Finite Group Databasebeta

← #35566488

Subgroups — #35566488

Order 21248.

2960 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128184C211yesmaximal, characteristic, complemented
2C2×#19128179C211yesmaximal, characteristic, complemented
3#24910538C213yesmaximal, complemented
4#24910505C213yesmaximal, complemented
5#24910472C213yesmaximal, complemented
6#24910392C213yesmaximal, complemented
7#24910384C211yesmaximal, characteristic, complemented
8#24910304C211yesmaximal, characteristic, complemented
9Q8×C22⋊Dic83C211yesmaximal, characteristic, complemented
10#24910158C213yesmaximal, complemented
11#24910147C213yesmaximal, complemented
12#124×Dic83C213yesmaximal, complemented
13#24909928C213yesmaximal, complemented
14C4×Q8×Dic83C211yesmaximal, characteristic, complemented
15C2×#548×C83C211yesmaximal, characteristic
16#19128184—28no—
17#19128179—28no—
18C2×Q8×Dic83C2211yescharacteristic
19C2×Q8×Dic83C2211yescharacteristic
20C2×Q8×Dic83C2213yes—
21C2×Q8×Dic83C2213yes—
22C2×Q8×Dic83C2211yescharacteristic
23C2×Q8×Dic83C2211yescharacteristic
24C2×D4×Dic83C2213yes—
25C2×D4×Dic83C2213yes—
26C2×#14581751C2213yes—
27C2×#14581751C2213yes—
28C2×#14581751C2213yes—
29C22×C4⋊Dic83C2213yes—
30#19127608C2213yes—
31#19127608C2213yes—
32#19127604C2213yes—
33#19127604C2213yes—
34#19127604C2213yes—
35#19127604C2213yes—
36#19127600C2213yes—
37#19127600C2213yes—
38#19127600C2213yes—
39#19127600C2213yes—
40#19127594C2213yes—
41#19127594C2216yes—
42#19127594C2216yes—
43#19127594C2211yescharacteristic
44#19127594C2213yes—
45#19127594C2213yes—
46#19127587C2216yes—
47#19127581C2216yes—
48#19127576C2216yes—
49#19127567C2213yes—
50#19127561C2213yes—
51#19127561C2213yes—
52C4⋊C4×Dic83C2213yes—
53C4⋊C4×Dic83C2213yes—
54C4×C4⋊Dic83C2213yes—
55C4×C4⋊Dic83C2213yes—
56C22⋊C4×Dic83C2216yes—
57C22⋊C4×Dic83C2216yes—
58C4×C22⋊Dic83C2213yes—
59C4×C22⋊Dic83C2213yes—
60C42×Dic83C2213yes—
61C22×#41×C83C2211yescharacteristic
62C23×Q8×C83C2211yescharacteristic
63#548×C83C4116yescomplemented
64C2×#124×C83C2213yes—
65C2×C22⋊Q8×C83C2213yes—
66C2×#121×C83C2213yes—
67C2×C4×Q8×C83C2211yescharacteristic
68C2×C4×D4×C83C2213yes—
69C23×Dic83C2316yes—
70Q8×Dic83—212no—
71Q8×Dic83—212no—
72Q8×Dic83—24no—
73Q8×Dic83—24no—
74Q8×Dic83—24no—
75Q8×Dic83D418yescomplemented
76D4×Dic83—212no—
77D4×Dic83—212no—
78#14581751—212no—
79#14581751—212no—
80#14581751—212no—
81C2×C4⋊Dic83—212no—
82C2×C4⋊Dic83C2313yes—
83C2×C4⋊Dic83C2313yes—
84C2×C4⋊Dic83C2313yes—
85C2×C4⋊Dic83C2313yes—
86C2×C4⋊Dic83C2313yes—
87C2×C4⋊Dic83C2313yes—
88C2×C4⋊Dic83C2311yescharacteristic
89C2×C4⋊Dic83C2313yes—
90C2×C4⋊Dic83C2313yes—
91C2×C4⋊Dic83C2313yes—
92C2×C22⋊Dic83C2313yes—
93C2×C22⋊Dic83C2313yes—
94C2×C22⋊Dic83C2313yes—
95C2×C22⋊Dic83C2311yescharacteristic
96C2×C22⋊Dic83C2311yescharacteristic
97C2×C22⋊Dic83C2313yes—
98C2×C22⋊Dic83C2313yes—
99C2×C22⋊Dic83C2316yes—
100C2×C22⋊Dic83C2313yes—
101C2×C4×Dic83C2313yes—
102C2×C4×Dic83C2313yes—
103C2×C4×Dic83C2313yes—
104C2×C4×Dic83C2316yes—
105C2×C4×Dic83C2311yescharacteristic
106C2×C4×Dic83C2313yes—
107C2×C4×Dic83C2313yes—
108C2×C4×Dic83C2313yes—
109C2×C4×Dic83C2313yes—
110C83⋊#95C2311yescharacteristic
111C83⋊#95C2316yes—
112C83⋊#95C2313yes—
113C83⋊#95C2313yes—
114C83⋊#95C2316yes—
115C83⋊#95C2316yes—
116C83⋊#95C2313yes—
117C83⋊#95C2311yescharacteristic
118C83⋊#95C2313yes—
119C2×#41×C83—24no—
120C2×#41×C83—24no—
121C2×#41×C83C2×C418yes—
122C22×Q8×C83—24no—
123C22×Q8×C83—24no—
124C22×Q8×C83C2×C418yescomplemented
125C22×Q8×C83C2313yes—
126C22×Q8×C83C2311yescharacteristic
127C22×D4×C83C2313yes—
128C22×D4×C83C2313yes—
129C23×C4×C83C2313yes—
130C23×C4×C83C2313yes—
131#124×C83C2×C4124yescomplemented
132C22⋊Q8×C83C2×C4124yescomplemented
133#121×C83C2×C4124yescomplemented
134C4×Q8×C83C2×C418yescomplemented
135C4×D4×C83C2×C4124yescomplemented
136C2×C4⋊C4×C83C2313yes—
137C2×C4⋊C4×C83C2313yes—
138C2×C22⋊C4×C83C2316yes—
139C2×C22⋊C4×C83C2311yescharacteristic
140C2×C22⋊C4×C83C2313yes—
141C2×C42×C83C2313yes—
142C22×Dic83—212no—
143C22×Dic83—46no—
144C22×Dic83—46no—
145C22×Dic83C2413yes—
146C22×Dic83C2413yes—
147C22×Dic83C2411yescharacteristic
148C22×Dic83C2413yes—
149C22×Dic83C2411yescharacteristic
150C22×Dic83C2413yes—
151C22×Dic83C2411yescharacteristic
152C22×Dic83C2411yescharacteristic
153C4⋊Dic83—26no—
154C4⋊Dic83—26no—
155C4⋊Dic83—26no—
156C4⋊Dic83—26no—
157C4⋊Dic83—22no—
158C4⋊Dic83—26no—
159C4⋊Dic83—26no—
160C4⋊Dic83—43no—
161C4⋊Dic83—43no—
162C4⋊Dic83C2×D4112yes—
163C22⋊Dic83—26no—
164C22⋊Dic83—22no—
165C22⋊Dic83—26no—
166C22⋊Dic83—26no—
167C22⋊Dic83—22no—
168C22⋊Dic83—26no—
169C22⋊Dic83—43no—
170C22⋊Dic83—46no—
171C22⋊Dic83—43no—
172C4×Dic83—26no—
173C4×Dic83—26no—
174C4×Dic83—26no—
175C4×Dic83—212no—
176C4×Dic83—22no—
177C4×Dic83—43no—
178C4×Dic83—43no—
179C4×Dic83—43no—
180C4×Dic83C2×D4112yes—
181C24×C83C2413yes—
182C24×C83C2411yescharacteristic
183#41×C83—232no—
184C2×Q8×C83—232no—
185C2×Q8×C83—26no—
186C2×Q8×C83—26no—
187C2×Q8×C83C22×C4112yes—
188C2×Q8×C83C22×C414yes—
189C2×Q8×C83C2×D414yes—
190C2×Q8×C83#4114yes—
191C2×D4×C83—26no—
192C2×D4×C83—26no—
193C2×D4×C83—43no—
194C2×D4×C83—43no—
195C2×D4×C83C22×C4112yes—
196C2×D4×C83C22×C4112yes—
197C22×C4×C83—26no—
198C22×C4×C83—26no—
199C22×C4×C83—26no—
200C22×C4×C83—26no—
201C22×C4×C83C22×C4112yes—
202C22×C4×C83C22×C4112yes—
203C22×C4×C83C2411yescharacteristic
204C22×C4×C83C2413yes—
205C22×C4×C83C2413yes—
206C22×C4×C83C2411yescharacteristic
207C22×C4×C83C2413yes—
208C4⋊C4×C83C22×C4112yes—
209C4⋊C4×C83C22×C4112yes—
210C22⋊C4×C83C22×C414yes—
211C22⋊C4×C83C22×C4112yes—
212C22⋊C4×C83C22×C4124yes—
213C42×C83C22×C4112yes—
214C23×C83—21no—
215C23×C83—23no—
216C23×C83—21no—
217C23×C83—23no—
218C23×C83—23no—
219C23×C83—21no—
220C23×C83—21no—
221C23×C83—21no—
222C23×C83—23no—
223C23×C83—23no—
224C23×C83—21no—
225C23×C83—23no—
226C23×C83C23×C416yes—
227C23×C83C23×C412yes—
228C23×C83C2511yescharacteristic
229Q8×C83—224no—
230Q8×C83C4×D4116yes—
231D4×C83—224no—
232D4×C83—412no—
233C2×C4×C83—21no—
234C2×C4×C83—23no—
235C2×C4×C83—21no—
236C2×C4×C83—23no—
237C2×C4×C83—21no—
238C2×C4×C83—23no—
239C2×C4×C83—23no—
240C2×C4×C83—21no—
241C2×C4×C83—224no—
242C2×C4×C83—224no—
243C2×C4×C83C23×C412yes—
244C2×C4×C83C23×C416yes—
245C2×C4×C83C23×C416yes—
246C2×C4×C83C23×C412yes—
247C2×C4×C83C23×C416yes—
248C2×C4×C83C22×D416yes—
249C2×C4×C83C2×#4116yes—
250C2×Dic83—21no—
251C2×Dic83—21no—
252C2×Dic83—21no—
253C2×Dic83—21no—
254C2×Dic83—23no—
255C2×Dic83—23no—
256C2×Dic83—21no—
257C2×Dic83—23no—
258C2×Dic83—21no—
259C2×Dic83—23no—
260C2×Dic83—23no—
261C2×Dic83—21no—
262C2×Dic83—23no—
263C2×Dic83—23no—
264C2×Dic83—23no—
265C2×Dic83—21no—
266C2×Dic83—23no—
267C2×Dic83—23no—
268C2×Dic83—21no—
269C2×Dic83—21no—
270C2×Dic83—23no—
271C2×Dic83—812no—
272C2×Dic83C22×D416yes—
273C2×Dic83C22×D412yes—
274C2×Dic832−1+412yes—
275C22×C83—22no—
276C22×C83—26no—
277C22×C83—22no—
278C22×C83—26no—
279C22×C83—22no—
280C22×C83—26no—
281C22×C83—41no—
282C22×C83—43no—
283C22×C83—43no—
284C22×C83—41no—
285C22×C83—41no—
286C22×C83—43no—
287C22×C83—41no—
288C22×C83—43no—
289C22×C83C24×C411yescharacteristic
290C22×C83C23×D411yescharacteristic
291C22×C83C22×#4111yescharacteristic
292C22×C83C2×2−1+411yescharacteristic
293C22×C83#58511yescharacteristic
294C22×C83#58511yescharacteristic
295C22×C83#58511yescharacteristic
296C4×C83—22no—
297C4×C83—26no—
298C4×C83—26no—
299C4×C83—22no—
300C4×C83C2×C4×D4112yes—
301Dic83—22no—
302Dic83—26no—
303Dic83—22no—
304Dic83—41no—
305Dic83—43no—
306Dic83—41no—
307Dic83—43no—
308Dic83—43no—
309#34312—831nomaximal, Sylow
310C2×C83—42no—
311C2×C83—46no—
312C2×C83C22×C4×D411yescharacteristic
313C2×C83#342811yescharacteristic
314C2×C83C4×2−1+411yescharacteristic
315C2×C83#349711yescharacteristic
316C2×C83#350011yescharacteristic
317C2×C83#351611yescharacteristic
318C2×C83#352411yescharacteristic
319C2×#548D8311yescharacteristic
320C2×#520—831no—
321C2×#519—831no—
322#2367—833no—
323#2363—833no—
324#2360—833no—
325#2352—833no—
326#2351—831no—
327#2341—831no—
328Q8×C22⋊C4—831no—
329#2324—833no—
330#2323—833no—
331C4×#124—833no—
332#2293—833no—
333C42×Q8—831no—
334C83#3431211yescharacteristic, complemented, Sylow
335C22×#41C2×D8311yescharacteristic
336C23×Q8C2×D8311yescharacteristic
337#548Dic83116yescomplemented
338C2×#124C2×D8313yes—
339C2×C22⋊Q8C2×D8313yes—
340C2×#121C2×D8313yes—
341#520—1668no—
342#519—1668no—
343C2×C4×Q8—831no—
344C2×C4×Q8—833no—
345C2×C4×Q8—831no—
346C2×C4×Q8—831no—
347C2×C4×Q8—831no—
348C2×C4×Q8—833no—
349C2×C4×Q8C2×D8311yescharacteristic
350C2×C4×D4—833no—
351C2×C4×D4—833no—
352C2×C4×D4C2×D8313yes—
353C2×#117—833no—
354C2×#117—833no—
355C2×#117—833no—
356C22×C4⋊C4—833no—
357#391—833no—
358#391—833no—
359#390—833no—
360#390—833no—
361#390—833no—
362#390—833no—
363#389—833no—
364#389—833no—
365#389—833no—
366#389—833no—
367#388—836no—
368#388—836no—
369#388—833no—
370#388—833no—
371#388—831no—
372#388—833no—
373#387—836no—
374#386—836no—
375#385—836no—
376#382—833no—
377#380—833no—
378#380—833no—
379C4×C4⋊C4—833no—
380C4×C4⋊C4—833no—
381C4×C4⋊C4—833no—
382C4×C4⋊C4—833no—
383C4×C22⋊C4—833no—
384C4×C22⋊C4—833no—
385C4×C22⋊C4—836no—
386C4×C22⋊C4—836no—
387C43—833no—
388C2×#41—24no—
389C2×#41—24no—
390C2×#41C2×Dic8318yes—
391C22×Q8—24no—
392C22×Q8—24no—
393C22×Q8C2×Dic8318yescomplemented
394C22×Q8C22×D8313yes—
395C22×Q8C22×D8311yescharacteristic
396C22×D4C22×D8313yes—
397C22×D4C22×D8313yes—
398C23×C4—836no—
399C23×C4C22×D8313yes—
400C23×C4C22×D8313yes—
401#124C2×Dic83124yescomplemented
402C22⋊Q8C2×Dic83124yescomplemented
403#121C2×Dic83124yescomplemented
404C4×Q8—838no—
405C4×Q8—16612no—
406C4×Q8—16612no—
407C4×Q8—1664no—
408C4×Q8—1664no—
409C4×Q8—1664no—
410C4×Q8C2×Dic8318yescomplemented
411C4×D4—16612no—
412C4×D4—16612no—
413C4×D4C2×Dic83124yescomplemented
414#117—16612no—
415#117—16612no—
416#117—16612no—
417C2×C4⋊C4—833no—
418C2×C4⋊C4—833no—
419C2×C4⋊C4—833no—
420C2×C4⋊C4—833no—
421C2×C4⋊C4—833no—
422C2×C4⋊C4—831no—
423C2×C4⋊C4—833no—
424C2×C4⋊C4—833no—
425C2×C4⋊C4—833no—
426C2×C4⋊C4—833no—
427C2×C4⋊C4—16612no—
428C2×C4⋊C4C22×D8313yes—
429C2×C4⋊C4C22×D8313yes—
430C2×C22⋊C4—833no—
431C2×C22⋊C4—836no—
432C2×C22⋊C4—833no—
433C2×C22⋊C4—833no—
434C2×C22⋊C4—831no—
435C2×C22⋊C4—833no—
436C2×C22⋊C4—833no—
437C2×C22⋊C4—831no—
438C2×C22⋊C4—833no—
439C2×C22⋊C4C22×D8311yescharacteristic
440C2×C22⋊C4C22×D8313yes—
441C2×C22⋊C4C22×D8316yes—
442C2×C42—833no—
443C2×C42—833no—
444C2×C42—833no—
445C2×C42—833no—
446C2×C42—833no—
447C2×C42—836no—
448C2×C42—833no—
449C2×C42—833no—
450C2×C42—831no—
451C2×C42C22×D8313yes—
452#95—831no—
453#95—833no—
454#95—833no—
455#95—833no—
456#95—836no—
457#95—836no—
458#95—833no—
459#95—836no—
460#95—831no—
461C24C23×D8311yescharacteristic
462C24C23×D8313yes—
463#41—232no—
464C2×Q8—232no—
465C2×Q8—26no—
466C2×Q8—26no—
467C2×Q8C22×Dic83112yes—
468C2×Q8C22×Dic8314yes—
469C2×Q8D4×D8314yes—
470C2×Q8#1305058914yes—
471C2×D4—26no—
472C2×D4—26no—
473C2×D4—43no—
474C2×D4—43no—
475C2×D4C22×Dic83112yes—
476C2×D4C22×Dic83112yes—
477C22×C4—26no—
478C22×C4—26no—
479C22×C4—26no—
480C22×C4—26no—
481C22×C4—3326no—
482C22×C4—3326no—
483C22×C4—831no—
484C22×C4—833no—
485C22×C4—831no—
486C22×C4—833no—
487C22×C4—833no—
488C22×C4—831no—
489C22×C4—833no—
490C22×C4—831no—
491C22×C4—16612no—
492C22×C4C22×Dic83112yes—
493C22×C4C22×Dic83112yes—
494C22×C4C23×D8311yescharacteristic
495C22×C4C23×D8313yes—
496C22×C4C23×D8313yes—
497C22×C4C23×D8311yescharacteristic
498C22×C4C23×D8313yes—
499C4⋊C4—3323no—
500C4⋊C4—3323no—
501C4⋊C4—8312no—
502C4⋊C4—1666no—
503C4⋊C4—1662no—
504C4⋊C4—1666no—
505C4⋊C4—1666no—
506C4⋊C4—1666no—
507C4⋊C4—1666no—
508C4⋊C4—1666no—
509C4⋊C4C22×Dic83112yes—
510C4⋊C4C22×Dic83112yes—
511C22⋊C4—3326no—
512C22⋊C4—3323no—
513C22⋊C4—3323no—
514C22⋊C4—1662no—
515C22⋊C4—1666no—
516C22⋊C4—1666no—
517C22⋊C4—1666no—
518C22⋊C4—1662no—
519C22⋊C4—1666no—
520C22⋊C4C22×Dic8314yes—
521C22⋊C4C22×Dic83112yes—
522C22⋊C4C22×Dic83124yes—
523C42—3323no—
524C42—3323no—
525C42—3323no—
526C42—8312no—
527C42—16612no—
528C42—1662no—
529C42—1666no—
530C42—1666no—
531C42—1666no—
532C42C22×Dic83112yes—
533C23—21no—
534C23—23no—
535C23—21no—
536C23—23no—
537C23—21no—
538C23—23no—
539C23—23no—
540C23—23no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—23no—
545C23C23×Dic8316yes—
546C23C23×Dic8312yes—
547C23C24×D8311yescharacteristic
548Q8—224no—
549Q8D4×Dic83116yes—
550D4—224no—
551D4—412no—
552C2×C4—23no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—23no—
556C2×C4—21no—
557C2×C4—23no—
558C2×C4—23no—
559C2×C4—21no—
560C2×C4—224no—
561C2×C4—224no—
562C2×C4—832no—
563C2×C4—836no—
564C2×C4—832no—
565C2×C4—66412no—
566C2×C4—1661no—
567C2×C4—1663no—
568C2×C4—1661no—
569C2×C4—1663no—
570C2×C4—1663no—
571C2×C4—1661no—
572C2×C4—1663no—
573C2×C4—1663no—
574C2×C4—1663no—
575C2×C4—1661no—
576C2×C4—1661no—
577C2×C4—1663no—
578C2×C4—1663no—
579C2×C4—1661no—
580C2×C4—1663no—
581C2×C4—1661no—
582C2×C4—1661no—
583C2×C4—1661no—
584C2×C4—1661no—
585C2×C4—1663no—
586C2×C4—1663no—
587C2×C4C23×Dic8312yes—
588C2×C4C23×Dic8316yes—
589C2×C4C23×Dic8316yes—
590C2×C4C23×Dic8312yes—
591C2×C4C23×Dic8316yes—
592C2×C4C2×D4×D8316yes—
593C2×C4C2×#1305058916yes—
594C22—22no—
595C22—26no—
596C22—22no—
597C22—26no—
598C22—22no—
599C22—26no—
600C22—41no—
601C22—43no—
602C22—41no—
603C22—43no—
604C22—43no—
605C22—41no—
606C22—41no—
607C22—43no—
608C22C24×Dic8311yescharacteristic
609C22C22×D4×D8311yescharacteristic
610C22C22×#1305058911yescharacteristic
611C222−1+4×D8311yescharacteristic
612C22#1912853711yescharacteristic
613C22#1912853711yescharacteristic
614C22#1912853811yescharacteristic
615C4—22no—
616C4—26no—
617C4—26no—
618C4—22no—
619C4—3321no—
620C4—3323no—
621C4—3323no—
622C4—3321no—
623C4—3323no—
624C4—1662no—
625C4—1662no—
626C4—1666no—
627C4C2×D4×Dic83112yes—
628C2—42no—
629C2—46no—
630C2C22×D4×Dic8311yescharacteristic
631C2#2492066611yescharacteristic
632C22−1+4×Dic8311yescharacteristic
633C2#2492123111yescharacteristic
634C2#2492125411yescharacteristic
635C2#2492144911yescharacteristic
636C2#2492156511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
Fitting, prime radicalC2×#548×C83C2
socle, cosocleC23×C83C25
hypercenterC2×#548D83
hypocenterC83#34312
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#19128184C2semidirect
2. C2×#19128179C2semidirect
3. #24910538C2semidirect
4. #24910505C2semidirect
5. #24910472C2semidirect
6. #24910392C2semidirect
7. #24910384C2semidirect
8. #24910304C2semidirect
9. Q8×C22⋊Dic83C2semidirect
10. #24910158C2semidirect
11. #24910147C2semidirect
12. #124×Dic83C2semidirect
13. #24909928C2semidirect
14. C4×Q8×Dic83C2semidirect
63. #548×C83C4semidirect
75. Q8×Dic83D4semidirect
124. C22×Q8×C83C2×C4semidirect
131. #124×C83C2×C4semidirect
132. C22⋊Q8×C83C2×C4semidirect
133. #121×C83C2×C4semidirect
134. C4×Q8×C83C2×C4semidirect
135. C4×D4×C83C2×C4semidirect
334. C83#34312semidirect
337. #548Dic83semidirect
393. C22×Q8C2×Dic83semidirect
401. #124C2×Dic83semidirect
402. C22⋊Q8C2×Dic83semidirect
403. #121C2×Dic83semidirect
410. C4×Q8C2×Dic83semidirect
413. C4×D4C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281842
C2×#191281792
#249105382
#249105052
#249104722
#249103922
#249103842
#249103042
Q8×C22⋊Dic832
#249101582
#249101472
#124×Dic832
#249099282
C4×Q8×Dic832
C2×#548×C832
#3431283