Finite Group Databasebeta

← #35566489

Subgroups — #35566489

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×#19128358C211yesmaximal, characteristic
2C2×#19128177C211yesmaximal, characteristic, complemented
3#24910536C213yesmaximal, complemented
4#24910497C213yesmaximal, complemented
5#24910468C213yesmaximal, complemented
6#24910388C213yesmaximal, complemented
7#24910381C211yesmaximal, characteristic, complemented
8#24910304C211yesmaximal, characteristic, complemented
9Q8×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
10#24910150C213yesmaximal, complemented
11#24910145C213yesmaximal, complemented
12C4×#14581779C213yesmaximal, complemented
13#24909922C213yesmaximal, complemented
14C4×Q8×Dic83C211yesmaximal, characteristic, complemented
15C2×#520×C83C211yesmaximal, characteristic, complemented
16C22×Q8×D83C2211yescharacteristic
17#19128358C4116yescomplemented
18C2×#14581779C2213yes—
19C2×#14581771C2213yes—
20C2×#14581763C2213yes—
21#19128177—28no—
22C2×Q8×Dic83C2211yescharacteristic
23C2×Q8×Dic83C2211yescharacteristic
24C2×Q8×Dic83C2211yescharacteristic
25C2×C4×Dic166C2213yes—
26C2×#14581757C2213yes—
27C2×C4×C22⋊D83C2213yes—
28C2×#14581750C2213yes—
29C2×#14581749C2213yes—
30#19127607C2213yes—
31#19127606C2213yes—
32#19127604C2213yes—
33#19127604C2213yes—
34#19127601C2213yes—
35#19127601C2213yes—
36#19127599C2213yes—
37#19127597C2213yes—
38#19127596C2213yes—
39#19127596C2213yes—
40#19127595C2216yes—
41#19127595C2213yes—
42#19127595C2213yes—
43#19127591C2213yes—
44#19127591C2211yescharacteristic
45#19127589C2216yes—
46#19127582C2216yes—
47#19127580C2216yes—
48#19127574C2216yes—
49#19127566C2213yes—
50#19127561C2213yes—
51#19127558C2213yes—
52C4⋊C4×Dic83C2213yes—
53C4⋊C4×Dic83C2213yes—
54C4×Dic83⋊C4C2213yes—
55C4×C4⋊Dic83C2213yes—
56C4×(C2×D83)⋊C4C2213yes—
57C4×C22⋊Dic83C2216yes—
58C4×C22⋊Dic83C2213yes—
59C4×C22⋊Dic83C2216yes—
60C42×Dic83C2213yes—
61C22×#41×C83C2211yescharacteristic
62#520×C83—28no—
63C2×C4×Q8×C83C2211yescharacteristic
64C2×C4×Q8×C83C2211yescharacteristic
65C2×C4×D4×C83C2213yes—
66C2×C4×D4×C83C2213yes—
67C2×#117×C83C2213yes—
68C22×C4⋊C4×C83C2213yes—
69C22×Dic166C2313yes—
70C2×Q8×D83—24no—
71C2×Q8×D83—24no—
72C2×Q8×D83C2×C418yescomplemented
73C22×C22⋊D83C2313yes—
74C22×C4×D83C2313yes—
75#14581779C2×C4124yescomplemented
76#14581771C2×C4124yescomplemented
77#14581763C2×C4124yescomplemented
78Q8×Dic83—24no—
79Q8×Dic83—24no—
80Q8×Dic83C2×C418yescomplemented
81C4×Dic166—212no—
82#14581757—212no—
83C4×C22⋊D83C2×C4124yescomplemented
84#14581750—212no—
85#14581749—212no—
86C2×C4⋊Dic83C2313yes—
87C2×C4⋊Dic83C2313yes—
88C2×C4⋊Dic83C2313yes—
89C2×Dic83⋊C4C2313yes—
90C2×Dic83⋊C4C2313yes—
91C2×Dic83⋊C4C2313yes—
92C2×Dic83⋊C4C2313yes—
93C2×Dic83⋊C4C2313yes—
94C2×C22⋊Dic83C2313yes—
95C2×C22⋊Dic83C2316yes—
96C2×C22⋊Dic83C2313yes—
97C2×C22⋊Dic83C2316yes—
98C2×C22⋊Dic83C2313yes—
99C2×(C2×D83)⋊C4C2313yes—
100C2×(C2×D83)⋊C4C2311yescharacteristic
101C2×(C2×D83)⋊C4C2311yescharacteristic
102C2×(C2×D83)⋊C4C2313yes—
103C2×(C2×D83)⋊C4C2311yescharacteristic
104C2×C4×Dic83C2313yes—
105C2×C4×Dic83C2313yes—
106C2×C4×Dic83C2316yes—
107C2×C4×Dic83C2311yescharacteristic
108C2×C4×Dic83C2313yes—
109C2×C4×Dic83C2313yes—
110C2×C4×Dic83C2313yes—
111C2×C4×Dic83C2313yes—
112C83⋊#95C2316yes—
113C83⋊#95C2311yescharacteristic
114C83⋊#95C2313yes—
115C83⋊#95C2316yes—
116C83⋊#95C2313yes—
117C83⋊#95C2316yes—
118C83⋊#95C2313yes—
119C83⋊#95C2311yescharacteristic
120C83⋊#95C2313yes—
121C2×#41×C83—24no—
122C2×#41×C83—24no—
123C2×#41×C83C2×C418yes—
124C22×Q8×C83C2311yescharacteristic
125C22×D4×C83C2313yes—
126C23×C4×C83C2313yes—
127C23×C4×C83C2316yes—
128C4×Q8×C83—24no—
129C4×Q8×C83D418yescomplemented
130C4×D4×C83—212no—
131C4×D4×C83—212no—
132#117×C83—212no—
133C2×C4⋊C4×C83—212no—
134C2×C4⋊C4×C83C2313yes—
135C2×C4⋊C4×C83C2313yes—
136C2×C4⋊C4×C83C2311yescharacteristic
137C2×C4⋊C4×C83C2313yes—
138C2×C22⋊C4×C83C2313yes—
139C2×C22⋊C4×C83C2313yes—
140C2×C42×C83C2313yes—
141C2×C42×C83C2313yes—
142C23×D83C2411yescharacteristic
143C2×Dic166—26no—
144C2×Dic166—26no—
145C2×Dic166C22×C4112yes—
146Q8×D83—232no—
147C2×C22⋊D83—43no—
148C2×C22⋊D83—43no—
149C2×C22⋊D83C22×C4112yes—
150C22×Dic83C2413yes—
151C22×Dic83C2413yes—
152C22×Dic83C2411yescharacteristic
153C22×Dic83C2413yes—
154C22×Dic83C2413yes—
155C22×Dic83C2411yescharacteristic
156C22×Dic83C2411yescharacteristic
157C2×C4×D83—26no—
158C2×C4×D83—26no—
159C2×C4×D83C22×C4112yes—
160C4⋊Dic83—26no—
161C4⋊Dic83—26no—
162C4⋊Dic83C22×C4112yes—
163Dic83⋊C4—26no—
164Dic83⋊C4—26no—
165Dic83⋊C4—43no—
166Dic83⋊C4—43no—
167Dic83⋊C4C22×C4112yes—
168C22⋊Dic83—26no—
169C22⋊Dic83—43no—
170C22⋊Dic83—46no—
171C22⋊Dic83C22×C4124yes—
172C22⋊Dic83C22×C4112yes—
173(C2×D83)⋊C4—22no—
174(C2×D83)⋊C4—26no—
175(C2×D83)⋊C4—22no—
176(C2×D83)⋊C4—43no—
177(C2×D83)⋊C4C22×C414yes—
178C4×Dic83—26no—
179C4×Dic83—212no—
180C4×Dic83—26no—
181C4×Dic83—22no—
182C4×Dic83—43no—
183C4×Dic83—43no—
184C4×Dic83—43no—
185C4×Dic83C22×C4112yes—
186C24×C83C2413yes—
187#41×C83—232no—
188C2×Q8×C83C22×C414yes—
189C2×Q8×C83C2×D414yes—
190C2×Q8×C83#4114yes—
191C2×D4×C83—26no—
192C2×D4×C83—26no—
193C2×D4×C83C22×C4112yes—
194C22×C4×C83—26no—
195C22×C4×C83—26no—
196C22×C4×C83—212no—
197C22×C4×C83—46no—
198C22×C4×C83—46no—
199C22×C4×C83C22×C4112yes—
200C22×C4×C83C2411yescharacteristic
201C22×C4×C83C2413yes—
202C22×C4×C83C2411yescharacteristic
203C22×C4×C83C2411yescharacteristic
204C22×C4×C83C2413yes—
205C22×C4×C83C2413yes—
206C4⋊C4×C83—26no—
207C4⋊C4×C83—26no—
208C4⋊C4×C83—22no—
209C4⋊C4×C83C2×D4112yes—
210C22⋊C4×C83—26no—
211C22⋊C4×C83—26no—
212C42×C83—26no—
213C42×C83C2×D4112yes—
214C23×C83—23no—
215C23×C83—23no—
216C23×C83—23no—
217C23×C83—23no—
218C23×C83—23no—
219C23×C83—23no—
220C23×C83C23×C416yes—
221C23×C83C2511yescharacteristic
222C22×D83—21no—
223C22×D83—21no—
224C22×D83—21no—
225C22×D83—21no—
226C22×D83—21no—
227C22×D83—21no—
228C22×D83C23×C412yes—
229Q8×C83C4×D4116yes—
230D4×C83—224no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—23no—
235C2×C4×C83—23no—
236C2×C4×C83—21no—
237C2×C4×C83—224no—
238C2×C4×C83—21no—
239C2×C4×C83—23no—
240C2×C4×C83—21no—
241C2×C4×C83—23no—
242C2×C4×C83—23no—
243C2×C4×C83—21no—
244C2×C4×C83—812no—
245C2×C4×C83C23×C412yes—
246C2×C4×C83C23×C416yes—
247C2×C4×C83C22×D412yes—
248C2×C4×C83C22×D416yes—
249C2×C4×C83C22×D416yes—
250C2×C4×C83C2×#4116yes—
251C22⋊D83—412no—
252C2×Dic83—23no—
253C2×Dic83—21no—
254C2×Dic83—23no—
255C2×Dic83—21no—
256C2×Dic83—21no—
257C2×Dic83—21no—
258C2×Dic83—23no—
259C2×Dic83—23no—
260C2×Dic83—21no—
261C2×Dic83—23no—
262C2×Dic83—23no—
263C2×Dic83—23no—
264C2×Dic83—23no—
265C2×Dic83—23no—
266C2×Dic83—21no—
267C2×Dic83—23no—
268C2×Dic83—21no—
269C2×Dic83C23×C412yes—
270C2×Dic83C23×C416yes—
271C2×Dic83C23×C416yes—
272C2×Dic832−1+412yes—
273C4×D83—224no—
274Dic166—224no—
275C22×C83—26no—
276C22×C83—26no—
277C22×C83—26no—
278C22×C83—43no—
279C22×C83—43no—
280C22×C83—43no—
281C22×C83—43no—
282C22×C83C24×C411yescharacteristic
283C22×C83C23×D411yescharacteristic
284C22×C83C22×#4111yescharacteristic
285C22×C83C2×2−1+411yescharacteristic
286C22×C83#58511yescharacteristic
287C22×C83#58511yescharacteristic
288C22×C83#58511yescharacteristic
289C2×D83—22no—
290C2×D83—22no—
291C2×D83—22no—
292C2×D83—41no—
293C2×D83—41no—
294C2×D83—41no—
295C2×D83—41no—
296C4×C83—22no—
297C4×C83—22no—
298C4×C83—26no—
299C4×C83—41no—
300C4×C83—43no—
301C4×C83C2×C4×D4112yes—
302Dic83—26no—
303Dic83—26no—
304Dic83—22no—
305Dic83—22no—
306Dic83—43no—
307Dic83—41no—
308Dic83—43no—
309#34312—831nomaximal, Sylow
310C2×C83—46no—
311C2×C83C22×C4×D411yescharacteristic
312C2×C83#342811yescharacteristic
313C2×C83C4×2−1+411yescharacteristic
314C2×C83#349711yescharacteristic
315C2×C83#350011yescharacteristic
316C2×C83#351611yescharacteristic
317C2×C83#352411yescharacteristic
318D83—42no—
319C2×#548—831no—
320C2×#520D8311yescharacteristic, complemented
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×Q8—831no—
337#548—8316no—
338C2×#124—833no—
339C2×C22⋊Q8—833no—
340C2×#121—833no—
341#520—28no—
342#519—1668no—
343C2×C4×Q8—831no—
344C2×C4×Q8—831no—
345C2×C4×Q8—831no—
346C2×C4×Q8—833no—
347C2×C4×Q8—833no—
348C2×C4×Q8C2×D8311yescharacteristic
349C2×C4×Q8C2×D8311yescharacteristic
350C2×C4×D4—833no—
351C2×C4×D4C2×D8313yes—
352C2×C4×D4C2×D8313yes—
353C2×#117—833no—
354C2×#117—833no—
355C2×#117C2×D8313yes—
356C22×C4⋊C4C2×D8313yes—
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—833no—
368#388—831no—
369#388—836no—
370#388—836no—
371#388—833no—
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—836no—
385C4×C22⋊C4—836no—
386C4×C22⋊C4—833no—
387C43—833no—
388C2×#41—24no—
389C2×#41—24no—
390C2×#41C4×D8318yes—
391C22×Q8—838no—
392C22×Q8—833no—
393C22×Q8—1664no—
394C22×Q8—1664no—
395C22×Q8C22×D8311yescharacteristic
396C22×D4—833no—
397C22×D4C22×D8313yes—
398C23×C4—833no—
399C23×C4C22×D8313yes—
400C23×C4C22×D8316yes—
401#124—8324no—
402C22⋊Q8—8324no—
403#121—8324no—
404C4×Q8—24no—
405C4×Q8—838no—
406C4×Q8—1664no—
407C4×Q8—1664no—
408C4×Q8—16612no—
409C4×Q8—16612no—
410C4×Q8C22⋊D8318yescomplemented
411C4×D4—212no—
412C4×D4—212no—
413C4×D4—8324no—
414#117—212no—
415#117—16612no—
416#117—16612no—
417C2×C4⋊C4—212no—
418C2×C4⋊C4—833no—
419C2×C4⋊C4—833no—
420C2×C4⋊C4—833no—
421C2×C4⋊C4—833no—
422C2×C4⋊C4—833no—
423C2×C4⋊C4—833no—
424C2×C4⋊C4—833no—
425C2×C4⋊C4—833no—
426C2×C4⋊C4C22×D8313yes—
427C2×C4⋊C4C22×D8313yes—
428C2×C4⋊C4C22×D8311yescharacteristic
429C2×C4⋊C4C22×D8313yes—
430C2×C22⋊C4—836no—
431C2×C22⋊C4—833no—
432C2×C22⋊C4—833no—
433C2×C22⋊C4—833no—
434C2×C22⋊C4—836no—
435C2×C22⋊C4—831no—
436C2×C22⋊C4—833no—
437C2×C22⋊C4—833no—
438C2×C22⋊C4—831no—
439C2×C22⋊C4—831no—
440C2×C22⋊C4C22×D8313yes—
441C2×C22⋊C4C22×D8313yes—
442C2×C42—833no—
443C2×C42—833no—
444C2×C42—833no—
445C2×C42—833no—
446C2×C42—836no—
447C2×C42—833no—
448C2×C42—833no—
449C2×C42—831no—
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—836no—
460#95—831no—
461C24—831no—
462C24C23×D8313yes—
463#41—232no—
464C2×Q8—8312no—
465C2×Q8—16632no—
466C2×Q8—1666no—
467C2×Q8—1666no—
468C2×Q8C2×C4×D8314yes—
469C2×Q8C2×C22⋊D8314yes—
470C2×Q8#1305059014yes—
471C2×D4—26no—
472C2×D4—26no—
473C2×D4—3323no—
474C2×D4—3323no—
475C2×D4—8312no—
476C2×D4C2×C4×D83112yes—
477C22×C4—26no—
478C22×C4—26no—
479C22×C4—212no—
480C22×C4—46no—
481C22×C4—46no—
482C22×C4—831no—
483C22×C4—831no—
484C22×C4—833no—
485C22×C4—833no—
486C22×C4—8312no—
487C22×C4—833no—
488C22×C4—833no—
489C22×C4—831no—
490C22×C4—1666no—
491C22×C4—1666no—
492C22×C4C2×C4×D83112yes—
493C22×C4C23×D8313yes—
494C22×C4C23×D8311yescharacteristic
495C22×C4C23×D8311yescharacteristic
496C22×C4C23×D8313yes—
497C22×C4C23×D8311yescharacteristic
498C22×C4C23×D8313yes—
499C4⋊C4—26no—
500C4⋊C4—26no—
501C4⋊C4—22no—
502C4⋊C4—3323no—
503C4⋊C4—3323no—
504C4⋊C4—8312no—
505C4⋊C4—8312no—
506C4⋊C4—1666no—
507C4⋊C4—1666no—
508C4⋊C4—1666no—
509C4⋊C4—1666no—
510C4⋊C4C2×C22⋊D83112yes—
511C22⋊C4—26no—
512C22⋊C4—26no—
513C22⋊C4—3323no—
514C22⋊C4—3323no—
515C22⋊C4—3326no—
516C22⋊C4—8312no—
517C22⋊C4—8324no—
518C22⋊C4—834no—
519C22⋊C4—1662no—
520C22⋊C4—1666no—
521C22⋊C4—1666no—
522C22⋊C4—1662no—
523C42—26no—
524C42—3323no—
525C42—3323no—
526C42—3323no—
527C42—8312no—
528C42—1662no—
529C42—1666no—
530C42—16612no—
531C42—1666no—
532C42C2×C22⋊D83112yes—
533C23—23no—
534C23—23no—
535C23—23no—
536C23—23no—
537C23—23no—
538C23—23no—
539C23—832no—
540C23—1661no—
541C23—1661no—
542C23—1661no—
543C23—1661no—
544C23—1661no—
545C23—1661no—
546C23C22×C4×D8316yes—
547C23C24×D8311yescharacteristic
548Q8—16624no—
549Q8C4×C22⋊D83116yes—
550D4—224no—
551D4—33212no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—23no—
556C2×C4—23no—
557C2×C4—21no—
558C2×C4—224no—
559C2×C4—21no—
560C2×C4—23no—
561C2×C4—21no—
562C2×C4—23no—
563C2×C4—23no—
564C2×C4—21no—
565C2×C4—812no—
566C2×C4—832no—
567C2×C4—836no—
568C2×C4—836no—
569C2×C4—832no—
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—1663no—
580C2×C4—1661no—
581C2×C4—1663no—
582C2×C4—1661no—
583C2×C4—1661no—
584C2×C4—1661no—
585C2×C4—1663no—
586C2×C4—1663no—
587C2×C4—16624no—
588C2×C4C22×C4×D8312yes—
589C2×C4C22×C4×D8316yes—
590C2×C4C22×C22⋊D8316yes—
591C2×C4C22×C22⋊D8312yes—
592C2×C4C22×C22⋊D8316yes—
593C2×C4C2×#1305059016yes—
594C22—26no—
595C22—26no—
596C22—26no—
597C22—43no—
598C22—43no—
599C22—43no—
600C22—43no—
601C22—3321no—
602C22—3321no—
603C22—3321no—
604C22—3321no—
605C22—1662no—
606C22—1662no—
607C22—1662no—
608C22C23×C4×D8311yescharacteristic
609C22C23×C22⋊D8311yescharacteristic
610C22C22×#1305059011yescharacteristic
611C222−1+4×D8311yescharacteristic
612C22#1912853611yescharacteristic
613C22#1912853611yescharacteristic
614C22#1912853811yescharacteristic
615C4—22no—
616C4—22no—
617C4—26no—
618C4—41no—
619C4—43no—
620C4—3323no—
621C4—3321no—
622C4—3323no—
623C4—1666no—
624C4—1666no—
625C4—1662no—
626C4—1662no—
627C4C2×C4×C22⋊D83112yes—
628C2—46no—
629C2—3322no—
630C2C22×C4×C22⋊D8311yescharacteristic
631C2#2492066311yescharacteristic
632C2#2492067411yescharacteristic
633C2#2492123011yescharacteristic
634C2#2492125111yescharacteristic
635C2#2492144811yescharacteristic
636C2#2492156411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC2×#520×C83C2
socle, cosocleC23×C83C25
prime radicalC2×#19128358C2
hypercenterC2×#520D83
hypocenterC83#34312
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. C2×#19128177C2semidirect
3. #24910536C2semidirect
4. #24910497C2semidirect
5. #24910468C2semidirect
6. #24910388C2semidirect
7. #24910381C2semidirect
8. #24910304C2semidirect
9. Q8×(C2×D83)⋊C4C2semidirect
10. #24910150C2semidirect
11. #24910145C2semidirect
12. C4×#14581779C2semidirect
13. #24909922C2semidirect
14. C4×Q8×Dic83C2semidirect
15. C2×#520×C83C2semidirect
17. #19128358C4semidirect
72. C2×Q8×D83C2×C4semidirect
75. #14581779C2×C4semidirect
76. #14581771C2×C4semidirect
77. #14581763C2×C4semidirect
80. Q8×Dic83C2×C4semidirect
83. C4×C22⋊D83C2×C4semidirect
129. C4×Q8×C83D4semidirect
320. C2×#520D83semidirect
334. C83#34312semidirect
410. C4×Q8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191283582
C2×#191281772
#249105362
#249104972
#249104682
#249103882
#249103812
#249103042
Q8×(C2×D83)⋊C42
#249101502
#249101452
C4×#145817792
#249099222
C4×Q8×Dic832
C2×#520×C832
#3431283