Finite Group Databasebeta

← #35416660

Subgroups — #35416660

Order 21248.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127978C211yesmaximal, characteristic
2#24910107C211yesmaximal, characteristic
3#24908412C211yesmaximal, characteristic, complemented
4#24908229C211yesmaximal, characteristic, complemented
5#24908198C211yesmaximal, characteristic, complemented
6#24908193C211yesmaximal, characteristic, complemented
7#24908159C211yesmaximal, characteristic
8#24908145C211yesmaximal, characteristic
9#24908090C211yesmaximal, characteristic
10#24908083C211yesmaximal, characteristic
11#24908055C211yesmaximal, characteristic, complemented
12#24908034C211yesmaximal, characteristic, complemented
13#24907470C211yesmaximal, characteristic
14#24907320C211yesmaximal, characteristic, complemented
15C83×#2331C211yesmaximal, characteristic, complemented
16C2×C4×D332C2211yescharacteristic
17#19127978C418yescomplemented
18C2×#14581725C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×C4⋊(C83⋊C8)C2211yescharacteristic
24C2×#14581712C2211yescharacteristic
25C2×#14581712C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29C2×C4×C83⋊C8C2211yescharacteristic
30#19127603C2211yescharacteristic
31#19127603C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127599C2211yescharacteristic
34#19127590—21no—
35#19127590—21no—
36#19127563—21no—
37#19127560—21no—
38C4×C4⋊Dic83C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49C2×C4⋊Q8×C83C2211yescharacteristic
50#391×C83—21no—
51#391×C83—21no—
52#389×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#387×C83—21no—
55#387×C83—21no—
56#382×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59#376×C83C2211yescharacteristic
60C22×D332C2311yescharacteristic
61C22×C4×D83—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C4×D332—22no—
67C4×D332C2×C414yes—
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×Dic83⋊C4—21no—
72C2×Dic83⋊C4—21no—
73C2×(C2×D83)⋊C4—21no—
74C2×(C2×D83)⋊C4—21no—
75C2×(C2×D83)⋊C4—21no—
76C2×(C2×D83)⋊C4—21no—
77C2×(C2×D83)⋊C4—21no—
78C2×C4×Dic83—21no—
79#14581725—41no—
80#14581725C2×C414yes—
81#14581725C2×C414yes—
82#14581723—41no—
83C4⋊(C83⋊C8)—22no—
84C4⋊(C83⋊C8)C2×C414yes—
85#14581712—41no—
86#14581712—41no—
87#14581712C2×C414yes—
88#14581712C2×C414yes—
89#14581703—41no—
90C4×C83⋊C8—41no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C22×Q8×C83—21no—
95C4⋊Q8×C83—22no—
96C4⋊Q8×C83C2×C414yes—
97C2×C4⋊C4×C83—21no—
98C2×C4⋊C4×C83—21no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83C2311yescharacteristic
101C2×C4⋊C4×C83C2311yescharacteristic
102C2×C4⋊C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83C2311yescharacteristic
104C2×C42×C83—21no—
105C2×C42×C83—21no—
106C2×C42×C83C2311yescharacteristic
107C2×C42×C83C2311yescharacteristic
108C2×C42×C83C2311yescharacteristic
109#95×C83—21no—
110#95×C83—21no—
111#95×C83—21no—
112#95×C83—21no—
113#95×C83—21no—
114#95×C83—21no—
115#95×C83D411yescharacteristic
116#95×C83D411yescharacteristic
117#95×C83D411yescharacteristic
118#95×C83D411yescharacteristic
119C23×D83—21no—
120C2×D332—21no—
121C2×D332—21no—
122C2×D332—21no—
123C2×D332—21no—
124C2×D332—21no—
125C2×D332C22×C412yes—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C2×C4×D83—24no—
130C2×C4×D83—81no—
131C2×C4×D83—81no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8C22×C412yes—
143C2×C83⋊C8C22×C412yes—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83C22×C412yes—
150Dic83⋊C4—81no—
151Dic83⋊C4—81no—
152(C2×D83)⋊C4—24no—
153(C2×D83)⋊C4—81no—
154(C2×D83)⋊C4—81no—
155(C2×D83)⋊C4—81no—
156(C2×D83)⋊C4—81no—
157C4×Dic83—81no—
158C2×Q8×C83—24no—
159C2×Q8×C83—81no—
160C2×Q8×C83—81no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83C2×D411yescharacteristic
166C22×C4×C83C2×D411yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2411yescharacteristic
172C4⋊C4×C83—24no—
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—81no—
180C4⋊C4×C83—81no—
181C4⋊C4×C83C22×C412yes—
182C4⋊C4×C83C22×C412yes—
183C42×C83—21no—
184C42×C83—21no—
185C42×C83—21no—
186C42×C83—21no—
187C42×C83—81no—
188C42×C83—81no—
189C42×C83C22×C412yes—
190C42×C83#4112yes—
191C23×C83C22×D411yescharacteristic
192C22×D83—22no—
193C22×D83—41no—
194C22×D83—41no—
195C22×D83—41no—
196C22×D83—41no—
197C22×D83—41no—
198C22×D83—41no—
199Q8×C83—84no—
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83C4×D412yes—
229C2×C4×C83C4×D412yes—
230C2×C4×C83C23×C411yescharacteristic
231C2×C4×C83C2×#4111yescharacteristic
232C2×C4×C83H(5,𝔽2)11yescharacteristic
233C2×C4×C832−1+411yescharacteristic
234C2×C4×C832−1+411yescharacteristic
235C2×C4×C832−1+411yescharacteristic
236C2×Dic83—22no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245D332—22no—
246D332—22no—
247D332—41no—
248D332—41no—
249C4×D83—84no—
250C83⋊C8—22no—
251C83⋊C8—22no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C22×C83C2×C4×D411yescharacteristic
255C22×C83#53611yescharacteristic
256C22×C83#53711yescharacteristic
257C22×C83#57611yescharacteristic
258C22×C83#57611yescharacteristic
259C22×C83#57711yescharacteristic
260C22×C83#57811yescharacteristic
261C2×D83—42no—
262C2×D83—42no—
263C2×D83—42no—
264C2×D83—81no—
265C2×D83—81no—
266C2×D83—81no—
267C2×D83—81no—
268C4×C83—22no—
269C4×C83—22no—
270C4×C83—42no—
271C4×C83—42no—
272C4×C83—41no—
273C4×C83—41no—
274C4×C83—41no—
275C4×C83—41no—
276C4×C83—81no—
277C4×C83—81no—
278C4×C83#51911yescharacteristic
279C4×C83#52011yescharacteristic
280C4×C83#54111yescharacteristic
281C4×C83#54211yescharacteristic
282Dic83—42no—
283Dic83—81no—
284Dic83—81no—
285#22127—831nomaximal, Sylow
286C2×C83#232311yescharacteristic
287C2×C83#294711yescharacteristic
288C2×C83#294811yescharacteristic
289C2×C83#324111yescharacteristic
290C2×C83#324311yescharacteristic
291C2×C83#325711yescharacteristic
292C2×C83#326211yescharacteristic
293D83—82no—
294C2×#474—831no—
295#2331D8311yescharacteristic, complemented
296#2319—831no—
297#1987—831no—
298#1949—831no—
299#1943—831no—
300#1942—831no—
301#1936—831no—
302#1934—831no—
303#1926—831no—
304#1925—831no—
305#1921—831no—
306#1918—831no—
307#1804—831no—
308#1768—831no—
309C83#2212711yescharacteristic, complemented, Sylow
310C2×C4⋊Q8C2×D8311yescharacteristic
311C2×C4×D4—831no—
312#474—838no—
313C2×#107—831no—
314C2×#107—831no—
315C2×#107—831no—
316C2×#106—831no—
317C2×C4⋊C8—831no—
318C2×C4⋊C8—831no—
319C2×#102—831no—
320C2×#102—831no—
321C2×#102—831no—
322C2×#102—831no—
323C2×#97—831no—
324C2×C4×C8—831no—
325#391—21no—
326#391—21no—
327#390—831no—
328#390—831no—
329#389—831no—
330#389—831no—
331#389C2×D8311yescharacteristic
332#389C2×D8311yescharacteristic
333#388—1661no—
334#388—1661no—
335#387—21no—
336#387—21no—
337#382C2×D8311yescharacteristic
338#381—1661no—
339#380—1661no—
340C4×C4⋊C4—831no—
341C4×C4⋊C4C2×D8311yescharacteristic
342C4×C4⋊C4C2×D8311yescharacteristic
343#376C2×D8311yescharacteristic
344#340—831no—
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#336—831no—
353#336—831no—
354C22×Q8—21no—
355C22×D4—831no—
356C23×C4—1661no—
357C22×C8—831no—
358C22×C8—831no—
359C22×C8—831no—
360C22×C8—831no—
361C4⋊Q8—22no—
362C4⋊Q8C4×D8314yes—
363C4×D4—834no—
364C4×D4—1662no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—1661no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4C22×D8311yescharacteristic
374C2×C4⋊C4C22×D8311yescharacteristic
375C2×C4⋊C4C22×D8311yescharacteristic
376C2×C4⋊C4C22×D8311yescharacteristic
377C2×C22⋊C4—1661no—
378C2×C22⋊C4—1661no—
379C2×C22⋊C4—1661no—
380C2×C22⋊C4—1661no—
381C2×C22⋊C4—1661no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—1661no—
385C2×C42C22×D8311yescharacteristic
386C2×C42C22×D8311yescharacteristic
387C2×C42C22×D8311yescharacteristic
388#107—3321no—
389#107—834no—
390#107—834no—
391#106—3321no—
392C4⋊C8—834no—
393C4⋊C8—1662no—
394#102—3321no—
395#102—3321no—
396#102—834no—
397#102—834no—
398#97—3321no—
399C4×C8—3321no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95C22⋊D8311yescharacteristic
410#95C22⋊D8311yescharacteristic
411#95C22⋊D8311yescharacteristic
412#95C22⋊D8311yescharacteristic
413C24—1661no—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—832no—
418C2×D4—1661no—
419C2×D4—1661no—
420C2×D4—1661no—
421C2×D4—1661no—
422C2×D4—1661no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—6641no—
428C22×C4—6641no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1664no—
433C22×C4C2×C22⋊D8311yescharacteristic
434C22×C4C2×C22⋊D8311yescharacteristic
435C22×C4C2×C22⋊D8311yescharacteristic
436C22×C4C2×C22⋊D8311yescharacteristic
437C22×C4C2×C22⋊D8311yescharacteristic
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4C23×D8311yescharacteristic
440C2×C8—832no—
441C2×C8—832no—
442C2×C8—1661no—
443C2×C8—1661no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8—1661no—
451C2×C8—1661no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—832no—
462C4⋊C4—6641no—
463C4⋊C4—6641no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4C2×C4×D8312yes—
470C4⋊C4C2×C4×D8312yes—
471C22⋊C4—6641no—
472C22⋊C4—6641no—
473C22⋊C4—6641no—
474C22⋊C4—6641no—
475C22⋊C4—1664no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—81no—
481C42—81no—
482C42—6641no—
483C42C2×C4×D8312yes—
484C42#1305059012yes—
485C23—3321no—
486C23—3321no—
487C23—3321no—
488C23—3321no—
489C23—3321no—
490C23—3321no—
491C23—1662no—
492C23C22×C22⋊D8311yescharacteristic
493Q8—84no—
494D4—3321no—
495D4—3321no—
496D4—1662no—
497D4—1662no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—3321no—
534C2×C4—6644no—
535C2×C4—1662no—
536C2×C4C4×C22⋊D8312yes—
537C2×C4C4×C22⋊D8312yes—
538C2×C4C22×C4×D8311yescharacteristic
539C2×C4C2×#1305059011yescharacteristic
540C2×C4#1458183811yescharacteristic
541C2×C4#1458183911yescharacteristic
542C2×C4#1458184011yescharacteristic
543C2×C4#1458184011yescharacteristic
544C8—3321no—
545C8—3321no—
546C8—1662no—
547C8—1662no—
548C22—3322no—
549C22—3322no—
550C22—3322no—
551C22—6641no—
552C22—6641no—
553C22—6641no—
554C22—6641no—
555C22C2×C4×C22⋊D8311yescharacteristic
556C22#1912827511yescharacteristic
557C22#1912827911yescharacteristic
558C22#1912850411yescharacteristic
559C22#1912850411yescharacteristic
560C22#1912850911yescharacteristic
561C22#1912851611yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—3322no—
573C4—6641no—
574C4—6641no—
575C4#1912817711yescharacteristic
576C4#1912818311yescharacteristic
577C4#1912830911yescharacteristic
578C4#1912832311yescharacteristic
579C2—6642no—
580C2#2491013911yescharacteristic
581C2#2491630611yescharacteristic
582C2#2491631311yescharacteristic
583C2#2491871611yescharacteristic
584C2#2491874611yescharacteristic
585C2#2491887411yescharacteristic
586C2#2491891911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2331C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenter#2331D83
hypocenterC83#22127
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #24908412C2semidirect
4. #24908229C2semidirect
5. #24908198C2semidirect
6. #24908193C2semidirect
11. #24908055C2semidirect
12. #24908034C2semidirect
14. #24907320C2semidirect
15. C83×#2331C2semidirect
17. #19127978C4semidirect
295. #2331D83semidirect
309. C83#22127semidirect

Maximal subgroups

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

subgroupindex
C2×#191279782
#249101072
#249084122
#249082292
#249081982
#249081932
#249081592
#249081452
#249080902
#249080832
#249080552
#249080342
#249074702
#249073202
C83×#23312
#2212783