Finite Group Databasebeta

← #26376548

Subgroups — #26376548

Order 12032.

3190 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094007C211yesmaximal, characteristic
2C2×#15689596C211yesmaximal, characteristic
3#20246380C218yesmaximal, complemented
4#20246258C214yesmaximal, complemented
5#20246238C214yesmaximal, complemented
6#20246057C214yesmaximal, complemented
7#20246000C211yesmaximal, characteristic, complemented
8#20245963C211yesmaximal, characteristic
9C4×#13094007C214yesmaximal, complemented
10C2×#15688986C211yesmaximal, characteristic, complemented
11#20245741C211yesmaximal, characteristic, complemented
12C22×#117×C47C211yesmaximal, characteristic, complemented
13C23×C22⋊D47C2211yescharacteristic
14C2×#13094007C4116yescomplemented
15C2×#13094007C2212yes—
16C2×#13094007C2214yes—
17C2×#13094007C2212yes—
18#15689596—28no—
19C2×C4×C22⋊D47C2214yes—
20C2×(C22⋊D47)⋊C4C2214yes—
21C2×#13093980C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×C22⋊Dic47C2211yescharacteristic
24C22×C22⋊Dic47C2211yescharacteristic
25C22×C22⋊Dic47C2211yescharacteristic
26C22×(C2×D47)⋊C4C2212yes—
27C2×C22⋊C4×D47C2212yes—
28#15689018C2218yescomplemented
29#15689014C2218yescomplemented
30#15689013C2218yescomplemented
31#15689007C2218yescomplemented
32#15689004C2218yescomplemented
33#15689002C2214yes—
34#15689000C2214yes—
35#15688996C2214yescomplemented
36#15688995C2214yescomplemented
37#15688994C2218yescomplemented
38#15688986C2214yes—
39#15688986C2212yescomplemented
40#15688986C2212yescomplemented
41#15688984C2212yes—
42#15688981C2214yes—
43#15688978C2212yes—
44C4×Dic47⋊C4C2218yescomplemented
45C22⋊C4×Dic47C2214yes—
46C22⋊C4×Dic47C2214yescomplemented
47C4×(C2×D47)⋊C4C2218yescomplemented
48C4×C22⋊Dic47C2214yescomplemented
49C2×C47⋊#95C2214yescomplemented
50C24×C4×C47C2211yescharacteristic
51C2×#117×C47—28no—
52C2×#117×C47C2214yes—
53C2×#117×C47C2214yescomplemented
54C22×C4⋊C4×C47C2212yescomplemented
55C22×C22⋊C4×C47C2211yescharacteristic
56C22×C22⋊C4×C47C2211yescharacteristic
57C22×C42×C47C2212yescomplemented
58C24×D47C2311yescharacteristic
59C22×C22⋊D47—24no—
60C22×C22⋊D47—24no—
61C22×C22⋊D47C2×C418yes—
62C22×C22⋊D47C2312yes—
63C22×C22⋊D47C2312yes—
64C23×Dic47C2311yescharacteristic
65C23×Dic47C2312yes—
66C23×Dic47C2312yes—
67C23×Dic47C2312yes—
68C22×C4×D47C2314yes—
69#13094007C2×C4116yescomplemented
70#13094007C2×C4132yescomplemented
71#13094007C2×C4116yescomplemented
72C4×C22⋊D47—216no—
73(C22⋊D47)⋊C4—216no—
74#13093980—28no—
75C2×Dic47⋊C4C2×C4116yescomplemented
76C2×Dic47⋊C4C2314yes—
77C2×Dic47⋊C4C2318yes—
78C2×Dic47⋊C4C2314yes—
79C2×Dic47⋊C4C2314yes—
80C2×C22⋊Dic47—24no—
81C2×C22⋊Dic47—24no—
82C2×C22⋊Dic47C2×C418yescomplemented
83C2×C22⋊Dic47C2312yes—
84C2×C22⋊Dic47C2312yes—
85C2×C22⋊Dic47C2312yes—
86C2×C22⋊Dic47C2312yes—
87C2×C22⋊Dic47C2312yes—
88C2×C22⋊Dic47C2312yes—
89C2×(C2×D47)⋊C4C2×C4116yescomplemented
90C2×(C2×D47)⋊C4C2314yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2314yes—
93C2×(C2×D47)⋊C4C2314yes—
94C22⋊C4×D47—28no—
95C2×C4×Dic47C2314yes—
96C2×C4×Dic47C2318yes—
97C47⋊#95C2318yes—
98C47⋊#95C2318yes—
99C47⋊#95C2318yes—
100C47⋊#95C2318yes—
101C47⋊#95C2318yes—
102C25×C47C2311yescharacteristic
103C23×C4×C47—24no—
104C23×C4×C47—24no—
105C23×C4×C47C2×C418yescomplemented
106C23×C4×C47C2312yes—
107C23×C4×C47C2312yes—
108C23×C4×C47C2311yescharacteristic
109C23×C4×C47C2311yescharacteristic
110C23×C4×C47C2312yes—
111C23×C4×C47C2312yes—
112#117×C47—216no—
113#117×C47—216no—
114C2×C4⋊C4×C47—28no—
115C2×C4⋊C4×C47C2318yes—
116C2×C22⋊C4×C47—24no—
117C2×C22⋊C4×C47D418yes—
118C2×C22⋊C4×C47C2312yes—
119C2×C22⋊C4×C47C2312yes—
120C2×C22⋊C4×C47C2312yes—
121C2×C22⋊C4×C47C2312yes—
122C2×C42×C47—28no—
123C2×C42×C47C2318yes—
124C24×C47—44no—
125C24×C47C22×C414yes—
126C24×C47C2×D414yes—
127C24×C47#4114yes—
128C24×C47C2411yescharacteristic
129C24×C47C2411yescharacteristic
130C24×C47C2411yescharacteristic
131C23×D47—22no—
132C23×D47—22no—
133C23×D47—44no—
134C23×D47C22×C414yes—
135C23×D47C2412yes—
136C22×C4×C47—232no—
137C22×C4×C47—22no—
138C22×C4×C47—22no—
139C22×C4×C47—24no—
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—24no—
145C22×C4×C47—24no—
146C22×C4×C47—44no—
147C22×C4×C47—44no—
148C22×C4×C47C22×C418yes—
149C22×C4×C47C22×C418yes—
150C22×C4×C47C22×C414yes—
151C22×C4×C47C22×C414yes—
152C22×C4×C47C2×D418yes—
153C22×C4×C47C2414yes—
154C22×C4×C47C2412yes—
155C22×C4×C47C2412yes—
156C22×C4×C47C2414yes—
157C2×C22⋊D47—232no—
158C2×C22⋊D47—24no—
159C2×C22⋊D47—24no—
160C2×C22⋊D47—24no—
161C2×C22⋊D47—24no—
162C2×C22⋊D47C22×C418yes—
163C2×C22⋊D47C22×C418yes—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—24no—
167C22×Dic47—24no—
168C22×Dic47—44no—
169C22×Dic47—44no—
170C22×Dic47—44no—
171C22×Dic47C22×C414yes—
172C22×Dic47C22×C418yes—
173C22×Dic47C2414yes—
174C22×Dic47C2414yes—
175C22×Dic47C2414yes—
176C22×Dic47C2412yes—
177C2×C4×D47—28no—
178C2×C4×D47—48no—
179C4⋊C4×C47—216no—
180C22⋊C4×C47—24no—
181C22⋊C4×C47—24no—
182C22⋊C4×C47C2×D418yes—
183C22⋊C4×C47C2×D418yes—
184C42×C47—216no—
185C22⋊Dic47—24no—
186C22⋊Dic47—24no—
187C22⋊Dic47—24no—
188C22⋊Dic47—24no—
189C22⋊Dic47C22×C418yes—
190C22⋊Dic47C22×C418yes—
191(C2×D47)⋊C4—28no—
192(C2×D47)⋊C4—48no—
193(C2×D47)⋊C4C22×C4116yes—
194(C2×D47)⋊C4C22×C4116yes—
195Dic47⋊C4—28no—
196Dic47⋊C4—48no—
197Dic47⋊C4C22×C4116yes—
198Dic47⋊C4C22×C4116yes—
199C4×Dic47—28no—
200C4×Dic47—48no—
201C23×C47—24no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—48no—
205C23×C47—48no—
206C23×C47—48no—
207C23×C47C4×D4116yes—
208C23×C47C23×C412yes—
209C23×C47C23×C412yes—
210C23×C47C23×C412yes—
211C23×C47C22×D412yes—
212C23×C47C22×D412yes—
213C23×C47C22×D412yes—
214C23×C47C2×#4112yes—
215C23×C47C2×#4112yes—
216C23×C47C2×#4112yes—
217C23×C47C2511yescharacteristic
218C22×D47—22no—
219C22×D47—22no—
220C22×D47—28no—
221C22×D47—28no—
222C22×D47—48no—
223C22×D47—84no—
224C22×D47—84no—
225C22×D47C23×C414yes—
226C2×C4×C47—216no—
227C2×C4×C47—28no—
228C2×C4×C47—28no—
229C2×C4×C47—216no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—28no—
235C2×C4×C47—24no—
236C2×C4×C47—28no—
237C2×C4×C47—48no—
238C2×C4×C47—48no—
239C2×C4×C47C23×C414yes—
240C2×C4×C47C23×C414yes—
241C2×C4×C47C22×D418yes—
242C22⋊D47—216no—
243C22⋊D47—216no—
244C2×Dic47—28no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—28no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—28no—
251C2×Dic47—28no—
252C2×Dic47—48no—
253C2×Dic47—84no—
254C2×Dic47—84no—
255C2×Dic47C23×C414yes—
256C2×Dic47C23×C418yes—
257C4×D47—88no—
258#34199—471nomaximal, Sylow
259C22×C47—24no—
260C22×C47—24no—
261C22×C47—24no—
262C22×C47—24no—
263C22×C47—24no—
264C22×C47—24no—
265C22×C47—24no—
266C22×C47—24no—
267C22×C47—24no—
268C22×C47—416no—
269C22×C47C2×C4×D414yes—
270C22×C47C2×C4×D414yes—
271C22×C47C2×C4×D414yes—
272C22×C47C24×C411yescharacteristic
273C22×C47C23×D411yescharacteristic
274C22×C47C22×#4111yescharacteristic
275C22×C47#58514yes—
276C2×D47—24no—
277C2×D47—28no—
278C2×D47—44no—
279C2×D47—44no—
280C2×D47—88no—
281C4×C47—24no—
282C4×C47—24no—
283C4×C47—28no—
284C4×C47—44no—
285Dic47—24no—
286Dic47—28no—
287Dic47—44no—
288Dic47—44no—
289C22×#123—471no—
290C2×#518—471no—
291C22×#117D4711yescharacteristic, complemented
292#2359—478no—
293#2344—474no—
294#2342—474no—
295#2322—474no—
296#2315—471no—
297#2308—471no—
298C4×#123—474no—
299C2×#381—471no—
300#2273—471no—
301C2×C47—24no—
302C2×C47—24no—
303C2×C47—24no—
304C2×C47C22×C4×D411yescharacteristic
305C2×C47#342812yes—
306C2×C47#348512yes—
307C2×C47#349112yes—
308D47—44no—
309C23×D4—471no—
310C24×C4C2×D4711yescharacteristic
311C2×#123—4716no—
312C2×#123—472no—
313C2×#123—474no—
314C2×#123—472no—
315#518—948no—
316C2×C4×D4—474no—
317C2×C4×D4—474no—
318C2×#117—28no—
319C2×#117—472no—
320C2×#117C2×D4714yes—
321C2×#117C2×D4714yescomplemented
322C22×C4⋊C4—472no—
323C22×C4⋊C4C2×D4712yescomplemented
324C22×C22⋊C4—471no—
325C22×C22⋊C4—472no—
326C22×C22⋊C4—471no—
327C22×C22⋊C4—471no—
328C22×C22⋊C4—472no—
329C22×C22⋊C4C2×D4711yescharacteristic
330C22×C22⋊C4C2×D4711yescharacteristic
331C22×C42C2×D4712yescomplemented
332#389—478no—
333#388—478no—
334#388—478no—
335#387—478no—
336#387—478no—
337#386—474no—
338#386—474no—
339#385—474no—
340#385—478no—
341#385—474no—
342#381—472no—
343#381—474no—
344#381—472no—
345#381—472no—
346#380—474no—
347#379—472no—
348C4×C4⋊C4—478no—
349C4×C22⋊C4—474no—
350C4×C22⋊C4—474no—
351C4×C22⋊C4—474no—
352C4×C22⋊C4—478no—
353C2×#95—474no—
354C47#3419911yescharacteristic, complemented, Sylow
355C25—471no—
356C25C22×D4711yescharacteristic
357C22×D4—478no—
358C22×D4—472no—
359C22×D4—472no—
360C22×D4—944no—
361C22×D4—944no—
362C23×C4—24no—
363C23×C4—24no—
364C23×C4—471no—
365C23×C4—472no—
366C23×C4—472no—
367C23×C4—472no—
368C23×C4—474no—
369C23×C4C4×D4718yescomplemented
370C23×C4C22×D4712yes—
371C23×C4C22×D4711yescharacteristic
372C23×C4C22×D4712yes—
373C23×C4C22×D4711yescharacteristic
374C23×C4C22×D4712yes—
375C23×C4C22×D4712yes—
376#123—4716no—
377#123—4732no—
378#123—4716no—
379C4×D4—9416no—
380C4×D4—9416no—
381#117—216no—
382#117—216no—
383#117—948no—
384C2×C4⋊C4—28no—
385C2×C4⋊C4—4716no—
386C2×C4⋊C4—474no—
387C2×C4⋊C4—474no—
388C2×C4⋊C4—474no—
389C2×C4⋊C4—478no—
390C2×C4⋊C4C22×D4718yes—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—478no—
393C2×C22⋊C4—472no—
394C2×C22⋊C4—472no—
395C2×C22⋊C4—4716no—
396C2×C22⋊C4—474no—
397C2×C22⋊C4—474no—
398C2×C22⋊C4—472no—
399C2×C22⋊C4—472no—
400C2×C22⋊C4—472no—
401C2×C22⋊C4—472no—
402C2×C22⋊C4—474no—
403C2×C22⋊C4—478no—
404C2×C22⋊C4—944no—
405C2×C22⋊C4—944no—
406C2×C22⋊C4—948no—
407C2×C22⋊C4C22⋊D4718yes—
408C2×C22⋊C4C22×D4712yes—
409C2×C22⋊C4C22×D4712yes—
410C2×C22⋊C4C22×D4712yes—
411C2×C22⋊C4C22×D4712yes—
412C2×C42—28no—
413C2×C42—474no—
414C2×C42—478no—
415C2×C42C22×D4718yes—
416#95—478no—
417#95—478no—
418#95—478no—
419#95—478no—
420#95—478no—
421C24—44no—
422C24—472no—
423C24—474no—
424C24—942no—
425C24—942no—
426C24—1884no—
427C24C2×C4×D4714yes—
428C24#1062535014yes—
429C24C2×C22⋊D4714yes—
430C24C23×D4711yescharacteristic
431C24C23×D4711yescharacteristic
432C24C23×D4711yescharacteristic
433C2×D4—478no—
434C2×D4—478no—
435C2×D4—9432no—
436C2×D4—944no—
437C2×D4—944no—
438C2×D4—944no—
439C2×D4—944no—
440C22×C4—232no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—474no—
453C22×C4—474no—
454C22×C4—472no—
455C22×C4—478no—
456C22×C4—474no—
457C22×C4—474no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—944no—
461C22×C4—944no—
462C22×C4—948no—
463C22×C4—1888no—
464C22×C4—1884no—
465C22×C4—1884no—
466C22×C4—1884no—
467C22×C4C2×C4×D4718yes—
468C22×C4C2×C4×D4714yes—
469C22×C4C2×C4×D4718yes—
470C22×C4C2×C4×D4714yes—
471C22×C4C2×C22⋊D4718yes—
472C22×C4C23×D4714yes—
473C22×C4C23×D4712yes—
474C22×C4C23×D4712yes—
475C22×C4C23×D4714yes—
476C4⋊C4—216no—
477C4⋊C4—4716no—
478C4⋊C4—4716no—
479C4⋊C4—948no—
480C4⋊C4—1888no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—478no—
484C22⋊C4—478no—
485C22⋊C4—4716no—
486C22⋊C4—4716no—
487C22⋊C4—944no—
488C22⋊C4—944no—
489C22⋊C4—944no—
490C22⋊C4—944no—
491C22⋊C4—948no—
492C22⋊C4—1888no—
493C22⋊C4C2×C22⋊D4718yes—
494C22⋊C4C2×C22⋊D4718yes—
495C42—216no—
496C42—948no—
497C42—1888no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—474no—
505C23—948no—
506C23—942no—
507C23—942no—
508C23—948no—
509C23—3764no—
510C23—3764no—
511C23—1888no—
512C23C4×C22⋊D47116yes—
513C23C22×C4×D4712yes—
514C23C22×C4×D4712yes—
515C23C22×C4×D4712yes—
516C23C22×C22⋊D4712yes—
517C23C22×C22⋊D4712yes—
518C23C22×C22⋊D4712yes—
519C23C2×#1062535012yes—
520C23C2×#1062535012yes—
521C23C2×#1062535012yes—
522C23C24×D4711yescharacteristic
523D4—9416no—
524D4—9416no—
525C2×C4—216no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—216no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—478no—
539C2×C4—474no—
540C2×C4—948no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—948no—
544C2×C4—944no—
545C2×C4—944no—
546C2×C4—948no—
547C2×C4—948no—
548C2×C4—3768no—
549C2×C4—3764no—
550C2×C4—3764no—
551C2×C4—1888no—
552C2×C4C22×C4×D4714yes—
553C2×C4C22×C4×D4714yes—
554C2×C4C22×C22⋊D4718yes—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—416no—
565C22—944no—
566C22—948no—
567C22—3768no—
568C22—1884no—
569C22—1884no—
570C22C2×C4×C22⋊D4714yes—
571C22C2×C4×C22⋊D4714yes—
572C22C2×C4×C22⋊D4714yes—
573C22C23×C4×D4711yescharacteristic
574C22C23×C22⋊D4711yescharacteristic
575C22C22×#1062535011yescharacteristic
576C22#1568995914yes—
577C4—28no—
578C4—24no—
579C4—24no—
580C4—44no—
581C4—944no—
582C4—948no—
583C4—1884no—
584C4—1884no—
585C2—24no—
586C2—24no—
587C2—24no—
588C2—1884no—
589C2C22×C4×C22⋊D4711yescharacteristic
590C2#2025658912yes—
591C2#2025703212yes—
592C2#2025708612yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC22×#117×C47C2
socle, cosocleC23×C47C25
prime radicalC23×C22⋊D47C22
hypercenterC22×#117D47
hypocenterC47#34199
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. #20246380C2semidirect
4. #20246258C2semidirect
5. #20246238C2semidirect
6. #20246057C2semidirect
7. #20246000C2semidirect
9. C4×#13094007C2semidirect
10. C2×#15688986C2semidirect
11. #20245741C2semidirect
12. C22×#117×C47C2semidirect
14. C2×#13094007C4semidirect
28. #15689018C22semidirect
29. #15689014C22semidirect
30. #15689013C22semidirect
31. #15689007C22semidirect
32. #15689004C22semidirect
35. #15688996C22semidirect
36. #15688995C22semidirect
37. #15688994C22semidirect
39. #15688986C22semidirect
40. #15688986C22semidirect
44. C4×Dic47⋊C4C22semidirect
46. C22⋊C4×Dic47C22semidirect
47. C4×(C2×D47)⋊C4C22semidirect
48. C4×C22⋊Dic47C22semidirect
49. C2×C47⋊#95C22semidirect
53. C2×#117×C47C22semidirect
54. C22×C4⋊C4×C47C22semidirect
57. C22×C42×C47C22semidirect
69. #13094007C2×C4semidirect
70. #13094007C2×C4semidirect
71. #13094007C2×C4semidirect
75. C2×Dic47⋊C4C2×C4semidirect
82. C2×C22⋊Dic47C2×C4semidirect
89. C2×(C2×D47)⋊C4C2×C4semidirect
105. C23×C4×C47C2×C4semidirect
291. C22×#117D47semidirect
321. C2×#117C2×D47semidirect
323. C22×C4⋊C4C2×D47semidirect
331. C22×C42C2×D47semidirect
354. C47#34199semidirect
369. C23×C4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940072
C2×#156895962
#202463802
#202462582
#202462382
#202460572
#202460002
#202459632
C4×#130940072
C2×#156889862
#202457412
C22×#117×C472
#3419947