Finite Group Databasebeta

← #35565352

Subgroups — #35565352

Order 21248.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128183C211yesmaximal, characteristic
2C2×#19128183C211yesmaximal, characteristic
3C2×C4×#13050591C211yesmaximal, characteristic
4#24910384C211yesmaximal, characteristic, complemented
5#24910381C211yesmaximal, characteristic, complemented
6Q8×Dic83⋊C4C211yesmaximal, characteristic, complemented
7#24910374C213yesmaximal, complemented
8#24910372C213yesmaximal, complemented
9#24910364C213yesmaximal, complemented
10#24910363C213yesmaximal, complemented
11#24910084C213yesmaximal, complemented
12#24910078C213yesmaximal, complemented
13#24909943C213yesmaximal, complemented
14#389×D83C213yesmaximal, complemented
15Q8×C4⋊C4×C83C211yesmaximal, characteristic, complemented
16C22×#13050591C2211yescharacteristic
17#19128183—28no—
18#19128183C4116yescomplemented
19C4×#13050591—28no—
20C2×Q8×Dic83C2211yescharacteristic
21C2×Q8×Dic83C2211yescharacteristic
22C2×Q8×Dic83C2211yescharacteristic
23C2×C4×D332C2213yes—
24C2×C4×D332C2213yes—
25C2×C4×D332C2213yes—
26C2×#14581754C2213yes—
27C2×#14581754C2213yes—
28C2×#14581754C2213yes—
29C2×#14581750C2213yes—
30C2×#14581749C2213yes—
31C2×#14581749C2213yes—
32C2×C4⋊C4×D83C2213yes—
33C2×C4⋊C4×D83C2216yes—
34C2×C4⋊C4×D83C2213yes—
35C2×C42×D83C2213yes—
36#19127600C2213yes—
37#19127600C2213yes—
38#19127599C2213yes—
39#19127599C2213yes—
40#19127598C2213yes—
41#19127598C2213yes—
42#19127597C2213yes—
43#19127597C2213yes—
44#19127596C2213yes—
45#19127596C2216yes—
46#19127596C2211yescharacteristic
47#19127575C2213yes—
48#19127575C2213yes—
49#19127572C2216yes—
50#19127572C2216yes—
51#19127572C2213yes—
52#19127572C2213yes—
53#19127570C2213yes—
54#19127567C2213yes—
55#19127566C2213yes—
56#19127560C2216yes—
57#19127558C2216yes—
58C4×Dic83⋊C4C2213yes—
59C4×Dic83⋊C4C2213yes—
60C4×C4⋊Dic83C2213yes—
61#95×D83C2216yes—
62C2×C4×Q8×C83C2211yescharacteristic
63C2×C4×Q8×C83C2211yescharacteristic
64C2×C4×Q8×C83C2211yescharacteristic
65#389×C83C2213yes—
66#389×C83C2213yes—
67#384×C83C2213yes—
68C4×C4⋊C4×C83C2213yes—
69C2×#13050591C2×C418yes—
70C2×#13050591D418yes—
71C2×#13050591Q818yes—
72C22×D332C2313yes—
73C22×C4×D83C2313yes—
74C22×C4×D83C2313yes—
75C22×C4×D83C2316yes—
76C22×C4×D83C2316yes—
77C22×C4×D83C2313yes—
78Q8×Dic83—24no—
79Q8×Dic83—24no—
80Q8×Dic83C2×C418yescomplemented
81C4×D332—212no—
82C4×D332—212no—
83C4×D332C2×C4124yescomplemented
84#14581754—212no—
85#14581754—212no—
86#14581754C2×C4124yescomplemented
87#14581750—212no—
88#14581749—212no—
89#14581749C2×C4124yescomplemented
90C2×C4⋊Dic83C2313yes—
91C2×C4⋊Dic83C2313yes—
92C2×C4⋊Dic83C2313yes—
93C2×C4⋊Dic83C2313yes—
94C2×C4⋊Dic83C2313yes—
95C2×Dic83⋊C4C2311yescharacteristic
96C2×Dic83⋊C4C2313yes—
97C2×Dic83⋊C4C2313yes—
98C2×Dic83⋊C4C2311yescharacteristic
99C2×Dic83⋊C4C2313yes—
100C2×Dic83⋊C4C2313yes—
101C2×Dic83⋊C4C2316yes—
102C2×Dic83⋊C4C2311yescharacteristic
103C2×Dic83⋊C4C2313yes—
104C4⋊C4×D83—212no—
105C4⋊C4×D83—412no—
106C4⋊C4×D83C2×C4124yescomplemented
107C2×(C2×D83)⋊C4C2313yes—
108C2×(C2×D83)⋊C4C2316yes—
109C2×(C2×D83)⋊C4C2313yes—
110C2×(C2×D83)⋊C4C2313yes—
111C2×(C2×D83)⋊C4C2313yes—
112C2×C4×Dic83C2313yes—
113C2×C4×Dic83C2313yes—
114C2×C4×Dic83C2313yes—
115C2×C4×Dic83C2311yescharacteristic
116C2×C4×Dic83C2313yes—
117C42×D83—212no—
118C83⋊#95C2311yescharacteristic
119C83⋊#95C2313yes—
120C83⋊#95C2311yescharacteristic
121C83⋊#95C2313yes—
122C83⋊#95C2313yes—
123C83⋊#95C2313yes—
124C83⋊#95C2316yes—
125C83⋊#95C2316yes—
126C22×Q8×C83C2311yescharacteristic
127C4×Q8×C83—24no—
128C4×Q8×C83—24no—
129C4×Q8×C83C2×C418yescomplemented
130C2×C4⋊C4×C83C2313yes—
131C2×C4⋊C4×C83C2313yes—
132C2×C4⋊C4×C83C2311yescharacteristic
133C2×C4⋊C4×C83C2316yes—
134C2×C4⋊C4×C83C2313yes—
135C2×C4⋊C4×C83C2313yes—
136C2×C42×C83C2313yes—
137C2×C42×C83C2313yes—
138C2×C42×C83C2313yes—
139#95×C83C2313yes—
140#95×C83C2313yes—
141C23×D83C2413yes—
142#13050591C4⋊C4164yescomplemented
143C2×D332C22×C4112yes—
144C2×D332C2×D4112yes—
145C2×D332C2×Q8112yes—
146C22×Dic83C2413yes—
147C22×Dic83C2413yes—
148C22×Dic83C2411yescharacteristic
149C22×Dic83C2413yes—
150C22×Dic83C2411yescharacteristic
151C22×Dic83C2411yescharacteristic
152C22×Dic83C2411yescharacteristic
153C2×C4×D83—26no—
154C2×C4×D83—26no—
155C2×C4×D83—26no—
156C2×C4×D83—26no—
157C2×C4×D83—212no—
158C2×C4×D83—46no—
159C2×C4×D83—43no—
160C2×C4×D83—43no—
161C2×C4×D83—46no—
162C2×C4×D83—46no—
163C2×C4×D83—46no—
164C2×C4×D83C22×C4124yes—
165C2×C4×D83C22×C4112yes—
166C2×C4×D83C2×D4112yes—
167C2×C4×D83C2×Q8112yes—
168C4⋊Dic83—26no—
169C4⋊Dic83—26no—
170C4⋊Dic83—26no—
171C4⋊Dic83—43no—
172C4⋊Dic83C22×C4112yes—
173Dic83⋊C4—22no—
174Dic83⋊C4—26no—
175Dic83⋊C4—22no—
176Dic83⋊C4—26no—
177Dic83⋊C4—46no—
178Dic83⋊C4—43no—
179Dic83⋊C4—43no—
180Dic83⋊C4C22×C4112yes—
181Dic83⋊C4C22×C414yes—
182(C2×D83)⋊C4—26no—
183(C2×D83)⋊C4—26no—
184(C2×D83)⋊C4—212no—
185(C2×D83)⋊C4C22×C4112yes—
186(C2×D83)⋊C4C22×C4112yes—
187C4×Dic83—26no—
188C4×Dic83—26no—
189C4×Dic83—22no—
190C4×Dic83—26no—
191C4×Dic83C22×C4112yes—
192C2×Q8×C83C22×C414yes—
193C2×Q8×C83C2×D414yes—
194C2×Q8×C83C2×Q814yes—
195C22×C4×C83C2413yes—
196C22×C4×C83C2413yes—
197C22×C4×C83C2411yescharacteristic
198C22×C4×C83C2413yes—
199C22×C4×C83C2411yescharacteristic
200C22×C4×C83C2411yescharacteristic
201C22×C4×C83C2413yes—
202C4⋊C4×C83—26no—
203C4⋊C4×C83—26no—
204C4⋊C4×C83—22no—
205C4⋊C4×C83—43no—
206C4⋊C4×C83—46no—
207C4⋊C4×C83C22×C4112yes—
208C42×C83—26no—
209C42×C83—26no—
210C42×C83C22×C4112yes—
211C23×C83C2511yescharacteristic
212C22×D83—23no—
213C22×D83—23no—
214C22×D83—23no—
215C22×D83—23no—
216C22×D83C23×C416yes—
217C22×D83C22×D416yes—
218C22×D83C22×Q816yes—
219Q8×C83C2×C4⋊C4116yescomplemented
220C2×C4×C83—21no—
221C2×C4×C83—23no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—23no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—23no—
228C2×C4×C83—23no—
229C2×C4×C83—23no—
230C2×C4×C83—23no—
231C2×C4×C83—21no—
232C2×C4×C83—23no—
233C2×C4×C83—23no—
234C2×C4×C83—21no—
235C2×C4×C83C23×C412yes—
236C2×C4×C83C23×C416yes—
237C2×C4×C83C23×C416yes—
238C2×C4×C83C22×D416yes—
239C2×C4×C83C22×Q816yes—
240C2×C4×C832−1+412yes—
241C2×Dic83—21no—
242C2×Dic83—23no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—23no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—23no—
249C2×Dic83—23no—
250C2×Dic83—23no—
251C2×Dic83—21no—
252C2×Dic83—23no—
253C2×Dic83—23no—
254C2×Dic83—23no—
255C2×Dic83—21no—
256C2×Dic83C23×C412yes—
257C2×Dic83C23×C416yes—
258C2×Dic83C23×C412yes—
259C2×Dic83C22×D412yes—
260C2×Dic83C22×Q812yes—
261C2×Dic83H(5,𝔽2)12yes—
262D332C2×C4⋊C4148yescomplemented
263C4×D83—424no—
264C4×D83—412no—
265C4×D83—412no—
266C4×D83—812no—
267C4×D83C2×C4⋊C4148yescomplemented
268C22×C83C24×C411yescharacteristic
269C22×C83C23×D411yescharacteristic
270C22×C83C23×Q811yescharacteristic
271C22×C83C2×H(5,𝔽2)11yescharacteristic
272C22×C83C2×2−1+411yescharacteristic
273C22×C83#58511yescharacteristic
274C22×C83#58511yescharacteristic
275C2×D83—26no—
276C2×D83—26no—
277C2×D83—26no—
278C2×D83—26no—
279C2×D83—26no—
280C2×D83—26no—
281C2×D83C22×C4⋊C4112yes—
282C4×C83—26no—
283C4×C83—22no—
284C4×C83—22no—
285C4×C83—43no—
286C4×C83—43no—
287C4×C83—41no—
288C4×C83C22×C4⋊C4112yes—
289Dic83—26no—
290Dic83—22no—
291Dic83—22no—
292Dic83—43no—
293Dic83—43no—
294Dic83—41no—
295Dic83C22×C4⋊C414yes—
296#34250—831nomaximal, Sylow
297C2×C83C23×C4⋊C411yescharacteristic
298C2×C83C4×H(5,𝔽2)11yescharacteristic
299C2×C83C4×2−1+411yescharacteristic
300C2×C83#348811yescharacteristic
301C2×C83#348911yescharacteristic
302C2×C83#350311yescharacteristic
303C2×C83#350411yescharacteristic
304D83—212no—
305C2×#520—831no—
306C2×#520—831no—
307C2×C4×#41—831no—
308#2351—831no—
309#2351—831no—
310Q8×C4⋊C4—831no—
311Q8×C4⋊C4D8311yescharacteristic, complemented
312#2349—833no—
313#2349—833no—
314D4×C4⋊C4—833no—
315D4×C4⋊C4—833no—
316#2317—833no—
317#2316—833no—
318#2296—833no—
319C2×#389—833no—
320C83#3425011yescharacteristic, complemented, Sylow
321C22×#41—831no—
322#520—8316no—
323#520—1668no—
324C4×#41—1668no—
325C2×C4×Q8—831no—
326C2×C4×Q8—831no—
327C2×C4×Q8—831no—
328C2×C4×Q8C2×D8311yescharacteristic
329C2×C4×Q8C2×D8311yescharacteristic
330C2×C4×Q8C2×D8311yescharacteristic
331C2×C4×D4—833no—
332C2×C4×D4—833no—
333C2×C4×D4—833no—
334C2×C4×D4—833no—
335C2×C4×D4—833no—
336C2×C4×D4—833no—
337C2×#117—833no—
338C2×#117—833no—
339C2×#117—833no—
340C22×C4⋊C4—836no—
341C22×C4⋊C4—833no—
342C22×C4⋊C4—833no—
343C22×C42—833no—
344#389—836no—
345#389—833no—
346#389—833no—
347#389—833no—
348#389—833no—
349#389—833no—
350#389—831no—
351#389—833no—
352#389—833no—
353#389—833no—
354#389—833no—
355#389C2×D8313yes—
356#389C2×D8313yes—
357#385—836no—
358#385—836no—
359#385—833no—
360#385—833no—
361#385—833no—
362#385—833no—
363#384—833no—
364#384C2×D8313yes—
365#382—833no—
366#382—833no—
367#380—836no—
368#380—836no—
369C4×C4⋊C4—833no—
370C4×C4⋊C4—833no—
371C4×C4⋊C4—833no—
372C4×C4⋊C4C2×D8313yes—
373C2×#95—836no—
374C2×#41—838no—
375C2×#41—838no—
376C2×#41—838no—
377C22×Q8C22×D8311yescharacteristic
378C22×D4—833no—
379C23×C4—836no—
380C23×C4—836no—
381C23×C4—833no—
382C23×C4—833no—
383C23×C4—833no—
384C4×Q8—24no—
385C4×Q8—24no—
386C4×Q8—838no—
387C4×Q8—1664no—
388C4×Q8—1664no—
389C4×Q8C4×D8318yescomplemented
390C4×D4—8324no—
391C4×D4—8324no—
392C4×D4—16612no—
393C4×D4—16612no—
394C4×D4—16612no—
395C4×D4—16612no—
396#117—8324no—
397#117—16612no—
398#117—16612no—
399C2×C4⋊C4—33212no—
400C2×C4⋊C4—836no—
401C2×C4⋊C4—833no—
402C2×C4⋊C4—833no—
403C2×C4⋊C4—833no—
404C2×C4⋊C4—833no—
405C2×C4⋊C4—833no—
406C2×C4⋊C4—831no—
407C2×C4⋊C4—8324no—
408C2×C4⋊C4—833no—
409C2×C4⋊C4—833no—
410C2×C4⋊C4—833no—
411C2×C4⋊C4—833no—
412C2×C4⋊C4—831no—
413C2×C4⋊C4—833no—
414C2×C4⋊C4—831no—
415C2×C4⋊C4—16612no—
416C2×C4⋊C4C22×D8313yes—
417C2×C4⋊C4C22×D8313yes—
418C2×C4⋊C4C22×D8311yescharacteristic
419C2×C4⋊C4C22×D8316yes—
420C2×C4⋊C4C22×D8313yes—
421C2×C4⋊C4C22×D8313yes—
422C2×C22⋊C4—833no—
423C2×C22⋊C4—833no—
424C2×C22⋊C4—836no—
425C2×C22⋊C4—833no—
426C2×C22⋊C4—833no—
427C2×C42—833no—
428C2×C42—833no—
429C2×C42—833no—
430C2×C42—833no—
431C2×C42—831no—
432C2×C42—16612no—
433C2×C42C22×D8313yes—
434C2×C42C22×D8313yes—
435C2×C42C22×D8313yes—
436#95—833no—
437#95—836no—
438#95—836no—
439#95—831no—
440#95—831no—
441#95—833no—
442#95—833no—
443#95—833no—
444#95C22×D8313yes—
445#95C22×D8313yes—
446C24—833no—
447#41—8364no—
448C2×Q8C2×C4×D8314yes—
449C2×Q8D4×D8314yes—
450C2×Q8Q8×D8314yes—
451C2×D4—8312no—
452C2×D4—8312no—
453C2×D4—8312no—
454C22×C4—3326no—
455C22×C4—3326no—
456C22×C4—3323no—
457C22×C4—3323no—
458C22×C4—3326no—
459C22×C4—3326no—
460C22×C4—8324no—
461C22×C4—831no—
462C22×C4—8312no—
463C22×C4—8312no—
464C22×C4—8312no—
465C22×C4—831no—
466C22×C4—833no—
467C22×C4—831no—
468C22×C4—833no—
469C22×C4—831no—
470C22×C4—833no—
471C22×C4—1666no—
472C22×C4—1666no—
473C22×C4—1666no—
474C22×C4—16612no—
475C22×C4—1666no—
476C22×C4C23×D8313yes—
477C22×C4C23×D8311yescharacteristic
478C22×C4C23×D8313yes—
479C22×C4C23×D8311yescharacteristic
480C22×C4C23×D8313yes—
481C22×C4C23×D8311yescharacteristic
482C22×C4C23×D8313yes—
483C4⋊C4—26no—
484C4⋊C4—26no—
485C4⋊C4—22no—
486C4⋊C4—46no—
487C4⋊C4—43no—
488C4⋊C4—3326no—
489C4⋊C4—3323no—
490C4⋊C4—3323no—
491C4⋊C4—3323no—
492C4⋊C4—8312no—
493C4⋊C4—8312no—
494C4⋊C4—834no—
495C4⋊C4—1666no—
496C4⋊C4—1666no—
497C4⋊C4—1666no—
498C4⋊C4—1662no—
499C4⋊C4—1666no—
500C4⋊C4—1666no—
501C4⋊C4—1662no—
502C4⋊C4C2×C4×D83112yes—
503C22⋊C4—8312no—
504C22⋊C4—8312no—
505C22⋊C4—1666no—
506C22⋊C4—16612no—
507C22⋊C4—1666no—
508C42—26no—
509C42—26no—
510C42—8312no—
511C42—1662no—
512C42—1666no—
513C42—1666no—
514C42—1666no—
515C42C2×C4×D83112yes—
516C23—836no—
517C23—836no—
518C23—836no—
519C23—1663no—
520C23—1663no—
521C23—1663no—
522C23—1663no—
523C23C24×D8311yescharacteristic
524Q8C4⋊C4×D83116yescomplemented
525D4—8348no—
526C2×C4—21no—
527C2×C4—23no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—23no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—23no—
534C2×C4—23no—
535C2×C4—23no—
536C2×C4—23no—
537C2×C4—21no—
538C2×C4—23no—
539C2×C4—23no—
540C2×C4—21no—
541C2×C4—33212no—
542C2×C4—33224no—
543C2×C4—33212no—
544C2×C4—832no—
545C2×C4—832no—
546C2×C4—832no—
547C2×C4—8348no—
548C2×C4—832no—
549C2×C4—836no—
550C2×C4—832no—
551C2×C4—66412no—
552C2×C4—1661no—
553C2×C4—1663no—
554C2×C4—1663no—
555C2×C4—1663no—
556C2×C4—1661no—
557C2×C4—1663no—
558C2×C4—1663no—
559C2×C4—1661no—
560C2×C4—1663no—
561C2×C4—1661no—
562C2×C4—1663no—
563C2×C4—1661no—
564C2×C4—1661no—
565C2×C4—1661no—
566C2×C4—1663no—
567C2×C4C22×C4×D8312yes—
568C2×C4C22×C4×D8316yes—
569C2×C4C22×C4×D8316yes—
570C2×C4C2×D4×D8316yes—
571C2×C4C2×Q8×D8316yes—
572C2×C4#1458184012yes—
573C22—8312no—
574C22—1666no—
575C22—1666no—
576C22—1666no—
577C22—1666no—
578C22—1666no—
579C22—1666no—
580C22C23×C4×D8311yescharacteristic
581C22C22×D4×D8311yescharacteristic
582C22C22×Q8×D8311yescharacteristic
583C22H(5,𝔽2)×D8311yescharacteristic
584C22C2×#1458184011yescharacteristic
585C22#1912853711yescharacteristic
586C22#1912853811yescharacteristic
587C4—22no—
588C4—26no—
589C4—22no—
590C4—43no—
591C4—43no—
592C4—41no—
593C4—3323no—
594C4—3323no—
595C4—3321no—
596C4—834no—
597C4—1662no—
598C4—1666no—
599C4—1662no—
600C4C2×C4⋊C4×D83112yes—
601C2—16612no—
602C2C22×C4⋊C4×D8311yescharacteristic
603C2#2492066911yescharacteristic
604C2C4×#1458184011yescharacteristic
605C2#2492113111yescharacteristic
606C2#2492114011yescharacteristic
607C2#2492128111yescharacteristic
608C2#2492128811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingQ8×C4⋊C4×C83C2
socle, cosocleC23×C83C25
prime radicalC22×#13050591C22
hypercenterQ8×C4⋊C4D83
hypocenterC83#34250
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #24910384C2semidirect
5. #24910381C2semidirect
6. Q8×Dic83⋊C4C2semidirect
7. #24910374C2semidirect
8. #24910372C2semidirect
9. #24910364C2semidirect
10. #24910363C2semidirect
11. #24910084C2semidirect
12. #24910078C2semidirect
13. #24909943C2semidirect
14. #389×D83C2semidirect
15. Q8×C4⋊C4×C83C2semidirect
18. #19128183C4semidirect
80. Q8×Dic83C2×C4semidirect
83. C4×D332C2×C4semidirect
86. #14581754C2×C4semidirect
89. #14581749C2×C4semidirect
106. C4⋊C4×D83C2×C4semidirect
129. C4×Q8×C83C2×C4semidirect
142. #13050591C4⋊C4semidirect
219. Q8×C83C2×C4⋊C4semidirect
262. D332C2×C4⋊C4semidirect
267. C4×D83C2×C4⋊C4semidirect
311. Q8×C4⋊C4D83semidirect
320. C83#34250semidirect
389. C4×Q8C4×D83semidirect
524. Q8C4⋊C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281832
C2×C4×#130505912
#249103842
#249103812
Q8×Dic83⋊C42
#249103742
#249103722
#249103642
#249103632
#249100842
#249100782
#249099432
#389×D832
Q8×C4⋊C4×C832
#3425083