Finite Group Databasebeta

← #35446346

Subgroups — #35446346

Order 21248.

1044 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×#19127945C212yesmaximal
2C2×#19127906C211yesmaximal, characteristic, complemented
3C2×#19127725C211yesmaximal, characteristic, complemented
4#24915815C211yesmaximal, characteristic, complemented
5#24908855C212yesmaximal, complemented
6#24908780C212yesmaximal, complemented
7#24908386C211yesmaximal, characteristic, complemented
8#24908221C211yesmaximal, characteristic, complemented
9#24907894C212yesmaximal, complemented
10#24907463C211yesmaximal, characteristic, complemented
11C4×#121×C83C211yesmaximal, characteristic
12C22×#13050578C2212yes—
13C22×#13050574C2212yes—
14C2×#14581792C2211yescharacteristic
15C2×#14581771C2212yes—
16#19127945—28no—
17#19127906—24no—
18C2×#14581725C2212yes—
19#19127725—24no—
20C2×C4⋊(C83⋊C8)C2212yes—
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×#14581715C2212yes—
24C2×#14581715C2212yes—
25C2×#14581715C2212yes—
26C2×#14581713C2212yes—
27#19127607C2212yes—
28#19127599—21no—
29#19127599—21no—
30#19127571C2211yescharacteristic
31#19127561—21no—
32#19127561—21no—
33#19127463C2212yes—
34#19127463C2212yes—
35C2×#121×C83C2211yescharacteristic
36C2×C4×D4×C83—21no—
37C2×C4×D4×C83C2211yescharacteristic
38C22×C42×C83C2211yescharacteristic
39#390×C83C2211yescharacteristic
40#389×C83C2211yescharacteristic
41#388×C83—21no—
42#386×C83—21no—
43#380×C83C2211yescharacteristic
44C4×C4⋊C4×C83C2211yescharacteristic
45C4×C22⋊C4×C83—21no—
46C22×Dic166C2312yes—
47C2×#13050578—24no—
48C2×#13050578—24no—
49C2×#13050578—42no—
50C2×#13050574—24no—
51C2×#13050574—24no—
52C2×#13050574—42no—
53C22×C83⋊C8C2312yes—
54C22×C83⋊C8C2312yes—
55#14581792—22no—
56#14581792—22no—
57#14581771—24no—
58#14581771—24no—
59C2×C4⋊Dic83—21no—
60C2×C4⋊Dic83—21no—
61C2×C4⋊Dic83—21no—
62C2×C4⋊Dic83—21no—
63C2×C4⋊Dic83C2312yes—
64C2×Dic83⋊C4—22no—
65C2×C22⋊Dic83—22no—
66#14581725—24no—
67C4⋊(C83⋊C8)—24no—
68C4⋊(C83⋊C8)—22no—
69C4⋊(C83⋊C8)D414yescomplemented
70#14581715—24no—
71#14581715—24no—
72#14581715—42no—
73#14581713—24no—
74C83⋊#95—22no—
75C83⋊#95—22no—
76C22×D4×C83—21no—
77C22×D4×C83C2311yescharacteristic
78C23×C4×C83—21no—
79C23×C4×C83D411yescharacteristic
80C23×C4×C83D411yescharacteristic
81C23×C4×C83C2311yescharacteristic
82#121×C83—22no—
83#121×C83D414yes—
84C4×D4×C83—22no—
85C4×D4×C83—22no—
86C4×D4×C83—44no—
87C2×C4⋊C4×C83—21no—
88C2×C4⋊C4×C83C2311yescharacteristic
89C2×C4⋊C4×C83C2311yescharacteristic
90C2×C4⋊C4×C83C2311yescharacteristic
91C2×C22⋊C4×C83—21no—
92C2×C22⋊C4×C83—21no—
93C2×C22⋊C4×C83—21no—
94C2×C22⋊C4×C83—21no—
95C2×C22⋊C4×C83—21no—
96C2×C42×C83—22no—
97C2×C42×C83—22no—
98C2×C42×C83—21no—
99C2×C42×C83D411yescharacteristic
100C2×C42×C83D411yescharacteristic
101C2×C42×C83C2311yescharacteristic
102C2×C42×C83C2311yescharacteristic
103#95×C83—21no—
104#95×C83—21no—
105#95×C83—21no—
106C2×Dic166—22no—
107C2×Dic166—22no—
108C2×Dic166—22no—
109C2×Dic166—22no—
110C2×Dic166—22no—
111C2×Dic166—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114#13050578—48no—
115#13050574—48no—
116C2×C83⋊C8—22no—
117C2×C83⋊C8—22no—
118C2×C83⋊C8—22no—
119C2×C83⋊C8—22no—
120C2×C83⋊C8—22no—
121C2×C83⋊C8C2×D414yes—
122C4⋊Dic83—22no—
123C4⋊Dic83—22no—
124C4⋊Dic83—42no—
125C4⋊Dic83—42no—
126C4⋊Dic83—42no—
127C4⋊Dic83—42no—
128Dic83⋊C4—44no—
129C22⋊Dic83—44no—
130C24×C83—21no—
131C24×C83C2×D411yescharacteristic
132C2×D4×C83—24no—
133C2×D4×C83—21no—
134C2×D4×C83—21no—
135C2×D4×C83—21no—
136C2×D4×C83—21no—
137C2×D4×C83—21no—
138C2×D4×C83—42no—
139C2×D4×C83—42no—
140C2×D4×C83C2×D412yes—
141C22×C4×C83—21no—
142C22×C4×C83—21no—
143C22×C4×C83—21no—
144C22×C4×C83—21no—
145C22×C4×C83—21no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—21no—
152C22×C4×C83—21no—
153C22×C4×C83—41no—
154C22×C4×C83—42no—
155C22×C4×C83—42no—
156C22×C4×C83—42no—
157C22×C4×C83C2×D411yescharacteristic
158C22×C4×C83C2×D411yescharacteristic
159C22×C4×C83C2×D411yescharacteristic
160C22×C4×C83C2×D411yescharacteristic
161C22×C4×C83C2×D411yescharacteristic
162C22×C4×C83C2×D412yes—
163C22×C4×C83#4112yes—
164C22×C4×C83C2411yescharacteristic
165C4⋊C4×C83—21no—
166C4⋊C4×C83—21no—
167C4⋊C4×C83—21no—
168C4⋊C4×C83—21no—
169C4⋊C4×C83—21no—
170C4⋊C4×C83—42no—
171C4⋊C4×C83C2×D412yes—
172C22⋊C4×C83—24no—
173C22⋊C4×C83—42no—
174C22⋊C4×C83—42no—
175C22⋊C4×C83—42no—
176C22⋊C4×C83—42no—
177C42×C83—22no—
178C42×C83—22no—
179C42×C83—21no—
180C42×C83—21no—
181C42×C83—42no—
182C42×C83C2×D412yes—
183C42×C83#4112yes—
184C23×C83—21no—
185C23×C83—21no—
186C23×C83—21no—
187C23×C83—21no—
188C23×C83—21no—
189C23×C83—21no—
190C23×C83—22no—
191C23×C83—41no—
192C23×C83—41no—
193C23×C83—41no—
194C23×C83—41no—
195C23×C83—41no—
196C23×C83—41no—
197C23×C83#12112yes—
198C23×C83C22×D411yescharacteristic
199D4×C83—22no—
200D4×C83—22no—
201D4×C83—48no—
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—21no—
209C2×C4×C83—21no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—22no—
213C2×C4×C83—22no—
214C2×C4×C83—21no—
215C2×C4×C83—22no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—44no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—44no—
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×C83—84no—
229C2×C4×C83#12112yes—
230C2×C4×C83#12112yes—
231C2×C4×C83#12112yes—
232C2×C4×C83AGL(1,ℤ/8)12yes—
233C2×C4×C83#13712yes—
234C2×C4×C83#13712yes—
235C2×C4×C83C22×D411yescharacteristic
236C2×C4×C83C22×D411yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83H(5,𝔽2)11yescharacteristic
240C2×C4×C832−1+411yescharacteristic
241C2×Dic83—42no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247Dic166—42no—
248Dic166—42no—
249Dic166—42no—
250Dic166—42no—
251C83⋊C8—24no—
252C83⋊C8—42no—
253C22×C83—22no—
254C22×C83—22no—
255C22×C83—22no—
256C22×C83—41no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83—42no—
261C22×C83—42no—
262C22×C83—42no—
263C22×C83—81no—
264C22×C83—81no—
265C22×C83—81no—
266C22×C83—81no—
267C22×C83C2×#12111yescharacteristic
268C22×C83C2×#12111yescharacteristic
269C22×C83#53711yescharacteristic
270C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C83C2×#13711yescharacteristic
272C22×C83C2×#13711yescharacteristic
273C22×C83#57511yescharacteristic
274C4×C83—22no—
275C4×C83—22no—
276C4×C83—22no—
277C4×C83—42no—
278C4×C83—41no—
279C4×C83—81no—
280C4×C83—81no—
281C4×C83#46814yescomplemented
282C4×C83D4211yescharacteristic
283C4×C83#54711yescharacteristic
284C4×C83#54811yescharacteristic
285C4×C83#55311yescharacteristic
286Dic83—82no—
287Dic83—82no—
288#24759—831nomaximal, Sylow
289C2×C83—42no—
290C2×C83—82no—
291C2×C83#241611yescharacteristic
292C2×C83#303611yescharacteristic
293C2×C83#303911yescharacteristic
294C2×C83C2×#46811yescharacteristic
295C2×C83#305611yescharacteristic
296C2×C83#325211yescharacteristic
297C2×C83#325411yescharacteristic
298C2×#468—832no—
299C2×#461—831no—
300C2×#423—831no—
301#2857—831no—
302C4×#121D8311yescharacteristic
303#2068—832no—
304#2055—832no—
305#1981—831no—
306#1947—831no—
307#1894—832no—
308#1802—831no—
309C83#2475911yescharacteristic, complemented, Sylow
310C22×SD16—832no—
311C22×M16—832no—
312C2×C4⋊Q8—831no—
313C2×C22⋊Q8—832no—
314C2×#121C2×D8311yescharacteristic
315C2×C4×D4—21no—
316C2×C4×D4C2×D8311yescharacteristic
317C22×C42C2×D8311yescharacteristic
318#468—1668no—
319#461—1664no—
320C2×#107—832no—
321#423—1664no—
322C2×C4⋊C8—831no—
323C2×C4⋊C8—832no—
324C2×C4⋊C8—831no—
325C2×#103—832no—
326C2×#103—832no—
327C2×#103—832no—
328C2×#102—832no—
329#391—832no—
330#390C2×D8311yescharacteristic
331#389—1661no—
332#389—1661no—
333#389C2×D8311yescharacteristic
334#388—21no—
335#386—21no—
336#384—831no—
337#380—1661no—
338#380—1661no—
339#380C2×D8311yescharacteristic
340C4×C4⋊C4C2×D8311yescharacteristic
341C4×C22⋊C4—21no—
342#340—832no—
343#340—832no—
344C22×Q8—832no—
345C22×D4—21no—
346C22×D4C22×D8311yescharacteristic
347C23×C4—21no—
348C23×C4C22⋊D8311yescharacteristic
349C23×C4C22⋊D8311yescharacteristic
350C23×C4C22×D8311yescharacteristic
351C2×SD16—3322no—
352C2×SD16—1664no—
353C2×SD16—1664no—
354C2×M16—3322no—
355C2×M16—1664no—
356C2×M16—1664no—
357C22×C8—832no—
358C22×C8—832no—
359C4⋊Q8—1662no—
360C4⋊Q8—1662no—
361C22⋊Q8—1664no—
362C22⋊Q8—1664no—
363#121—22no—
364#121D33214yes—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—44no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—832no—
370C2×C4⋊C4—1662no—
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—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1662no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—21no—
387C2×C42C22⋊D8311yescharacteristic
388C2×C42C22⋊D8311yescharacteristic
389C2×C42C22×D8311yescharacteristic
390C2×C42C22×D8311yescharacteristic
391#107—1664no—
392C4⋊C8—834no—
393C4⋊C8—1664no—
394C4⋊C8—1662no—
395#103—3322no—
396#103—1664no—
397#103—1664no—
398#102—1664no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—1662no—
403#95—1662no—
404C24—21no—
405C24C2×C22⋊D8311yescharacteristic
406C2×Q8—1662no—
407C2×Q8—1662no—
408C2×Q8—1662no—
409C2×Q8—1662no—
410C2×Q8—1662no—
411C2×Q8—1662no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4C2×D33212yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4C2×C22⋊D8311yescharacteristic
440C22×C4C2×C22⋊D8311yescharacteristic
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4C2×C22⋊D8311yescharacteristic
443C22×C4C2×C22⋊D8311yescharacteristic
444C22×C4C2×D33212yes—
445C22×C4#1305059012yes—
446C22×C4C23×D8311yescharacteristic
447SD16—3328no—
448M16—3328no—
449C2×C8—834no—
450C2×C8—1662no—
451C2×C8—1662no—
452C2×C8—1662no—
453C2×C8—1662no—
454C2×C8—1662no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—3322no—
462C4⋊C4—3322no—
463C4⋊C4—3322no—
464C4⋊C4—3324no—
465C4⋊C4—3322no—
466C4⋊C4—1662no—
467C4⋊C4—1662no—
468C4⋊C4C2×D33212yes—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—3324no—
475C42—22no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42D4×D8312yes—
481C42#1305058912yes—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23#1458176612yes—
496C23C22×C22⋊D8311yescharacteristic
497Q8—3322no—
498Q8—3322no—
499Q8—3322no—
500Q8—3322no—
501D4—22no—
502D4—22no—
503D4—48no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—84no—
531C2×C4—3322no—
532C2×C4—3322no—
533C2×C4—3322no—
534C2×C4—3322no—
535C2×C4—3322no—
536C2×C4—3322no—
537C2×C4#1458176412yes—
538C2×C4#1458176412yes—
539C2×C4#1458176612yes—
540C2×C4#1458181812yes—
541C2×C4#1458182212yes—
542C2×C4#1458182212yes—
543C2×C4C2×D4×D8311yescharacteristic
544C2×C4C22×D33211yescharacteristic
545C2×C4C2×#1305058911yescharacteristic
546C2×C4C2×#1305059011yescharacteristic
547C2×C4#1458183811yescharacteristic
548C2×C4#1458183911yescharacteristic
549C8—3322no—
550C8—1664no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C2×#1458176411yescharacteristic
566C22C2×#1458176611yescharacteristic
567C22#1912827911yescharacteristic
568C22C2×#1458181811yescharacteristic
569C22C2×#1458182211yescharacteristic
570C22C2×#1458182211yescharacteristic
571C22#1912849911yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—6642no—
580C4—6642no—
581C4#1912794514yescomplemented
582C4D4×D33211yescharacteristic
583C4#1912835511yescharacteristic
584C4#1912836011yescharacteristic
585C4#1912839711yescharacteristic
586C2—42no—
587C2—82no—
588C2#2491090311yescharacteristic
589C2#2491692711yescharacteristic
590C2#2491695011yescharacteristic
591C2C2×#1912794511yescharacteristic
592C2#2491709511yescharacteristic
593C2#2491883611yescharacteristic
594C2#2491885011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC4×#121×C83C2
socleC23×C83C22×D4
prime radicalC2×#121×C83C22
hypercenterC4×#121D83
hypocenterC83#24759
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#19127906C2semidirect
3. C2×#19127725C2semidirect
4. #24915815C2semidirect
5. #24908855C2semidirect
6. #24908780C2semidirect
7. #24908386C2semidirect
8. #24908221C2semidirect
9. #24907894C2semidirect
10. #24907463C2semidirect
69. C4⋊(C83⋊C8)D4semidirect
281. C4×C83#468semidirect
309. C83#24759semidirect
581. C4#19127945semidirect

Maximal subgroups

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

subgroupindex
C2×#191279452
C2×#191279062
C2×#191277252
#249158152
#249088552
#249087802
#249083862
#249082212
#249078942
#249074632
C4×#121×C832
#2475983