Finite Group Databasebeta

← #35416595

Subgroups — #35416595

Order 21248.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910205C211yesmaximal, characteristic, complemented
2#24910106C211yesmaximal, characteristic
3#24908421C211yesmaximal, characteristic, complemented
4#24908227C211yesmaximal, characteristic, complemented
5#24908202C211yesmaximal, characteristic, complemented
6#24908192C211yesmaximal, characteristic, complemented
7#24908162C211yesmaximal, characteristic
8#24908148C211yesmaximal, characteristic
9#24908092C211yesmaximal, characteristic
10#24908092C211yesmaximal, characteristic
11#24908050C211yesmaximal, characteristic, complemented
12#24908036C211yesmaximal, characteristic, complemented
13#24907463C211yesmaximal, characteristic
14#24907322C211yesmaximal, characteristic, complemented
15C2×#478×C83C211yesmaximal, characteristic
16C2×#14581724C2211yescharacteristic
17C2×#14581724C2211yescharacteristic
18C2×C4⋊(C83⋊C8)C2211yescharacteristic
19C2×#14581713C2211yescharacteristic
20C2×#14581713C2211yescharacteristic
21C2×#14581702C2211yescharacteristic
22C2×C8×Dic83C2211yescharacteristic
23#19127604C2211yescharacteristic
24#19127604C2211yescharacteristic
25#19127600—21no—
26#19127600—21no—
27#19127600C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127600C2211yescharacteristic
31#19127594—21no—
32#19127594—21no—
33#19127587—21no—
34#19127587—21no—
35#19127571C2211yescharacteristic
36#19127564—21no—
37#19127561—21no—
38C4⋊C4×Dic83C2211yescharacteristic
39C4⋊C4×Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41#19127547C2211yescharacteristic
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C2×#125×C83C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54#478×C83C418yescomplemented
55C2×#107×C83C2211yescharacteristic
56C2×#106×C83C2211yescharacteristic
57C2×C4⋊C8×C83C2211yescharacteristic
58C2×#102×C83C2211yescharacteristic
59C2×#102×C83C2211yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C2×C4⋊Dic83—21no—
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C22⋊Dic83—21no—
73C2×C22⋊Dic83—21no—
74C2×C22⋊Dic83—21no—
75C2×C22⋊Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83C2311yescharacteristic
80C2×C4×Dic83C2311yescharacteristic
81#14581724—41no—
82#14581724—41no—
83C4⋊(C83⋊C8)—22no—
84#14581713—41no—
85#14581713—41no—
86#14581702—41no—
87C8×Dic83—41no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95D411yescharacteristic
96C83⋊#95D411yescharacteristic
97C83⋊#95D411yescharacteristic
98C83⋊#95D411yescharacteristic
99C22×D4×C83C2311yescharacteristic
100C23×C4×C83—21no—
101C22×C8×C83C2311yescharacteristic
102C22×C8×C83C2311yescharacteristic
103#125×C83—22no—
104#125×C83C2×C414yes—
105C4×D4×C83—22no—
106C4×D4×C83C2×C414yes—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C22⋊C4×C83—21no—
113C2×C42×C83C2311yescharacteristic
114#107×C83C2×C414yes—
115#106×C83C2×C414yes—
116C4⋊C8×C83C2×C414yes—
117#102×C83C2×C414yes—
118#102×C83C2×C414yes—
119C22×Dic83—21no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83C2×D411yescharacteristic
124C22×Dic83C2×D411yescharacteristic
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C4⋊Dic83—21no—
134C4⋊Dic83—21no—
135C4⋊Dic83—21no—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—81no—
142C4⋊Dic83—81no—
143C4⋊Dic83—81no—
144C4⋊Dic83—81no—
145C4⋊Dic83—81no—
146C4⋊Dic83—81no—
147C22⋊Dic83—81no—
148C22⋊Dic83—81no—
149C22⋊Dic83—81no—
150C22⋊Dic83—81no—
151C4×Dic83—21no—
152C4×Dic83—21no—
153C4×Dic83—21no—
154C4×Dic83—21no—
155C4×Dic83—81no—
156C4×Dic83—81no—
157C4×Dic83—81no—
158C24×C83—21no—
159C2×D4×C83—21no—
160C2×D4×C83—21no—
161C2×D4×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83—21no—
164C2×D4×C83C22×C412yes—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—24no—
169C22×C4×C83—81no—
170C22×C4×C83—81no—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2411yescharacteristic
174C2×C8×C83—21no—
175C2×C8×C83—21no—
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83C22×C412yes—
179C2×C8×C83C22×C412yes—
180C4⋊C4×C83—24no—
181C4⋊C4×C83—24no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83C22×C412yes—
186C4⋊C4×C83C22×C412yes—
187C4⋊C4×C83C22×C412yes—
188C22⋊C4×C83—24no—
189C42×C83C22×C412yes—
190C42×C83#4112yes—
191C23×C83—22no—
192C23×C83—41no—
193C23×C83—41no—
194C23×C83—41no—
195C23×C83—41no—
196C23×C83—41no—
197C23×C83—41no—
198C23×C83C22×D411yescharacteristic
199D4×C83—22no—
200D4×C83—22no—
201D4×C83—41no—
202D4×C83—41no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—22no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—84no—
217C2×C4×C83C4×D412yes—
218C2×C4×C83C4×D412yes—
219C2×C4×C83C23×C411yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×C4×C83H(5,𝔽2)11yescharacteristic
222C2×C4×C832−1+411yescharacteristic
223C2×C4×C832−1+411yescharacteristic
224C2×C4×C832−1+411yescharacteristic
225C2×Dic83—21no—
226C2×Dic83—21no—
227C2×Dic83—21no—
228C2×Dic83—21no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C8×C83—22no—
250C8×C83—22no—
251C83⋊C8—41no—
252C83⋊C8—41no—
253C22×C83—42no—
254C22×C83—42no—
255C22×C83—42no—
256C22×C83—81no—
257C22×C83—81no—
258C22×C83—81no—
259C22×C83—81no—
260C22×C83C2×C4×D411yescharacteristic
261C22×C83#53611yescharacteristic
262C22×C83#53711yescharacteristic
263C22×C83#57611yescharacteristic
264C22×C83#57711yescharacteristic
265C22×C83#57711yescharacteristic
266C22×C83#57711yescharacteristic
267C4×C83—22no—
268C4×C83—22no—
269C4×C83—42no—
270C4×C83—42no—
271C4×C83—42no—
272C4×C83#51911yescharacteristic
273C4×C83#52011yescharacteristic
274C4×C83#54111yescharacteristic
275C4×C83#54211yescharacteristic
276Dic83—41no—
277Dic83—41no—
278Dic83—41no—
279Dic83—41no—
280Dic83—81no—
281Dic83—81no—
282Dic83—81no—
283Dic83—81no—
284#22122—831nomaximal, Sylow
285C2×C83—82no—
286C2×C83#232311yescharacteristic
287C2×C83#294611yescharacteristic
288C2×C83#295211yescharacteristic
289C2×C83#324111yescharacteristic
290C2×C83#324411yescharacteristic
291C2×C83#325711yescharacteristic
292C2×C83#326011yescharacteristic
293C2×#478D8311yescharacteristic
294#2330—831no—
295#2319—831no—
296#1989—831no—
297#1948—831no—
298#1943—831no—
299#1941—831no—
300#1936—831no—
301#1934—831no—
302#1926—831no—
303#1926—831no—
304#1920—831no—
305#1918—831no—
306#1802—831no—
307#1768—831no—
308C83#2212211yescharacteristic, complemented, Sylow
309C2×#125C2×D8311yescharacteristic
310C2×C4×D4C2×D8311yescharacteristic
311#478Dic8318yescomplemented
312C2×#107C2×D8311yescharacteristic
313C2×#106—831no—
314C2×#106—831no—
315C2×#106C2×D8311yescharacteristic
316C2×C4⋊C8—831no—
317C2×C4⋊C8C2×D8311yescharacteristic
318C2×#102—831no—
319C2×#102—831no—
320C2×#102C2×D8311yescharacteristic
321C2×#102C2×D8311yescharacteristic
322C2×#97—831no—
323C2×C4×C8—831no—
324#390—831no—
325#390—831no—
326#389—831no—
327#389—831no—
328#389—831no—
329#389—831no—
330#389—1661no—
331#389—1661no—
332#388—1661no—
333#388—1661no—
334#387—1661no—
335#387—1661no—
336#384—831no—
337#381—1661no—
338#380—1661no—
339C4×C4⋊C4—831no—
340C4×C4⋊C4—831no—
341C4×C4⋊C4—831no—
342#376—831no—
343#340—831no—
344#340—831no—
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#336—831no—
352#336—831no—
353C22×D4C22×D8311yescharacteristic
354C23×C4—21no—
355C22×C8—831no—
356C22×C8—831no—
357C22×C8C22×D8311yescharacteristic
358C22×C8C22×D8311yescharacteristic
359#125—22no—
360#125C2×Dic8314yes—
361C4×D4—22no—
362C4×D4C2×Dic8314yes—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—831no—
366C2×C4⋊C4—831no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—1661no—
370C2×C4⋊C4—1661no—
371C2×C4⋊C4—1661no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4C22×D8311yescharacteristic
376C2×C4⋊C4C22×D8311yescharacteristic
377C2×C4⋊C4C22×D8311yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—1661no—
380C2×C22⋊C4—1661no—
381C2×C22⋊C4—1661no—
382C2×C22⋊C4—1661no—
383C2×C42—831no—
384C2×C42—831no—
385C2×C42—1661no—
386C2×C42—1661no—
387C2×C42—1661no—
388C2×C42C22×D8311yescharacteristic
389#107C2×Dic8314yes—
390#106—3321no—
391#106—3321no—
392#106C2×Dic8314yes—
393C4⋊C8—1662no—
394C4⋊C8C2×Dic8314yes—
395#102—3321no—
396#102—3321no—
397#102C2×Dic8314yes—
398#102C2×Dic8314yes—
399#97—3321no—
400C4×C8—3321no—
401#95—831no—
402#95—831no—
403#95—831no—
404#95—831no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C22×Dic8312yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—831no—
426C22×C4—831no—
427C22×C4—831no—
428C22×C4—831no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1661no—
433C22×C4D4×D8311yescharacteristic
434C22×C4D4×D8311yescharacteristic
435C22×C4C23×D8311yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—1661no—
441C2×C8—1661no—
442C2×C8—1661no—
443C2×C8—1661no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8C22×Dic8312yes—
447C2×C8C22×Dic8312yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—6641no—
454C4⋊C4—6641no—
455C4⋊C4—6641no—
456C4⋊C4—6641no—
457C4⋊C4—6641no—
458C4⋊C4—6641no—
459C4⋊C4—1661no—
460C4⋊C4—1661no—
461C4⋊C4—1661no—
462C4⋊C4—1661no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4C22×Dic8312yes—
468C4⋊C4C22×Dic8312yes—
469C4⋊C4C22×Dic8312yes—
470C22⋊C4—24no—
471C22⋊C4—6641no—
472C22⋊C4—6641no—
473C22⋊C4—6641no—
474C22⋊C4—6641no—
475C42—6641no—
476C42—6641no—
477C42—6641no—
478C42—1661no—
479C42—1661no—
480C42—1661no—
481C42—1661no—
482C42C22×Dic8312yes—
483C42#1305058912yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D8311yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—84no—
510C2×C4—3321no—
511C2×C4—3321no—
512C2×C4—3321no—
513C2×C4—3321no—
514C2×C4—3321no—
515C2×C4—3321no—
516C2×C4—3321no—
517C2×C4—3321no—
518C2×C4—3321no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—1661no—
523C2×C4—1661no—
524C2×C4—1661no—
525C2×C4—1661no—
526C2×C4—1661no—
527C2×C4—1661no—
528C2×C4—1661no—
529C2×C4—1661no—
530C2×C4—1661no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4—1661no—
534C2×C4D4×Dic8312yes—
535C2×C4D4×Dic8312yes—
536C2×C4C23×Dic8311yescharacteristic
537C2×C4C2×#1305058911yescharacteristic
538C2×C4#1458183811yescharacteristic
539C2×C4#1458183911yescharacteristic
540C2×C4#1458184011yescharacteristic
541C2×C4#1458184011yescharacteristic
542C8—22no—
543C8—22no—
544C8—3321no—
545C8—3321no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×D4×Dic8311yescharacteristic
554C22#1912827411yescharacteristic
555C22#1912828111yescharacteristic
556C22#1912850511yescharacteristic
557C22#1912850811yescharacteristic
558C22#1912850811yescharacteristic
559C22#1912851111yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—3321no—
566C4—3321no—
567C4—3321no—
568C4—3321no—
569C4—6641no—
570C4—6641no—
571C4—6641no—
572C4—6641no—
573C4#1912817911yescharacteristic
574C4#1912818411yescharacteristic
575C4#1912831111yescharacteristic
576C4#1912832411yescharacteristic
577C2—82no—
578C2#2491014711yescharacteristic
579C2#2491629611yescharacteristic
580C2#2491634711yescharacteristic
581C2#2491872111yescharacteristic
582C2#2491876011yescharacteristic
583C2#2491887811yescharacteristic
584C2#2491891111yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#478×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#478D83
hypocenterC83#22122
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #24910205C2semidirect
3. #24908421C2semidirect
4. #24908227C2semidirect
5. #24908202C2semidirect
6. #24908192C2semidirect
11. #24908050C2semidirect
12. #24908036C2semidirect
14. #24907322C2semidirect
54. #478×C83C4semidirect
308. C83#22122semidirect
311. #478Dic83semidirect

Maximal subgroups

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

subgroupindex
#249102052
#249101062
#249084212
#249082272
#249082022
#249081922
#249081622
#249081482
#249080922
#249080502
#249080362
#249074632
#249073222
C2×#478×C832
#2212283