Finite Group Databasebeta

← #26279349

Subgroups — #26279349

Order 12032.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689413C211yesmaximal, characteristic
2C2×C4×#10625335C211yesmaximal, characteristic
3#20247227C211yesmaximal, characteristic, complemented
4#20244932C211yesmaximal, characteristic, complemented
5#20244865C211yesmaximal, characteristic, complemented
6#20244766C211yesmaximal, characteristic, complemented
7#20244721C211yesmaximal, characteristic, complemented
8#20244113C211yesmaximal, characteristic, complemented
9#20243947C211yesmaximal, characteristic, complemented
10#20243722C211yesmaximal, characteristic
11#20243722C211yesmaximal, characteristic
12#20243655C211yesmaximal, characteristic, complemented
13#20243649C211yesmaximal, characteristic
14#20243649C211yesmaximal, characteristic
15D4×C4⋊C4×C47C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C2×#13094024C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19#15689413—24no—
20C4×#10625335—24no—
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28C2×#13093947C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093945C2211yescharacteristic
31C2×#13093945C2211yescharacteristic
32C2×#13093945C2211yescharacteristic
33C2×C4×C47⋊C8C2211yescharacteristic
34#15689057—21no—
35#15689043—21no—
36#15689029C2211yescharacteristic
37#15689029C2211yescharacteristic
38#15689027—21no—
39#15689021C2211yescharacteristic
40#15689021C2211yescharacteristic
41#15689009—21no—
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49#15688878C2211yescharacteristic
50C2×C4×D4×C47C2211yescharacteristic
51C2×C4×D4×C47C2211yescharacteristic
52C2×C4×D4×C47C2211yescharacteristic
53C22×C4⋊C4×C47—21no—
54C47×#389C2211yescharacteristic
55C47×#389C2211yescharacteristic
56C47×#385—21no—
57C47×#385—21no—
58C47×#384C2211yescharacteristic
59C47×#380—21no—
60C4×C4⋊C4×C47C2211yescharacteristic
61C22×Dic94—21no—
62C22×Dic94C2311yescharacteristic
63C2×#10625335—22no—
64C2×#10625335—22no—
65C2×#10625335Q814yescomplemented
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70#13094024—22no—
71#13094024—41no—
72C4×Dic94—22no—
73C4×Dic94—41no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×C4×Dic47—21no—
82#13093955—22no—
83#13093955—22no—
84#13093955—22no—
85#13093955—22no—
86#13093955—41no—
87C4⋊(C47⋊C8)—22no—
88#13093947—22no—
89#13093947—22no—
90#13093947—41no—
91#13093945—22no—
92#13093945—22no—
93#13093945D414yes—
94C4×C47⋊C8—22no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C22×D4×C47C2311yescharacteristic
102C23×C4×C47—21no—
103C23×C4×C47—21no—
104C23×C4×C47—21no—
105C4×D4×C47—22no—
106C4×D4×C47—22no—
107C4×D4×C47—22no—
108C4×D4×C47—22no—
109C4×D4×C47—41no—
110C4×D4×C47—41no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47—44no—
116C2×C4⋊C4×C47D411yescharacteristic
117C2×C4⋊C4×C47D411yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C4⋊C4×C47C2311yescharacteristic
121C2×C22⋊C4×C47—21no—
122C2×C22⋊C4×C47—21no—
123C2×C22⋊C4×C47—21no—
124C2×C42×C47C2311yescharacteristic
125C2×C42×C47C2311yescharacteristic
126C2×C42×C47C2311yescharacteristic
127#95×C47—21no—
128#95×C47—21no—
129#95×C47D411yescharacteristic
130#95×C47D411yescharacteristic
131C24×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47C2×D412yes—
136C2×D4×C47C2×Q812yes—
137C2×D4×C47#4112yes—
138C22×C4×C47—24no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—42no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47—42no—
146C22×C4×C47—42no—
147C22×C4×C47—42no—
148C22×C4×C47—81no—
149C22×C4×C47—81no—
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2411yescharacteristic
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94—21no—
165C2×Dic94—21no—
166C2×Dic94—42no—
167C2×Dic94—42no—
168C2×Dic94—81no—
169C2×Dic94C2×Q812yes—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—41no—
176C4⋊C4×C47—42no—
177C4⋊C4×C47—42no—
178C4⋊C4×C47—42no—
179C4⋊C4×C47—81no—
180C4⋊C4×C47SD1614yescomplemented
181C4⋊C4×C47#4112yes—
182C22⋊C4×C47—24no—
183C22⋊C4×C47—42no—
184C22⋊C4×C47—42no—
185C42×C47—21no—
186C42×C47—21no—
187C42×C47—21no—
188C42×C47—21no—
189C42×C47—21no—
190C42×C47—21no—
191#10625335—28no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—42no—
198C4⋊Dic47C2×D412yes—
199Dic47⋊C4—42no—
200Dic47⋊C4—81no—
201Dic47⋊C4—81no—
202C4×Dic47—42no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8C2×D412yes—
214C2×C47⋊C8C2×Q812yes—
215C23×C47—22no—
216C23×C47—22no—
217C23×C47—22no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47—41no—
222C23×C47C22×D411yescharacteristic
223D4×C47—22no—
224D4×C47—22no—
225D4×C47—22no—
226D4×C47C22⋊Q814yes—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—22no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—48no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—84no—
253C2×C4×C47—84no—
254C2×C4×C47C2×SD1612yes—
255C2×C4×C47C2×SD1612yes—
256C2×C4×C47C2×SD1612yes—
257C2×C4×C47C22×D411yescharacteristic
258C2×C4×C47C22×Q811yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47H(5,𝔽2)11yescharacteristic
262C2×C4×C472−1+411yescharacteristic
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275Dic94—22no—
276Dic94—22no—
277Dic94—41no—
278Dic94—41no—
279Dic94—84no—
280C47⋊C8—22no—
281C47⋊C8—22no—
282C47⋊C8—41no—
283C47⋊C8—41no—
284#26376—471nomaximal, Sylow
285C22×C47—24no—
286C22×C47—42no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47—42no—
291C22×C47—42no—
292C22×C47#52511yescharacteristic
293C22×C47D4211yescharacteristic
294C22×C47D4×Q811yescharacteristic
295C22×C47C22×SD1611yescharacteristic
296C22×C47#57511yescharacteristic
297C22×C47#57711yescharacteristic
298C22×C47#57711yescharacteristic
299C4×C47—22no—
300C4×C47—22no—
301C4×C47—22no—
302C4×C47—42no—
303C4×C47—41no—
304C4×C47—41no—
305C4×C47—41no—
306C4×C47—81no—
307C4×C47—81no—
308C4×C47C2×C22⋊Q811yescharacteristic
309C4×C47#53911yescharacteristic
310C4×C47#54711yescharacteristic
311C4×C47#55411yescharacteristic
312Dic47—42no—
313Dic47—42no—
314Dic47—81no—
315Dic47—81no—
316C2×#476—471no—
317C2×C4×SD16—471no—
318#2451—471no—
319D4×C4⋊C4D4711yescharacteristic
320#2096—471no—
321#2084—471no—
322#2066—471no—
323#2058—471no—
324#1941—471no—
325#1916—471no—
326#1877—471no—
327#1877—471no—
328#1865—471no—
329#1864—471no—
330#1864—471no—
331C2×C47—44no—
332C2×C47#244911yescharacteristic
333C2×C47#310111yescharacteristic
334C2×C47#312511yescharacteristic
335C2×C47#326711yescharacteristic
336C2×C47D4×SD1611yescharacteristic
337C2×C47Q8×SD1611yescharacteristic
338C2×C47#338811yescharacteristic
339C22×SD16—471no—
340C2×C4⋊Q8—471no—
341C2×C4×Q8—471no—
342C2×C4×D4C2×D4711yescharacteristic
343C2×C4×D4C2×D4711yescharacteristic
344C2×C4×D4C2×D4711yescharacteristic
345C22×C4⋊C4—21no—
346#476—944no—
347C4×SD16—944no—
348C2×#106—471no—
349C2×#106—471no—
350C2×#106—471no—
351C2×#106—471no—
352C2×#106—471no—
353C2×C4⋊C8—471no—
354C2×#103—471no—
355C2×#103—471no—
356C2×#103—471no—
357C2×#102—471no—
358C2×#102—471no—
359C2×#102—471no—
360C2×C4×C8—471no—
361#398—941no—
362#395—941no—
363#391—471no—
364#391—471no—
365#391—941no—
366#389—471no—
367#389—471no—
368#389C2×D4711yescharacteristic
369#389C2×D4711yescharacteristic
370#387—941no—
371#385—21no—
372#385—21no—
373#384C2×D4711yescharacteristic
374#380—21no—
375C4×C4⋊C4C2×D4711yescharacteristic
376#340—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#336—471no—
381#336—471no—
382#336—471no—
383#336—471no—
384C47#2637611yescharacteristic, complemented, Sylow
385C22×Q8—471no—
386C22×Q8—941no—
387C22×D4C22×D4711yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—21no—
391C2×SD16—474no—
392C2×SD16—942no—
393C2×SD16—942no—
394C22×C8—471no—
395C22×C8—471no—
396C22×C8—471no—
397C22×C8—471no—
398C4⋊Q8—942no—
399C4⋊Q8—1881no—
400C4×Q8—942no—
401C4×Q8—1881no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—41no—
407C4×D4—41no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—44no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4C22⋊D4711yescharacteristic
421C2×C4⋊C4C22⋊D4711yescharacteristic
422C2×C4⋊C4C22×D4711yescharacteristic
423C2×C4⋊C4C22×D4711yescharacteristic
424C2×C4⋊C4C22×D4711yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C42—941no—
429C2×C42C22×D4711yescharacteristic
430C2×C42C22×D4711yescharacteristic
431C2×C42C22×D4711yescharacteristic
432#106—942no—
433#106—942no—
434#106—942no—
435#106—942no—
436#106—1881no—
437C4⋊C8—942no—
438#103—942no—
439#103—942no—
440#103—1881no—
441#102—474no—
442#102—942no—
443#102—942no—
444C4×C8—942no—
445#95—21no—
446#95—21no—
447#95—941no—
448#95—941no—
449#95—941no—
450#95—941no—
451#95—941no—
452#95—941no—
453#95C22⋊D4711yescharacteristic
454#95C22⋊D4711yescharacteristic
455C24—21no—
456C2×Q8—472no—
457C2×Q8—941no—
458C2×Q8—941no—
459C2×Q8—941no—
460C2×Q8—941no—
461C2×Q8—941no—
462C2×Q8—3761no—
463C2×Q8—1882no—
464C2×Q8—1882no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4#1062535012yes—
469C2×D4D4×D4712yes—
470C2×D4Q8×D4712yes—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—941no—
484C22×C4—941no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4C2×C22⋊D4711yescharacteristic
488C22×C4C2×C22⋊D4711yescharacteristic
489C22×C4C2×C22⋊D4711yescharacteristic
490C22×C4C2×C22⋊D4711yescharacteristic
491C22×C4C2×C22⋊D4711yescharacteristic
492C22×C4C2×C22⋊D4711yescharacteristic
493C22×C4C23×D4711yescharacteristic
494SD16—948no—
495C2×C8—472no—
496C2×C8—472no—
497C2×C8—941no—
498C2×C8—941no—
499C2×C8—941no—
500C2×C8—941no—
501C2×C8—941no—
502C2×C8—941no—
503C2×C8—941no—
504C2×C8—941no—
505C2×C8—941no—
506C2×C8—941no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—472no—
518C4⋊C4—941no—
519C4⋊C4—941no—
520C4⋊C4—941no—
521C4⋊C4—941no—
522C4⋊C4—941no—
523C4⋊C4—3761no—
524C4⋊C4—3761no—
525C4⋊C4—1882no—
526C4⋊C4—1882no—
527C4⋊C4#1062533514yescomplemented
528C4⋊C4#1062535012yes—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—1882no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊D4711yescharacteristic
547Q8—942no—
548Q8—942no—
549Q8—3764no—
550Q8—1881no—
551Q8—1881no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4#1309400214yes—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—942no—
584C2×C4—942no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4C2×#1062533512yes—
596C2×C4C2×#1062533512yes—
597C2×C4C2×#1062533512yes—
598C2×C4C2×D4×D4711yescharacteristic
599C2×C4C2×Q8×D4711yescharacteristic
600C2×C4C2×#1062535011yescharacteristic
601C2×C4C2×#1062535011yescharacteristic
602C2×C4#1309406911yescharacteristic
603C2×C4#1309407211yescharacteristic
604C8—942no—
605C8—942no—
606C8—1881no—
607C8—1881no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1568963411yescharacteristic
616C22D4×C22⋊D4711yescharacteristic
617C22Q8×C22⋊D4711yescharacteristic
618C22C22×#1062533511yescharacteristic
619C22#1568992111yescharacteristic
620C22#1568993111yescharacteristic
621C22#1568993411yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—3761no—
632C4—3761no—
633C4—1882no—
634C4—1882no—
635C4C2×#1309400211yescharacteristic
636C4#1568972011yescharacteristic
637C4#1568977411yescharacteristic
638C4#1568982811yescharacteristic
639C2—44no—
640C2#2024719711yescharacteristic
641C2#2025339611yescharacteristic
642C2#2025366511yescharacteristic
643C2#2025491411yescharacteristic
644C2D4×#1062533511yescharacteristic
645C2Q8×#1062533511yescharacteristic
646C2#2025633311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingD4×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenterD4×C4⋊C4D47
hypocenterC47#26376
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20247227C2semidirect
4. #20244932C2semidirect
5. #20244865C2semidirect
6. #20244766C2semidirect
7. #20244721C2semidirect
8. #20244113C2semidirect
9. #20243947C2semidirect
12. #20243655C2semidirect
65. C2×#10625335Q8semidirect
180. C4⋊C4×C47SD16semidirect
384. C47#26376semidirect
527. C4⋊C4#10625335semidirect

Maximal subgroups

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

subgroupindex
C2×#156894132
C2×C4×#106253352
#202472272
#202449322
#202448652
#202447662
#202447212
#202441132
#202439472
#202437222
#202436552
#202436492
D4×C4⋊C4×C472
#2637647