Finite Group Databasebeta

← #35416670

Subgroups — #35416670

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
1#24910212C211yesmaximal, characteristic, complemented
2#24910106C211yesmaximal, characteristic
3#24908412C211yesmaximal, characteristic, complemented
4#24908233C211yesmaximal, characteristic, complemented
5#24908202C211yesmaximal, characteristic, complemented
6#24908197C211yesmaximal, characteristic, complemented
7#24908162C211yesmaximal, characteristic
8#24908148C211yesmaximal, characteristic
9#24908092C211yesmaximal, characteristic
10#24908085C211yesmaximal, characteristic
11#24908057C211yesmaximal, characteristic, complemented
12#24908036C211yesmaximal, characteristic, complemented
13#24907473C211yesmaximal, characteristic
14#24907322C211yesmaximal, characteristic, complemented
15C2×#474×C83C211yesmaximal, characteristic
16C2×#14581726C2211yescharacteristic
17C2×#14581724C2211yescharacteristic
18C2×C4⋊(C83⋊C8)C2211yescharacteristic
19C2×#14581713C2211yescharacteristic
20C2×#14581713C2211yescharacteristic
21C2×#14581702C2211yescharacteristic
22C2×C8×Dic83C2211yescharacteristic
23#19127608—21no—
24#19127608—21no—
25#19127604C2211yescharacteristic
26#19127604C2211yescharacteristic
27#19127600C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127600C2211yescharacteristic
31#19127594—21no—
32#19127594—21no—
33#19127587—21no—
34#19127587—21no—
35#19127567C2211yescharacteristic
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×C4⋊Q8×C83C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54#474×C83C418yescomplemented
55C2×#107×C83C2211yescharacteristic
56C2×#107×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⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C22⋊Dic83—21no—
71C2×C22⋊Dic83—21no—
72C2×C22⋊Dic83—21no—
73C2×C22⋊Dic83—21no—
74C2×C4×Dic83—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83C2311yescharacteristic
78C2×C4×Dic83C2311yescharacteristic
79#14581726—41no—
80#14581724—41no—
81C4⋊(C83⋊C8)—22no—
82#14581713—41no—
83#14581713—41no—
84#14581702—41no—
85C8×Dic83—41no—
86C83⋊#95—21no—
87C83⋊#95—21no—
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×Q8×C83—21no—
100C22×D4×C83C2311yescharacteristic
101C23×C4×C83—21no—
102C22×C8×C83C2311yescharacteristic
103C22×C8×C83C2311yescharacteristic
104C4⋊Q8×C83—22no—
105C4⋊Q8×C83C2×C414yes—
106C4×D4×C83—22no—
107C4×D4×C83C2×C414yes—
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#107×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—
145C22⋊Dic83—81no—
146C22⋊Dic83—81no—
147C22⋊Dic83—81no—
148C22⋊Dic83—81no—
149C4×Dic83—21no—
150C4×Dic83—21no—
151C4×Dic83—21no—
152C4×Dic83—21no—
153C4×Dic83—81no—
154C4×Dic83—81no—
155C4×Dic83—81no—
156C24×C83—21no—
157C2×Q8×C83—24no—
158C2×Q8×C83—81no—
159C2×Q8×C83—81no—
160C2×D4×C83—21no—
161C2×D4×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83—21no—
164C2×D4×C83—21no—
165C2×D4×C83C22×C412yes—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—24no—
170C22×C4×C83—81no—
171C22×C4×C83—81no—
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2411yescharacteristic
175C2×C8×C83—21no—
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83C22×C412yes—
180C2×C8×C83C22×C412yes—
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
199Q8×C83—84no—
200D4×C83—22no—
201D4×C83—22no—
202D4×C83—41no—
203D4×C83—41no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—22no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83—84no—
218C2×C4×C83C4×D412yes—
219C2×C4×C83C4×D412yes—
220C2×C4×C83C23×C411yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C83H(5,𝔽2)11yescharacteristic
223C2×C4×C832−1+411yescharacteristic
224C2×C4×C832−1+411yescharacteristic
225C2×C4×C832−1+411yescharacteristic
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—21no—
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—
249C2×Dic83—41no—
250C8×C83—22no—
251C8×C83—22no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C22×C83—42no—
255C22×C83—42no—
256C22×C83—42no—
257C22×C83—81no—
258C22×C83—81no—
259C22×C83—81no—
260C22×C83—81no—
261C22×C83C2×C4×D411yescharacteristic
262C22×C83#53611yescharacteristic
263C22×C83#53711yescharacteristic
264C22×C83#57611yescharacteristic
265C22×C83#57611yescharacteristic
266C22×C83#57711yescharacteristic
267C22×C83#57811yescharacteristic
268C4×C83—22no—
269C4×C83—22no—
270C4×C83—42no—
271C4×C83—42no—
272C4×C83—42no—
273C4×C83#51911yescharacteristic
274C4×C83#52011yescharacteristic
275C4×C83#54111yescharacteristic
276C4×C83#54211yescharacteristic
277Dic83—41no—
278Dic83—41no—
279Dic83—41no—
280Dic83—41no—
281Dic83—81no—
282Dic83—81no—
283Dic83—81no—
284Dic83—81no—
285#22127—831nomaximal, Sylow
286C2×C83—82no—
287C2×C83#232311yescharacteristic
288C2×C83#294711yescharacteristic
289C2×C83#294811yescharacteristic
290C2×C83#324111yescharacteristic
291C2×C83#324311yescharacteristic
292C2×C83#325711yescharacteristic
293C2×C83#326211yescharacteristic
294C2×#474D8311yescharacteristic
295#2331—831no—
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×D4C2×D8311yescharacteristic
312#474Dic8318yescomplemented
313C2×#107—831no—
314C2×#107C2×D8311yescharacteristic
315C2×#107C2×D8311yescharacteristic
316C2×#106—831no—
317C2×C4⋊C8—831no—
318C2×C4⋊C8C2×D8311yescharacteristic
319C2×#102—831no—
320C2×#102—831no—
321C2×#102C2×D8311yescharacteristic
322C2×#102C2×D8311yescharacteristic
323C2×#97—831no—
324C2×C4×C8—831no—
325#391—1661no—
326#391—1661no—
327#390—831no—
328#390—831no—
329#389—831no—
330#389—831no—
331#389—831no—
332#389—831no—
333#388—1661no—
334#388—1661no—
335#387—1661no—
336#387—1661no—
337#382—831no—
338#381—1661no—
339#380—1661no—
340C4×C4⋊C4—831no—
341C4×C4⋊C4—831no—
342C4×C4⋊C4—831no—
343#376—831no—
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×D4C22×D8311yescharacteristic
356C23×C4—21no—
357C22×C8—831no—
358C22×C8—831no—
359C22×C8C22×D8311yescharacteristic
360C22×C8C22×D8311yescharacteristic
361C4⋊Q8—22no—
362C4⋊Q8C2×Dic8314yes—
363C4×D4—22no—
364C4×D4C2×Dic8314yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—831no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—1661no—
371C2×C4⋊C4—1661no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4C22×D8311yescharacteristic
375C2×C4⋊C4C22×D8311yescharacteristic
376C2×C4⋊C4C22×D8311yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1661no—
379C2×C22⋊C4—1661no—
380C2×C22⋊C4—1661no—
381C2×C22⋊C4—1661no—
382C2×C42—831no—
383C2×C42—831no—
384C2×C42—1661no—
385C2×C42—1661no—
386C2×C42—1661no—
387C2×C42C22×D8311yescharacteristic
388#107—3321no—
389#107C2×Dic8314yes—
390#107C2×Dic8314yes—
391#106—3321no—
392C4⋊C8—1662no—
393C4⋊C8C2×Dic8314yes—
394#102—3321no—
395#102—3321no—
396#102C2×Dic8314yes—
397#102C2×Dic8314yes—
398#97—3321no—
399C4×C8—3321no—
400#95—831no—
401#95—831no—
402#95—831no—
403#95—831no—
404#95—1661no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412#95—1661no—
413C24—21no—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C22×Dic8312yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—831no—
430C22×C4—831no—
431C22×C4—831no—
432C22×C4—831no—
433C22×C4—1661no—
434C22×C4—1661no—
435C22×C4—1661no—
436C22×C4—1661no—
437C22×C4D4×D8311yescharacteristic
438C22×C4D4×D8311yescharacteristic
439C22×C4C23×D8311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8C22×Dic8312yes—
451C2×C8C22×Dic8312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—6641no—
457C4⋊C4—6641no—
458C4⋊C4—6641no—
459C4⋊C4—6641no—
460C4⋊C4—1661no—
461C4⋊C4—1661no—
462C4⋊C4—1661no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4C22×Dic8312yes—
469C4⋊C4C22×Dic8312yes—
470C4⋊C4C22×Dic8312yes—
471C22⋊C4—24no—
472C22⋊C4—6641no—
473C22⋊C4—6641no—
474C22⋊C4—6641no—
475C22⋊C4—6641no—
476C42—6641no—
477C42—6641no—
478C42—6641no—
479C42—1661no—
480C42—1661no—
481C42—1661no—
482C42—1661no—
483C42C22×Dic8312yes—
484C42#1305058912yes—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C2×D4×D8311yescharacteristic
493Q8—84no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
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—3321no—
523C2×C4—3321no—
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×C4—1661no—
535C2×C4—1661no—
536C2×C4D4×Dic8312yes—
537C2×C4D4×Dic8312yes—
538C2×C4C23×Dic8311yescharacteristic
539C2×C4C2×#1305058911yescharacteristic
540C2×C4#1458183811yescharacteristic
541C2×C4#1458183911yescharacteristic
542C2×C4#1458184011yescharacteristic
543C2×C4#1458184011yescharacteristic
544C8—22no—
545C8—22no—
546C8—3321no—
547C8—3321no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C2×D4×Dic8311yescharacteristic
556C22#1912827411yescharacteristic
557C22#1912828111yescharacteristic
558C22#1912850211yescharacteristic
559C22#1912850211yescharacteristic
560C22#1912851011yescharacteristic
561C22#1912851711yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—3321no—
568C4—3321no—
569C4—3321no—
570C4—3321no—
571C4—6641no—
572C4—6641no—
573C4—6641no—
574C4—6641no—
575C4#1912817911yescharacteristic
576C4#1912818411yescharacteristic
577C4#1912831111yescharacteristic
578C4#1912832411yescharacteristic
579C2—82no—
580C2#2491014711yescharacteristic
581C2#2491630711yescharacteristic
582C2#2491631511yescharacteristic
583C2#2491872311yescharacteristic
584C2#2491874311yescharacteristic
585C2#2491888111yescharacteristic
586C2#2491892311yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#474×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#474D83
hypocenterC83#22127
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. #24910212C2semidirect
3. #24908412C2semidirect
4. #24908233C2semidirect
5. #24908202C2semidirect
6. #24908197C2semidirect
11. #24908057C2semidirect
12. #24908036C2semidirect
14. #24907322C2semidirect
54. #474×C83C4semidirect
309. C83#22127semidirect
312. #474Dic83semidirect

Maximal subgroups

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

subgroupindex
#249102122
#249101062
#249084122
#249082332
#249082022
#249081972
#249081622
#249081482
#249080922
#249080852
#249080572
#249080362
#249074732
#249073222
C2×#474×C832
#2212783