177C2×D188—42no—
178C2×D188C22×C412yes—
179C2×C4×D47—22no—
180C2×C4×D47—22no—
181C2×C4×D47—44no—
182C2×Dic94—21no—
183C2×Dic94—42no—
184C2×Dic94C22×C412yes—
185C4⋊C4×C47—42no—
186C22⋊C4×C47—44no—
187C22⋊C4×C47—44no—
188C22⋊C4×C47—42no—
189C42×C47—24no—
190C42×C47—24no—
191C42×C47—42no—
192C22⋊Dic47—22no—
193C22⋊Dic47—24no—
194C22⋊Dic47—24no—
195C22⋊Dic47—22no—
196(C2×D47)⋊C4—24no—
197(C2×D47)⋊C4—24no—
198C4⋊Dic47—22no—
199C4⋊Dic47C22×C414yes—
200Dic47⋊C4—24no—
201Dic47⋊C4—24no—
202C4×Dic47—22no—
203C4×Dic47—22no—
204#10625329—24no—
205#10625329—24no—
206#10625329—44no—
207C2×C47⋊C8—24no—
208C2×C47⋊C8—44no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—42no—
216C23×C47—42no—
217C23×C47—42no—
218C23×C47C2×C22⋊C411yescharacteristic
219C23×C47C2×C22⋊C412yes—
220C23×C47C2×C22⋊C412yes—
221C23×C47C2×C22⋊C411yescharacteristic
222C23×C47C22×D411yescharacteristic
223C22×D47—22no—
224C22×D47—42no—
225C22×D47—82no—
226Q8×C47—42no—
227Q8×C47—42no—
228D4×C47—42no—
229D4×C47—42no—
230D4×C47—42no—
231D4×C47—42no—
232D4×C47—42no—
233D4×C47—42no—
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—24no—
243C2×C4×C47—24no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—22no—
249C2×C4×C47—22no—
250C2×C4×C47—24no—
251C2×C4×C47—42no—
252C2×C4×C47—48no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47—42no—
256C2×C4×C47—42no—
257C2×C4×C47C2×C22⋊C411yescharacteristic
258C2×C4×C47C2×C22⋊C412yes—
259C2×C4×C47C2×C22⋊C412yes—
260C2×C4×C47C2×C22⋊C411yescharacteristic
261C2×C4×C47C23×C411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47H(5,𝔽2)12yes—
264C2×C4×C47H(5,𝔽2)12yes—
265C22⋊D47—44no—
266C22⋊D47—42no—
267C22⋊D47—42no—
268C22⋊D47—84no—
269C22⋊D47—84no—
270C2×Dic47—24no—
271C2×Dic47—22no—
272C2×Dic47—42no—
273C2×Dic47—82no—
274D188—24no—
275D188—42no—
276C4×D47—44no—
277C4×D47—44no—
278Dic94—24no—
279Dic94—42no—
280C47⋊C8—44no—
281#21775—471nomaximal, Sylow
282C22×C47—22no—
283C22×C47—21no—
284C22×C47—21no—
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—24no—
289C22×C47—24no—
290C22×C47—22no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—81no—
294C22×C47—81no—
295C22×C47—81no—
296C22×C47—81no—
297C22×C47C22×C22⋊C411yescharacteristic
298C22×C47#53411yescharacteristic
299C22×C47#53411yescharacteristic
300C2×D47—42no—
301C2×D47—44no—
302C2×D47—84no—
303C4×C47—21no—
304C4×C47—21no—
305C4×C47—22no—
306C4×C47—22no—
307C4×C47—44no—
308C4×C47—82no—
309C4×C47#51812yes—
310C4×C47#53412yes—
311Dic47—42no—
312Dic47—44no—
313#3449—471no—
314#2879—471no—
315#2297D4711yescharacteristic, complemented
316#1897—472no—
317#1896—472no—
318#1871—474no—
319#1826—472no—
320#1789—472no—
321C2×C47—21no—
322C2×C47—21no—
323C2×C47—41no—
324C2×C47—42no—
325C2×C47—41no—
326C2×C47—42no—
327C2×C47#230811yescharacteristic
328C2×C47#290112yes—
329D47—82no—
330C22×#41—471no—
331C22×#41C2×D4711yescharacteristic
332C22×M16—472no—
333#537—474no—
334#536—472no—
335#536—941no—
336#535—472no—
337#535—941no—
338#534—941no—
339#534—941no—
340#532—942no—
341#525—472no—
342#525—472no—
343#525—942no—
344C2×#117—471no—
345C2×#117C2×D4711yescharacteristic, complemented
346C22×C42C2×D4711yescharacteristic, complemented
347C2×#104—474no—
348C2×#104—474no—
349#408—942no—
350#408—942no—
351#407—472no—
352#407—472no—
353#391—21no—
354#390—21no—
355#386—21no—
356#386—21no—
357C4×C22⋊C4C2×D4712yes—
358C4×C22⋊C4C2×D4712yescomplemented
359#339—474no—
360#339—474no—
361C47#2177511yescharacteristic, complemented, Sylow
362C2×#41—42no—
363C2×#41—42no—
364C2×#41—472no—
365C2×#41—472no—
366C2×#41—941no—
367C2×#41—941no—
368C2×#41—1882no—
369C2×#41—1882no—
370C2×#41C4×D4712yes—
371C2×#41C4×D4712yes—
372C2×#41D18812yes—
373C2×#41D18812yes—
374C22×Q8—471no—
375C22×Q8C22⋊D4711yescharacteristic
376C22×D4—471no—
377C22×D4—941no—
378C22×D4C22⋊D4711yescharacteristic
379C22×D4C22⋊D4711yescharacteristic
380C22×D4C22⋊D4711yescharacteristic
381C23×C4—21no—
382C23×C4—941no—
383C23×C4C22×D4711yescharacteristic
384C23×C4C22×D4711yescharacteristic
385C23×C4C22×D4711yescharacteristic
386C2×M16—474no—
387C2×M16—942no—
388C2×M16—1882no—
389C2×M16—1882no—
390C22×C8—942no—
391C4⋊Q8—942no—
392#127—942no—
393#124—942no—
394#124—942no—
395#123—942no—
396#123—942no—
397#123—942no—
398#123—942no—
399C22⋊Q8—474no—
400C22⋊Q8—474no—
401C22⋊Q8—944no—
402#121—474no—
403#121—474no—
404#121—944no—
405#121—944no—
406#121—944no—
407#120—942no—
408#120—942no—
409#120—942no—
410#120—942no—
411C4×D4—944no—
412C4×D4—944no—
413#117—42no—
414#117—472no—
415#117—472no—
416#117—941no—
417#117—941no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—472no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—941no—
424C2×C42—21no—
425C2×C42—42no—
426C2×C42—941no—
427C2×C42C22×D4712yes—
428C2×C42C22×D4712yes—
429#104—1884no—
430#104—1884no—
431C22⋊C8—942no—
432C22⋊C8—942no—
433C22⋊C8—944no—
434#95—22no—
435#95—22no—
436C24—941no—
437C24C2×C22⋊D4711yescharacteristic
438C24C2×C22⋊D4711yescharacteristic
439C24C2×C22⋊D4711yescharacteristic
440#41—44no—
441#41—44no—
442#41—42no—
443#41—42no—
444#41—42no—
445#41—42no—
446#41—944no—
447#41—944no—
448#41—1884no—
449#41—1882no—
450#41—1882no—
451#41—1884no—
452C2×Q8—41no—
453C2×Q8—41no—
454C2×Q8—472no—
455C2×Q8—941no—
456C2×Q8—1882no—
457C2×Q8(C2×D47)⋊C412yes—
458C2×Q8(C2×D47)⋊C412yes—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—472no—
466C2×D4—944no—
467C2×D4—944no—
468C2×D4—942no—
469C2×D4—942no—
470C2×D4—941no—
471C2×D4—1882no—
472C2×D4—1882no—
473C2×D4—1882no—
474C2×D4(C2×D47)⋊C412yes—
475C2×D4(C2×D47)⋊C412yes—
476C2×D4(C2×D47)⋊C412yes—
477C2×D4(C2×D47)⋊C412yes—
478C2×D4(C2×D47)⋊C412yes—
479C2×D4(C2×D47)⋊C412yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—942no—
491C22×C4—941no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—1884no—
495C22×C4C2×C4×D4711yescharacteristic
496C22×C4C2×C4×D4711yescharacteristic
497C22×C4C2×C4×D4712yes—
498C22×C4C2×C4×D4712yes—
499C22×C4C2×D18811yescharacteristic
500C22×C4C2×D18811yescharacteristic
501C22×C4C2×D18812yes—
502C22×C4C2×D18812yes—
503C22×C4C2×C22⋊D4711yescharacteristic
504C22×C4C2×C22⋊D4711yescharacteristic
505C22×C4C2×C22⋊D4711yescharacteristic
506C22×C4C23×D4711yescharacteristic
507M16—944no—
508M16—944no—
509M16—1884no—
510C2×C8—944no—
511C2×C8—1884no—
512C4⋊C4—42no—
513C4⋊C4—474no—
514C4⋊C4—944no—
515C4⋊C4—944no—
516C4⋊C4—942no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—942no—
521C22⋊C4—944no—
522C22⋊C4—944no—
523C22⋊C4—942no—
524C22⋊C4—944no—
525C22⋊C4—944no—
526C42—24no—
527C42—24no—
528C42—42no—
529C42—942no—
530C42—942no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—942no—
541C23—3762no—
542C23—1882no—
543C23C2×(C2×D47)⋊C411yescharacteristic
544C23C2×(C2×D47)⋊C412yes—
545C23C2×(C2×D47)⋊C412yes—
546C23C2×(C2×D47)⋊C411yescharacteristic
547C23C22×C22⋊D4711yescharacteristic
548Q8—42no—
549Q8—42no—
550Q8—944no—
551Q8—1882no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—944no—
559D4—3764no—
560D4—3764no—
561D4—1882no—
562D4—1884no—
563D4—1882no—
564D4—1882no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—944no—
589C2×C4—942no—
590C2×C4—3762no—
591C2×C4—1884no—
592C2×C4—1882no—
593C2×C4—1884no—
594C2×C4C2×(C2×D47)⋊C411yescharacteristic
595C2×C4C2×(C2×D47)⋊C412yes—
596C2×C4C2×(C2×D47)⋊C412yes—
597C2×C4C2×(C2×D47)⋊C411yescharacteristic
598C2×C4C22×C4×D4711yescharacteristic
599C2×C4C22×D18811yescharacteristic
600C2×C4#1309406912yes—
601C2×C4#1309406912yes—
602C8—1884no—
603C22—22no—
604C22—22no—
605C22—24no—
606C22—24no—
607C22—22no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—3764no—
619C22—1884no—
620C22—1882no—
621C22C22×(C2×D47)⋊C411yescharacteristic
622C22#1568968211yescharacteristic
623C22#1568968211yescharacteristic
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—44no—
629C4—82no—
630C4—1884no—
631C4—1882no—
632C4#1568959612yes—
633C4#1568968512yes—
634C2—21no—
635C2—21no—
636C2—42no—
637C2—41no—
638C2—41no—
639C2—42no—
640C2—3762no—
641C2#2024596311yescharacteristic
642C2#2025193612yes—
subgroupG/H
centerC2×C4C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2297C2
socleC22×C47C22×C22⋊C4
prime radical#20256717C2
cosocleC22×C4×C47C24
hypercenter#2297D47
hypocenterC47#21775
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #20251833C2semidirect
3. #20243836C2semidirect
4. #20243831C2semidirect
5. #20243688C2semidirect
6. #20243488C2semidirect
7. #20243333C2semidirect
8. C47×#2297C2semidirect
10. C22×#10625329C22semidirect
11. #15689701C4semidirect
13. #15689694C4semidirect
15. #15689687C4semidirect
28. #15689087C22semidirect
29. #15688883C22semidirect
30. #15688883C22semidirect
32. C2×#117×C47C22semidirect
33. C22×C42×C47C22semidirect
39. C4×C22⋊C4×C47C22semidirect
44. C2×#10625350C2×C4semidirect
46. C22×Dic94C2×C4semidirect
47. C22×D188C2×C4semidirect
64. #13094003C2×C4semidirect
67. #13093998C2×C4semidirect
79. #13093983C2×C4semidirect
81. C2×C4⋊Dic47C2×C4semidirect
315. #2297D47semidirect
345. C2×#117C2×D47semidirect
346. C22×C42C2×D47semidirect
358. C4×C22⋊C4C2×D47semidirect
361. C47#21775semidirect

Maximal subgroups

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

subgroupindex
#202567172
#202518332
#202438362
#202438312
#202436882
#202434882
#202433332
C47×#22972
#2177547