Finite Group Databasebeta

← #35565341

Subgroups — #35565341

Order 21248.

2858 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128177C211yesmaximal, characteristic
2C2×#19128177C211yesmaximal, characteristic
3C2×C4×Q8×D83C211yesmaximal, characteristic
4#24910384C211yesmaximal, characteristic, complemented
5#24910383C213yesmaximal, complemented
6#24910382C213yesmaximal, complemented
7#24910381C211yesmaximal, characteristic, complemented
8#24910377C213yesmaximal, complemented
9Q8×Dic83⋊C4C211yesmaximal, characteristic, complemented
10#24910375C213yesmaximal, complemented
11#24910085C213yesmaximal
12#24910084C213yesmaximal
13#24909944C213yesmaximal
14#24909943C213yesmaximal
15Q8×C4⋊C4×C83C211yesmaximal, characteristic, complemented
16C22×Q8×D83C2211yescharacteristic
17#19128177—28no—
18#19128177C4116yescomplemented
19C2×Q8×Dic83C2211yescharacteristic
20C2×Q8×Dic83C2211yescharacteristic
21C2×Q8×Dic83C2211yescharacteristic
22C2×C4×Dic166C2213yes—
23C2×C4×Dic166C2213yes—
24C2×C4×Dic166C2213yes—
25C2×#14581757C2213yes—
26C2×#14581757C2213yes—
27C2×#14581757C2213yes—
28C4×Q8×D83—28no—
29C2×#14581750C2213yes—
30C2×#14581750C2213yes—
31C2×#14581749C2213yes—
32C2×#14581749C2213yes—
33C2×C4⋊C4×D83C2213yes—
34C2×C42×D83C2213yes—
35#19127600C2213yes—
36#19127600C2213yes—
37#19127599C2213yes—
38#19127599C2213yes—
39#19127598C2213yes—
40#19127598C2213yes—
41#19127597C2213yes—
42#19127597C2213yes—
43#19127596C2216yes—
44#19127596C2216yes—
45#19127596C2213yes—
46#19127596C2213yes—
47#19127575C2213yes—
48#19127575C2213yes—
49#19127572C2213yes—
50#19127572C2216yes—
51#19127572C2211yescharacteristic
52#19127570C2213yes—
53#19127570C2216yes—
54#19127567C2213yes—
55#19127566C2213yes—
56#19127565C2216yes—
57C4⋊C4×Dic83C2216yes—
58C4×Dic83⋊C4C2213yes—
59C4×Dic83⋊C4C2213yes—
60C4×C4⋊Dic83C2213yes—
61#19127552C2216yes—
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—
69C22×Dic166C2313yes—
70C2×Q8×D83C2×C418yes—
71C2×Q8×D83D418yes—
72C2×Q8×D83Q818yes—
73C22×C4×D83C2311yescharacteristic
74C22×C4×D83C2313yes—
75C22×C4×D83C2313yes—
76Q8×Dic83—24no—
77Q8×Dic83—24no—
78Q8×Dic83C2×C418yescomplemented
79C4×Dic166—212no—
80C4×Dic166—212no—
81C4×Dic166C2×C4124yescomplemented
82#14581757—212no—
83#14581757—212no—
84#14581757C2×C4124yescomplemented
85#14581750—212no—
86#14581750C2×C4124yes—
87#14581749—212no—
88#14581749C2×C4124yes—
89C2×C4⋊Dic83C2313yes—
90C2×C4⋊Dic83C2313yes—
91C2×C4⋊Dic83C2313yes—
92C2×C4⋊Dic83C2313yes—
93C2×C4⋊Dic83C2313yes—
94C2×Dic83⋊C4C2313yes—
95C2×Dic83⋊C4C2311yescharacteristic
96C2×Dic83⋊C4C2313yes—
97C2×Dic83⋊C4C2316yes—
98C2×Dic83⋊C4C2313yes—
99C2×Dic83⋊C4C2316yes—
100C2×Dic83⋊C4C2313yes—
101C2×Dic83⋊C4C2313yes—
102C2×Dic83⋊C4C2313yes—
103C2×Dic83⋊C4C2313yes—
104C4⋊C4×D83—212no—
105C2×(C2×D83)⋊C4C2313yes—
106C2×(C2×D83)⋊C4C2311yescharacteristic
107C2×(C2×D83)⋊C4C2311yescharacteristic
108C2×(C2×D83)⋊C4C2313yes—
109C2×C4×Dic83C2313yes—
110C2×C4×Dic83C2316yes—
111C2×C4×Dic83C2313yes—
112C2×C4×Dic83C2313yes—
113C2×C4×Dic83C2313yes—
114C2×C4×Dic83C2316yes—
115C2×C4×Dic83C2313yes—
116C42×D83—212no—
117C83⋊#95C2316yes—
118C83⋊#95C2311yescharacteristic
119C83⋊#95C2313yes—
120C83⋊#95C2311yescharacteristic
121C83⋊#95C2313yes—
122C83⋊#95C2313yes—
123C83⋊#95C2313yes—
124C83⋊#95C2316yes—
125C22×Q8×C83C2311yescharacteristic
126C4×Q8×C83—24no—
127C4×Q8×C83—24no—
128C4×Q8×C83C2×C418yescomplemented
129C2×C4⋊C4×C83C2313yes—
130C2×C4⋊C4×C83C2313yes—
131C2×C4⋊C4×C83C2311yescharacteristic
132C2×C4⋊C4×C83C2316yes—
133C2×C4⋊C4×C83C2313yes—
134C2×C4⋊C4×C83C2313yes—
135C2×C42×C83C2313yes—
136C2×C42×C83C2313yes—
137C2×C42×C83C2313yes—
138#95×C83C2313yes—
139#95×C83C2313yes—
140C23×D83C2411yescharacteristic
141C2×Dic166C22×C4112yes—
142C2×Dic166C2×D4112yes—
143C2×Dic166C2×Q8112yes—
144Q8×D83C4⋊C4164yescomplemented
145C22×Dic83C2413yes—
146C22×Dic83C2413yes—
147C22×Dic83C2411yescharacteristic
148C22×Dic83C2411yescharacteristic
149C22×Dic83C2413yes—
150C22×Dic83C2411yescharacteristic
151C22×Dic83C2413yes—
152C2×C4×D83—22no—
153C2×C4×D83—22no—
154C2×C4×D83—22no—
155C2×C4×D83—26no—
156C2×C4×D83—43no—
157C2×C4×D83—43no—
158C2×C4×D83C22×C4112yes—
159C2×C4×D83C2×D4112yes—
160C2×C4×D83C2×Q8112yes—
161C4⋊Dic83—26no—
162C4⋊Dic83—26no—
163C4⋊Dic83—26no—
164C4⋊Dic83—43no—
165C4⋊Dic83C22×C4112yes—
166Dic83⋊C4—26no—
167Dic83⋊C4—212no—
168Dic83⋊C4—26no—
169Dic83⋊C4—26no—
170Dic83⋊C4—22no—
171Dic83⋊C4—43no—
172Dic83⋊C4—46no—
173Dic83⋊C4—43no—
174Dic83⋊C4C22×C4112yes—
175Dic83⋊C4C22×C4112yes—
176(C2×D83)⋊C4—22no—
177(C2×D83)⋊C4—26no—
178(C2×D83)⋊C4C22×C414yes—
179(C2×D83)⋊C4C22×C4112yes—
180C4×Dic83—26no—
181C4×Dic83—26no—
182C4×Dic83—26no—
183C4×Dic83—26no—
184C4×Dic83—212no—
185C4×Dic83C22×C4112yes—
186C4×Dic83C22×C4124yes—
187C2×Q8×C83C22×C414yes—
188C2×Q8×C83C2×D414yes—
189C2×Q8×C83C2×Q814yes—
190C22×C4×C83C2413yes—
191C22×C4×C83C2411yescharacteristic
192C22×C4×C83C2413yes—
193C22×C4×C83C2413yes—
194C22×C4×C83C2411yescharacteristic
195C22×C4×C83C2411yescharacteristic
196C22×C4×C83C2413yes—
197C4⋊C4×C83—26no—
198C4⋊C4×C83—26no—
199C4⋊C4×C83—22no—
200C4⋊C4×C83—43no—
201C4⋊C4×C83—46no—
202C4⋊C4×C83C22×C4112yes—
203C42×C83—26no—
204C42×C83—26no—
205C42×C83C22×C4112yes—
206C23×C83C2511yescharacteristic
207C22×D83—21no—
208C22×D83—21no—
209C22×D83—21no—
210C22×D83—21no—
211C22×D83C23×C412yes—
212C22×D83C22×D412yes—
213C22×D83C22×Q812yes—
214Q8×C83C2×C4⋊C4116yes—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—23no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—23no—
221C2×C4×C83—21no—
222C2×C4×C83—23no—
223C2×C4×C83—23no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—23no—
227C2×C4×C83—23no—
228C2×C4×C83—23no—
229C2×C4×C83—23no—
230C2×C4×C83C23×C412yes—
231C2×C4×C83C23×C416yes—
232C2×C4×C83C23×C416yes—
233C2×C4×C83C22×D416yes—
234C2×C4×C83C22×Q816yes—
235C2×C4×C832−1+412yes—
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—23no—
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—23no—
242C2×Dic83—21no—
243C2×Dic83—23no—
244C2×Dic83—23no—
245C2×Dic83—23no—
246C2×Dic83—23no—
247C2×Dic83—21no—
248C2×Dic83—23no—
249C2×Dic83—23no—
250C2×Dic83—21no—
251C2×Dic83C23×C412yes—
252C2×Dic83C23×C416yes—
253C2×Dic83C23×C416yes—
254C2×Dic83C22×D416yes—
255C2×Dic83C22×Q816yes—
256C2×Dic832−1+412yes—
257C4×D83—28no—
258C4×D83—412no—
259C4×D83C2×C4⋊C4148yes—
260Dic166C2×C4⋊C4148yes—
261C22×C83C24×C411yescharacteristic
262C22×C83C23×D411yescharacteristic
263C22×C83C23×Q811yescharacteristic
264C22×C83C2×2−1+411yescharacteristic
265C22×C83C2×2−1+411yescharacteristic
266C22×C83#58511yescharacteristic
267C22×C83#58511yescharacteristic
268C2×D83—22no—
269C2×D83—22no—
270C2×D83—22no—
271C2×D83—22no—
272C2×D83—22no—
273C2×D83—22no—
274C2×D83C22×C4⋊C414yes—
275C4×C83—26no—
276C4×C83—22no—
277C4×C83—22no—
278C4×C83—41no—
279C4×C83—43no—
280C4×C83—43no—
281C4×C83C22×C4⋊C4112yes—
282Dic83—22no—
283Dic83—26no—
284Dic83—22no—
285Dic83—43no—
286Dic83—43no—
287Dic83—41no—
288Dic83C22×C4⋊C4112yes—
289#34249—831nomaximal, Sylow
290C2×C83C23×C4⋊C411yescharacteristic
291C2×C83C4×2−1+411yescharacteristic
292C2×C83C4×2−1+411yescharacteristic
293C2×C83#348611yescharacteristic
294C2×C83#348711yescharacteristic
295C2×C83#350211yescharacteristic
296C2×C83#350211yescharacteristic
297D83—24no—
298C2×#519—831no—
299C2×#519—831no—
300C22×C4×Q8—831no—
301#2351—833no—
302#2351—833no—
303#2351—831no—
304#2351—831no—
305Q8×C4⋊C4—833no—
306Q8×C4⋊C4—833no—
307Q8×C4⋊C4—831no—
308Q8×C4⋊C4D8311yescharacteristic, complemented
309#2317—833no—
310#2317—833no—
311#2296—833no—
312#2296—833no—
313C83#3424911yescharacteristic, complemented, Sylow
314C23×Q8—831no—
315#519—8316no—
316#519—1668no—
317C2×C4×Q8—831no—
318C2×C4×Q8—831no—
319C2×C4×Q8—831no—
320C2×C4×Q8—833no—
321C2×C4×Q8—833no—
322C2×C4×Q8—833no—
323C2×C4×Q8—833no—
324C2×C4×Q8—833no—
325C2×C4×Q8—833no—
326C2×C4×Q8—1668no—
327C2×C4×Q8C2×D8311yescharacteristic
328C2×C4×Q8C2×D8311yescharacteristic
329C2×C4×Q8C2×D8311yescharacteristic
330C2×#117—833no—
331C2×#117—833no—
332C2×#117—833no—
333C2×#117—833no—
334C22×C4⋊C4—833no—
335C22×C42—833no—
336#389—836no—
337#389—836no—
338#389—833no—
339#389—833no—
340#389—833no—
341#389—833no—
342#389—833no—
343#389—833no—
344#389—833no—
345#389—833no—
346#389—833no—
347#389—833no—
348#389C2×D8313yes—
349#389C2×D8313yes—
350#385—836no—
351#385—833no—
352#385—831no—
353#385—833no—
354#385—833no—
355#384—833no—
356#384—836no—
357#384C2×D8313yes—
358#382—833no—
359#382—833no—
360#382—836no—
361C4×C4⋊C4—836no—
362C4×C4⋊C4—833no—
363C4×C4⋊C4—833no—
364C4×C4⋊C4—833no—
365C4×C4⋊C4—836no—
366C4×C4⋊C4C2×D8313yes—
367C22×Q8—838no—
368C22×Q8—838no—
369C22×Q8—838no—
370C22×Q8—833no—
371C22×Q8C22×D8311yescharacteristic
372C23×C4—833no—
373C23×C4—833no—
374C23×C4—831no—
375C4×Q8—24no—
376C4×Q8—24no—
377C4×Q8—838no—
378C4×Q8—8324no—
379C4×Q8—8324no—
380C4×Q8—16612no—
381C4×Q8—16612no—
382C4×Q8—16612no—
383C4×Q8—16612no—
384C4×Q8—1664no—
385C4×Q8—1664no—
386C4×Q8C4×D8318yescomplemented
387#117—8324no—
388#117—8324no—
389#117—16612no—
390#117—16612no—
391C2×C4⋊C4—833no—
392C2×C4⋊C4—836no—
393C2×C4⋊C4—833no—
394C2×C4⋊C4—833no—
395C2×C4⋊C4—833no—
396C2×C4⋊C4—833no—
397C2×C4⋊C4—833no—
398C2×C4⋊C4—836no—
399C2×C4⋊C4—833no—
400C2×C4⋊C4—833no—
401C2×C4⋊C4—833no—
402C2×C4⋊C4—833no—
403C2×C4⋊C4—831no—
404C2×C4⋊C4—833no—
405C2×C4⋊C4—833no—
406C2×C4⋊C4—16612no—
407C2×C4⋊C4C22×D8313yes—
408C2×C4⋊C4C22×D8313yes—
409C2×C4⋊C4C22×D8311yescharacteristic
410C2×C4⋊C4C22×D8316yes—
411C2×C4⋊C4C22×D8313yes—
412C2×C4⋊C4C22×D8313yes—
413C2×C22⋊C4—831no—
414C2×C22⋊C4—833no—
415C2×C22⋊C4—833no—
416C2×C22⋊C4—831no—
417C2×C42—836no—
418C2×C42—833no—
419C2×C42—836no—
420C2×C42—833no—
421C2×C42—833no—
422C2×C42—833no—
423C2×C42—833no—
424C2×C42—16612no—
425C2×C42C22×D8313yes—
426C2×C42C22×D8313yes—
427C2×C42C22×D8313yes—
428#95—833no—
429#95—831no—
430#95—831no—
431#95—836no—
432#95—833no—
433#95—833no—
434#95—833no—
435#95—836no—
436#95C22×D8313yes—
437#95C22×D8313yes—
438C24—831no—
439C2×Q8—8364no—
440C2×Q8—8312no—
441C2×Q8—8312no—
442C2×Q8—8312no—
443C2×Q8C2×C4×D8314yes—
444C2×Q8D4×D8314yes—
445C2×Q8Q8×D8314yes—
446C22×C4—3323no—
447C22×C4—3323no—
448C22×C4—833no—
449C22×C4—8312no—
450C22×C4—8312no—
451C22×C4—8312no—
452C22×C4—831no—
453C22×C4—831no—
454C22×C4—833no—
455C22×C4—833no—
456C22×C4—833no—
457C22×C4—831no—
458C22×C4—1666no—
459C22×C4—1662no—
460C22×C4—1662no—
461C22×C4—1662no—
462C22×C4C23×D8311yescharacteristic
463C22×C4C23×D8313yes—
464C22×C4C23×D8313yes—
465C22×C4C23×D8311yescharacteristic
466C22×C4C23×D8313yes—
467C22×C4C23×D8311yescharacteristic
468C22×C4C23×D8313yes—
469C4⋊C4—26no—
470C4⋊C4—26no—
471C4⋊C4—22no—
472C4⋊C4—46no—
473C4⋊C4—43no—
474C4⋊C4—3323no—
475C4⋊C4—3326no—
476C4⋊C4—3323no—
477C4⋊C4—3323no—
478C4⋊C4—8312no—
479C4⋊C4—8312no—
480C4⋊C4—8312no—
481C4⋊C4—16612no—
482C4⋊C4—1666no—
483C4⋊C4—1666no—
484C4⋊C4—1662no—
485C4⋊C4—1666no—
486C4⋊C4—1666no—
487C4⋊C4—1666no—
488C4⋊C4—1666no—
489C4⋊C4C2×C4×D83112yes—
490C22⋊C4—8312no—
491C22⋊C4—834no—
492C22⋊C4—1662no—
493C22⋊C4—1666no—
494C42—26no—
495C42—26no—
496C42—8324no—
497C42—8312no—
498C42—1666no—
499C42—1666no—
500C42—16612no—
501C42—1666no—
502C42—1666no—
503C42C2×C4×D83112yes—
504C23—832no—
505C23—832no—
506C23—832no—
507C23—1661no—
508C23—1661no—
509C23—1661no—
510C23—1661no—
511C23C24×D8311yescharacteristic
512Q8—8348no—
513Q8C4⋊C4×D83116yes—
514C2×C4—21no—
515C2×C4—23no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—23no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—23no—
526C2×C4—23no—
527C2×C4—23no—
528C2×C4—23no—
529C2×C4—33212no—
530C2×C4—836no—
531C2×C4—836no—
532C2×C4—836no—
533C2×C4—8348no—
534C2×C4—832no—
535C2×C4—836no—
536C2×C4—832no—
537C2×C4—1668no—
538C2×C4—1663no—
539C2×C4—1661no—
540C2×C4—1663no—
541C2×C4—1663no—
542C2×C4—1663no—
543C2×C4—1661no—
544C2×C4—1663no—
545C2×C4—1661no—
546C2×C4—1663no—
547C2×C4—1661no—
548C2×C4—1661no—
549C2×C4—1663no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1663no—
553C2×C4C22×C4×D8316yes—
554C2×C4C22×C4×D8312yes—
555C2×C4C22×C4×D8316yes—
556C2×C4C2×D4×D8316yes—
557C2×C4C2×Q8×D8316yes—
558C2×C4#1458184012yes—
559C22—834no—
560C22—1662no—
561C22—1662no—
562C22—1662no—
563C22—1662no—
564C22—1662no—
565C22—1662no—
566C22C23×C4×D8311yescharacteristic
567C22C22×D4×D8311yescharacteristic
568C22C22×Q8×D8311yescharacteristic
569C222−1+4×D8311yescharacteristic
570C22C2×#1458184011yescharacteristic
571C22#1912853611yescharacteristic
572C22#1912853711yescharacteristic
573C4—22no—
574C4—26no—
575C4—22no—
576C4—41no—
577C4—43no—
578C4—43no—
579C4—3323no—
580C4—3323no—
581C4—3321no—
582C4—8312no—
583C4—1662no—
584C4—1662no—
585C4—1666no—
586C4C2×C4⋊C4×D83112yes—
587C2—1664no—
588C2C22×C4⋊C4×D8311yescharacteristic
589C2#2492067411yescharacteristic
590C2C4×#1458184011yescharacteristic
591C2#2492111311yescharacteristic
592C2#2492111611yescharacteristic
593C2#2492127011yescharacteristic
594C2#2492127211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingQ8×C4⋊C4×C83C2
socle, cosocleC23×C83C25
prime radicalC23×D83C24
hypercenterQ8×C4⋊C4D83
hypocenterC83#34249
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. #24910383C2semidirect
6. #24910382C2semidirect
7. #24910381C2semidirect
8. #24910377C2semidirect
9. Q8×Dic83⋊C4C2semidirect
10. #24910375C2semidirect
15. Q8×C4⋊C4×C83C2semidirect
18. #19128177C4semidirect
78. Q8×Dic83C2×C4semidirect
81. C4×Dic166C2×C4semidirect
84. #14581757C2×C4semidirect
128. C4×Q8×C83C2×C4semidirect
144. Q8×D83C4⋊C4semidirect
308. Q8×C4⋊C4D83semidirect
313. C83#34249semidirect
386. C4×Q8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281772
C2×C4×Q8×D832
#249103842
#249103832
#249103822
#249103812
#249103772
Q8×Dic83⋊C42
#249103752
#249100852
#249100842
#249099442
#249099432
Q8×C4⋊C4×C832
#3424983