Finite Group Databasebeta

← #26187713

Subgroups — #26187713

Order 12032.

716 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#10625324C211yesmaximal, characteristic, complemented
2C4×SD16×D47C211yesmaximal, characteristic
3#20252124C211yesmaximal, characteristic
4#20242281C211yesmaximal, characteristic, complemented
5#20242276C211yesmaximal, characteristic, complemented
6#20242232C211yesmaximal, characteristic, complemented
7#20242226C211yesmaximal, characteristic, complemented
8#20242223C211yesmaximal, characteristic, complemented
9C8×#10625336C211yesmaximal, characteristic, complemented
10#20242185C211yesmaximal, characteristic, complemented
11#20242142C211yesmaximal, characteristic, complemented
12#20241815C211yesmaximal, characteristic, complemented
13#20241811C211yesmaximal, characteristic
14#20241667C211yesmaximal, characteristic, complemented
15C47×#1583C211yesmaximal, characteristic, complemented
16C2×SD16×D47C411yescharacteristic
17C2×M16×D47—21no—
18C4×Q8×D47C2211yescharacteristic
19C4×D4×D47C2211yescharacteristic
20#15689253—21no—
21#15689253C2211yescharacteristic, complemented
22#15689250—21no—
23#15689250C2211yescharacteristic, complemented
24Q8×C47⋊C8C2211yescharacteristic, complemented
25#15689214C411yescharacteristic
26C4×#10625325C2211yescharacteristic
27SD16×Dic47C411yescharacteristic
28C4×#10625336C2211yescharacteristic
29#15689210C411yescharacteristic
30C4×#10625335C2211yescharacteristic
31#15689208C411yescharacteristic
32#15689201C2211yescharacteristic
33#15689197—21no—
34#15689192—21no—
35C8×D188C2211yescharacteristic
36#15689188C2211yescharacteristic
37#15689174C2211yescharacteristic
38#106×D47C411yescharacteristic
39#15689145—21no—
40#15689145C2211yescharacteristic
41#15689145C2211yescharacteristic
42#103×D47C411yescharacteristic
43#102×D47C411yescharacteristic
44#15689085—21no—
45C4×#10625324—21no—
46C4×#10625324C2211yescharacteristic
47C4×C8×D47C2211yescharacteristic
48#15688874C2211yescharacteristic, complemented
49#15688873C2211yescharacteristic, complemented
50#15688873C2211yescharacteristic, complemented
51#15688855C2211yescharacteristic, complemented
52Q8⋊(C47⋊C8)C2211yescharacteristic, complemented
53#15688853C2211yescharacteristic, complemented
54#15688852C2211yescharacteristic
55D4⋊(C47⋊C8)C2211yescharacteristic
56#15688850C2211yescharacteristic
57#15688845C2211yescharacteristic, complemented
58C8×C47⋊C8C2211yescharacteristic, complemented
59C8×C47⋊C8C2211yescharacteristic, complemented
60C8×Q8×C47C2211yescharacteristic, complemented
61C4×SD16×C47C2211yescharacteristic
62C47×#436C2211yescharacteristic
63C47×#334C2211yescharacteristic, complemented
64C47×Q8⋊C8C2211yescharacteristic, complemented
65C47×D4⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic, complemented
67C2×Q8×D47C2×C411yescharacteristic
68C2×D4×D47C2×C411yescharacteristic
69C22×C4×D47—21no—
70C2×#10625325C2×C411yescharacteristic
71C2×#10625335C2×C411yescharacteristic
72C2×#10625336C2×C411yescharacteristic
73SD16×D47—24no—
74C2×#10625329—21no—
75C2×#10625324—21no—
76C2×#10625324—21no—
77C2×#10625324D411yescharacteristic
78C2×#10625324D411yescharacteristic
79M16×D47—44no—
80C2×C8×D47—21no—
81C2×C8×D47—21no—
82C2×C8×D47C2×C411yescharacteristic
83C2×C8×D47C2×C411yescharacteristic
84Q8×Dic47C2×C411yescharacteristic
85C4×Dic94—21no—
86C4×Dic94C2311yescharacteristic
87#13093989—21no—
88#13093989C2×C411yescharacteristic
89C4×D188C2311yescharacteristic
90D4×Dic47C2×C411yescharacteristic
91#13093986C2×C411yescharacteristic
92C4×C22⋊D47—21no—
93(C22⋊D47)⋊C4—21no—
94C4⋊C4×D47—21no—
95C4⋊C4×D47C2×C411yescharacteristic
96C4⋊C4×D47C2×C411yescharacteristic
97C22⋊C4×D47—21no—
98C42×D47—21no—
99C42×D47C2311yescharacteristic
100#13093956C2×C411yescharacteristic
101#13093955C2×C411yescharacteristic
102#13093955C2×C411yescharacteristic
103C4⋊(C47⋊C8)—21no—
104C4⋊(C47⋊C8)C2311yescharacteristic
105C4⋊(C47⋊C8)C2311yescharacteristic
106Dic47⋊C8—21no—
107Dic47⋊C8—21no—
108Dic47⋊C8—21no—
109Dic47⋊C8—21no—
110Dic47⋊C8—21no—
111#13093949C2×C411yescharacteristic
112#13093948C2×C411yescharacteristic
113#13093947C2×C411yescharacteristic
114#13093946C2×C411yescharacteristic
115#13093945C2×C411yescharacteristic
116#13093944C2×C411yescharacteristic
117C22⋊(C47⋊C8)—21no—
118(C2×D47)⋊C8—21no—
119(C2×D47)⋊C8—21no—
120(C2×D47)⋊C8—21no—
121#13093934—21no—
122#13093934—21no—
123#13093934D411yescharacteristic
124#13093934D411yescharacteristic
125C4×C47⋊C8—21no—
126C4×C47⋊C8C2311yescharacteristic
127C4×C47⋊C8C2311yescharacteristic
128C4×C47⋊C8C2311yescharacteristic
129C8×Dic47C2×C411yescharacteristic
130C8×Dic47C2×C411yescharacteristic
131C2×SD16×C47C2×C411yescharacteristic
132C2×M16×C47—21no—
133C4×Q8×C47C2311yescharacteristic
134C4×D4×C47C2311yescharacteristic
135#106×C47C2×C411yescharacteristic
136C4⋊C8×C47—21no—
137C4⋊C8×C47C2311yescharacteristic
138C4⋊C8×C47C2311yescharacteristic
139#103×C47C2×C411yescharacteristic
140#102×C47C2×C411yescharacteristic
141C22⋊C8×C47—21no—
142C4×C8×C47—21no—
143C4×C8×C47C2311yescharacteristic
144C4×C8×C47C2311yescharacteristic
145C4×C8×C47C2311yescharacteristic
146C23×D47—21no—
147C2×Q8×C47C22×C411yescharacteristic
148C2×D4×C47C22×C411yescharacteristic
149C22×C4×C47—21no—
150C2×C22⋊D47—21no—
151C22×Dic47—21no—
152Q8×D47—22no—
153Q8×D47—41no—
154D4×D47—22no—
155D4×D47—41no—
156C2×D188C22×C411yescharacteristic
157C2×C4×D47—21no—
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×C4×D47—21no—
161C2×C4×D47—21no—
162C2×C4×D47—42no—
163C2×C4×D47—42no—
164C2×C4×D47C22×C411yescharacteristic
165C2×C4×D47C2×D411yescharacteristic
166C2×C4×D47#4111yescharacteristic
167C2×Dic94—21no—
168C2×Dic94C22×C411yescharacteristic
169SD16×C47—22no—
170M16×C47—42no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47C22×C411yescharacteristic
175C2×C8×C47C22×C411yescharacteristic
176C2×C8×C47C2×D411yescharacteristic
177C2×C8×C47C2×D411yescharacteristic
178C4⋊C4×C47—21no—
179C4⋊C4×C47C22×C411yescharacteristic
180C4⋊C4×C47C22×C411yescharacteristic
181C22⋊C4×C47—21no—
182C42×C47—21no—
183C42×C47C2411yescharacteristic
184C22⋊Dic47—21no—
185#10625336M1614yescomplemented
186#10625335—22no—
187(C2×D47)⋊C4—21no—
188(C2×D47)⋊C4—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47C22×C411yescharacteristic
191C4⋊Dic47C22×C411yescharacteristic
192Dic47⋊C4—21no—
193Dic47⋊C4—21no—
194Dic47⋊C4—21no—
195Dic47⋊C4—21no—
196C4×Dic47—21no—
197C4×Dic47—21no—
198C4×Dic47—21no—
199C4×Dic47C22×C411yescharacteristic
200C4×Dic47C2×D411yescharacteristic
201C4×Dic47#4111yescharacteristic
202#10625329—42no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8C22×C411yescharacteristic
207C2×C47⋊C8C22×C411yescharacteristic
208C2×C47⋊C8C2×D411yescharacteristic
209C2×C47⋊C8C2×D411yescharacteristic
210#10625325—22no—
211#10625324—22no—
212#10625324—42no—
213#10625324—42no—
214#10625324SD1614yescomplemented
215C8×D47—22no—
216C8×D47—41no—
217C8×D47—42no—
218C8×D47—42no—
219C23×C47—21no—
220C22×D47—21no—
221C22×D47—42no—
222C22×D47—42no—
223C22×D47—81no—
224C22×D47C4×D411yescharacteristic
225Q8×C47—21no—
226Q8×C47C2×M1612yescomplemented
227D4×C47—21no—
228D4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47C23×C411yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C22⋊D47—42no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—42no—
242C2×Dic47C4×D411yescharacteristic
243C2×Dic47C4×D411yescharacteristic
244C2×Dic47C4×D411yescharacteristic
245D188—21no—
246D188C2×M1612yes—
247C4×D47—21no—
248C4×D47—21no—
249C4×D47—41no—
250C4×D47—42no—
251C4×D47—42no—
252C4×D47—41no—
253C4×D47—41no—
254C4×D47—42no—
255C4×D47—81no—
256C4×D47C2×SD1612yes—
257C4×D47#13512yes—
258Dic94—21no—
259Dic94—21no—
260Dic94—42no—
261C8×C47—21no—
262C8×C47—21no—
263C8×C47—21no—
264C8×C47—41no—
265C8×C47—41no—
266C8×C47—41no—
267C8×C47C2×SD1612yescomplemented
268C47⋊C8—21no—
269C47⋊C8—21no—
270C47⋊C8—41no—
271C47⋊C8—41no—
272C47⋊C8—41no—
273C47⋊C8C2×M1612yescomplemented
274C47⋊C8C2×SD1612yescomplemented
275M16×SD16—471nomaximal, Sylow
276C22×C47—41no—
277C22×C47—41no—
278C22×C47—41no—
279C22×C47C2×C4×D411yescharacteristic
280C2×D47—21no—
281C2×D47—21no—
282C2×D47—22no—
283C2×D47—41no—
284C2×D47—41no—
285C2×D47—84no—
286C2×D47C4×SD1612yes—
287C4×C47—22no—
288C4×C47—21no—
289C4×C47—41no—
290C4×C47—41no—
291C4×C47C22×M1611yescharacteristic
292C4×C47#56811yescharacteristic
293C4×C47C22×SD1611yescharacteristic
294C4×C47C2×#13511yescharacteristic
295Dic47—21no—
296Dic47—21no—
297Dic47—41no—
298Dic47—41no—
299Dic47—41no—
300Dic47C4×SD1612yes—
301Q8×M16—471no—
302C2×C4×SD16—471no—
303D4×M16—471no—
304#1590—471no—
305#1590—471no—
306#1583—471no—
307#1583D4711yescharacteristic, complemented
308#1582—471no—
309#1582—471no—
310C8×SD16—471no—
311C8×SD16—471no—
312#1566—471no—
313#1487—471no—
314#1486—471no—
315C8×M16—471no—
316C2×C47—81no—
317C2×C47D4×M1611yescharacteristic
318C2×C47C2×C4×SD1611yescharacteristic
319C2×C47#295711yescharacteristic
320D47—22no—
321D47—42no—
322C22×SD16—471no—
323C22×M16—941no—
324C2×C4×Q8—471no—
325C2×C4×D4—471no—
326#446—471no—
327#446—471no—
328#446—941no—
329#446—941no—
330C8×Q8—471no—
331C8×Q8C2×D4711yescharacteristic, complemented
332C4×SD16—471no—
333C4×SD16—471no—
334C4×SD16—471no—
335C4×SD16—471no—
336C4×SD16—471no—
337C4×SD16—471no—
338C4×SD16—471no—
339C4×SD16C2×D4711yescharacteristic
340#436—471no—
341#436C2×D4711yescharacteristic
342#435—941no—
343#435—941no—
344C8×D4—471no—
345C8×D4—471no—
346#431—471no—
347C2×#106—471no—
348#423—471no—
349#423—471no—
350#423—941no—
351C2×#103—471no—
352C2×#102—471no—
353#407—941no—
354C4×M16—471no—
355C4×M16—941no—
356C2×C4×C8—471no—
357#334—471no—
358#334—471no—
359#334—471no—
360#334C2×D4711yescharacteristic, complemented
361Q8⋊C8—471no—
362Q8⋊C8—471no—
363Q8⋊C8—471no—
364Q8⋊C8C2×D4711yescharacteristic, complemented
365D4⋊C8—471no—
366D4⋊C8—471no—
367D4⋊C8—471no—
368D4⋊C8C2×D4711yescharacteristic
369#322—471no—
370#322C2×D4711yescharacteristic, complemented
371C82—471no—
372C82—471no—
373C47M16×SD1611yescharacteristic, complemented, Sylow
374C22×Q8—471no—
375C22×D4—471no—
376C23×C4—941no—
377C2×SD16—471no—
378C2×SD16—471no—
379C2×SD16—471no—
380C2×SD16—944no—
381C2×SD16C4×D4711yescharacteristic
382C2×M16—21no—
383C2×M16—471no—
384C2×M16—471no—
385C2×M16—941no—
386C2×M16—941no—
387C2×M16—941no—
388C2×M16—1884no—
389C22×C8—471no—
390C22×C8—471no—
391C22×C8—941no—
392C22×C8—941no—
393C4×Q8—471no—
394C4×Q8—471no—
395C4×Q8—471no—
396C4×Q8—941no—
397C4×Q8—941no—
398C4×Q8C22×D4711yescharacteristic
399C4×D4—471no—
400C4×D4—471no—
401C4×D4—471no—
402C4×D4—941no—
403C4×D4—941no—
404C4×D4C22×D4711yescharacteristic
405C2×C4⋊C4—471no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—941no—
408C2×C22⋊C4—941no—
409C2×C42—471no—
410C2×C42—941no—
411#106—471no—
412#106—471no—
413#106—471no—
414#106C4×D4711yescharacteristic
415C4⋊C8—21no—
416C4⋊C8—471no—
417C4⋊C8—471no—
418C4⋊C8—941no—
419C4⋊C8—941no—
420C4⋊C8—941no—
421C4⋊C8—941no—
422C4⋊C8—941no—
423C4⋊C8—941no—
424C4⋊C8C22×D4711yescharacteristic
425C4⋊C8C22×D4711yescharacteristic
426#103—471no—
427#103—471no—
428#103—471no—
429#103C4×D4711yescharacteristic
430#102—471no—
431#102—471no—
432#102—471no—
433#102C4×D4711yescharacteristic
434C22⋊C8—21no—
435C22⋊C8—941no—
436C22⋊C8—941no—
437C22⋊C8—941no—
438C22⋊C8—941no—
439#97—471no—
440#97—471no—
441#97—941no—
442#97—941no—
443C4×C8—21no—
444C4×C8—471no—
445C4×C8—471no—
446C4×C8—471no—
447C4×C8—471no—
448C4×C8—471no—
449C4×C8—941no—
450C4×C8C22×D4711yescharacteristic
451C4×C8C22×D4711yescharacteristic
452C4×C8C22×D4711yescharacteristic
453C24—941no—
454C2×Q8—471no—
455C2×Q8—941no—
456C2×Q8—942no—
457C2×Q8—1881no—
458C2×Q8C2×C4×D4711yescharacteristic
459C2×D4—471no—
460C2×D4—941no—
461C2×D4—942no—
462C2×D4—1881no—
463C2×D4C2×C4×D4711yescharacteristic
464C22×C4—21no—
465C22×C4—471no—
466C22×C4—471no—
467C22×C4—471no—
468C22×C4—941no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—1882no—
475C22×C4—1882no—
476SD16—22no—
477SD16—474no—
478SD16—942no—
479SD16—942no—
480M16—42no—
481M16—474no—
482M16—942no—
483M16—1882no—
484M16—1882no—
485M16—1882no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—471no—
490C2×C8—471no—
491C2×C8—471no—
492C2×C8—471no—
493C2×C8—942no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—1881no—
498C2×C8—1882no—
499C2×C8—1882no—
500C2×C8C2×C4×D4711yescharacteristic
501C2×C8C2×C4×D4711yescharacteristic
502C2×C8D4×D4711yescharacteristic
503C2×C8D4×D4711yescharacteristic
504C4⋊C4—21no—
505C4⋊C4—471no—
506C4⋊C4—471no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4C2×C4×D4711yescharacteristic
513C4⋊C4C2×C4×D4711yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—941no—
516C22⋊C4—941no—
517C22⋊C4—941no—
518C42—21no—
519C42—471no—
520C42—471no—
521C42—471no—
522C42—941no—
523C42—941no—
524C42—941no—
525C42C23×D4711yescharacteristic
526C23—21no—
527C23—471no—
528C23—941no—
529C23—3761no—
530C23—1882no—
531C23—1882no—
532Q8—21no—
533Q8—941no—
534Q8—941no—
535Q8—1882no—
536Q8M16×D4712yescomplemented
537D4—21no—
538D4—21no—
539D4—472no—
540D4—941no—
541D4—1882no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—471no—
548C2×C4—472no—
549C2×C4—472no—
550C2×C4—471no—
551C2×C4—471no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—3761no—
558C2×C4—1881no—
559C2×C4—1882no—
560C2×C4—1882no—
561C2×C4—1882no—
562C2×C4—1882no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4C22×C4×D4711yescharacteristic
566C2×C4C2×D4×D4711yescharacteristic
567C2×C4#41×D4711yescharacteristic
568C8—21no—
569C8—21no—
570C8—21no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—472no—
575C8—472no—
576C8—941no—
577C8—941no—
578C8—1881no—
579C8—1881no—
580C8—1881no—
581C8SD16×D4712yescomplemented
582C22—41no—
583C22—41no—
584C22—41no—
585C22—472no—
586C22—941no—
587C22—941no—
588C22—942no—
589C22—3764no—
590C22—1881no—
591C22—1881no—
592C22C4×D4×D4711yescharacteristic
593C4—21no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—472no—
598C4—941no—
599C4—941no—
600C4—1881no—
601C4—1881no—
602C4—1881no—
603C4C2×M16×D4711yescharacteristic
604C4#1568988811yescharacteristic
605C4C2×SD16×D4711yescharacteristic
606C4#135×D4711yescharacteristic
607C2—81no—
608C2—942no—
609C2—1882no—
610C2#2025212111yescharacteristic
611C2C4×SD16×D4711yescharacteristic
612C2#2025230211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1583C2
socleC22×C47C2×C4×D4
prime radicalC2×D4×D47C2×C4
cosocleC42×C47C24
hypercenter#1583D47
hypocenterC47M16×SD16
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
1. Q8×#10625324C2semidirect
4. #20242281C2semidirect
5. #20242276C2semidirect
6. #20242232C2semidirect
7. #20242226C2semidirect
8. #20242223C2semidirect
9. C8×#10625336C2semidirect
10. #20242185C2semidirect
11. #20242142C2semidirect
12. #20241815C2semidirect
14. #20241667C2semidirect
15. C47×#1583C2semidirect
21. #15689253C22semidirect
23. #15689250C22semidirect
24. Q8×C47⋊C8C22semidirect
48. #15688874C22semidirect
49. #15688873C22semidirect
50. #15688873C22semidirect
51. #15688855C22semidirect
52. Q8⋊(C47⋊C8)C22semidirect
53. #15688853C22semidirect
57. #15688845C22semidirect
58. C8×C47⋊C8C22semidirect
59. C8×C47⋊C8C22semidirect
60. C8×Q8×C47C22semidirect
63. C47×#334C22semidirect
64. C47×Q8⋊C8C22semidirect
66. C47×#322C22semidirect
185. #10625336M16semidirect
214. #10625324SD16semidirect
226. Q8×C47C2×M16semidirect
267. C8×C47C2×SD16semidirect
273. C47⋊C8C2×M16semidirect
274. C47⋊C8C2×SD16semidirect
307. #1583D47semidirect
331. C8×Q8C2×D47semidirect
360. #334C2×D47semidirect
364. Q8⋊C8C2×D47semidirect
370. #322C2×D47semidirect
373. C47M16×SD16semidirect
536. Q8M16×D47semidirect
581. C8SD16×D47semidirect

Maximal subgroups

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

subgroupindex
Q8×#106253242
C4×SD16×D472
#202521242
#202422812
#202422762
#202422322
#202422262
#202422232
C8×#106253362
#202421852
#202421422
#202418152
#202418112
#202416672
C47×#15832
M16×SD1647