Finite Group Databasebeta

← #35446368

Subgroups — #35446368

Order 21248.

1100 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127949C212yesmaximal, complemented
2C2×#19127901C211yesmaximal, characteristic, complemented
3C2×#19127725C211yesmaximal, characteristic, complemented
4#24915808C211yesmaximal, characteristic
5#24908848C212yesmaximal, complemented
6#24908778C212yesmaximal, complemented
7#24908390C211yesmaximal, characteristic, complemented
8#24908221C211yesmaximal, characteristic, complemented
9#24907899C212yesmaximal, complemented
10#24907468C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C83C211yesmaximal, characteristic, complemented
12C22×#13050577C2212yes—
13C22×#13050574C2212yes—
14C2×#14581789C2211yescharacteristic
15C2×#14581766—21no—
16C2×#14581766—21no—
17C2×#14581766C2212yes—
18#19127949—28no—
19#19127901—24no—
20C2×#14581725C2212yes—
21#19127725—24no—
22C2×C4⋊(C83⋊C8)C2212yes—
23C2×C4⋊(C83⋊C8)C2211yescharacteristic
24C2×C4⋊(C83⋊C8)C2211yescharacteristic
25C2×#14581716C2212yes—
26C2×#14581712C2212yes—
27C2×#14581712C2212yes—
28C2×#14581712C2212yes—
29#19127603—21no—
30#19127603—21no—
31#19127603C2212yes—
32#19127571C2211yescharacteristic
33#19127463C2212yes—
34#19127463C2212yes—
35C2×C22⋊Q8×C83C2211yescharacteristic
36C2×C4×Q8×C83C2211yescharacteristic
37C22×C42×C83C2211yescharacteristic
38#391×C83C2211yescharacteristic
39#389×C83C2211yescharacteristic
40#387×C83—21no—
41#385×C83—21no—
42#380×C83C2211yescharacteristic
43C4×C4⋊C4×C83—21no—
44C4×C4⋊C4×C83C2211yescharacteristic
45C4×C22⋊C4×C83—21no—
46C22×D332—21no—
47C22×D332—21no—
48C22×D332C2312yes—
49C22×C22⋊D83—22no—
50C2×#13050577—24no—
51C2×#13050577—24no—
52C2×#13050577—42no—
53C2×#13050574—24no—
54C2×#13050574—24no—
55C2×#13050574—42no—
56C22×C83⋊C8C2312yes—
57C22×C83⋊C8C2312yes—
58#14581789—22no—
59#14581789—22no—
60#14581766—24no—
61#14581766—24no—
62#14581766—44no—
63#14581766—44no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83C2312yes—
67C2×(C2×D83)⋊C4—22no—
68C2×(C2×D83)⋊C4—22no—
69C2×(C2×D83)⋊C4—22no—
70#14581725—24no—
71C4⋊(C83⋊C8)—24no—
72C4⋊(C83⋊C8)—22no—
73C4⋊(C83⋊C8)D414yescomplemented
74#14581716—24no—
75#14581712—24no—
76#14581712—24no—
77#14581712—42no—
78C22×Q8×C83C2311yescharacteristic
79C23×C4×C83D411yescharacteristic
80C23×C4×C83D411yescharacteristic
81C23×C4×C83C2311yescharacteristic
82C22⋊Q8×C83—22no—
83C22⋊Q8×C83D414yescomplemented
84C4×Q8×C83—22no—
85C4×Q8×C83—22no—
86C2×C4⋊C4×C83—21no—
87C2×C4⋊C4×C83—21no—
88C2×C4⋊C4×C83—21no—
89C2×C4⋊C4×C83C2311yescharacteristic
90C2×C4⋊C4×C83C2311yescharacteristic
91C2×C4⋊C4×C83C2311yescharacteristic
92C2×C22⋊C4×C83—21no—
93C2×C22⋊C4×C83—21no—
94C2×C42×C83—22no—
95C2×C42×C83—22no—
96C2×C42×C83—21no—
97C2×C42×C83—21no—
98C2×C42×C83D411yescharacteristic
99C2×C42×C83D411yescharacteristic
100C2×C42×C83C2311yescharacteristic
101C2×C42×C83C2311yescharacteristic
102#95×C83—21no—
103#95×C83—21no—
104#95×C83—21no—
105#95×C83—21no—
106#95×C83—21no—
107C23×D83—22no—
108C2×D332—22no—
109C2×D332—22no—
110C2×D332—22no—
111C2×D332—22no—
112C2×D332—22no—
113C2×D332—22no—
114C2×D332—42no—
115C2×D332—42no—
116C2×D332—42no—
117C2×D332—42no—
118C2×D332—42no—
119C2×D332—42no—
120C2×C22⋊D83—44no—
121C2×C22⋊D83—44no—
122C2×C22⋊D83—44no—
123C22×Dic83—22no—
124#13050577—48no—
125#13050574—48no—
126C2×C83⋊C8—22no—
127C2×C83⋊C8—22no—
128C2×C83⋊C8—22no—
129C2×C83⋊C8—22no—
130C2×C83⋊C8—22no—
131C2×C83⋊C8C2×D414yes—
132C4⋊Dic83—22no—
133C4⋊Dic83—22no—
134C4⋊Dic83—42no—
135C4⋊Dic83—42no—
136(C2×D83)⋊C4—44no—
137(C2×D83)⋊C4—44no—
138(C2×D83)⋊C4—44no—
139C24×C83C2×D411yescharacteristic
140C2×Q8×C83—21no—
141C2×Q8×C83—21no—
142C2×Q8×C83—21no—
143C2×Q8×C83—21no—
144C2×Q8×C83—21no—
145C2×Q8×C83C2×D412yes—
146C22×C4×C83—21no—
147C22×C4×C83—21no—
148C22×C4×C83—21no—
149C22×C4×C83—21no—
150C22×C4×C83—21no—
151C22×C4×C83—21no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—21no—
158C22×C4×C83—21no—
159C22×C4×C83—41no—
160C22×C4×C83C2×D411yescharacteristic
161C22×C4×C83C2×D411yescharacteristic
162C22×C4×C83C2×D411yescharacteristic
163C22×C4×C83C2×D411yescharacteristic
164C22×C4×C83C2×D411yescharacteristic
165C22×C4×C83C2×D412yes—
166C22×C4×C83#4112yes—
167C22×C4×C83C2411yescharacteristic
168C4⋊C4×C83—24no—
169C4⋊C4×C83—21no—
170C4⋊C4×C83—21no—
171C4⋊C4×C83—21no—
172C4⋊C4×C83—21no—
173C4⋊C4×C83—21no—
174C4⋊C4×C83—42no—
175C4⋊C4×C83—42no—
176C4⋊C4×C83C2×D412yes—
177C22⋊C4×C83—24no—
178C22⋊C4×C83—42no—
179C42×C83—22no—
180C42×C83—22no—
181C42×C83—21no—
182C42×C83—21no—
183C42×C83—42no—
184C42×C83—42no—
185C42×C83C2×D412yes—
186C42×C83#4112yes—
187C23×C83—21no—
188C23×C83—21no—
189C23×C83—21no—
190C23×C83—21no—
191C23×C83—21no—
192C23×C83—21no—
193C23×C83#12112yes—
194C23×C83C22×D411yescharacteristic
195C22×D83—42no—
196C22×D83—42no—
197C22×D83—42no—
198C22×D83—42no—
199C22×D83—42no—
200C22×D83—42no—
201C22×D83—42no—
202Q8×C83—22no—
203Q8×C83—22no—
204Q8×C83—41no—
205Q8×C83—41no—
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—22no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—21no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—44no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—44no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83#12112yes—
234C2×C4×C83#12112yes—
235C2×C4×C83#12112yes—
236C2×C4×C83AGL(1,ℤ/8)12yes—
237C2×C4×C83AGL(1,ℤ/8)12yes—
238C2×C4×C83#13712yes—
239C2×C4×C83C22×D411yescharacteristic
240C2×C4×C83C22×D411yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83H(5,𝔽2)11yescharacteristic
244C2×C4×C832−1+411yescharacteristic
245C22⋊D83—88no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249D332—42no—
250D332—42no—
251D332—42no—
252D332—42no—
253D332—84no—
254D332—84no—
255C83⋊C8—24no—
256C83⋊C8—42no—
257C22×C83—22no—
258C22×C83—22no—
259C22×C83—22no—
260C22×C83—41no—
261C22×C83—41no—
262C22×C83—41no—
263C22×C83—41no—
264C22×C83C2×#12111yescharacteristic
265C22×C83C2×#12111yescharacteristic
266C22×C83#53711yescharacteristic
267C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C83C2×#13711yescharacteristic
270C22×C83#57511yescharacteristic
271C2×D83—82no—
272C2×D83—82no—
273C2×D83—82no—
274C2×D83—82no—
275C2×D83—82no—
276C2×D83—82no—
277C2×D83—82no—
278C4×C83—22no—
279C4×C83—22no—
280C4×C83—22no—
281C4×C83—42no—
282C4×C83—42no—
283C4×C83—41no—
284C4×C83—81no—
285C4×C83—81no—
286C4×C83#46814yescomplemented
287C4×C83D4211yescharacteristic
288C4×C83#54711yescharacteristic
289C4×C83#54811yescharacteristic
290C4×C83#55311yescharacteristic
291Dic83—82no—
292#24761—831nomaximal, Sylow
293C2×C83—42no—
294C2×C83#241611yescharacteristic
295C2×C83#303711yescharacteristic
296C2×C83#303811yescharacteristic
297C2×C83C2×#46811yescharacteristic
298C2×C83#305511yescharacteristic
299C2×C83#325111yescharacteristic
300C2×C83#325411yescharacteristic
301D83—162no—
302C2×#468—832no—
303C2×#460—831no—
304C2×#423—831no—
305#2856—831no—
306C4×C22⋊Q8D8311yescharacteristic, complemented
307#2067—832no—
308#2055—832no—
309#1982—831no—
310#1947—831no—
311#1895—832no—
312#1803—831no—
313C83#2476111yescharacteristic, complemented, Sylow
314C22×SD16—832no—
315C22×M16—832no—
316C2×#127—831no—
317C2×C22⋊Q8C2×D8311yescharacteristic
318C2×#121—832no—
319C2×#121—1661no—
320C2×#121—1661no—
321C2×C4×Q8C2×D8311yescharacteristic
322C22×C42C2×D8311yescharacteristic
323#468—1668no—
324#460—1664no—
325C2×#107—832no—
326#423—1664no—
327C2×C4⋊C8—831no—
328C2×C4⋊C8—832no—
329C2×C4⋊C8—831no—
330C2×#103—832no—
331C2×#102—832no—
332C2×#102—832no—
333C2×#102—832no—
334#391C2×D8311yescharacteristic
335#390—832no—
336#390—1661no—
337#390—1661no—
338#389C2×D8311yescharacteristic
339#387—21no—
340#385—21no—
341#384—831no—
342#380C2×D8311yescharacteristic
343C4×C4⋊C4—21no—
344C4×C4⋊C4C2×D8311yescharacteristic
345C4×C22⋊C4—21no—
346#340—832no—
347#340—832no—
348C22×Q8C22×D8311yescharacteristic
349C22×D4—832no—
350C22×D4—1662no—
351C22×D4—1661no—
352C22×D4—1661no—
353C23×C4C22⋊D8311yescharacteristic
354C23×C4C22⋊D8311yescharacteristic
355C23×C4C22×D8311yescharacteristic
356C2×SD16—3322no—
357C2×SD16—1664no—
358C2×SD16—1664no—
359C2×M16—3322no—
360C2×M16—1664no—
361C2×M16—1664no—
362C22×C8—832no—
363C22×C8—832no—
364#127—1662no—
365#127—1662no—
366C22⋊Q8—22no—
367C22⋊Q8D33214yescomplemented
368#121—3324no—
369#121—3324no—
370#121—1664no—
371#121—1664no—
372C4×Q8—22no—
373C4×Q8—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—832no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4C22×D8311yescharacteristic
381C2×C4⋊C4C22×D8311yescharacteristic
382C2×C4⋊C4C22×D8311yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—1662no—
386C2×C22⋊C4—1662no—
387C2×C22⋊C4—1662no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42C22⋊D8311yescharacteristic
393C2×C42C22⋊D8311yescharacteristic
394C2×C42C22×D8311yescharacteristic
395C2×C42C22×D8311yescharacteristic
396#107—1664no—
397C4⋊C8—834no—
398C4⋊C8—1664no—
399C4⋊C8—1662no—
400#103—1664no—
401#102—3322no—
402#102—1664no—
403#102—1664no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409C24—1662no—
410C24C2×C22⋊D8311yescharacteristic
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8C2×D33212yes—
417C2×D4—3322no—
418C2×D4—3322no—
419C2×D4—3324no—
420C2×D4—3324no—
421C2×D4—3324no—
422C2×D4—3322no—
423C2×D4—3322no—
424C2×D4—3322no—
425C2×D4—3322no—
426C2×D4—1662no—
427C2×D4—1662no—
428C2×D4—1662no—
429C2×D4—1662no—
430C2×D4—1662no—
431C2×D4—1662no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—1662no—
447C22×C4C2×C22⋊D8311yescharacteristic
448C22×C4C2×C22⋊D8311yescharacteristic
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×D33212yes—
453C22×C4#1305059012yes—
454C22×C4C23×D8311yescharacteristic
455SD16—3328no—
456M16—3328no—
457C2×C8—834no—
458C2×C8—1662no—
459C2×C8—1662no—
460C2×C8—1662no—
461C2×C8—1662no—
462C2×C8—1662no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—3322no—
472C4⋊C4—3322no—
473C4⋊C4—1662no—
474C4⋊C4—1662no—
475C4⋊C4C2×D33212yes—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—3324no—
479C22⋊C4—3324no—
480C22⋊C4—3324no—
481C42—22no—
482C42—22no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—42no—
487C42D4×D8312yes—
488C42#1305058912yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—3322no—
496C23—3322no—
497C23—3322no—
498C23—3322no—
499C23—3322no—
500C23—3322no—
501C23—3322no—
502C23#1458176612yes—
503C23C22×C22⋊D8311yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—41no—
507Q8—41no—
508D4—3322no—
509D4—3322no—
510D4—3322no—
511D4—3322no—
512D4—6644no—
513D4—6648no—
514D4—6644no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—3322no—
543C2×C4—3322no—
544C2×C4—3322no—
545C2×C4#1458176412yes—
546C2×C4#1458176412yes—
547C2×C4#1458176612yes—
548C2×C4#1458181712yes—
549C2×C4#1458181712yes—
550C2×C4#1458182312yes—
551C2×C4C2×D4×D8311yescharacteristic
552C2×C4C22×D33211yescharacteristic
553C2×C4C2×#1305058911yescharacteristic
554C2×C4C2×#1305059011yescharacteristic
555C2×C4#1458183811yescharacteristic
556C2×C4#1458183911yescharacteristic
557C8—3322no—
558C8—1664no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—6642no—
567C22—6642no—
568C22—6642no—
569C22—6642no—
570C22—6642no—
571C22—6642no—
572C22—6642no—
573C22C2×#1458176411yescharacteristic
574C22C2×#1458176611yescharacteristic
575C22#1912827911yescharacteristic
576C22C2×#1458181711yescharacteristic
577C22C2×#1458181711yescharacteristic
578C22C2×#1458182311yescharacteristic
579C22#1912849911yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—6642no—
589C4#1912794914yescomplemented
590C4D4×D33211yescharacteristic
591C4#1912835511yescharacteristic
592C4#1912836011yescharacteristic
593C4#1912839711yescharacteristic
594C2—42no—
595C2—13282no—
596C2#2491090311yescharacteristic
597C2#2491693511yescharacteristic
598C2#2491694311yescharacteristic
599C2C2×#1912794911yescharacteristic
600C2#2491708411yescharacteristic
601C2#2491882511yescharacteristic
602C2#2491884611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC4×C22⋊Q8×C83C2
socleC23×C83C22×D4
prime radical#24915808C2
hypercenterC4×C22⋊Q8D83
hypocenterC83#24761
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19127949C2semidirect
2. C2×#19127901C2semidirect
3. C2×#19127725C2semidirect
5. #24908848C2semidirect
6. #24908778C2semidirect
7. #24908390C2semidirect
8. #24908221C2semidirect
9. #24907899C2semidirect
10. #24907468C2semidirect
11. C4×C22⋊Q8×C83C2semidirect
73. C4⋊(C83⋊C8)D4semidirect
83. C22⋊Q8×C83D4semidirect
286. C4×C83#468semidirect
306. C4×C22⋊Q8D83semidirect
313. C83#24761semidirect
367. C22⋊Q8D332semidirect
589. C4#19127949semidirect

Maximal subgroups

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

subgroupindex
C2×#191279492
C2×#191279012
C2×#191277252
#249158082
#249088482
#249087782
#249083902
#249082212
#249078992
#249074682
C4×C22⋊Q8×C832
#2476183