Finite Group Databasebeta

← #26143292

Subgroups — #26143292

Order 12032.

1590 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689058C211yesmaximal, characteristic
2C2×#15689018C212yesmaximal, complemented
3C2×#15689014C211yesmaximal, characteristic
4C2×#15689013C212yesmaximal
5#20241656C211yesmaximal, characteristic, complemented
6#20241654C212yesmaximal, complemented
7#20241651C211yesmaximal, characteristic, complemented
8#20241632C212yesmaximal
9#20241630C211yesmaximal, characteristic
10C4×C47⋊#95C211yesmaximal, characteristic, complemented
11C2×C47×#389C211yesmaximal, characteristic, complemented
12C22×Dic47⋊C4C2211yescharacteristic
13C22×Dic47⋊C4C2212yes—
14C22×Dic47⋊C4C2212yes—
15C22×(C2×D47)⋊C4C2211yescharacteristic
16C22×(C2×D47)⋊C4C2212yes—
17C22×(C2×D47)⋊C4C2211yescharacteristic
18C22×(C2×D47)⋊C4C2212yes—
19C22×(C2×D47)⋊C4C2211yescharacteristic
20C22×C4×Dic47C2212yes—
21#15689058C418yescomplemented
22#15689018—28no—
23#15689014—24no—
24#15689013—28no—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2211yescharacteristic
34C22×C4⋊C4×C47C2212yes—
35C22×C4⋊C4×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C47×#389—24no—
39C2×#95×C47C2212yes—
40C24×D47C2311yescharacteristic
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2312yes—
45C23×Dic47C2312yes—
46C2×Dic47⋊C4—24no—
47C2×Dic47⋊C4—24no—
48C2×Dic47⋊C4—22no—
49C2×Dic47⋊C4—24no—
50C2×Dic47⋊C4—24no—
51C2×Dic47⋊C4—42no—
52C2×Dic47⋊C4—42no—
53C2×Dic47⋊C4C2×C418yes—
54C2×(C2×D47)⋊C4—22no—
55C2×(C2×D47)⋊C4—22no—
56C2×(C2×D47)⋊C4—24no—
57C2×(C2×D47)⋊C4—24no—
58C2×(C2×D47)⋊C4—24no—
59C2×(C2×D47)⋊C4—22no—
60C2×(C2×D47)⋊C4—22no—
61C2×(C2×D47)⋊C4—24no—
62C2×(C2×D47)⋊C4—42no—
63C2×(C2×D47)⋊C4—42no—
64C2×(C2×D47)⋊C4—42no—
65C2×(C2×D47)⋊C4C2×C414yes—
66C2×(C2×D47)⋊C4C2×C418yes—
67C2×C4×Dic47—42no—
68C2×C4×Dic47—42no—
69C2×C4×Dic47D418yes—
70C47⋊#95—22no—
71C47⋊#95—22no—
72C47⋊#95—22no—
73C47⋊#95—24no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95—24no—
77C47⋊#95—24no—
78C47⋊#95C2×C414yes—
79C23×C4×C47C2312yes—
80C23×C4×C47C2312yes—
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C23×C4×C47C2311yescharacteristic
84C2×C4⋊C4×C47—22no—
85C2×C4⋊C4×C47—22no—
86C2×C4⋊C4×C47—24no—
87C2×C4⋊C4×C47—24no—
88C2×C4⋊C4×C47—22no—
89C2×C4⋊C4×C47—24no—
90C2×C4⋊C4×C47—42no—
91C2×C4⋊C4×C47C2×C414yes—
92C2×C42×C47—42no—
93C2×C42×C47D414yes—
94#95×C47—24no—
95C24×C47C2411yescharacteristic
96C23×D47—22no—
97C23×D47—21no—
98C23×D47—22no—
99C23×D47—21no—
100C23×D47—22no—
101C23×D47—21no—
102C23×D47—21no—
103C23×D47—21no—
104C23×D47—22no—
105C23×D47—21no—
106C23×D47C22×C412yes—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—21no—
110C22×C4×C47—22no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47C22×C414yes—
130C22×C4×C47C22×C412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D414yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47#4112yes—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—21no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—22no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47C22×C412yes—
161C22×Dic47C22×C414yes—
162C22×Dic47C2×D414yes—
163C22×Dic47C2×D414yes—
164C22×Dic47#4114yes—
165C4⋊C4×C47—28no—
166C4⋊C4×C47—44no—
167C4⋊C4×C47—48no—
168C42×C47—44no—
169(C2×D47)⋊C4—28no—
170(C2×D47)⋊C4—216no—
171(C2×D47)⋊C4—48no—
172(C2×D47)⋊C4—44no—
173(C2×D47)⋊C4—44no—
174Dic47⋊C4—216no—
175Dic47⋊C4—48no—
176Dic47⋊C4—44no—
177C4×Dic47—48no—
178C23×C47C23×C411yescharacteristic
179C23×C47C22×D412yes—
180C23×C47C22×D411yescharacteristic
181C23×C47C2×#4112yes—
182C23×C47C2×#4111yescharacteristic
183C23×C47H(5,𝔽2)12yes—
184C23×C47H(5,𝔽2)11yescharacteristic
185C23×C472−1+411yescharacteristic
186C23×C472−1+411yescharacteristic
187C23×C472−1+411yescharacteristic
188C23×C472−1+412yes—
189C22×D47—24no—
190C22×D47—22no—
191C22×D47—22no—
192C22×D47—24no—
193C22×D47—22no—
194C22×D47—41no—
195C22×D47—41no—
196C22×D47—42no—
197C22×D47—42no—
198C22×D47—41no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—42no—
202C22×D47—42no—
203C22×D47—42no—
204C22×D47—41no—
205C22×D47—42no—
206C22×D47—42no—
207C22×D47—42no—
208C22×D47—41no—
209C22×D47—42no—
210C22×D47—42no—
211C22×D47—41no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—24no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47C4×D414yes—
240C2×C4×C47#12714yes—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—22no—
249C2×Dic47—24no—
250C2×Dic47—24no—
251C2×Dic47—24no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—42no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—41no—
269C2×Dic47C4×D418yes—
270#14322—471nomaximal, Sylow
271C22×C47C2×C4×D412yes—
272C22×C47C2×C4×D411yescharacteristic
273C22×C47#51911yescharacteristic
274C22×C47#52012yes—
275C22×C47#52011yescharacteristic
276C22×C47C2×#12711yescharacteristic
277C22×C47#53211yescharacteristic
278C22×C47#53212yes—
279C22×C47#53412yes—
280C22×C47#53411yescharacteristic
281C22×C47#53611yescharacteristic
282C22×C47#53611yescharacteristic
283C22×C47#53612yes—
284C22×C47#53612yes—
285C22×C47#53712yes—
286C22×C47#53711yescharacteristic
287C22×C47#53912yes—
288C22×C47#53912yes—
289C22×C47#53912yes—
290C22×C47#54111yescharacteristic
291C22×C47#54112yes—
292C22×C47#54212yes—
293C22×C47#54211yescharacteristic
294C2×D47—42no—
295C2×D47—44no—
296C2×D47—42no—
297C2×D47—42no—
298C2×D47—44no—
299C2×D47—81no—
300C2×D47—81no—
301C2×D47—81no—
302C2×D47—82no—
303C2×D47—81no—
304C2×D47—82no—
305C4×C47—24no—
306C4×C47—24no—
307C4×C47—44no—
308C4×C47—42no—
309C4×C47—82no—
310Dic47—28no—
311Dic47—42no—
312Dic47—44no—
313Dic47—81no—
314Dic47—81no—
315C2×#399—471no—
316C2×#389—472no—
317C2×#389D4711yescharacteristic, complemented
318C2×#388—471no—
319C2×#388—472no—
320#1445—472no—
321#1445—471no—
322#1444—471no—
323#1439—472no—
324#1439—471no—
325C4×#95—471no—
326C2×C47C4×#12711yescharacteristic
327C2×C47#232212yes—
328C2×C47#232211yescharacteristic
329C2×C47#232311yescharacteristic
330C2×C47#232312yes—
331C2×C47#265611yescharacteristic
332C2×C47#265811yescharacteristic
333C2×C47#266311yescharacteristic
334C2×C47#266312yes—
335C2×C47#266411yescharacteristic
336C2×C47#266412yes—
337D47—82no—
338C22×C4⋊C4—472no—
339C22×C4⋊C4—471no—
340C22×C4⋊C4—472no—
341C22×C4⋊C4C2×D4712yes—
342C22×C4⋊C4C2×D4711yescharacteristic
343C22×C4⋊C4C2×D4711yescharacteristic
344C22×C22⋊C4—472no—
345C22×C22⋊C4—471no—
346C22×C22⋊C4—471no—
347C22×C22⋊C4—472no—
348C22×C22⋊C4—471no—
349C22×C42—472no—
350C22×C42C2×D4711yescharacteristic
351#399—478no—
352#389—24no—
353#389—948no—
354#388—948no—
355#388—944no—
356C2×#95—471no—
357C2×#95—472no—
358C2×#95—471no—
359C2×#95—472no—
360C2×#95—472no—
361C2×#95—472no—
362C2×#95—471no—
363C2×#95—472no—
364C2×#95—471no—
365C2×#95C2×D4712yes—
366C47#1432211yescharacteristic, complemented, Sylow
367C25—471no—
368C23×C4—472no—
369C23×C4—471no—
370C23×C4—472no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4C22×D4712yes—
374C23×C4C22×D4712yes—
375C23×C4C22×D4711yescharacteristic
376C23×C4C22×D4711yescharacteristic
377C23×C4C22×D4711yescharacteristic
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—42no—
385C2×C4⋊C4—478no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4—944no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—1882no—
392C2×C4⋊C4—1882no—
393C2×C4⋊C4C4×D4714yes—
394C2×C22⋊C4—478no—
395C2×C22⋊C4—474no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4—942no—
400C2×C22⋊C4—944no—
401C2×C22⋊C4—944no—
402C2×C22⋊C4—942no—
403C2×C22⋊C4—942no—
404C2×C22⋊C4—1882no—
405C2×C22⋊C4—1882no—
406C2×C22⋊C4—1882no—
407C2×C42—42no—
408C2×C42—478no—
409C2×C42—1882no—
410C2×C42—1882no—
411C2×C42C22⋊D4714yes—
412#95—24no—
413#95—474no—
414#95—944no—
415#95—944no—
416#95—944no—
417#95—944no—
418#95—944no—
419#95—942no—
420#95—942no—
421#95—942no—
422C24—472no—
423C24—942no—
424C24—941no—
425C24—942no—
426C24—941no—
427C24—942no—
428C24—941no—
429C24—941no—
430C24—941no—
431C24—942no—
432C24—941no—
433C24C23×D4711yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—474no—
457C22×C4—474no—
458C22×C4—474no—
459C22×C4—472no—
460C22×C4—474no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—941no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—941no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—942no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—942no—
484C22×C4—941no—
485C22×C4C2×C4×D4714yes—
486C22×C4C2×C4×D4712yes—
487C22×C4#1062535012yes—
488C22×C4D4×D4714yes—
489C22×C4C2×C22⋊D4712yes—
490C22×C4C2×C22⋊D4712yes—
491C22×C4C2×C22⋊D4712yes—
492C4⋊C4—28no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—9416no—
496C4⋊C4—1888no—
497C4⋊C4—1884no—
498C22⋊C4—9416no—
499C22⋊C4—948no—
500C22⋊C4—1884no—
501C22⋊C4—1888no—
502C22⋊C4—1884no—
503C42—44no—
504C42—1888no—
505C23—944no—
506C23—942no—
507C23—942no—
508C23—944no—
509C23—942no—
510C23—1881no—
511C23—1882no—
512C23—1882no—
513C23—1881no—
514C23—1881no—
515C23—1881no—
516C23—1882no—
517C23—1882no—
518C23—1881no—
519C23—1881no—
520C23—1881no—
521C23—1882no—
522C23—1882no—
523C23—1882no—
524C23—1881no—
525C23—1882no—
526C23—1882no—
527C23—1882no—
528C23C22×C4×D4711yescharacteristic
529C23C2×D4×D4712yes—
530C23C22×C22⋊D4711yescharacteristic
531C23#41×D4712yes—
532C23C2×#1062535011yescharacteristic
533C23#1309406912yes—
534C23#1309407011yescharacteristic
535C23#1309407111yescharacteristic
536C23#1309407211yescharacteristic
537C23#1309407212yes—
538C23#1309407211yescharacteristic
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—478no—
567C2×C4—944no—
568C2×C4—944no—
569C2×C4—944no—
570C2×C4—944no—
571C2×C4—944no—
572C2×C4—944no—
573C2×C4—944no—
574C2×C4—942no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—944no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1882no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4—1881no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4—1882no—
595C2×C4C4×C22⋊D4714yes—
596C2×C4#1309402014yes—
597C22—3761no—
598C22—3761no—
599C22—3761no—
600C22—3762no—
601C22—3761no—
602C22—3762no—
603C22—1884no—
604C22—1882no—
605C22—1882no—
606C22—1884no—
607C22—1882no—
608C22C4×D4×D4712yes—
609C22C2×C4×C22⋊D4711yescharacteristic
610C22#1568959911yescharacteristic
611C22#1568960412yes—
612C22#1568960511yescharacteristic
613C22C2×#1309402011yescharacteristic
614C22#1568966911yescharacteristic
615C22#1568967012yes—
616C22#1568968112yes—
617C22#1568968211yescharacteristic
618C22#1568969711yescharacteristic
619C22#1568969711yescharacteristic
620C22#1568969812yes—
621C22#1568969812yes—
622C22#1568970012yes—
623C22#1568970111yescharacteristic
624C22#1568971912yes—
625C22#1568972012yes—
626C22#1568972112yes—
627C22#1568973111yescharacteristic
628C22#1568973212yes—
629C22#1568974312yes—
630C22#1568974511yescharacteristic
631C4—24no—
632C4—24no—
633C4—44no—
634C4—42no—
635C4—82no—
636C4—948no—
637C4—3761no—
638C4—3761no—
639C4—1882no—
640C4—1884no—
641C2—3762no—
642C2C4×#1309402011yescharacteristic
643C2#2024605212yes—
644C2#2024605711yescharacteristic
645C2#2024606411yescharacteristic
646C2#2024606512yes—
647C2#2024949211yescharacteristic
648C2#2024950011yescharacteristic
649C2#2024953911yescharacteristic
650C2#2024954212yes—
651C2#2024954911yescharacteristic
652C2#2024955112yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#389C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC2×#389D47
hypocenterC47#14322
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×#15689018C2semidirect
5. #20241656C2semidirect
6. #20241654C2semidirect
7. #20241651C2semidirect
10. C4×C47⋊#95C2semidirect
11. C2×C47×#389C2semidirect
21. #15689058C4semidirect
317. C2×#389D47semidirect
366. C47#14322semidirect

Maximal subgroups

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

subgroupindex
C2×#156890582
C2×#156890182
C2×#156890142
C2×#156890132
#202416562
#202416542
#202416512
#202416322
#202416302
C4×C47⋊#952
C2×C47×#3892
#1432247