Finite Group Databasebeta

← #35410069

Subgroups — #35410069

Order 21248.

1520 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×#19127840C211yesmaximal, characteristic
2C2×#19127799C212yesmaximal, complemented
3C4×#14581766C211yesmaximal, characteristic
4#24907818C212yesmaximal, complemented
5#24907780C211yesmaximal, characteristic
6#24907717C212yesmaximal, complemented
7#24907651C211yesmaximal, characteristic, complemented
8#24907398C211yesmaximal, characteristic, complemented
9#24907293C212yesmaximal, complemented
10#24907225C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C83C211yesmaximal, characteristic, complemented
12C22×#13050577C2212yes—
13C2×#14581766C2211yescharacteristic
14C2×C4×D332C2211yescharacteristic
15C2×C4×C22⋊D83—21no—
16#19127840C418yescomplemented
17#19127799—28no—
18C2×#14581725C2212yes—
19C2×#14581716C2212yes—
20C2×#14581716C2212yes—
21C2×#14581716C2212yes—
22C2×#14581712C2212yes—
23C2×C22⋊(C83⋊C8)C2211yescharacteristic
24C2×C22⋊(C83⋊C8)C2211yescharacteristic
25C2×#14581703C2212yes—
26#19127603C2211yescharacteristic
27#19127599C2211yescharacteristic
28#19127592—21no—
29#19127580—21no—
30#19127561C2211yescharacteristic
31C4×C4⋊Dic83C2211yescharacteristic
32C4×(C2×D83)⋊C4—21no—
33#19127463C2212yes—
34#19127463C2212yes—
35#19127456C2212yes—
36C23×Q8×C83C2211yescharacteristic
37C2×C4×Q8×C83—28no—
38C2×C4×Q8×C83—21no—
39C2×C4×Q8×C83—21no—
40C2×C4×Q8×C83C2212yes—
41C2×C4×Q8×C83C2212yes—
42C22×C4⋊C4×C83—21no—
43C22×C4⋊C4×C83C2211yescharacteristic
44C22×C42×C83—21no—
45C22×C42×C83C2211yescharacteristic
46C22×D332C2311yescharacteristic
47C22×C22⋊D83—21no—
48C22×C4×D83—21no—
49C2×#13050577—24no—
50C2×#13050577—24no—
51C2×#13050577C2×C418yescomplemented
52C22×C83⋊C8C2312yes—
53C22×C83⋊C8C2312yes—
54#14581766—22no—
55#14581766C2×C414yes—
56C4×D332—24no—
57C4×C22⋊D83—44no—
58C2×C4⋊Dic83C2311yescharacteristic
59C2×C4⋊Dic83C2311yescharacteristic
60C2×C4⋊Dic83C2311yescharacteristic
61C2×Dic83⋊C4—21no—
62C2×C22⋊Dic83—21no—
63C2×(C2×D83)⋊C4—21no—
64C2×(C2×D83)⋊C4—21no—
65C2×(C2×D83)⋊C4—21no—
66C2×(C2×D83)⋊C4—21no—
67C2×C4×Dic83—21no—
68#14581725—24no—
69#14581716—24no—
70#14581716—24no—
71#14581716C2×C418yescomplemented
72#14581712—24no—
73C22⋊(C83⋊C8)—22no—
74C22⋊(C83⋊C8)C2×C414yescomplemented
75#14581703—24no—
76C83⋊#95—21no—
77C83⋊#95—21no—
78C83⋊#95—21no—
79C22×Q8×C83—24no—
80C22×Q8×C83—24no—
81C22×Q8×C83—21no—
82C22×Q8×C83—22no—
83C22×Q8×C83C2×C414yescomplemented
84C22×Q8×C83C2312yes—
85C23×C4×C83—21no—
86C23×C4×C83—21no—
87C23×C4×C83D411yescharacteristic
88C23×C4×C83D411yescharacteristic
89C23×C4×C83C2311yescharacteristic
90C4×Q8×C83—28no—
91C4×Q8×C83—28no—
92C4×Q8×C83—24no—
93C4×Q8×C83—24no—
94C4×Q8×C83—24no—
95C4×Q8×C83D418yescomplemented
96C2×C4⋊C4×C83—28no—
97C2×C4⋊C4×C83—22no—
98C2×C4⋊C4×C83—22no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83—24no—
101C2×C4⋊C4×C83C2312yes—
102C2×C42×C83—28no—
103C2×C42×C83—22no—
104C2×C42×C83—22no—
105C2×C42×C83—24no—
106C2×C42×C83D411yescharacteristic
107C2×C42×C83D411yescharacteristic
108C2×C42×C83C2311yescharacteristic
109C2×C42×C83C2311yescharacteristic
110C23×D83—21no—
111C2×D332—22no—
112C2×D332—22no—
113C2×D332—21no—
114C2×D332C22×C412yes—
115C2×C22⋊D83—24no—
116C2×C22⋊D83—42no—
117C2×C22⋊D83—42no—
118C22×Dic83—21no—
119C22×Dic83—21no—
120C22×Dic83—21no—
121C2×C4×D83—42no—
122C2×C4×D83—42no—
123C2×C4×D83—42no—
124#13050577—216no—
125C2×C83⋊C8—22no—
126C2×C83⋊C8—22no—
127C2×C83⋊C8—22no—
128C2×C83⋊C8—22no—
129C2×C83⋊C8—22no—
130C2×C83⋊C8C22×C414yes—
131C4⋊Dic83—21no—
132C4⋊Dic83—22no—
133C4⋊Dic83—22no—
134C4⋊Dic83C22×C412yes—
135Dic83⋊C4—42no—
136C22⋊Dic83—42no—
137(C2×D83)⋊C4—24no—
138(C2×D83)⋊C4—42no—
139(C2×D83)⋊C4—42no—
140(C2×D83)⋊C4—42no—
141C4×Dic83—42no—
142C24×C83C2×D411yescharacteristic
143C2×Q8×C83—24no—
144C2×Q8×C83—216no—
145C2×Q8×C83—24no—
146C2×Q8×C83—216no—
147C2×Q8×C83—24no—
148C2×Q8×C83—22no—
149C2×Q8×C83—22no—
150C2×Q8×C83—22no—
151C2×Q8×C83C22×C414yes—
152C2×Q8×C83C2×D414yes—
153C2×Q8×C83#4114yes—
154C22×C4×C83—24no—
155C22×C4×C83—24no—
156C22×C4×C83—24no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—24no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—41no—
168C22×C4×C83—42no—
169C22×C4×C83—42no—
170C22×C4×C83C22×C412yes—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83#4112yes—
177C22×C4×C83C2411yescharacteristic
178C4⋊C4×C83—28no—
179C4⋊C4×C83—24no—
180C4⋊C4×C83—28no—
181C4⋊C4×C83—22no—
182C4⋊C4×C83C2×D414yes—
183C42×C83—28no—
184C42×C83—28no—
185C42×C83—22no—
186C42×C83—22no—
187C42×C83—22no—
188C42×C83C2×D414yes—
189C23×C83—21no—
190C23×C83—21no—
191C23×C83—21no—
192C23×C83—21no—
193C23×C83C4×D412yes—
194C23×C83AGL(1,ℤ/8)12yes—
195C23×C83#13712yes—
196C23×C83C22×D411yescharacteristic
197C22×D83—22no—
198C22×D83—41no—
199C22×D83—41no—
200C22×D83—41no—
201C22×D83—41no—
202C22×D83—41no—
203C22×D83—41no—
204Q8×C83—216no—
205Q8×C83—24no—
206Q8×C83—24no—
207Q8×C83—24no—
208Q8×C83C4×D418yescomplemented
209C2×C4×C83—28no—
210C2×C4×C83—216no—
211C2×C4×C83—24no—
212C2×C4×C83—21no—
213C2×C4×C83—24no—
214C2×C4×C83—21no—
215C2×C4×C83—24no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—24no—
223C2×C4×C83—24no—
224C2×C4×C83—21no—
225C2×C4×C83—24no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—44no—
229C2×C4×C83—48no—
230C2×C4×C83—42no—
231C2×C4×C83—42no—
232C2×C4×C83C4×D412yes—
233C2×C4×C83#12012yes—
234C2×C4×C83#12012yes—
235C2×C4×C83#13712yes—
236C2×C4×C83C23×C411yescharacteristic
237C2×C4×C83C22×D412yes—
238C2×C4×C83C2×#4112yes—
239C2×C4×C83C2×#4111yescharacteristic
240C22⋊D83—48no—
241C2×Dic83—22no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250D332—24no—
251D332—42no—
252C4×D83—84no—
253C83⋊C8—24no—
254C83⋊C8—42no—
255C22×C83—22no—
256C22×C83—22no—
257C22×C83—22no—
258C22×C83—22no—
259C22×C83—22no—
260C22×C83—22no—
261C22×C83#44014yescomplemented
262C22×C83C2×C4×D411yescharacteristic
263C22×C83C2×#12011yescharacteristic
264C22×C83#52511yescharacteristic
265C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C83C2×#13711yescharacteristic
267C22×C83C2×#13711yescharacteristic
268C22×C83#57511yescharacteristic
269C2×D83—42no—
270C2×D83—42no—
271C2×D83—42no—
272C2×D83—81no—
273C2×D83—81no—
274C2×D83—81no—
275C2×D83—81no—
276C4×C83—22no—
277C4×C83—22no—
278C4×C83—22no—
279C4×C83—28no—
280C4×C83—22no—
281C4×C83—41no—
282C4×C83—44no—
283C4×C83C2×C4×D412yes—
284C4×C83#52512yes—
285Dic83—42no—
286Dic83—81no—
287Dic83—81no—
288#21454—831nomaximal, Sylow
289C2×C83—24no—
290C2×C83C4×#12011yescharacteristic
291C2×C83C2×#44011yescharacteristic
292C2×C83C4×#13711yescharacteristic
293C2×C83#300411yescharacteristic
294C2×C83#300611yescharacteristic
295C2×C83#311211yescharacteristic
296C2×C83#311711yescharacteristic
297D83—82no—
298C22×C4×Q8D8311yescharacteristic, complemented
299C2×#448—831no—
300C2×#440—832no—
301C4×#121—831no—
302#1880—832no—
303#1874—831no—
304#1863—832no—
305#1852—831no—
306#1787—831no—
307#1763—832no—
308#1749—831no—
309C83#2145411yescharacteristic, complemented, Sylow
310C23×Q8C2×D8311yescharacteristic
311C22×SD16—832no—
312C2×#121—831no—
313C2×C4×Q8—28no—
314C2×C4×Q8—21no—
315C2×C4×Q8—21no—
316C2×C4×Q8C2×D8312yes—
317C2×C4×Q8C2×D8312yes—
318C2×C4×D4—831no—
319C2×C4×D4—1661no—
320C22×C4⋊C4—21no—
321C22×C4⋊C4C2×D8311yescharacteristic
322C22×C42—21no—
323C22×C42C2×D8311yescharacteristic
324#448—838no—
325#440—1668no—
326C2×#107—832no—
327C2×#103—832no—
328C2×#103—832no—
329C2×#103—832no—
330C2×#102—832no—
331C2×C22⋊C8—831no—
332C2×C22⋊C8—831no—
333C2×#97—832no—
334#390—831no—
335#389—831no—
336#388—1661no—
337#386—1661no—
338#380—831no—
339C4×C4⋊C4—831no—
340C4×C22⋊C4—1661no—
341#340—832no—
342#340—832no—
343#336—832no—
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8C4×D8314yescomplemented
349C22×Q8C22×D8312yes—
350C22×D4—831no—
351C22×D4—1661no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—1661no—
355C23×C4C22⋊D8311yescharacteristic
356C23×C4C22⋊D8311yescharacteristic
357C23×C4C22×D8311yescharacteristic
358C2×SD16—838no—
359C2×SD16—1664no—
360C2×SD16—1664no—
361C22×C8—832no—
362C22×C8—832no—
363#121—834no—
364#121—1662no—
365C4×Q8—28no—
366C4×Q8—28no—
367C4×Q8—24no—
368C4×Q8—24no—
369C4×Q8—24no—
370C4×Q8C22⋊D8318yescomplemented
371C4×D4—3324no—
372C4×D4—1664no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—831no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4C22×D8312yes—
383C2×C22⋊C4—1661no—
384C2×C22⋊C4—1661no—
385C2×C22⋊C4—1661no—
386C2×C22⋊C4—1661no—
387C2×C22⋊C4—1661no—
388C2×C42—28no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—24no—
392C2×C42—1661no—
393C2×C42C22⋊D8311yescharacteristic
394C2×C42C22⋊D8311yescharacteristic
395C2×C42C22×D8311yescharacteristic
396C2×C42C22×D8311yescharacteristic
397#107—1664no—
398#103—838no—
399#103—1664no—
400#103—1664no—
401#102—1664no—
402C22⋊C8—834no—
403C22⋊C8—1662no—
404#97—1664no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408C24—1661no—
409C24C2×C22⋊D8311yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8C2×C4×D8314yes—
419C2×Q8C2×C22⋊D8314yes—
420C2×Q8#1305059014yes—
421C2×D4—3322no—
422C2×D4—3322no—
423C2×D4—832no—
424C2×D4—1664no—
425C2×D4—1662no—
426C2×D4—1661no—
427C2×D4—1662no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—3322no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4C2×C4×D8312yes—
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C2×C22⋊D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4#1305059012yes—
457C22×C4C23×D8311yescharacteristic
458SD16—16616no—
459C2×C8—834no—
460C2×C8—1662no—
461C2×C8—1662no—
462C2×C8—1662no—
463C2×C8—1662no—
464C2×C8—1662no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—3322no—
470C4⋊C4—832no—
471C4⋊C4—1662no—
472C4⋊C4—1662no—
473C4⋊C4—1661no—
474C4⋊C4C2×C22⋊D8314yes—
475C22⋊C4—3322no—
476C22⋊C4—3322no—
477C22⋊C4—3322no—
478C22⋊C4—3322no—
479C22⋊C4—1664no—
480C42—28no—
481C42—28no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—3322no—
486C42C2×C22⋊D8314yes—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—3321no—
492C23—3321no—
493C23—3321no—
494C23—3321no—
495C23—3321no—
496C23—3321no—
497C23—1662no—
498C23C4×C22⋊D8312yes—
499C23#1458181712yes—
500C23#1458182312yes—
501C23C22×C22⋊D8311yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8C4×C22⋊D8318yescomplemented
507D4—3328no—
508D4—3322no—
509D4—1664no—
510C2×C4—216no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—6644no—
542C2×C4—1662no—
543C2×C4C4×C22⋊D8312yes—
544C2×C4#1458176112yes—
545C2×C4#1458176112yes—
546C2×C4#1458182312yes—
547C2×C4C22×C4×D8311yescharacteristic
548C2×C4C22×C22⋊D8312yes—
549C2×C4C2×#1305059012yes—
550C2×C4C2×#1305059011yescharacteristic
551C8—3322no—
552C8—1664no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—3322no—
560C22—3322no—
561C22—3322no—
562C22—6641no—
563C22—6641no—
564C22—6641no—
565C22—6641no—
566C22#1912779914yescomplemented
567C22C2×C4×C22⋊D8311yescharacteristic
568C22C2×#1458176111yescharacteristic
569C22#1912821211yescharacteristic
570C22C2×#1458181711yescharacteristic
571C22C2×#1458182311yescharacteristic
572C22C2×#1458182311yescharacteristic
573C22#1912849911yescharacteristic
574C4—28no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—44no—
581C4—3322no—
582C4—6641no—
583C4—6641no—
584C4C2×C4×C22⋊D8312yes—
585C4#1912821212yes—
586C2—24no—
587C2—6642no—
588C2C4×#1458176111yescharacteristic
589C2C2×#1912779911yescharacteristic
590C2C4×#1458182311yescharacteristic
591C2#2491669711yescharacteristic
592C2#2491671511yescharacteristic
593C2#2491758511yescharacteristic
594C2#2491763411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC22×C4×Q8×C83C2
socleC23×C83C22×D4
prime radicalC2×#14581766C22
cosocleC22×C4×C83C24
hypercenterC22×C4×Q8D83
hypocenterC83#21454
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#19127799C2semidirect
4. #24907818C2semidirect
6. #24907717C2semidirect
7. #24907651C2semidirect
8. #24907398C2semidirect
9. #24907293C2semidirect
10. #24907225C2semidirect
11. C22×C4×Q8×C83C2semidirect
16. #19127840C4semidirect
51. C2×#13050577C2×C4semidirect
71. #14581716C2×C4semidirect
74. C22⋊(C83⋊C8)C2×C4semidirect
83. C22×Q8×C83C2×C4semidirect
95. C4×Q8×C83D4semidirect
208. Q8×C83C4×D4semidirect
261. C22×C83#440semidirect
298. C22×C4×Q8D83semidirect
309. C83#21454semidirect
348. C22×Q8C4×D83semidirect
370. C4×Q8C22⋊D83semidirect
506. Q8C4×C22⋊D83semidirect
566. C22#19127799semidirect

Maximal subgroups

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

subgroupindex
C2×#191278402
C2×#191277992
C4×#145817662
#249078182
#249077802
#249077172
#249076512
#249073982
#249072932
#249072252
C22×C4×Q8×C832
#2145483