Finite Group Databasebeta

← #35334524

Subgroups — #35334524

Order 21248.

1498 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127594C212yesmaximal
2#24905720C212yesmaximal, complemented
3#24905720C212yesmaximal, complemented
4#24905708C211yesmaximal, characteristic
5#24905697C212yesmaximal, complemented
6#24905695C212yesmaximal, complemented
7#24905685C212yesmaximal
8#24905685C211yesmaximal, characteristic
9C2×#397×C83C211yesmaximal, characteristic
10C24×Dic83C2211yescharacteristic
11C22×C4⋊Dic83C2212yes—
12C22×C4⋊Dic83C2211yescharacteristic
13C22×C22⋊Dic83C2212yes—
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C4×Dic83C2212yes—
16#19127594—28no—
17C2×C83⋊#95C2212yes—
18C2×C83⋊#95C2212yes—
19C2×C83⋊#95C2212yes—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2211yescharacteristic
27C22×C4⋊C4×C83C2211yescharacteristic
28C22×C22⋊C4×C83C2212yes—
29C22×C22⋊C4×C83C2211yescharacteristic
30#397×C83C418yescomplemented
31C2×#95×C83C2212yes—
32C2×#95×C83C2211yescharacteristic
33C23×Dic83—24no—
34C23×Dic83—22no—
35C23×Dic83—41no—
36C23×Dic83—42no—
37C23×Dic83—41no—
38C23×Dic83C2312yes—
39C23×Dic83C2312yes—
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C2×C4⋊Dic83—24no—
43C2×C4⋊Dic83—24no—
44C2×C4⋊Dic83—41no—
45C2×C4⋊Dic83—42no—
46C2×C4⋊Dic83—42no—
47C2×C22⋊Dic83—24no—
48C2×C22⋊Dic83—24no—
49C2×C22⋊Dic83—41no—
50C2×C22⋊Dic83—42no—
51C2×C22⋊Dic83—41no—
52C2×C22⋊Dic83D414yes—
53C2×C4×Dic83—24no—
54C2×C4×Dic83—42no—
55C2×C4×Dic83—42no—
56C83⋊#95—24no—
57C83⋊#95—24no—
58C83⋊#95—24no—
59C83⋊#95—24no—
60C83⋊#95—24no—
61C83⋊#95—41no—
62C83⋊#95—42no—
63C83⋊#95—42no—
64C83⋊#95—42no—
65C83⋊#95—42no—
66C25×C83C2311yescharacteristic
67C23×C4×C83C2312yes—
68C23×C4×C83C2311yescharacteristic
69C23×C4×C83C2312yes—
70C23×C4×C83C2311yescharacteristic
71C2×C4⋊C4×C83—41no—
72C2×C4⋊C4×C83—41no—
73C2×C4⋊C4×C83C2×C414yescomplemented
74C2×C22⋊C4×C83—24no—
75C2×C22⋊C4×C83—24no—
76C2×C22⋊C4×C83—41no—
77C2×C22⋊C4×C83—41no—
78C2×C22⋊C4×C83C2×C418yes—
79C2×C22⋊C4×C83C2×C414yes—
80#95×C83C2×C418yescomplemented
81#95×C83C2×C414yescomplemented
82C22×Dic83—22no—
83C22×Dic83—22no—
84C22×Dic83—22no—
85C22×Dic83—22no—
86C22×Dic83—22no—
87C22×Dic83—22no—
88C22×Dic83—22no—
89C22×Dic83—22no—
90C22×Dic83—22no—
91C22×Dic83—22no—
92C22×Dic83—22no—
93C22×Dic83—22no—
94C22×Dic83—22no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—44no—
108C22×Dic83—48no—
109C22×Dic83—48no—
110C22×Dic83—44no—
111C22×Dic83—44no—
112C22×Dic83C2×D414yes—
113C22×Dic83#4114yes—
114C22×Dic83#4114yes—
115C4⋊Dic83—48no—
116C4⋊Dic83—44no—
117C22⋊Dic83—44no—
118C22⋊Dic83—48no—
119C4×Dic83—48no—
120C24×C83—22no—
121C24×C83—22no—
122C24×C83—21no—
123C24×C83—21no—
124C24×C83—22no—
125C24×C83—21no—
126C24×C83—21no—
127C24×C83—22no—
128C24×C83C22×C412yes—
129C24×C83C2×D412yes—
130C24×C83#4112yes—
131C24×C83C2411yescharacteristic
132C22×C4×C83—22no—
133C22×C4×C83—22no—
134C22×C4×C83—22no—
135C22×C4×C83—22no—
136C22×C4×C83—22no—
137C22×C4×C83—22no—
138C22×C4×C83—22no—
139C22×C4×C83—22no—
140C22×C4×C83—21no—
141C22×C4×C83—21no—
142C22×C4×C83—21no—
143C22×C4×C83—21no—
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83C22×C414yes—
153C22×C4×C83C22×C412yes—
154C22×C4×C83C22×C414yes—
155C22×C4×C83C22×C412yes—
156C4⋊C4×C83—44no—
157C22⋊C4×C83—216no—
158C22⋊C4×C83—44no—
159C23×C83—24no—
160C23×C83—22no—
161C23×C83—22no—
162C23×C83—24no—
163C23×C83—22no—
164C23×C83—22no—
165C23×C83—22no—
166C23×C83—22no—
167C23×C83—24no—
168C23×C83—24no—
169C23×C83—24no—
170C23×C83—24no—
171C23×C83—42no—
172C23×C83—42no—
173C23×C83—42no—
174C23×C83—42no—
175C23×C83—41no—
176C23×C83—42no—
177C23×C83—41no—
178C23×C83—41no—
179C23×C83—42no—
180C23×C83—41no—
181C23×C83C4×D414yes—
182C23×C83C23×C411yescharacteristic
183C23×C83C22×D411yescharacteristic
184C23×C83C2×#4112yes—
185C23×C83C2×#4112yes—
186C23×C83C2×#4111yescharacteristic
187C23×C83H(5,𝔽2)11yescharacteristic
188C23×C83H(5,𝔽2)11yescharacteristic
189C23×C83H(5,𝔽2)11yescharacteristic
190C23×C83H(5,𝔽2)12yes—
191C23×C832−1+411yescharacteristic
192C23×C832−1+412yes—
193C2×C4×C83—24no—
194C2×C4×C83—24no—
195C2×C4×C83—24no—
196C2×C4×C83—24no—
197C2×C4×C83—22no—
198C2×C4×C83—22no—
199C2×C4×C83—24no—
200C2×C4×C83—24no—
201C2×C4×C83—24no—
202C2×C4×C83—24no—
203C2×C4×C83—42no—
204C2×C4×C83—42no—
205C2×C4×C83—42no—
206C2×C4×C83—42no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83—41no—
211C2×C4×C83—42no—
212C2×C4×C83—42no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—42no—
216C2×C4×C83—41no—
217C2×Dic83—24no—
218C2×Dic83—24no—
219C2×Dic83—24no—
220C2×Dic83—24no—
221C2×Dic83—24no—
222C2×Dic83—24no—
223C2×Dic83—24no—
224C2×Dic83—24no—
225C2×Dic83—24no—
226C2×Dic83—416no—
227C2×Dic83—42no—
228C2×Dic83—42no—
229C2×Dic83—42no—
230C2×Dic83—42no—
231C2×Dic83—42no—
232C2×Dic83—42no—
233C2×Dic83—42no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83—42no—
237C2×Dic83—42no—
238C2×Dic83—42no—
239C2×Dic83—42no—
240C2×Dic83—42no—
241C2×Dic83—42no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C22×C83—24no—
247C22×C83—28no—
248C22×C83—44no—
249C22×C83—42no—
250C22×C83—42no—
251C22×C83—44no—
252C22×C83—42no—
253C22×C83—42no—
254C22×C83—42no—
255C22×C83—44no—
256C22×C83—42no—
257C22×C83C2×C4×D411yescharacteristic
258C22×C83C4×#4112yes—
259C22×C83#51811yescharacteristic
260C22×C83#52011yescharacteristic
261C22×C83#52012yes—
262C22×C83#53411yescharacteristic
263C22×C83#53711yescharacteristic
264C22×C83#53811yescharacteristic
265C22×C83#53812yes—
266C22×C83#53912yes—
267C22×C83#53911yescharacteristic
268C22×C83#54112yes—
269C22×C83#54212yes—
270C22×C83#54612yes—
271C22×C83#54712yes—
272C22×C83#55112yes—
273C22×C83#55111yescharacteristic
274C22×C83#55212yes—
275C22×C83#55312yes—
276C22×C83#55911yescharacteristic
277C22×C83H(3,𝔽4)11yescharacteristic
278C22×C83#56212yes—
279C22×C83#56311yescharacteristic
280C4×C83—44no—
281C4×C83—44no—
282C4×C83—42no—
283C4×C83—42no—
284Dic83—44no—
285Dic83—44no—
286Dic83—44no—
287Dic83—82no—
288#14577—831nomaximal, Sylow
289C2×C83—44no—
290C2×C83#232511yescharacteristic
291C2×C83#233412yes—
292C2×C83#235912yes—
293C2×C83#237411yescharacteristic
294C2×C83#237911yescharacteristic
295C2×C83#270812yes—
296C2×C83#271211yescharacteristic
297C2×C83#272211yescharacteristic
298C2×C83#273611yescharacteristic
299C2×C83#273912yes—
300C2×C83#277111yescharacteristic
301C2×#397D8311yescharacteristic
302C2×#388—832no—
303#1442—832no—
304#1442—832no—
305#1439—831no—
306#1436—832no—
307#1436—832no—
308C22⋊#95—831no—
309C22⋊#95—832no—
310C83#1457711yescharacteristic, complemented, Sylow
311C24×C4—831no—
312C22×C4⋊C4—831no—
313C22×C4⋊C4—832no—
314C22×C4⋊C4C2×D8311yescharacteristic
315C22×C22⋊C4—831no—
316C22×C22⋊C4—832no—
317C22×C22⋊C4C2×D8312yes—
318C22×C22⋊C4C2×D8311yescharacteristic
319C22×C42—832no—
320#397Dic8318yescomplemented
321#388—1668no—
322C2×#95—832no—
323C2×#95—832no—
324C2×#95—832no—
325C2×#95—832no—
326C2×#95—831no—
327C2×#95—832no—
328C2×#95—832no—
329C2×#95—832no—
330C2×#95—832no—
331C2×#95—832no—
332C2×#95C2×D8312yes—
333C2×#95C2×D8311yescharacteristic
334C25C22×D8311yescharacteristic
335C23×C4—3321no—
336C23×C4—3321no—
337C23×C4—3322no—
338C23×C4—832no—
339C23×C4—832no—
340C23×C4—832no—
341C23×C4—832no—
342C23×C4—1664no—
343C23×C4—1662no—
344C23×C4C22×D8312yes—
345C23×C4C22×D8311yescharacteristic
346C23×C4C22×D8311yescharacteristic
347C23×C4C22×D8312yes—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—3321no—
351C2×C4⋊C4—3322no—
352C2×C4⋊C4—3322no—
353C2×C4⋊C4—1664no—
354C2×C4⋊C4—1664no—
355C2×C4⋊C4C2×Dic8314yescomplemented
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—41no—
359C2×C22⋊C4—41no—
360C2×C22⋊C4—3322no—
361C2×C22⋊C4—3321no—
362C2×C22⋊C4—3321no—
363C2×C22⋊C4—834no—
364C2×C22⋊C4—1664no—
365C2×C22⋊C4—1664no—
366C2×C22⋊C4C2×Dic8318yes—
367C2×C22⋊C4C2×Dic8314yes—
368C2×C42—3322no—
369C2×C42—3322no—
370C2×C42—1664no—
371#95—3321no—
372#95—3322no—
373#95—3322no—
374#95—3322no—
375#95—3322no—
376#95—1664no—
377#95—1664no—
378#95—1664no—
379#95—1664no—
380#95—1664no—
381#95C2×Dic8318yescomplemented
382#95C2×Dic8314yescomplemented
383C24—22no—
384C24—22no—
385C24—21no—
386C24—21no—
387C24—22no—
388C24—21no—
389C24—22no—
390C24—21no—
391C24C22×Dic8312yes—
392C24D4×D8312yes—
393C24#1305058912yes—
394C24C23×D8311yescharacteristic
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—3324no—
416C22×C4—3324no—
417C22×C4—3328no—
418C22×C4—3328no—
419C22×C4—3324no—
420C22×C4—834no—
421C22×C4—834no—
422C22×C4—834no—
423C22×C4—1662no—
424C22×C4—1662no—
425C22×C4—1662no—
426C22×C4—1662no—
427C22×C4—1662no—
428C22×C4—1662no—
429C22×C4—1662no—
430C22×C4—1662no—
431C22×C4—1662no—
432C22×C4—1662no—
433C22×C4—1662no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4—1662no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4—1662no—
440C22×C4—1662no—
441C22×C4—1662no—
442C22×C4—1662no—
443C22×C4—1662no—
444C22×C4—1662no—
445C22×C4—1662no—
446C22×C4—1662no—
447C22×C4—1662no—
448C22×C4C22×Dic8314yes—
449C22×C4C22×Dic8312yes—
450C22×C4C22×Dic8314yes—
451C22×C4C22×Dic8312yes—
452C4⋊C4—44no—
453C4⋊C4—3324no—
454C4⋊C4—3328no—
455C22⋊C4—216no—
456C22⋊C4—44no—
457C22⋊C4—3328no—
458C22⋊C4—3324no—
459C42—3328no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23D4×Dic8314yes—
483C23C23×Dic8311yescharacteristic
484C23C2×D4×D8311yescharacteristic
485C23#41×D8312yes—
486C23#41×D8312yes—
487C23C2×#1305058911yescharacteristic
488C23#1458183711yescharacteristic
489C23#1458183711yescharacteristic
490C23#1458183812yes—
491C23#1458183811yescharacteristic
492C23#1458183912yes—
493C23#1458183911yescharacteristic
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—33216no—
519C2×C4—3322no—
520C2×C4—3322no—
521C2×C4—3322no—
522C2×C4—3322no—
523C2×C4—3322no—
524C2×C4—3322no—
525C2×C4—3322no—
526C2×C4—3322no—
527C2×C4—3322no—
528C2×C4—3322no—
529C2×C4—3322no—
530C2×C4—3322no—
531C2×C4—3322no—
532C2×C4—3322no—
533C2×C4—3322no—
534C2×C4—3322no—
535C2×C4—3322no—
536C2×C4—3322no—
537C2×C4—3322no—
538C2×C4—1664no—
539C2×C4—1664no—
540C2×C4—1664no—
541C2×C4—1664no—
542C2×C4—1664no—
543C2×C4—1664no—
544C2×C4—1664no—
545C2×C4—1664no—
546C2×C4—1664no—
547C22—24no—
548C22—28no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22C2×D4×Dic8311yescharacteristic
559C22#41×Dic8312yes—
560C22#1912817611yescharacteristic
561C22#1912818412yes—
562C22#1912818411yescharacteristic
563C22#1912825911yescharacteristic
564C22#1912827811yescharacteristic
565C22#1912828811yescharacteristic
566C22#1912829012yes—
567C22#1912829212yes—
568C22#1912829611yescharacteristic
569C22#1912830812yes—
570C22#1912831712yes—
571C22#1912834312yes—
572C22#1912834912yes—
573C22#1912837412yes—
574C22#1912837911yescharacteristic
575C22#1912838812yes—
576C22#1912839612yes—
577C22#1912843511yescharacteristic
578C22#1912844311yescharacteristic
579C22#1912844512yes—
580C22#1912845111yescharacteristic
581C4—42no—
582C4—42no—
583C4—44no—
584C4—44no—
585C4—3324no—
586C4—3324no—
587C4—3324no—
588C4—6642no—
589C2—44no—
590C2#2491016811yescharacteristic
591C2#2491024212yes—
592C2#2491045912yes—
593C2#2491059911yescharacteristic
594C2#2491063711yescharacteristic
595C2#2491418012yes—
596C2#2491423211yescharacteristic
597C2#2491436611yescharacteristic
598C2#2491451011yescharacteristic
599C2#2491455012yes—
600C2#2491498211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#397×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#397D83
hypocenterC83#14577
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. #24905720C2semidirect
3. #24905720C2semidirect
5. #24905697C2semidirect
6. #24905695C2semidirect
30. #397×C83C4semidirect
73. C2×C4⋊C4×C83C2×C4semidirect
80. #95×C83C2×C4semidirect
81. #95×C83C2×C4semidirect
310. C83#14577semidirect
320. #397Dic83semidirect
355. C2×C4⋊C4C2×Dic83semidirect
381. #95C2×Dic83semidirect
382. #95C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275942
#249057202
#249057082
#249056972
#249056952
#249056852
C2×#397×C832
#1457783