Finite Group Databasebeta

← #26238577

Subgroups — #26238577

Order 12032.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689345C211yesmaximal, characteristic
2C2×C4×#10625337C211yesmaximal, characteristic, complemented
3C2×C4×#10625336C211yesmaximal, characteristic
4#20246457C211yesmaximal, characteristic
5#20244197C211yesmaximal, characteristic
6#20244154C211yesmaximal, characteristic, complemented
7#20244070C211yesmaximal, characteristic
8#20244063C211yesmaximal, characteristic, complemented
9#20243743C211yesmaximal, characteristic
10#20243729C211yesmaximal, characteristic, complemented
11#20243655C211yesmaximal, characteristic, complemented
12#20243634C211yesmaximal, characteristic
13C4⋊C4×C47⋊C8C211yesmaximal, characteristic, complemented
14C4×#13093948C211yesmaximal, characteristic, complemented
15C47×#2351C211yesmaximal, characteristic, complemented
16C22×#10625337C2211yescharacteristic
17C22×#10625336C2211yescharacteristic
18C2×#13094012C2211yescharacteristic
19C2×C4×Dic94C2211yescharacteristic
20C2×C4×D188C2211yescharacteristic
21#15689345C418yescomplemented
22C4×#10625337—24no—
23C4×#10625336—24no—
24C2×#13093957C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C2×#13093944C2211yescharacteristic
33C2×#13093944C2211yescharacteristic
34C2×#13093944C2211yescharacteristic
35C2×C4×C47⋊C8C2211yescharacteristic
36C2×C4×C47⋊C8C2211yescharacteristic
37C2×C4×C47⋊C8C2211yescharacteristic
38#15689029C2211yescharacteristic
39#15689025C2211yescharacteristic
40#15689013—21no—
41#15689012—21no—
42#15688994—21no—
43C4×C4⋊Dic47C2211yescharacteristic
44#15688970—21no—
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688878C2211yescharacteristic
50#15688878C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C47×#389—21no—
55C47×#389—21no—
56C47×#389C2211yescharacteristic
57C47×#389C2211yescharacteristic
58C47×#382—21no—
59C47×#382C2211yescharacteristic
60C4×C4⋊C4×C47—21no—
61C4×C4⋊C4×C47C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C22×D188C2311yescharacteristic
64C22×C4×D47—21no—
65C2×#10625337—22no—
66C2×#10625337—22no—
67C2×#10625337C2×C414yescomplemented
68C2×#10625336—22no—
69C2×#10625336—22no—
70C2×#10625336C2×C414yes—
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75#13094012—22no—
76#13094012C2×C414yes—
77C4×Dic94—22no—
78C4×Dic94—41no—
79C4×D188—22no—
80C4×D188—41no—
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×Dic47⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×(C2×D47)⋊C4—21no—
88C2×(C2×D47)⋊C4—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91#13093957—22no—
92#13093955—22no—
93C4⋊(C47⋊C8)C2×C414yescomplemented
94#13093948—22no—
95#13093948D414yes—
96#13093947—22no—
97#13093947—41no—
98#13093947C2×C414yescomplemented
99#13093944—22no—
100#13093944—41no—
101#13093944C2×C414yes—
102C4×C47⋊C8—22no—
103C4×C47⋊C8—22no—
104C4×C47⋊C8—22no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×Q8×C47C2311yescharacteristic
109C4×Q8×C47—22no—
110C4×Q8×C47—22no—
111C4×Q8×C47—22no—
112C4×Q8×C47—41no—
113C4×Q8×C47—41no—
114C4×Q8×C47C2×C414yescomplemented
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47—21no—
119C2×C4⋊C4×C47—21no—
120C2×C4⋊C4×C47D411yescharacteristic
121C2×C4⋊C4×C47D411yescharacteristic
122C2×C4⋊C4×C47C2311yescharacteristic
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C42×C47—21no—
126C2×C42×C47—21no—
127C2×C42×C47—21no—
128C2×C42×C47C2311yescharacteristic
129C2×C42×C47C2311yescharacteristic
130C2×C42×C47C2311yescharacteristic
131#95×C47—21no—
132#95×C47—21no—
133#95×C47—21no—
134#95×C47—21no—
135#95×C47D411yescharacteristic
136#95×C47D411yescharacteristic
137C23×D47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47C22×C412yes—
142C2×Q8×C47C2×D412yes—
143C2×Q8×C47#4112yes—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188—21no—
161C2×D188—21no—
162C2×D188—21no—
163C2×D188C22×C412yes—
164C2×C4×D47—42no—
165C2×C4×D47—42no—
166C2×C4×D47—81no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94C22×C412yes—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—42no—
182C4⋊C4×C47—42no—
183C4⋊C4×C47—42no—
184C4⋊C4×C47—42no—
185C4⋊C4×C47C22×C412yes—
186C42×C47—24no—
187C42×C47—21no—
188C42×C47—21no—
189C42×C47—21no—
190C42×C47—21no—
191C42×C47—42no—
192C42×C47—42no—
193C42×C47C22×C412yes—
194C42×C47#4112yes—
195#10625337—28no—
196#10625336—28no—
197(C2×D47)⋊C4—24no—
198(C2×D47)⋊C4—24no—
199(C2×D47)⋊C4—42no—
200(C2×D47)⋊C4—42no—
201(C2×D47)⋊C4—81no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47—21no—
205C4⋊Dic47C2×D412yes—
206Dic47⋊C4—42no—
207C4×Dic47—42no—
208C4×Dic47—81no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8C22×C412yes—
219C2×C47⋊C8C22×C412yes—
220C2×C47⋊C8C2×D412yes—
221C23×C47C22×D411yescharacteristic
222C22×D47—22no—
223C22×D47—41no—
224C22×D47—41no—
225C22×D47—41no—
226C22×D47—41no—
227C22×D47—41no—
228C22×D47—41no—
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47—22no—
232Q8×C47C4×D414yes—
233C2×C4×C47—22no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—22no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
250C2×C4×C47—21no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47C4×D412yes—
260C2×C4×C47C4×D412yes—
261C2×C4×C47#13512yes—
262C2×C4×C47#13512yes—
263C2×C4×C47C23×C411yescharacteristic
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C2×C4×C47H(5,𝔽2)11yescharacteristic
268C2×C4×C472−1+411yescharacteristic
269C2×Dic47—22no—
270C2×Dic47—22no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278D188—22no—
279D188—22no—
280D188—41no—
281D188—41no—
282C4×D47—84no—
283Dic94—22no—
284Dic94—22no—
285Dic94—41no—
286Dic94—41no—
287C47⋊C8—22no—
288C47⋊C8—22no—
289C47⋊C8—22no—
290C47⋊C8—41no—
291#23000—471nomaximal, Sylow
292C22×C47C2×C4×D411yescharacteristic
293C22×C47D4211yescharacteristic
294C22×C47#54811yescharacteristic
295C22×C47C2×#13511yescharacteristic
296C22×C47C2×#13511yescharacteristic
297C22×C47#57711yescharacteristic
298C22×C47#57811yescharacteristic
299C2×D47—42no—
300C2×D47—42no—
301C2×D47—42no—
302C2×D47—81no—
303C2×D47—81no—
304C2×D47—81no—
305C2×D47—81no—
306C4×C47—24no—
307C4×C47—22no—
308C4×C47—22no—
309C4×C47—22no—
310C4×C47—22no—
311C4×C47—42no—
312C4×C47—41no—
313C4×C47—41no—
314C4×C47—81no—
315C4×C47—81no—
316C4×C47C2×C4×D411yescharacteristic
317C4×C47#52011yescharacteristic
318C4×C47#54711yescharacteristic
319C4×C47#55311yescharacteristic
320Dic47—42no—
321Dic47—42no—
322Dic47—81no—
323C2×#464—471no—
324C2×C4×Q16—471no—
325C2×C4×SD16—471no—
326#2367—471no—
327#2351D4711yescharacteristic, complemented
328#1958—471no—
329#1949—471no—
330#1934—471no—
331#1933—471no—
332#1880—471no—
333#1878—471no—
334#1865—471no—
335#1862—471no—
336C8×C4⋊C4—471no—
337C4×#103—471no—
338C2×C47#236111yescharacteristic
339C2×C47C4×#13511yescharacteristic
340C2×C47#294711yescharacteristic
341C2×C47#328211yescharacteristic
342C2×C47#328511yescharacteristic
343C2×C47#336411yescharacteristic
344C2×C47#336511yescharacteristic
345D47—82no—
346C22×Q16—471no—
347C22×SD16—471no—
348C2×#124—471no—
349C2×C4×Q8—471no—
350C2×C4×Q8C2×D4711yescharacteristic
351C2×C4×Q8C2×D4711yescharacteristic
352C2×C4×Q8C2×D4711yescharacteristic
353C2×C4×D4—471no—
354#464—478no—
355C4×Q16—944no—
356C4×SD16—944no—
357C2×#107—471no—
358C2×#106—471no—
359C2×C4⋊C8—471no—
360C2×#103—471no—
361C2×#103—471no—
362C2×#103—471no—
363C2×#103—471no—
364C2×#103—471no—
365C2×#102—471no—
366C2×#102—471no—
367C2×#102—471no—
368C2×C4×C8—471no—
369C2×C4×C8—471no—
370C2×C4×C8—471no—
371#391—471no—
372#390—471no—
373#389—21no—
374#389—21no—
375#389C2×D4711yescharacteristic
376#389C2×D4711yescharacteristic
377#388—941no—
378#388—941no—
379#385—941no—
380#382—21no—
381#382C2×D4711yescharacteristic
382C4×C4⋊C4—21no—
383C4×C4⋊C4—471no—
384C4×C4⋊C4C2×D4711yescharacteristic
385C4×C22⋊C4—941no—
386#340—471no—
387#340—471no—
388#340—471no—
389#340—471no—
390#336—471no—
391#336—471no—
392C47#2300011yescharacteristic, complemented, Sylow
393C22×Q8—471no—
394C22×Q8C22×D4711yescharacteristic
395C22×D4—471no—
396C23×C4—941no—
397C2×Q16—474no—
398C2×Q16—942no—
399C2×Q16—942no—
400C2×SD16—474no—
401C2×SD16—942no—
402C2×SD16—942no—
403C22×C8—471no—
404C22×C8—471no—
405C22×C8—471no—
406C22×C8—471no—
407#124—474no—
408#124—942no—
409C4×Q8—22no—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—41no—
413C4×Q8—41no—
414C4×Q8—942no—
415C4×Q8—1881no—
416C4×Q8C4×D4714yescomplemented
417C4×D4—942no—
418C4×D4—1881no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4C22⋊D4711yescharacteristic
428C2×C4⋊C4C22⋊D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C4⋊C4C22×D4711yescharacteristic
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C22⋊C4—941no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C22⋊C4—941no—
436C2×C22⋊C4—941no—
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—941no—
441C2×C42—941no—
442C2×C42C22×D4711yescharacteristic
443C2×C42C22×D4711yescharacteristic
444C2×C42C22×D4711yescharacteristic
445#107—942no—
446#106—942no—
447C4⋊C8—474no—
448#103—474no—
449#103—474no—
450#103—942no—
451#103—942no—
452#103—1881no—
453#102—474no—
454#102—942no—
455#102—1881no—
456C4×C8—942no—
457C4×C8—942no—
458C4×C8—942no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—941no—
464#95—941no—
465#95—941no—
466#95C22⋊D4711yescharacteristic
467#95C22⋊D4711yescharacteristic
468C24—941no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—472no—
473C2×Q8—941no—
474C2×Q8—941no—
475C2×Q8—941no—
476C2×Q8—941no—
477C2×Q8—941no—
478C2×Q8C2×C4×D4712yes—
479C2×Q8D4×D4712yes—
480C2×Q8#1062535412yes—
481C2×D4—472no—
482C2×D4—941no—
483C2×D4—941no—
484C2×D4—941no—
485C2×D4—941no—
486C2×D4—941no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—941no—
492C22×C4—941no—
493C22×C4—941no—
494C22×C4—3761no—
495C22×C4—1882no—
496C22×C4—1882no—
497C22×C4C2×C22⋊D4711yescharacteristic
498C22×C4C2×C22⋊D4711yescharacteristic
499C22×C4C2×C22⋊D4711yescharacteristic
500C22×C4C2×C22⋊D4711yescharacteristic
501C22×C4C2×C22⋊D4711yescharacteristic
502C22×C4C2×C22⋊D4711yescharacteristic
503C22×C4C23×D4711yescharacteristic
504Q16—948no—
505SD16—948no—
506C2×C8—472no—
507C2×C8—472no—
508C2×C8—472no—
509C2×C8—941no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—941no—
515C2×C8—941no—
516C2×C8—941no—
517C2×C8—941no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—472no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4—941no—
534C4⋊C4—1882no—
535C4⋊C4C2×C4×D4712yes—
536C22⋊C4—944no—
537C22⋊C4—944no—
538C22⋊C4—3761no—
539C22⋊C4—1882no—
540C22⋊C4—1882no—
541C42—24no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—42no—
547C42—42no—
548C42—3761no—
549C42—1882no—
550C42C2×C4×D4712yes—
551C42#1062535012yes—
552C23—942no—
553C23—1881no—
554C23—1881no—
555C23—1881no—
556C23—1881no—
557C23—1881no—
558C23—1881no—
559C23C22×C22⋊D4711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—942no—
564Q8—942no—
565Q8—1881no—
566Q8—1881no—
567Q8#1309398614yes—
568D4—942no—
569D4—942no—
570D4—1881no—
571D4—1881no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—3764no—
601C2×C4—1881no—
602C2×C4—1881no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4C4×C22⋊D4712yes—
609C2×C4C4×C22⋊D4712yes—
610C2×C4#1309404412yes—
611C2×C4#1309404412yes—
612C2×C4C22×C4×D4711yescharacteristic
613C2×C4C2×D4×D4711yescharacteristic
614C2×C4C2×#1062535011yescharacteristic
615C2×C4C2×#1062535411yescharacteristic
616C2×C4#1309406911yescharacteristic
617C2×C4#1309407211yescharacteristic
618C8—942no—
619C8—942no—
620C8—942no—
621C8—1881no—
622C22—3761no—
623C22—3761no—
624C22—3761no—
625C22—3761no—
626C22—1882no—
627C22—1882no—
628C22—1882no—
629C22C2×C4×C22⋊D4711yescharacteristic
630C22D4×C22⋊D4711yescharacteristic
631C22#1568978411yescharacteristic
632C22C2×#1309404411yescharacteristic
633C22C2×#1309404411yescharacteristic
634C22#1568993111yescharacteristic
635C22#1568993811yescharacteristic
636C4—22no—
637C4—22no—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—42no—
644C4—81no—
645C4—81no—
646C4—3761no—
647C4—1882no—
648C4—1882no—
649C4C2×#1309398611yescharacteristic
650C4#1568960411yescharacteristic
651C4#1568977411yescharacteristic
652C4#1568982111yescharacteristic
653C2—3762no—
654C2#2024640411yescharacteristic
655C2C4×#1309404411yescharacteristic
656C2#2025223111yescharacteristic
657C2#2025509611yescharacteristic
658C2#2025512511yescharacteristic
659C2#2025607311yescharacteristic
660C2#2025608411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2351C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#2351D47
hypocenterC47#23000
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. C2×C4×#10625337C2semidirect
6. #20244154C2semidirect
8. #20244063C2semidirect
10. #20243729C2semidirect
11. #20243655C2semidirect
13. C4⋊C4×C47⋊C8C2semidirect
14. C4×#13093948C2semidirect
15. C47×#2351C2semidirect
21. #15689345C4semidirect
67. C2×#10625337C2×C4semidirect
93. C4⋊(C47⋊C8)C2×C4semidirect
98. #13093947C2×C4semidirect
114. C4×Q8×C47C2×C4semidirect
327. #2351D47semidirect
392. C47#23000semidirect
416. C4×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156893452
C2×C4×#106253372
C2×C4×#106253362
#202464572
#202441972
#202441542
#202440702
#202440632
#202437432
#202437292
#202436552
#202436342
C4⋊C4×C47⋊C82
C4×#130939482
C47×#23512
#2300047