Finite Group Databasebeta

← #26376607

Subgroups — #26376607

Order 12032.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094000C211yesmaximal, characteristic
2C2×#15689601C211yesmaximal, characteristic, complemented
3C2×#117×D47C211yesmaximal, characteristic
4#20246420C214yesmaximal, complemented
5#20246398C214yesmaximal, complemented
6#20246252C214yesmaximal, complemented
7#20246230C214yesmaximal
8#20246073C214yesmaximal, complemented
9#20246004C211yesmaximal, characteristic, complemented
10#20245926C214yesmaximal, complemented
11#20245767C211yesmaximal, characteristic
12#20245744C211yesmaximal, characteristic, complemented
13C47×#2310C211yesmaximal, characteristic, complemented
14C23×C4×D47C2211yescharacteristic
15C2×#13094000C4116yescomplemented
16C2×#13094000C2212yes—
17C2×#13094000C2212yes—
18C2×#13094000C2212yes—
19C2×#13094000C2212yes—
20#15689601—28no—
21C2×Q8×Dic47C2214yescomplemented
22C2×Q8×Dic47C2214yescomplemented
23C2×#13093983C2211yescharacteristic
24C2×#13093983C2214yes—
25C2×#13093983C2211yescharacteristic
26C2×#13093982C2212yes—
27C2×#13093981C2212yes—
28C2×#13093980C2212yes—
29#117×D47—28no—
30C22×C4⋊Dic47C2211yescharacteristic
31C22×Dic47⋊C4C2212yes—
32C2×C4⋊C4×D47C2212yes—
33C22×(C2×D47)⋊C4C2212yes—
34C2×C22⋊C4×D47C2212yes—
35C2×C42×D47C2212yes—
36#15689021C2214yescomplemented
37#15689020C2218yescomplemented
38#15689019C2214yescomplemented
39#15689015C2218yes—
40#15689012C2218yes—
41#15689010C2218yescomplemented
42#15689005C2218yescomplemented
43#15688997C2218yes—
44#15688984C2212yes—
45#15688981C2214yes—
46#15688980C2212yes—
47#15688980C2212yes—
48#15688978C2212yes—
49C4⋊C4×Dic47C2214yescomplemented
50C4×C4⋊Dic47C2214yescomplemented
51#15688974C2218yescomplemented
52C22⋊C4×Dic47C2214yes—
53C4×(C2×D47)⋊C4C2212yes—
54C4×(C2×D47)⋊C4C2212yes—
55#15688970C2218yes—
56C23×Q8×C47C2211yescharacteristic
57C2×#117×C47C2211yescharacteristic
58C2×#117×C47C2211yescharacteristic
59C47×#391C2214yescomplemented
60C47×#391C2214yescomplemented
61C47×#381C2214yes—
62C24×D47C2311yescharacteristic
63C23×Dic47C2311yescharacteristic
64C23×Dic47C2312yes—
65C22×C4×D47—24no—
66C22×C4×D47—24no—
67C22×C4×D47C2×C418yes—
68C22×C4×D47C2314yes—
69C22×C4×D47C2311yescharacteristic
70C22×C4×D47C2311yescharacteristic
71C22×C4×D47C2311yescharacteristic
72C22×C4×D47C2311yescharacteristic
73#13094000C2×C4116yes—
74#13094000C2×C4116yes—
75#13094000C2×C4116yescomplemented
76#13094000C2×C4116yescomplemented
77Q8×Dic47—216no—
78Q8×Dic47—216no—
79#13093983—24no—
80#13093983—216no—
81#13093983D418yes—
82#13093982—28no—
83#13093981—28no—
84#13093980—28no—
85C2×C4⋊Dic47C2×C418yescomplemented
86C2×C4⋊Dic47C2312yes—
87C2×C4⋊Dic47C2318yes—
88C2×C4⋊Dic47C2312yes—
89C2×C4⋊Dic47C2312yes—
90C2×C4⋊Dic47C2312yes—
91C2×Dic47⋊C4C2×C4116yescomplemented
92C2×Dic47⋊C4C2314yes—
93C2×Dic47⋊C4C2318yes—
94C2×Dic47⋊C4C2314yes—
95C2×Dic47⋊C4C2314yes—
96C4⋊C4×D47—28no—
97C2×C22⋊Dic47C2312yes—
98C2×C22⋊Dic47C2312yes—
99C2×(C2×D47)⋊C4C2×C4116yes—
100C2×(C2×D47)⋊C4C2314yes—
101C2×(C2×D47)⋊C4C2314yes—
102C2×(C2×D47)⋊C4C2314yes—
103C2×(C2×D47)⋊C4C2314yes—
104C2×(C2×D47)⋊C4C2314yes—
105C22⋊C4×D47—28no—
106C2×C4×Dic47C2318yes—
107C2×C4×Dic47C2312yes—
108C2×C4×Dic47C2314yes—
109C2×C4×Dic47C2318yes—
110C2×C4×Dic47C2312yes—
111C42×D47—28no—
112C47⋊#95C2318yes—
113C47⋊#95C2314yes—
114C47⋊#95C2314yes—
115C47⋊#95C2318yes—
116C22×Q8×C47—24no—
117C22×Q8×C47—24no—
118C22×Q8×C47C2×C418yescomplemented
119C22×Q8×C47C2312yes—
120C22×Q8×C47C2312yes—
121C23×C4×C47C2312yes—
122C23×C4×C47C2311yescharacteristic
123#117×C47—24no—
124#117×C47—24no—
125C2×C4⋊C4×C47C2312yes—
126C2×C4⋊C4×C47C2312yes—
127C2×C22⋊C4×C47C2312yes—
128C2×C22⋊C4×C47C2312yes—
129C2×C42×C47C2312yes—
130C2×C42×C47C2312yes—
131#95×C47C2318yes—
132#95×C47C2318yes—
133C24×C47C2411yescharacteristic
134C23×D47—22no—
135C23×D47—22no—
136C23×D47—24no—
137C23×D47—42no—
138C23×D47C22×C414yes—
139C23×D47C2411yescharacteristic
140C23×D47C2411yescharacteristic
141C2×Q8×C47—232no—
142C2×Q8×C47—24no—
143C2×Q8×C47—24no—
144C2×Q8×C47—24no—
145C2×Q8×C47—24no—
146C2×Q8×C47C22×C418yes—
147C2×Q8×C47C22×C418yes—
148C22×C4×C47—24no—
149C22×C4×C47—24no—
150C22×C4×C47C22×C418yes—
151C22×C4×C47C22×C414yes—
152C22×C4×C47C2×D414yes—
153C22×C4×C47#4114yes—
154C22×C4×C47C2414yes—
155C22×C4×C47C2414yes—
156C22×C4×C47C2414yes—
157C22×C4×C47C2411yescharacteristic
158C22×C4×C47C2411yescharacteristic
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—44no—
162C22×Dic47C22×C414yes—
163C22×Dic47C22×C418yes—
164C22×Dic47C2414yes—
165C22×Dic47C2411yescharacteristic
166C22×Dic47C2414yes—
167C22×Dic47C2411yescharacteristic
168C22×Dic47C2414yes—
169C2×C4×D47—232no—
170C2×C4×D47—22no—
171C2×C4×D47—22no—
172C2×C4×D47—22no—
173C2×C4×D47—22no—
174C2×C4×D47—22no—
175C2×C4×D47—22no—
176C2×C4×D47—22no—
177C2×C4×D47—22no—
178C2×C4×D47—28no—
179C2×C4×D47—28no—
180C2×C4×D47—44no—
181C2×C4×D47C22×C414yes—
182C2×C4×D47C22×C414yes—
183C2×C4×D47C22×C414yes—
184C2×C4×D47C22×C414yes—
185C4⋊C4×C47—24no—
186C4⋊C4×C47—24no—
187C22⋊C4×C47—24no—
188C22⋊C4×C47—24no—
189C42×C47—24no—
190C42×C47—24no—
191C22⋊Dic47—24no—
192C22⋊Dic47C2×D418yes—
193(C2×D47)⋊C4—28no—
194(C2×D47)⋊C4—28no—
195(C2×D47)⋊C4—44no—
196(C2×D47)⋊C4C22×C4116yes—
197(C2×D47)⋊C4C22×C4116yes—
198C4⋊Dic47—24no—
199C4⋊Dic47—216no—
200C4⋊Dic47C22×C418yes—
201C4⋊Dic47C22×C418yes—
202C4⋊Dic47C2×D418yes—
203Dic47⋊C4—28no—
204Dic47⋊C4—48no—
205Dic47⋊C4C22×C4116yes—
206Dic47⋊C4C22×C4116yes—
207C4×Dic47—24no—
208C4×Dic47—28no—
209C4×Dic47—216no—
210C4×Dic47—48no—
211C4×Dic47C2×D418yes—
212C23×C47—22no—
213C23×C47—22no—
214C23×C47C23×C412yes—
215C23×C47C22×D412yes—
216C23×C47C2×#4112yes—
217C23×C47C2511yescharacteristic
218C22×D47—21no—
219C22×D47—21no—
220C22×D47—22no—
221C22×D47—28no—
222C22×D47—21no—
223C22×D47—21no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—28no—
227C22×D47—44no—
228C22×D47—48no—
229C22×D47—48no—
230C22×D47C23×C412yes—
231C22×D47C23×C412yes—
232C22×D472−1+414yes—
233Q8×C47—216no—
234Q8×C47—216no—
235C2×C4×C47—216no—
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×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47C4×D4116yes—
245C2×C4×C47C23×C418yes—
246C2×C4×C47C23×C412yes—
247C2×C4×C47C23×C412yes—
248C2×C4×C47C22×D412yes—
249C2×C4×C47C22×D412yes—
250C2×C4×C47C2×#4112yes—
251C2×C4×C47C2×#4112yes—
252C2×Dic47—28no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—28no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—48no—
264C2×Dic47C23×C412yes—
265C2×Dic47C23×C412yes—
266C2×Dic47C23×C418yes—
267C2×Dic47C22×D418yes—
268C4×D47—28no—
269C4×D47—28no—
270C4×D47—28no—
271C4×D47—28no—
272C4×D47—416no—
273#34203—471nomaximal, Sylow
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—24no—
277C22×C47—24no—
278C22×C47C2×C4×D414yes—
279C22×C47C24×C411yescharacteristic
280C22×C47C23×D411yescharacteristic
281C22×C47C22×#4111yescharacteristic
282C22×C47C2×2−1+412yes—
283C22×C47#58512yes—
284C2×D47—22no—
285C2×D47—22no—
286C2×D47—24no—
287C2×D47—24no—
288C2×D47—22no—
289C2×D47—22no—
290C2×D47—22no—
291C2×D47—416no—
292C2×D47—42no—
293C2×D47—42no—
294C2×D47#51914yescomplemented
295C4×C47—28no—
296C4×C47—44no—
297C4×C47—44no—
298C4×C47C2×C4×D414yes—
299C4×C47C2×C4×D414yes—
300Dic47—22no—
301Dic47—22no—
302Dic47—28no—
303Dic47—28no—
304Dic47—44no—
305C22×C22⋊Q8—471no—
306C2×#519—471no—
307C22×#117—471no—
308#2362—474no—
309#2360—474no—
310#2344—474no—
311#2342—474no—
312#2323—474no—
313#2316—471no—
314#2310D4711yescharacteristic, complemented
315C4×C22⋊Q8—474no—
316#2280—471no—
317#2274—471no—
318C2×C47—24no—
319C2×C47C22×C4×D411yescharacteristic
320C2×C47C2×#51911yescharacteristic
321C2×C47#342811yescharacteristic
322C2×C47#348912yes—
323C2×C47#349512yes—
324D47—24no—
325D47—42no—
326C23×Q8C2×D4711yescharacteristic
327C24×C4—471no—
328C2×C22⋊Q8—4716no—
329C2×C22⋊Q8—472no—
330C2×C22⋊Q8—472no—
331C2×C22⋊Q8—472no—
332C2×C22⋊Q8—472no—
333#519—948no—
334C2×C4×Q8—474no—
335C2×C4×Q8—474no—
336C2×#117—471no—
337C2×#117—471no—
338C2×#117—474no—
339C2×#117—472no—
340C2×#117—472no—
341C2×#117—472no—
342C2×#117—948no—
343C2×#117C2×D4711yescharacteristic
344C2×#117C2×D4711yescharacteristic
345C22×C4⋊C4—471no—
346C22×C4⋊C4—472no—
347C22×C4⋊C4—472no—
348C22×C22⋊C4—472no—
349C22×C22⋊C4—472no—
350C22×C42—472no—
351#391C2×D4714yescomplemented
352#391C2×D4714yescomplemented
353#389—474no—
354#389—474no—
355#389—478no—
356#388—478no—
357#388—478no—
358#387—478no—
359#387—478no—
360#385—478no—
361#381—472no—
362#381C2×D4714yes—
363#380—472no—
364#380—472no—
365#380—474no—
366#379—472no—
367C4×C4⋊C4—474no—
368C4×C4⋊C4—474no—
369C4×C4⋊C4—478no—
370C4×C22⋊C4—474no—
371C4×C22⋊C4—472no—
372C4×C22⋊C4—472no—
373C4×C22⋊C4—478no—
374C47#3420311yescharacteristic, complemented, Sylow
375C25—471no—
376C22×Q8—24no—
377C22×Q8—24no—
378C22×Q8C4×D4718yescomplemented
379C22×Q8C22×D4712yes—
380C22×Q8C22×D4712yes—
381C23×C4—471no—
382C23×C4—472no—
383C23×C4—478no—
384C23×C4—471no—
385C23×C4—471no—
386C23×C4—471no—
387C23×C4—471no—
388C23×C4—474no—
389C23×C4—944no—
390C23×C4—944no—
391C23×C4C22×D4712yes—
392C23×C4C22×D4711yescharacteristic
393C22⋊Q8—4716no—
394C22⋊Q8—4716no—
395C22⋊Q8—4716no—
396C22⋊Q8—4716no—
397C4×Q8—9416no—
398C4×Q8—9416no—
399#117—24no—
400#117—24no—
401#117—478no—
402#117—948no—
403#117—948no—
404#117—9416no—
405#117—948no—
406#117—944no—
407C2×C4⋊C4—4716no—
408C2×C4⋊C4—478no—
409C2×C4⋊C4—472no—
410C2×C4⋊C4—474no—
411C2×C4⋊C4—474no—
412C2×C4⋊C4—472no—
413C2×C4⋊C4—472no—
414C2×C4⋊C4—474no—
415C2×C4⋊C4—472no—
416C2×C4⋊C4—478no—
417C2×C4⋊C4—478no—
418C2×C4⋊C4—948no—
419C2×C4⋊C4C22×D4712yes—
420C2×C4⋊C4C22×D4712yes—
421C2×C22⋊C4—472no—
422C2×C22⋊C4—4716no—
423C2×C22⋊C4—474no—
424C2×C22⋊C4—474no—
425C2×C22⋊C4—472no—
426C2×C22⋊C4—474no—
427C2×C22⋊C4—474no—
428C2×C22⋊C4—474no—
429C2×C22⋊C4—948no—
430C2×C22⋊C4C22×D4712yes—
431C2×C22⋊C4C22×D4712yes—
432C2×C42—472no—
433C2×C42—474no—
434C2×C42—472no—
435C2×C42—478no—
436C2×C42—478no—
437C2×C42—948no—
438C2×C42C22×D4712yes—
439C2×C42C22×D4712yes—
440#95—474no—
441#95—478no—
442#95—474no—
443#95—478no—
444#95C22×D4718yes—
445#95C22×D4718yes—
446C24—471no—
447C24—471no—
448C24—474no—
449C24—944no—
450C24—942no—
451C24—942no—
452C24—1882no—
453C24C23×D4711yescharacteristic
454C2×Q8—232no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8C2×C4×D4718yes—
460C2×Q8C2×C4×D4718yes—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—474no—
464C22×C4—471no—
465C22×C4—474no—
466C22×C4—471no—
467C22×C4—474no—
468C22×C4—478no—
469C22×C4—474no—
470C22×C4—474no—
471C22×C4—474no—
472C22×C4—474no—
473C22×C4—474no—
474C22×C4—948no—
475C22×C4—948no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—9432no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—1884no—
488C22×C4—1884no—
489C22×C4C2×C4×D4718yes—
490C22×C4C2×C4×D4714yes—
491C22×C4D4×D4714yes—
492C22×C4#1062535214yes—
493C22×C4C23×D4714yes—
494C22×C4C23×D4714yes—
495C22×C4C23×D4714yes—
496C22×C4C23×D4711yescharacteristic
497C22×C4C23×D4711yescharacteristic
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—4716no—
501C4⋊C4—4716no—
502C4⋊C4—478no—
503C4⋊C4—478no—
504C4⋊C4—478no—
505C4⋊C4—948no—
506C4⋊C4—9416no—
507C4⋊C4—944no—
508C4⋊C4—1888no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—478no—
512C22⋊C4—4716no—
513C22⋊C4—4716no—
514C22⋊C4—948no—
515C22⋊C4—948no—
516C22⋊C4—944no—
517C22⋊C4—1884no—
518C42—24no—
519C42—24no—
520C42—478no—
521C42—948no—
522C42—9416no—
523C42—944no—
524C42—1888no—
525C23—22no—
526C23—22no—
527C23—472no—
528C23—474no—
529C23—472no—
530C23—948no—
531C23—941no—
532C23—941no—
533C23—942no—
534C23—948no—
535C23—941no—
536C23—941no—
537C23—942no—
538C23—942no—
539C23—1884no—
540C23—1888no—
541C23—1888no—
542C23C22×C4×D4712yes—
543C23C2×D4×D4712yes—
544C23C2×#1062535212yes—
545C23C24×D4711yescharacteristic
546Q8—216no—
547Q8—216no—
548C2×C4—216no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—478no—
558C2×C4—472no—
559C2×C4—472no—
560C2×C4—478no—
561C2×C4—948no—
562C2×C4—941no—
563C2×C4—941no—
564C2×C4—941no—
565C2×C4—941no—
566C2×C4—944no—
567C2×C4—944no—
568C2×C4—948no—
569C2×C4—948no—
570C2×C4—948no—
571C2×C4—948no—
572C2×C4—948no—
573C2×C4—944no—
574C2×C4—944no—
575C2×C4—944no—
576C2×C4—18816no—
577C2×C4—1888no—
578C2×C4(C22⋊D47)⋊C4116yes—
579C2×C4C22×C4×D4718yes—
580C2×C4C22×C4×D4712yes—
581C2×C4C22×C4×D4712yes—
582C2×C4C2×D4×D4712yes—
583C2×C4C2×D4×D4712yes—
584C2×C4C2×#1062535212yes—
585C2×C4C2×#1062535212yes—
586C22—24no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—474no—
591C22—944no—
592C22—942no—
593C22—942no—
594C22—944no—
595C22—942no—
596C22—942no—
597C22—942no—
598C22—18816no—
599C22—1882no—
600C22—1882no—
601C22C2×(C22⋊D47)⋊C414yes—
602C22C23×C4×D4711yescharacteristic
603C22C22×D4×D4711yescharacteristic
604C22C22×#1062535211yescharacteristic
605C222−1+4×D4712yes—
606C22#1568996012yes—
607C4—28no—
608C4—44no—
609C4—44no—
610C4—948no—
611C4—942no—
612C4—942no—
613C4—948no—
614C4—1884no—
615C4C2×(C22⋊D47)⋊C414yes—
616C4C2×(C22⋊D47)⋊C414yes—
617C2—24no—
618C2—944no—
619C2—1882no—
620C2C22×(C22⋊D47)⋊C411yescharacteristic
621C2#519×D4711yescharacteristic
622C2#2025659011yescharacteristic
623C2#2025706612yes—
624C2#2025713712yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2310C2
socle, cosocleC23×C47C25
prime radicalC24×D47C23
hypercenter#2310D47
hypocenterC47#34203
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C2×#15689601C2semidirect
4. #20246420C2semidirect
5. #20246398C2semidirect
6. #20246252C2semidirect
8. #20246073C2semidirect
9. #20246004C2semidirect
10. #20245926C2semidirect
12. #20245744C2semidirect
13. C47×#2310C2semidirect
15. C2×#13094000C4semidirect
21. C2×Q8×Dic47C22semidirect
22. C2×Q8×Dic47C22semidirect
36. #15689021C22semidirect
37. #15689020C22semidirect
38. #15689019C22semidirect
41. #15689010C22semidirect
42. #15689005C22semidirect
49. C4⋊C4×Dic47C22semidirect
50. C4×C4⋊Dic47C22semidirect
51. #15688974C22semidirect
59. C47×#391C22semidirect
60. C47×#391C22semidirect
75. #13094000C2×C4semidirect
76. #13094000C2×C4semidirect
85. C2×C4⋊Dic47C2×C4semidirect
91. C2×Dic47⋊C4C2×C4semidirect
118. C22×Q8×C47C2×C4semidirect
294. C2×D47#519semidirect
314. #2310D47semidirect
351. #391C2×D47semidirect
352. #391C2×D47semidirect
374. C47#34203semidirect
378. C22×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940002
C2×#156896012
C2×#117×D472
#202464202
#202463982
#202462522
#202462302
#202460732
#202460042
#202459262
#202457672
#202457442
C47×#23102
#3420347