Finite Group Databasebeta

← #35566487

Subgroups — #35566487

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×#19128363C211yesmaximal, characteristic
2C2×#19128183C211yesmaximal, characteristic, complemented
3C2×#19128177C211yesmaximal, characteristic, complemented
4#24910534C213yesmaximal, complemented
5#24910503C213yesmaximal, complemented
6#24910470C213yesmaximal, complemented
7#24910395C213yesmaximal, complemented
8#24910384C211yesmaximal, characteristic, complemented
9#24910301C211yesmaximal, characteristic, complemented
10Q8×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
11#24910157C213yesmaximal, complemented
12#24910146C213yesmaximal, complemented
13C4×#14581780C213yesmaximal, complemented
14#24909927C213yesmaximal, complemented
15C42×Q8×C83C211yesmaximal, characteristic, complemented
16C22×#13050591C2211yescharacteristic
17C22×Q8×D83C2211yescharacteristic
18#19128363C4116yescomplemented
19C2×#14581780C2213yes—
20C2×#14581772C2213yes—
21C2×#14581767C2213yes—
22#19128183—28no—
23#19128177—28no—
24C2×Q8×Dic83C2211yescharacteristic
25C2×Q8×Dic83C2211yescharacteristic
26C2×C4×Dic166C2213yes—
27C2×#14581757C2213yes—
28C2×C4×D332C2213yes—
29C2×C4×D332C2213yes—
30C2×#14581754C2213yes—
31C2×#14581750C2213yes—
32C2×#14581749C2213yes—
33C2×#14581749C2213yes—
34C2×C4⋊C4×D83C2213yes—
35#19127607C2213yes—
36#19127606C2213yes—
37#19127603C2213yes—
38#19127603C2213yes—
39#19127603C2213yes—
40#19127602C2213yes—
41#19127600C2213yes—
42#19127600C2213yes—
43#19127599C2213yes—
44#19127599C2213yes—
45#19127593C2216yes—
46#19127592C2213yes—
47#19127592C2216yes—
48#19127592C2211yescharacteristic
49#19127592C2213yes—
50#19127592C2213yes—
51#19127586C2216yes—
52#19127579C2216yes—
53#19127575C2216yes—
54#19127567C2213yes—
55#19127558C2213yes—
56#19127558C2213yes—
57C4×C4⋊Dic83C2213yes—
58C4×(C2×D83)⋊C4C2216yes—
59C4×(C2×D83)⋊C4C2213yes—
60C4×(C2×D83)⋊C4C2213yes—
61C4×(C2×D83)⋊C4C2216yes—
62C2×C4×Q8×C83C2211yescharacteristic
63C2×C4×Q8×C83C2211yescharacteristic
64C2×C4×Q8×C83C2211yescharacteristic
65C4×C4⋊C4×C83C2213yes—
66C4×C4⋊C4×C83C2213yes—
67C4×C4⋊C4×C83C2213yes—
68C43×C83C2213yes—
69C2×#13050591—24no—
70C2×#13050591—24no—
71C2×#13050591C2×C418yes—
72C22×Dic166C2313yes—
73C2×Q8×D83—24no—
74C2×Q8×D83—24no—
75C2×Q8×D83C2×C418yescomplemented
76C22×D332C2313yes—
77C22×D332C2313yes—
78C22×C4×D83C2313yes—
79C22×C4×D83C2313yes—
80C22×C4×D83C2316yes—
81#14581780C2×C4124yescomplemented
82#14581772C2×C4124yescomplemented
83#14581767C2×C4124yescomplemented
84Q8×Dic83—24no—
85Q8×Dic83—24no—
86C4×Dic166—212no—
87#14581757—212no—
88C4×D332—212no—
89C4×D332C2×C4124yescomplemented
90#14581754—212no—
91#14581750—212no—
92#14581749—212no—
93#14581749—212no—
94C2×C4⋊Dic83C2313yes—
95C2×C4⋊Dic83C2313yes—
96C2×C4⋊Dic83C2313yes—
97C2×C4⋊Dic83C2313yes—
98C2×C4⋊Dic83C2313yes—
99C2×Dic83⋊C4C2313yes—
100C2×Dic83⋊C4C2313yes—
101C2×Dic83⋊C4C2313yes—
102C2×Dic83⋊C4C2311yescharacteristic
103C4⋊C4×D83—212no—
104C2×(C2×D83)⋊C4C2313yes—
105C2×(C2×D83)⋊C4C2313yes—
106C2×(C2×D83)⋊C4C2313yes—
107C2×(C2×D83)⋊C4C2311yescharacteristic
108C2×(C2×D83)⋊C4C2311yescharacteristic
109C2×(C2×D83)⋊C4C2313yes—
110C2×(C2×D83)⋊C4C2313yes—
111C2×(C2×D83)⋊C4C2316yes—
112C2×(C2×D83)⋊C4C2313yes—
113C2×(C2×D83)⋊C4C2316yes—
114C2×(C2×D83)⋊C4C2311yescharacteristic
115C2×(C2×D83)⋊C4C2313yes—
116C2×C4×Dic83C2313yes—
117C2×C4×Dic83C2313yes—
118C2×C4×Dic83C2316yes—
119C83⋊#95C2311yescharacteristic
120C83⋊#95C2313yes—
121C83⋊#95C2313yes—
122C83⋊#95C2316yes—
123C83⋊#95C2316yes—
124C83⋊#95C2313yes—
125C83⋊#95C2311yescharacteristic
126C83⋊#95C2313yes—
127C22×Q8×C83C2311yescharacteristic
128C4×Q8×C83—24no—
129C4×Q8×C83C2×C418yescomplemented
130C4×Q8×C83D418yescomplemented
131C2×C4⋊C4×C83C2313yes—
132C2×C4⋊C4×C83C2313yes—
133C2×C4⋊C4×C83C2313yes—
134C2×C42×C83C2311yescharacteristic
135C2×C42×C83C2313yes—
136C2×C42×C83C2313yes—
137C2×C42×C83C2313yes—
138C2×C42×C83C2313yes—
139C2×C42×C83C2313yes—
140C2×C42×C83C2313yes—
141#95×C83C2316yes—
142C23×D83C2411yescharacteristic
143C23×D83C2413yes—
144#13050591—232no—
145C2×Dic166—26no—
146C2×Dic166—26no—
147C2×Dic166C22×C4112yes—
148Q8×D83—232no—
149C2×D332—26no—
150C2×D332—26no—
151C2×D332—43no—
152C2×D332—43no—
153C2×D332C22×C4112yes—
154C2×D332C22×C4112yes—
155C22×Dic83C2413yes—
156C22×Dic83C2411yescharacteristic
157C22×Dic83C2413yes—
158C22×Dic83C2411yescharacteristic
159C22×Dic83C2411yescharacteristic
160C22×Dic83C2413yes—
161C2×C4×D83—26no—
162C2×C4×D83—26no—
163C2×C4×D83—26no—
164C2×C4×D83—26no—
165C2×C4×D83—212no—
166C2×C4×D83—46no—
167C2×C4×D83—46no—
168C2×C4×D83C22×C4112yes—
169C2×C4×D83C22×C4112yes—
170C4⋊Dic83—26no—
171C4⋊Dic83—26no—
172C4⋊Dic83—43no—
173C4⋊Dic83—43no—
174C4⋊Dic83C22×C4112yes—
175Dic83⋊C4—26no—
176Dic83⋊C4—22no—
177Dic83⋊C4—26no—
178Dic83⋊C4—26no—
179(C2×D83)⋊C4—22no—
180(C2×D83)⋊C4—26no—
181(C2×D83)⋊C4—26no—
182(C2×D83)⋊C4—26no—
183(C2×D83)⋊C4—22no—
184(C2×D83)⋊C4—26no—
185(C2×D83)⋊C4—43no—
186(C2×D83)⋊C4—46no—
187(C2×D83)⋊C4—43no—
188(C2×D83)⋊C4C22×C4124yes—
189(C2×D83)⋊C4C22×C4112yes—
190(C2×D83)⋊C4C22×C414yes—
191C4×Dic83—26no—
192C4×Dic83—26no—
193C4×Dic83—212no—
194C2×Q8×C83C22×C414yes—
195C2×Q8×C83C2×D414yes—
196C2×Q8×C83#4114yes—
197C22×C4×C83C2413yes—
198C22×C4×C83C2413yes—
199C22×C4×C83C2411yescharacteristic
200C22×C4×C83C2413yes—
201C22×C4×C83C2411yescharacteristic
202C22×C4×C83C2411yescharacteristic
203C22×C4×C83C2413yes—
204C4⋊C4×C83—26no—
205C4⋊C4×C83C22×C4112yes—
206C4⋊C4×C83C2×D4112yes—
207C42×C83—26no—
208C42×C83—22no—
209C42×C83—43no—
210C42×C83—43no—
211C42×C83—43no—
212C42×C83C22×C4112yes—
213C42×C83C2×D4112yes—
214C23×C83C2511yescharacteristic
215C22×D83—21no—
216C22×D83—23no—
217C22×D83—21no—
218C22×D83—23no—
219C22×D83—21no—
220C22×D83—23no—
221C22×D83—21no—
222C22×D83—23no—
223C22×D83—23no—
224C22×D83—23no—
225C22×D83—21no—
226C22×D83—21no—
227C22×D83C23×C416yes—
228C22×D83C23×C412yes—
229Q8×C83C4×D4116yes—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—23no—
237C2×C4×C83—23no—
238C2×C4×C83—23no—
239C2×C4×C83—23no—
240C2×C4×C83—23no—
241C2×C4×C83—23no—
242C2×C4×C83—23no—
243C2×C4×C83C23×C412yes—
244C2×C4×C83C23×C416yes—
245C2×C4×C83C23×C416yes—
246C2×C4×C83C22×D412yes—
247C2×C4×C83C22×D416yes—
248C2×C4×C83C22×D416yes—
249C2×C4×C83C2×#4116yes—
250C2×C4×C832−1+412yes—
251C2×Dic83—23no—
252C2×Dic83—21no—
253C2×Dic83—21no—
254C2×Dic83—23no—
255C2×Dic83—21no—
256C2×Dic83—21no—
257C2×Dic83—23no—
258C2×Dic83—23no—
259C2×Dic83—21no—
260C2×Dic83—23no—
261C2×Dic83—21no—
262C2×Dic83—23no—
263C2×Dic83—23no—
264C2×Dic83—21no—
265C2×Dic83—21no—
266C2×Dic83—23no—
267C2×Dic83C23×C416yes—
268C2×Dic83C23×C412yes—
269D332—224no—
270D332—412no—
271C4×D83—224no—
272C4×D83—224no—
273C4×D83—812no—
274Dic166—224no—
275C22×C83C24×C411yescharacteristic
276C22×C83C23×D411yescharacteristic
277C22×C83C22×#4111yescharacteristic
278C22×C83C2×2−1+411yescharacteristic
279C22×C83#58511yescharacteristic
280C22×C83#58511yescharacteristic
281C22×C83#58511yescharacteristic
282C2×D83—22no—
283C2×D83—26no—
284C2×D83—22no—
285C2×D83—26no—
286C2×D83—22no—
287C2×D83—26no—
288C2×D83—41no—
289C2×D83—43no—
290C2×D83—43no—
291C2×D83—41no—
292C2×D83—41no—
293C2×D83—43no—
294C2×D83—41no—
295C2×D83—43no—
296C4×C83—26no—
297C4×C83—22no—
298C4×C83—22no—
299C4×C83—26no—
300C4×C83—22no—
301C4×C83—43no—
302C4×C83C2×C4×D4112yes—
303Dic83—26no—
304Dic83—22no—
305Dic83—41no—
306Dic83—43no—
307Dic83—43no—
308Dic83—41no—
309#34312—831nomaximal, Sylow
310C2×C83C22×C4×D411yescharacteristic
311C2×C83#342811yescharacteristic
312C2×C83C4×2−1+411yescharacteristic
313C2×C83#349711yescharacteristic
314C2×C83#350011yescharacteristic
315C2×C83#351611yescharacteristic
316C2×C83#352411yescharacteristic
317D83—42no—
318D83—46no—
319C2×#548—831no—
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×Q8D8311yescharacteristic, complemented
334C83#3431211yescharacteristic, complemented, Sylow
335C22×#41—831no—
336C23×Q8—831no—
337#548—8316no—
338C2×#124—833no—
339C2×C22⋊Q8—833no—
340C2×#121—833no—
341#520—1668no—
342#519—1668no—
343C2×C4×Q8—831no—
344C2×C4×Q8—831no—
345C2×C4×Q8—833no—
346C2×C4×Q8—833no—
347C2×C4×Q8C2×D8311yescharacteristic
348C2×C4×Q8C2×D8311yescharacteristic
349C2×C4×Q8C2×D8311yescharacteristic
350C2×C4×D4—833no—
351C2×C4×D4—833no—
352C2×C4×D4—833no—
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—833no—
369#388—833no—
370#388—831no—
371#388—833no—
372#388—836no—
373#387—836no—
374#386—836no—
375#385—836no—
376#382—833no—
377#380—833no—
378#380—833no—
379C4×C4⋊C4—833no—
380C4×C4⋊C4C2×D8313yes—
381C4×C4⋊C4C2×D8313yes—
382C4×C4⋊C4C2×D8313yes—
383C4×C22⋊C4—833no—
384C4×C22⋊C4—833no—
385C4×C22⋊C4—836no—
386C4×C22⋊C4—836no—
387C43C2×D8313yes—
388C2×#41—838no—
389C2×#41—1664no—
390C2×#41—1664no—
391C22×Q8—838no—
392C22×Q8—833no—
393C22×Q8—1664no—
394C22×Q8—1664no—
395C22×Q8C22×D8311yescharacteristic
396C22×D4—833no—
397C22×D4—833no—
398C23×C4—836no—
399C23×C4—833no—
400C23×C4—833no—
401#124—8324no—
402C22⋊Q8—8324no—
403#121—8324no—
404C4×Q8—24no—
405C4×Q8—1664no—
406C4×Q8—1664no—
407C4×Q8—16612no—
408C4×Q8—16612no—
409C4×Q8C4×D8318yescomplemented
410C4×Q8D33218yescomplemented
411C4×D4—8324no—
412C4×D4—16612no—
413C4×D4—16612no—
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—16612no—
427C2×C4⋊C4C22×D8313yes—
428C2×C4⋊C4C22×D8313yes—
429C2×C4⋊C4C22×D8313yes—
430C2×C22⋊C4—833no—
431C2×C22⋊C4—836no—
432C2×C22⋊C4—833no—
433C2×C22⋊C4—831no—
434C2×C22⋊C4—833no—
435C2×C22⋊C4—836no—
436C2×C22⋊C4—833no—
437C2×C22⋊C4—831no—
438C2×C22⋊C4—833no—
439C2×C22⋊C4—833no—
440C2×C22⋊C4—831no—
441C2×C22⋊C4—833no—
442C2×C42—833no—
443C2×C42—836no—
444C2×C42—833no—
445C2×C42C22×D8313yes—
446C2×C42C22×D8311yescharacteristic
447C2×C42C22×D8313yes—
448C2×C42C22×D8313yes—
449C2×C42C22×D8313yes—
450C2×C42C22×D8313yes—
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—831no—
460#95C22×D8316yes—
461C24—833no—
462C24—831no—
463#41—16632no—
464C2×Q8—8312no—
465C2×Q8—16632no—
466C2×Q8—1666no—
467C2×Q8—1666no—
468C2×Q8C2×C4×D8314yes—
469C2×Q8C2×D33214yes—
470C2×Q8#1305059014yes—
471C2×D4—3323no—
472C2×D4—3323no—
473C2×D4—8312no—
474C2×D4—8312no—
475C2×D4—1666no—
476C2×D4—1666no—
477C22×C4—3326no—
478C22×C4—3326no—
479C22×C4—833no—
480C22×C4—831no—
481C22×C4—8312no—
482C22×C4—8312no—
483C22×C4—833no—
484C22×C4—831no—
485C22×C4—833no—
486C22×C4—831no—
487C22×C4—16612no—
488C22×C4—1666no—
489C22×C4—1666no—
490C22×C4—1666no—
491C22×C4—1666no—
492C22×C4C23×D8313yes—
493C22×C4C23×D8311yescharacteristic
494C22×C4C23×D8313yes—
495C22×C4C23×D8313yes—
496C22×C4C23×D8311yescharacteristic
497C22×C4C23×D8311yescharacteristic
498C22×C4C23×D8313yes—
499C4⋊C4—26no—
500C4⋊C4—3323no—
501C4⋊C4—3323no—
502C4⋊C4—8312no—
503C4⋊C4—1666no—
504C4⋊C4—1666no—
505C4⋊C4—1662no—
506C4⋊C4—1666no—
507C4⋊C4—1666no—
508C4⋊C4—1666no—
509C4⋊C4C2×C4×D83112yes—
510C4⋊C4C2×D332112yes—
511C22⋊C4—3323no—
512C22⋊C4—3326no—
513C22⋊C4—3323no—
514C22⋊C4—834no—
515C22⋊C4—8312no—
516C22⋊C4—8324no—
517C22⋊C4—1666no—
518C22⋊C4—1662no—
519C22⋊C4—1666no—
520C22⋊C4—1666no—
521C22⋊C4—1662no—
522C22⋊C4—1666no—
523C42—26no—
524C42—22no—
525C42—43no—
526C42—43no—
527C42—43no—
528C42—1666no—
529C42—1666no—
530C42—16612no—
531C42C2×C4×D83112yes—
532C42C2×D332112yes—
533C23—836no—
534C23—832no—
535C23—1661no—
536C23—1663no—
537C23—1661no—
538C23—1663no—
539C23—1661no—
540C23—1663no—
541C23—1661no—
542C23—1663no—
543C23—1663no—
544C23—1663no—
545C23—1661no—
546C23—1661no—
547C23C24×D8311yescharacteristic
548Q8—16624no—
549Q8C4×D332116yes—
550D4—33212no—
551D4—16624no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—23no—
559C2×C4—23no—
560C2×C4—23no—
561C2×C4—23no—
562C2×C4—23no—
563C2×C4—23no—
564C2×C4—23no—
565C2×C4—836no—
566C2×C4—832no—
567C2×C4—66412no—
568C2×C4—1661no—
569C2×C4—1663no—
570C2×C4—1661no—
571C2×C4—1663no—
572C2×C4—1661no—
573C2×C4—1663no—
574C2×C4—1661no—
575C2×C4—1663no—
576C2×C4—1663no—
577C2×C4—1661no—
578C2×C4—1661no—
579C2×C4—1663no—
580C2×C4—1663no—
581C2×C4—1661no—
582C2×C4—1661no—
583C2×C4—1663no—
584C2×C4—16624no—
585C2×C4—16624no—
586C2×C4C22×C4×D8312yes—
587C2×C4C22×C4×D8316yes—
588C2×C4C22×C4×D8316yes—
589C2×C4C22×D33216yes—
590C2×C4C22×D33212yes—
591C2×C4C22×D33216yes—
592C2×C4C2×#1305059016yes—
593C2×C4#1458184012yes—
594C22—3321no—
595C22—3323no—
596C22—3321no—
597C22—3323no—
598C22—3323no—
599C22—3321no—
600C22—3321no—
601C22—3323no—
602C22—1662no—
603C22—1666no—
604C22—1662no—
605C22—1666no—
606C22—1666no—
607C22—1662no—
608C22C23×C4×D8311yescharacteristic
609C22C23×D33211yescharacteristic
610C22C22×#1305059011yescharacteristic
611C22C2×#1458184011yescharacteristic
612C22#1912853711yescharacteristic
613C22#1912853711yescharacteristic
614C22#1912853711yescharacteristic
615C4—26no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—26no—
620C4—43no—
621C4—3321no—
622C4—3323no—
623C4—3323no—
624C4—3321no—
625C4—1662no—
626C4—1666no—
627C4C2×C4×D332112yes—
628C2—3326no—
629C2—3322no—
630C2C22×C4×D33211yescharacteristic
631C2#2492066211yescharacteristic
632C2C4×#1458184011yescharacteristic
633C2#2492123411yescharacteristic
634C2#2492125611yescharacteristic
635C2#2492145611yescharacteristic
636C2#2492157211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC42×Q8×C83C2
socle, cosocleC23×C83C25
prime radicalC2×#19128363C2
hypercenterC42×Q8D83
hypocenterC83#34312
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#19128183C2semidirect
3. C2×#19128177C2semidirect
4. #24910534C2semidirect
5. #24910503C2semidirect
6. #24910470C2semidirect
7. #24910395C2semidirect
8. #24910384C2semidirect
9. #24910301C2semidirect
10. Q8×(C2×D83)⋊C4C2semidirect
11. #24910157C2semidirect
12. #24910146C2semidirect
13. C4×#14581780C2semidirect
14. #24909927C2semidirect
15. C42×Q8×C83C2semidirect
18. #19128363C4semidirect
75. C2×Q8×D83C2×C4semidirect
81. #14581780C2×C4semidirect
82. #14581772C2×C4semidirect
83. #14581767C2×C4semidirect
89. C4×D332C2×C4semidirect
129. C4×Q8×C83C2×C4semidirect
130. C4×Q8×C83D4semidirect
333. C42×Q8D83semidirect
334. C83#34312semidirect
409. C4×Q8C4×D83semidirect
410. C4×Q8D332semidirect

Maximal subgroups

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

subgroupindex
C2×#191283632
C2×#191281832
C2×#191281772
#249105342
#249105032
#249104702
#249103952
#249103842
#249103012
Q8×(C2×D83)⋊C42
#249101572
#249101462
C4×#145817802
#249099272
C42×Q8×C832
#3431283