Finite Group Databasebeta

← #26375376

Subgroups — #26375376

Order 12032.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689728C211yesmaximal, characteristic
2#20246184C214yesmaximal, complemented
3#20246151C214yesmaximal, complemented
4#20246127C211yesmaximal, characteristic, complemented
5#20246111C211yesmaximal, characteristic, complemented
6#20246077C212yesmaximal, complemented
7#20246026C212yesmaximal, complemented
8#20245950C211yesmaximal, characteristic, complemented
9#20245943C211yesmaximal, characteristic, complemented
10#20245909C214yesmaximal, complemented
11#20245900C214yesmaximal, complemented
12#20245894C212yesmaximal, complemented
13C4×#13093986C212yesmaximal, complemented
14#117×Dic47C211yesmaximal, characteristic, complemented
15C47×#2314C211yesmaximal, characteristic, complemented
16#15689728C4116yescomplemented
17C2×#13094021C2211yescharacteristic
18C2×#13094016C2211yescharacteristic
19C2×#13094009C2214yes—
20C2×#13093998C2212yes—
21C2×#13093997C2214yes—
22C2×C4×D188C2212yes—
23C2×#13093986C2212yes—
24C2×#13093986C2212yes—
25C2×#13093986C2214yes—
26C2×(C22⋊D47)⋊C4C2214yes—
27C2×#13093983C2211yescharacteristic
28C22×C4×Dic47C2211yescharacteristic
29C2×C42×D47C2214yes—
30#15689024C2212yes—
31#15689024C2212yes—
32#15689024C2214yes—
33#15689023C2214yes—
34#15689019C2212yes—
35#15689019C2212yes—
36#15689018C2214yes—
37#15689015C2214yes—
38#15689014C2214yes—
39#15689014C2214yes—
40#15689012C2214yes—
41#15689005C2214yes—
42#15689004C2214yes—
43#15689001C2214yes—
44#15689001C2214yes—
45#15689000C2214yes—
46#15688999C2214yes—
47#15688997C2214yes—
48#15688985C2214yes—
49#15688982C2214yes—
50C4⋊C4×Dic47C2212yes—
51C4⋊C4×Dic47C2212yes—
52C4⋊C4×Dic47C2212yes—
53C4⋊C4×Dic47C2212yes—
54#15688974C2214yes—
55C22⋊C4×Dic47C2212yes—
56C22⋊C4×Dic47C2212yes—
57C4×(C2×D47)⋊C4C2214yes—
58#15688970C2214yes—
59#15688970C2214yes—
60#15688970C2214yes—
61#15688969C2211yescharacteristic
62#15688969C2211yescharacteristic
63C42×Dic47C2211yescharacteristic
64C42×Dic47C2211yescharacteristic
65C2×#117×C47C2211yescharacteristic
66C47×#388C2214yes—
67C47×#387C2214yes—
68C47×#384C2211yescharacteristic
69C47×#382C2211yescharacteristic
70C47×#380C2212yes—
71C4×C4⋊C4×C47C2212yes—
72C22×D188C2312yes—
73C22×D188C2314yes—
74C22×C22⋊D47C2314yes—
75C23×Dic47C2311yescharacteristic
76C23×Dic47C2311yescharacteristic
77C22×C4×D47C2314yes—
78C22×C4×D47C2314yes—
79C22×C4×D47C2314yes—
80#13094021C2×C418yescomplemented
81#13094016C2×C418yescomplemented
82#13094009C2×C4132yescomplemented
83#13093998C2×C4116yescomplemented
84#13093997C2×C4132yescomplemented
85C4×D188C2×C4116yescomplemented
86#13093986—28no—
87#13093986—28no—
88#13093986—48no—
89(C22⋊D47)⋊C4—48no—
90#13093983—24no—
91C2×C4⋊Dic47C2312yes—
92C2×C4⋊Dic47C2312yes—
93C2×Dic47⋊C4C2314yes—
94C2×Dic47⋊C4C2314yes—
95C2×Dic47⋊C4C2314yes—
96C2×C22⋊Dic47C2312yes—
97C2×(C2×D47)⋊C4C2314yes—
98C2×(C2×D47)⋊C4C2314yes—
99C2×(C2×D47)⋊C4C2314yes—
100C2×(C2×D47)⋊C4C2314yes—
101C2×(C2×D47)⋊C4C2314yes—
102C2×(C2×D47)⋊C4C2314yes—
103C2×(C2×D47)⋊C4C2314yes—
104C2×(C2×D47)⋊C4C2314yes—
105C2×C4×Dic47—24no—
106C2×C4×Dic47C2311yescharacteristic
107C2×C4×Dic47C2312yes—
108C2×C4×Dic47C2312yes—
109C2×C4×Dic47C2311yescharacteristic
110C2×C4×Dic47C2311yescharacteristic
111C2×C4×Dic47C2314yes—
112C2×C4×Dic47C2311yescharacteristic
113C2×C4×Dic47C2312yes—
114C2×C4×Dic47C2312yes—
115C2×C4×Dic47C2314yes—
116C2×C4×Dic47C2311yescharacteristic
117C2×C4×Dic47C2311yescharacteristic
118C42×D47—48no—
119C47⋊#95C2312yes—
120C47⋊#95C2312yes—
121C47⋊#95C2312yes—
122C47⋊#95C2312yes—
123C47⋊#95C2314yes—
124C47⋊#95C2314yes—
125C47⋊#95C2314yes—
126C47⋊#95C2314yes—
127C23×C4×C47C2311yescharacteristic
128#117×C47C2×C418yescomplemented
129C2×C4⋊C4×C47C2312yes—
130C2×C4⋊C4×C47C2312yes—
131C2×C4⋊C4×C47C2314yes—
132C2×C4⋊C4×C47C2312yes—
133C2×C22⋊C4×C47C2314yes—
134C2×C22⋊C4×C47C2312yes—
135C2×C42×C47C2314yes—
136C2×C42×C47C2311yescharacteristic
137C2×C42×C47C2311yescharacteristic
138#95×C47C2314yes—
139#95×C47C2314yes—
140#95×C47C2314yes—
141C24×C47C2411yescharacteristic
142C23×D47C2414yes—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47C22×C414yes—
146C22×C4×C47C2414yes—
147C22×C4×C47C2414yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2412yes—
150C22×C4×C47C2411yescharacteristic
151C22×C4×C47C2411yescharacteristic
152C2×C22⋊D47—44no—
153C2×C22⋊D47—44no—
154C2×C22⋊D47C22×C4116yes—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—24no—
158C22×Dic47—42no—
159C22×Dic47C2411yescharacteristic
160C22×Dic47C2412yes—
161C22×Dic47C2411yescharacteristic
162C22×Dic47C2414yes—
163C22×Dic47C2411yescharacteristic
164C22×Dic47C2412yes—
165C22×Dic47C2411yescharacteristic
166C2×D188—24no—
167C2×D188—24no—
168C2×D188—44no—
169C2×D188—44no—
170C2×D188C22×C418yes—
171C2×D188C22×C4116yes—
172C2×C4×D47—28no—
173C2×C4×D47—28no—
174C2×C4×D47—28no—
175C2×C4×D47—28no—
176C2×C4×D47—44no—
177C2×C4×D47—44no—
178C2×C4×D47—44no—
179C2×C4×D47—44no—
180C2×C4×D47C22×C4116yes—
181C4⋊C4×C47—24no—
182C4⋊C4×C47—24no—
183C4⋊C4×C47—44no—
184C4⋊C4×C47C22×C418yes—
185C22⋊C4×C47—44no—
186C22⋊C4×C47C22×C418yes—
187C42×C47—44no—
188C42×C47C22×C414yes—
189C42×C47C22×C414yes—
190C22⋊Dic47—24no—
191(C2×D47)⋊C4—28no—
192(C2×D47)⋊C4—28no—
193(C2×D47)⋊C4—44no—
194(C2×D47)⋊C4—44no—
195(C2×D47)⋊C4—44no—
196(C2×D47)⋊C4—44no—
197(C2×D47)⋊C4C22×C4116yes—
198(C2×D47)⋊C4C22×C4116yes—
199C4⋊Dic47—24no—
200C4⋊Dic47C22×C418yes—
201Dic47⋊C4—44no—
202Dic47⋊C4—44no—
203Dic47⋊C4C22×C4116yes—
204C4×Dic47—22no—
205C4×Dic47—22no—
206C4×Dic47—22no—
207C4×Dic47—22no—
208C4×Dic47—24no—
209C4×Dic47—28no—
210C4×Dic47—24no—
211C4×Dic47—44no—
212C4×Dic47—42no—
213C4×Dic47—42no—
214C4×Dic47#4114yes—
215C4×Dic47#4114yes—
216C23×C47—21no—
217C23×C47—21no—
218C23×C47—22no—
219C23×C47—22no—
220C23×C47C23×C412yes—
221C23×C47C2511yescharacteristic
222C22×D47—24no—
223C22×D47—24no—
224C22×D47—24no—
225C22×D47—24no—
226C22×D47—24no—
227C22×D47—24no—
228C22×D47C23×C418yes—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—22no—
235C2×C4×C47—28no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—24no—
241C2×C4×C47—24no—
242C2×C4×C47C23×C414yes—
243C2×C4×C47C23×C412yes—
244C2×C4×C47C23×C414yes—
245C2×C4×C47C23×C412yes—
246C2×C4×C47C2×#4112yes—
247C2×C4×C47C2×#4112yes—
248C22⋊D47—416no—
249C2×Dic47—22no—
250C2×Dic47—24no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—24no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—44no—
262C2×Dic47—44no—
263C2×Dic47C23×C418yes—
264C2×Dic47C2×#4112yes—
265C2×Dic47C2×#4114yes—
266C2×Dic47C2×#4112yes—
267C2×Dic47H(5,𝔽2)14yes—
268D188—216no—
269D188—416no—
270C4×D47—416no—
271C4×D47—416no—
272C4×D47—416no—
273C4×#540—471nomaximal, Sylow
274C22×C47—22no—
275C22×C47—24no—
276C22×C47—41no—
277C22×C47—41no—
278C22×C47—42no—
279C22×C47C24×C411yescharacteristic
280C22×C47C22×#4111yescharacteristic
281C22×C47C22×#4111yescharacteristic
282C22×C47C2×H(5,𝔽2)12yes—
283C22×C47#58512yes—
284C2×D47—28no—
285C2×D47—28no—
286C2×D47—28no—
287C2×D47—44no—
288C2×D47—44no—
289C2×D47—44no—
290C2×D47—44no—
291C4×C47—22no—
292C4×C47—24no—
293C4×C47—24no—
294C4×C47—44no—
295C4×C47—44no—
296C4×C47C4×#4114yes—
297Dic47—28no—
298Dic47—22no—
299Dic47—24no—
300Dic47—41no—
301Dic47—41no—
302Dic47—42no—
303Dic47#54014yescomplemented
304C2×#540—471no—
305#2335—474no—
306#2333—474no—
307#2330—471no—
308#2328—471no—
309#2324—472no—
310#2319—472no—
311#2314D4711yescharacteristic, complemented
312C4×#127—471no—
313C4×#125—471no—
314C4×#123—474no—
315C4×#121—474no—
316C4×#121—472no—
317C42×D4—472no—
318C4×#117—471no—
319C2×C47—42no—
320C2×C47C2×C4×#4111yescharacteristic
321C2×C47C4×H(5,𝔽2)12yes—
322C2×C47C2×#54011yescharacteristic
323C2×C47#345511yescharacteristic
324C2×C47#346012yes—
325D47—48no—
326#540—4716no—
327C2×#127—471no—
328C2×#125—471no—
329C2×#123—474no—
330C2×#121—472no—
331C2×#121—474no—
332C2×C4×D4—472no—
333C2×C4×D4—472no—
334C2×C4×D4—474no—
335C2×C4×D4—474no—
336C2×C4×D4—472no—
337C2×#117—471no—
338C2×#117C2×D4711yescharacteristic
339C22×C42—471no—
340C22×C42—474no—
341#390—472no—
342#390—474no—
343#390—472no—
344#390—474no—
345#389—472no—
346#389—472no—
347#389—474no—
348#388—474no—
349#388—474no—
350#388—474no—
351#388—474no—
352#388C2×D4714yes—
353#387—474no—
354#387—474no—
355#387C2×D4714yes—
356#386—474no—
357#386—474no—
358#386—474no—
359#386—474no—
360#385—474no—
361#384C2×D4711yescharacteristic
362#382C2×D4711yescharacteristic
363#381—474no—
364#380—474no—
365#380C2×D4712yes—
366C4×C4⋊C4—472no—
367C4×C4⋊C4—472no—
368C4×C4⋊C4—472no—
369C4×C4⋊C4—472no—
370C4×C4⋊C4—474no—
371C4×C4⋊C4C2×D4712yes—
372C4×C22⋊C4—472no—
373C4×C22⋊C4—472no—
374C4×C22⋊C4—474no—
375C4×C22⋊C4—474no—
376C4×C22⋊C4—474no—
377C4×C22⋊C4—474no—
378#376—471no—
379#376—471no—
380C43—471no—
381C43—471no—
382C47C4×#54011yescharacteristic, complemented, Sylow
383C22×D4—474no—
384C22×D4—472no—
385C22×D4—474no—
386C23×C4—474no—
387C23×C4—471no—
388C23×C4—471no—
389C23×C4—474no—
390C23×C4—474no—
391C23×C4C22×D4711yescharacteristic
392#127—478no—
393#125—478no—
394#123—4732no—
395#121—4716no—
396#121—4732no—
397C4×D4—4716no—
398C4×D4—948no—
399C4×D4—948no—
400C4×D4—1888no—
401C4×D4—1888no—
402#117—944no—
403#117C4×D4718yescomplemented
404C2×C4⋊C4—472no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—472no—
407C2×C4⋊C4—474no—
408C2×C4⋊C4—474no—
409C2×C4⋊C4C22×D4712yes—
410C2×C4⋊C4C22×D4712yes—
411C2×C4⋊C4C22×D4714yes—
412C2×C4⋊C4C22×D4712yes—
413C2×C22⋊C4—474no—
414C2×C22⋊C4—474no—
415C2×C22⋊C4—472no—
416C2×C22⋊C4—474no—
417C2×C22⋊C4—474no—
418C2×C22⋊C4—474no—
419C2×C22⋊C4—474no—
420C2×C22⋊C4—474no—
421C2×C22⋊C4—474no—
422C2×C22⋊C4C22×D4714yes—
423C2×C22⋊C4C22×D4712yes—
424C2×C42—471no—
425C2×C42—471no—
426C2×C42—471no—
427C2×C42—472no—
428C2×C42—474no—
429C2×C42—472no—
430C2×C42—471no—
431C2×C42—471no—
432C2×C42—471no—
433C2×C42—472no—
434C2×C42—472no—
435C2×C42—474no—
436C2×C42—944no—
437C2×C42—1888no—
438C2×C42C22×D4714yes—
439C2×C42C22×D4711yescharacteristic
440C2×C42C22×D4711yescharacteristic
441#95—472no—
442#95—472no—
443#95—472no—
444#95—472no—
445#95—474no—
446#95—474no—
447#95—474no—
448#95—474no—
449#95C22×D4714yes—
450#95C22×D4714yes—
451#95C22×D4714yes—
452C24—474no—
453C24C23×D4711yescharacteristic
454C2×D4—4716no—
455C2×D4—478no—
456C2×D4—4716no—
457C2×D4—944no—
458C2×D4—944no—
459C2×D4—1884no—
460C2×D4—1884no—
461C2×D4—1884no—
462C2×D4—1884no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—471no—
466C22×C4—471no—
467C22×C4—472no—
468C22×C4—474no—
469C22×C4—4716no—
470C22×C4—471no—
471C22×C4—472no—
472C22×C4—471no—
473C22×C4—948no—
474C22×C4—948no—
475C22×C4—948no—
476C22×C4—948no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—944no—
480C22×C4—1884no—
481C22×C4—1884no—
482C22×C4—1884no—
483C22×C4—1884no—
484C22×C4—1882no—
485C22×C4C2×C4×D4714yes—
486C22×C4C23×D4714yes—
487C22×C4C23×D4714yes—
488C22×C4C23×D4712yes—
489C22×C4C23×D4712yes—
490C22×C4C23×D4711yescharacteristic
491C22×C4C23×D4711yescharacteristic
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—44no—
495C4⋊C4—4716no—
496C4⋊C4—478no—
497C4⋊C4—944no—
498C4⋊C4—1884no—
499C4⋊C4—1884no—
500C4⋊C4C2×C4×D4718yes—
501C22⋊C4—44no—
502C22⋊C4—4716no—
503C22⋊C4—4716no—
504C22⋊C4—948no—
505C22⋊C4—948no—
506C22⋊C4—944no—
507C22⋊C4—1884no—
508C22⋊C4—1884no—
509C22⋊C4—1884no—
510C22⋊C4—1884no—
511C22⋊C4C2×C4×D4718yes—
512C42—44no—
513C42—474no—
514C42—474no—
515C42—942no—
516C42—942no—
517C42—944no—
518C42—948no—
519C42—944no—
520C42—942no—
521C42—942no—
522C42—1882no—
523C42—1882no—
524C42—1884no—
525C42C2×C4×D4714yes—
526C42C2×C4×D4714yes—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—478no—
532C23—944no—
533C23—944no—
534C23—944no—
535C23—944no—
536C23—944no—
537C23—944no—
538C23C22×C4×D4712yes—
539C23C24×D4711yescharacteristic
540D4—9416no—
541D4—18816no—
542D4—18816no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—472no—
557C2×C4—474no—
558C2×C4—478no—
559C2×C4—472no—
560C2×C4—474no—
561C2×C4—942no—
562C2×C4—944no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—944no—
566C2×C4—941no—
567C2×C4—941no—
568C2×C4—942no—
569C2×C4—942no—
570C2×C4—942no—
571C2×C4—942no—
572C2×C4—942no—
573C2×C4—18816no—
574C2×C4—18816no—
575C2×C4—18816no—
576C2×C4—1884no—
577C2×C4—1884no—
578C2×C4C22×C4×D4714yes—
579C2×C4C22×C4×D4714yes—
580C2×C4C22×C4×D4712yes—
581C2×C4C22×C4×D4712yes—
582C2×C4#41×D4712yes—
583C2×C4#41×D4712yes—
584C22—22no—
585C22—24no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—948no—
590C22—948no—
591C22—948no—
592C22—1884no—
593C22—1884no—
594C22—1884no—
595C22—1884no—
596C22C23×C4×D4711yescharacteristic
597C22C2×#41×D4711yescharacteristic
598C22C2×#41×D4711yescharacteristic
599C22H(5,𝔽2)×D4712yes—
600C22#1568996012yes—
601C4—22no—
602C4—24no—
603C4—24no—
604C4—44no—
605C4—44no—
606C4—474no—
607C4—942no—
608C4—944no—
609C4—948no—
610C4—1881no—
611C4—1882no—
612C4—1881no—
613C4#1568959114yes—
614C2—42no—
615C2—1888no—
616C2C2×#1568959111yescharacteristic
617C2#2025659412yes—
618C2#540×D4711yescharacteristic
619C2#2025677911yescharacteristic
620C2#2025680812yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2314C2
socle, cosocleC23×C47C25
prime radicalC2×#15689728C2
hypercenter#2314D47
hypocenterC47C4×#540
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #20246184C2semidirect
3. #20246151C2semidirect
4. #20246127C2semidirect
5. #20246111C2semidirect
6. #20246077C2semidirect
7. #20246026C2semidirect
8. #20245950C2semidirect
9. #20245943C2semidirect
10. #20245909C2semidirect
11. #20245900C2semidirect
12. #20245894C2semidirect
13. C4×#13093986C2semidirect
14. #117×Dic47C2semidirect
15. C47×#2314C2semidirect
16. #15689728C4semidirect
80. #13094021C2×C4semidirect
81. #13094016C2×C4semidirect
82. #13094009C2×C4semidirect
83. #13093998C2×C4semidirect
84. #13093997C2×C4semidirect
85. C4×D188C2×C4semidirect
128. #117×C47C2×C4semidirect
303. Dic47#540semidirect
311. #2314D47semidirect
382. C47C4×#540semidirect
403. #117C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897282
#202461842
#202461512
#202461272
#202461112
#202460772
#202460262
#202459502
#202459432
#202459092
#202459002
#202458942
C4×#130939862
#117×Dic472
C47×#23142
C4×#54047