Finite Group Databasebeta

← #35422762

Subgroups — #35422762

Order 21248.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127804C211yesmaximal, characteristic
2C2×#19127799C211yesmaximal, characteristic
3C2×#19127786C211yesmaximal, characteristic
4Q8×C4⋊Dic83C211yesmaximal, characteristic, complemented
5#24910371C211yesmaximal, characteristic
6#24908200C211yesmaximal, characteristic, complemented
7#24908195C211yesmaximal, characteristic, complemented
8#24907729C211yesmaximal, characteristic, complemented
9#24907725C211yesmaximal, characteristic
10#24907715C211yesmaximal, characteristic, complemented
11#24907711C211yesmaximal, characteristic
12#24907632C211yesmaximal, characteristic, complemented
13#24907464C211yesmaximal, characteristic
14#24907340C211yesmaximal, characteristic, complemented
15C83×#1805C211yesmaximal, characteristic, complemented
16C22×#13050577C2211yescharacteristic
17C2×Q8×Dic83C2211yescharacteristic
18C2×Q8×Dic83C2211yescharacteristic
19C2×C4×D332C2211yescharacteristic
20C2×#14581754C2211yescharacteristic
21C2×#14581754C2211yescharacteristic
22#19127804—24no—
23#19127799C418yescomplemented
24#19127786—24no—
25C2×#14581725C2211yescharacteristic
26C2×#14581725C2211yescharacteristic
27C2×#14581725C2211yescharacteristic
28C2×#14581723C2211yescharacteristic
29C2×#14581723C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581714C2211yescharacteristic
32C2×#14581714C2211yescharacteristic
33C2×#14581712C2211yescharacteristic
34C2×#14581703C2211yescharacteristic
35C2×#14581702C2211yescharacteristic
36C2×C8×Dic83C2211yescharacteristic
37#19127600—21no—
38#19127600—21no—
39#19127600C2211yescharacteristic
40#19127600C2211yescharacteristic
41#19127597C2211yescharacteristic
42#19127597C2211yescharacteristic
43#19127575—21no—
44#19127575—21no—
45#19127571—21no—
46#19127571C2211yescharacteristic
47#19127560—21no—
48C4×C4⋊Dic83—21no—
49C4×C4⋊Dic83C2211yescharacteristic
50#95×D83—21no—
51#19127464C2211yescharacteristic
52#19127464C2211yescharacteristic
53#19127455C2211yescharacteristic
54#19127455C2211yescharacteristic
55C2×C4×Q8×C83C2211yescharacteristic
56C2×C4⋊C8×C83C2211yescharacteristic
57C2×#103×C83C2211yescharacteristic
58C2×#103×C83C2211yescharacteristic
59#382×C83C2211yescharacteristic
60#340×C83C2211yescharacteristic
61#340×C83C2211yescharacteristic
62C22×D332C2311yescharacteristic
63C22×C4×D83—21no—
64C22×C4×D83—21no—
65C22×C4×D83—21no—
66C2×#13050577C2×C414yes—
67C2×#13050577D414yes—
68C2×#13050577Q814yes—
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71Q8×Dic83—22no—
72Q8×Dic83—22no—
73Q8×Dic83—41no—
74Q8×Dic83—41no—
75C4×D332—22no—
76C4×D332C2×C414yes—
77#14581754—22no—
78#14581754—22no—
79#14581754—41no—
80#14581754—41no—
81C2×C4⋊Dic83—21no—
82C2×C4⋊Dic83—21no—
83C2×C4⋊Dic83—21no—
84C2×C4⋊Dic83—21no—
85C2×C4⋊Dic83—21no—
86C2×C4⋊Dic83—21no—
87C2×C4⋊Dic83D411yescharacteristic
88C2×C4⋊Dic83D411yescharacteristic
89C2×C4⋊Dic83C2311yescharacteristic
90C2×C4⋊Dic83C2311yescharacteristic
91C2×C4⋊Dic83C2311yescharacteristic
92C2×Dic83⋊C4—21no—
93C2×Dic83⋊C4—21no—
94C2×(C2×D83)⋊C4—21no—
95C2×(C2×D83)⋊C4—21no—
96C2×(C2×D83)⋊C4—21no—
97C2×C4×Dic83—21no—
98C2×C4×Dic83—21no—
99C2×C4×Dic83C2311yescharacteristic
100C2×C4×Dic83C2311yescharacteristic
101#14581725—22no—
102#14581725—41no—
103#14581725C2×C414yescomplemented
104#14581723—22no—
105#14581723—41no—
106#14581716C2×C414yescomplemented
107#14581714—22no—
108#14581714—22no—
109#14581712C2×C414yes—
110#14581703C2×C414yescomplemented
111#14581702—22no—
112C8×Dic83—22no—
113C83⋊#95—21no—
114C83⋊#95—21no—
115C83⋊#95—21no—
116C83⋊#95—21no—
117C83⋊#95D411yescharacteristic
118C83⋊#95D411yescharacteristic
119C22×Q8×C83C2311yescharacteristic
120C22×C8×C83C2311yescharacteristic
121C22×C8×C83C2311yescharacteristic
122C4×Q8×C83—22no—
123C4×Q8×C83C2×C414yescomplemented
124C2×C4⋊C4×C83—21no—
125C2×C4⋊C4×C83C2311yescharacteristic
126C2×C4⋊C4×C83C2311yescharacteristic
127C2×C4⋊C4×C83C2311yescharacteristic
128C2×C42×C83—21no—
129C2×C42×C83C2311yescharacteristic
130C4⋊C8×C83—22no—
131#103×C83—22no—
132#103×C83—22no—
133#95×C83—21no—
134#95×C83—21no—
135C23×D83—21no—
136C2×D332—21no—
137C2×D332—21no—
138C2×D332—21no—
139C2×D332C22×C412yes—
140C2×D332C2×D412yes—
141C2×D332C2×Q812yes—
142C22×Dic83—21no—
143C22×Dic83—21no—
144C22×Dic83—21no—
145C22×Dic83C2×D411yescharacteristic
146C22×Dic83C2×D411yescharacteristic
147C22×Dic83C2×D411yescharacteristic
148C22×Dic83C2×D411yescharacteristic
149C2×C4×D83—24no—
150C2×C4×D83—42no—
151C2×C4×D83—42no—
152C2×C4×D83—42no—
153C2×C4×D83—42no—
154C2×C4×D83—81no—
155C2×C4×D83—81no—
156C2×C4×D83—81no—
157C2×C4×D83—81no—
158#13050577C4⋊C4116yescomplemented
159C2×C83⋊C8—21no—
160C2×C83⋊C8—21no—
161C2×C83⋊C8C22×C412yes—
162C2×C83⋊C8C22×C412yes—
163C2×C83⋊C8C2×D412yes—
164C2×C83⋊C8C2×Q812yes—
165C4⋊Dic83—21no—
166C4⋊Dic83—22no—
167C4⋊Dic83—21no—
168C4⋊Dic83—21no—
169C4⋊Dic83—21no—
170C4⋊Dic83—21no—
171C4⋊Dic83—41no—
172C4⋊Dic83—42no—
173C4⋊Dic83—42no—
174C4⋊Dic83—42no—
175C4⋊Dic83—81no—
176C4⋊Dic83—81no—
177C4⋊Dic83—81no—
178C4⋊Dic83C22×C412yes—
179Dic83⋊C4—81no—
180Dic83⋊C4—81no—
181(C2×D83)⋊C4—24no—
182(C2×D83)⋊C4—42no—
183(C2×D83)⋊C4—42no—
184C4×Dic83—21no—
185C4×Dic83—21no—
186C4×Dic83—21no—
187C4×Dic83—21no—
188C4×Dic83—42no—
189C4×Dic83—42no—
190C2×Q8×C83—21no—
191C2×Q8×C83—21no—
192C2×Q8×C83—21no—
193C2×Q8×C83C22×C412yes—
194C2×Q8×C83C2×D412yes—
195C2×Q8×C83C2×Q812yes—
196C22×C4×C83—21no—
197C22×C4×C83—21no—
198C22×C4×C83—21no—
199C22×C4×C83—21no—
200C22×C4×C83C2×D411yescharacteristic
201C22×C4×C83C2×D411yescharacteristic
202C22×C4×C83C2411yescharacteristic
203C2×C8×C83—21no—
204C2×C8×C83—21no—
205C2×C8×C83—21no—
206C2×C8×C83—21no—
207C2×C8×C83—21no—
208C2×C8×C83—21no—
209C4⋊C4×C83—24no—
210C4⋊C4×C83—21no—
211C4⋊C4×C83—21no—
212C4⋊C4×C83—21no—
213C4⋊C4×C83—21no—
214C4⋊C4×C83—21no—
215C4⋊C4×C83C22×C412yes—
216C42×C83—24no—
217C42×C83C22×C412yes—
218C42×C83#4112yes—
219C23×C83C22×D411yescharacteristic
220C22×D83—22no—
221C22×D83—22no—
222C22×D83—22no—
223C22×D83—41no—
224C22×D83—41no—
225C22×D83—41no—
226C22×D83—41no—
227Q8×C83—22no—
228Q8×C83—22no—
229Q8×C83—22no—
230Q8×C83C2×C4⋊C414yes—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—22no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—22no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83C4×D412yes—
248C2×C4×C83C4×D412yes—
249C2×C4×C83C23×C411yescharacteristic
250C2×C4×C83C22×D411yescharacteristic
251C2×C4×C83C22×Q811yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83H(5,𝔽2)11yescharacteristic
254C2×C4×C832−1+411yescharacteristic
255C2×Dic83—21no—
256C2×Dic83—21no—
257C2×Dic83—22no—
258C2×Dic83—21no—
259C2×Dic83—21no—
260C2×Dic83—21no—
261C2×Dic83—21no—
262C2×Dic83—21no—
263C2×Dic83—21no—
264C2×Dic83—21no—
265C2×Dic83—21no—
266C2×Dic83—21no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275C2×Dic83#13712yes—
276D332—22no—
277D332—22no—
278D332—22no—
279D332C2×C4⋊C414yes—
280C4×D83—84no—
281C4×D83—84no—
282C4×D83—84no—
283C8×C83—41no—
284C8×C83—41no—
285C83⋊C8—22no—
286C83⋊C8C2×C4⋊C414yes—
287C22×C83C2×C4×D411yescharacteristic
288C22×C83D4211yescharacteristic
289C22×C83D4×Q811yescharacteristic
290C22×C83C2×#13711yescharacteristic
291C22×C83#57511yescharacteristic
292C22×C83#57611yescharacteristic
293C22×C83#57711yescharacteristic
294C2×D83—24no—
295C2×D83—42no—
296C2×D83—42no—
297C2×D83—42no—
298C2×D83—42no—
299C2×D83—42no—
300C2×D83—42no—
301C4×C83—24no—
302C4×C83—22no—
303C4×C83—22no—
304C4×C83—42no—
305C4×C83—81no—
306C4×C83—81no—
307C4×C83C22×C4⋊C411yescharacteristic
308C4×C83#52011yescharacteristic
309C4×C83#54711yescharacteristic
310C4×C83#55411yescharacteristic
311Dic83—22no—
312Dic83—42no—
313Dic83—41no—
314Dic83—41no—
315Dic83—41no—
316Dic83—81no—
317Dic83—81no—
318#22678—831nomaximal, Sylow
319C2×C83D4×C4⋊C411yescharacteristic
320C2×C83C4×#13711yescharacteristic
321C2×C83#294611yescharacteristic
322C2×C83#326611yescharacteristic
323C2×C83#326811yescharacteristic
324C2×C83#338511yescharacteristic
325C2×C83#338811yescharacteristic
326D83—44no—
327C2×#440—831no—
328C2×#440—831no—
329C2×C4×SD16—831no—
330Q8×C4⋊C4—831no—
331#2349—831no—
332#1943—831no—
333#1942—831no—
334#1865—831no—
335#1864—831no—
336#1863—831no—
337#1862—831no—
338#1847—831no—
339#1805D8311yescharacteristic, complemented
340#1802—831no—
341#1771—831no—
342C83#2267811yescharacteristic, complemented, Sylow
343C22×SD16—831no—
344C2×C4×Q8—831no—
345C2×C4×Q8—831no—
346C2×C4×Q8C2×D8311yescharacteristic
347C2×C4×D4—831no—
348C2×C4×D4—831no—
349C2×C4×D4—831no—
350#440—838no—
351#440—1664no—
352C4×SD16—1664no—
353C2×#107—831no—
354C2×#107—831no—
355C2×#107—831no—
356C2×#106—831no—
357C2×#106—831no—
358C2×C4⋊C8C2×D8311yescharacteristic
359C2×#103—831no—
360C2×#103C2×D8311yescharacteristic
361C2×#103C2×D8311yescharacteristic
362C2×#102—831no—
363C2×#102—831no—
364C2×#102—831no—
365C2×#97—831no—
366C2×#97—831no—
367C2×C4×C8—831no—
368#389—831no—
369#389—831no—
370#389—831no—
371#389—831no—
372#389—1661no—
373#389—1661no—
374#385—1661no—
375#385—1661no—
376#384—831no—
377#384—1661no—
378#382C2×D8311yescharacteristic
379#380—1661no—
380C4×C4⋊C4—831no—
381C4×C4⋊C4—1661no—
382C2×#95—1661no—
383#340—831no—
384#340—831no—
385#340C2×D8311yescharacteristic
386#340C2×D8311yescharacteristic
387#336—831no—
388#336—831no—
389C22×Q8C22×D8311yescharacteristic
390C22×D4—831no—
391C23×C4—1661no—
392C23×C4—1661no—
393C23×C4—1661no—
394C2×SD16—834no—
395C2×SD16—834no—
396C2×SD16—834no—
397C22×C8—831no—
398C22×C8—831no—
399C22×C8C22×D8311yescharacteristic
400C22×C8C22×D8311yescharacteristic
401C4×Q8—22no—
402C4×Q8—3321no—
403C4×Q8—3321no—
404C4×Q8—1662no—
405C4×Q8—1662no—
406C4×Q8C4×D8314yescomplemented
407C4×D4—3321no—
408C4×D4—3321no—
409C4×D4—834no—
410C4×D4—1662no—
411C4×D4—1662no—
412C4×D4—1662no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—831no—
415C2×C4⋊C4—831no—
416C2×C4⋊C4—831no—
417C2×C4⋊C4—831no—
418C2×C4⋊C4—831no—
419C2×C4⋊C4—1661no—
420C2×C4⋊C4—1661no—
421C2×C4⋊C4—1661no—
422C2×C4⋊C4—1661no—
423C2×C4⋊C4—1661no—
424C2×C4⋊C4—1661no—
425C2×C4⋊C4—1661no—
426C2×C4⋊C4—1661no—
427C2×C4⋊C4C22×D8311yescharacteristic
428C2×C4⋊C4C22×D8311yescharacteristic
429C2×C4⋊C4C22×D8311yescharacteristic
430C2×C22⋊C4—1661no—
431C2×C22⋊C4—1661no—
432C2×C22⋊C4—1661no—
433C2×C42—21no—
434C2×C42—831no—
435C2×C42—831no—
436C2×C42—1661no—
437C2×C42—1661no—
438C2×C42C22×D8311yescharacteristic
439#107—3321no—
440#107—834no—
441#107—1662no—
442#106—3321no—
443#106—1662no—
444C4⋊C8—22no—
445#103—22no—
446#103—22no—
447#103—834no—
448#102—834no—
449#102—1662no—
450#102—1662no—
451#97—834no—
452#97—1662no—
453C4×C8—1662no—
454#95—21no—
455#95—21no—
456#95—831no—
457#95—831no—
458#95—1661no—
459#95—1661no—
460#95—1661no—
461#95—1661no—
462C24—1661no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×C4×D8312yes—
467C2×Q8D4×D8312yes—
468C2×Q8Q8×D8312yes—
469C2×D4—832no—
470C2×D4—832no—
471C2×D4—832no—
472C2×D4—1661no—
473C2×D4—1661no—
474C2×D4—1661no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—3322no—
480C22×C4—3322no—
481C22×C4—3322no—
482C22×C4—3322no—
483C22×C4—831no—
484C22×C4—831no—
485C22×C4—831no—
486C22×C4—831no—
487C22×C4—6641no—
488C22×C4—6641no—
489C22×C4—6641no—
490C22×C4—6641no—
491C22×C4—1661no—
492C22×C4—1661no—
493C22×C4—1661no—
494C22×C4—1664no—
495C22×C4D4×D8311yescharacteristic
496C22×C4D4×D8311yescharacteristic
497C22×C4C23×D8311yescharacteristic
498SD16—8316no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—832no—
506C2×C8—832no—
507C2×C8—832no—
508C2×C8—832no—
509C2×C8—1661no—
510C2×C8—1661no—
511C4⋊C4—21no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—3322no—
518C4⋊C4—3322no—
519C4⋊C4—3322no—
520C4⋊C4—3321no—
521C4⋊C4—832no—
522C4⋊C4—6641no—
523C4⋊C4—6641no—
524C4⋊C4—6641no—
525C4⋊C4—6641no—
526C4⋊C4—6641no—
527C4⋊C4—1662no—
528C4⋊C4—1661no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4C2×C4×D8312yes—
534C22⋊C4—3322no—
535C22⋊C4—3322no—
536C22⋊C4—1664no—
537C42—24no—
538C42—3322no—
539C42—3322no—
540C42—1661no—
541C42—1661no—
542C42—1661no—
543C42—1661no—
544C42C2×C4×D8312yes—
545C42#1305058912yes—
546C23—3321no—
547C23—3321no—
548C23—3321no—
549C23—3321no—
550C23—1662no—
551C23—1662no—
552C23—1662no—
553C23C2×D4×D8311yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C4⋊C4×D8314yes—
558D4—834no—
559D4—1662no—
560D4—1662no—
561D4—1662no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—3321no—
586C2×C4—832no—
587C2×C4—6644no—
588C2×C4—6644no—
589C2×C4—6644no—
590C2×C4—1661no—
591C2×C4—1661no—
592C2×C4—1661no—
593C2×C4—1662no—
594C2×C4—1661no—
595C2×C4—1661no—
596C2×C4—1661no—
597C2×C4—1661no—
598C2×C4—1661no—
599C2×C4—1661no—
600C2×C4—1661no—
601C2×C4—1661no—
602C2×C4(C22⋊D83)⋊C412yes—
603C2×C4(C22⋊D83)⋊C412yes—
604C2×C4C22×C4×D8311yescharacteristic
605C2×C4C2×D4×D8311yescharacteristic
606C2×C4C2×Q8×D8311yescharacteristic
607C2×C4C2×#1305058911yescharacteristic
608C2×C4#1458183811yescharacteristic
609C2×C4#1458183911yescharacteristic
610C8—41no—
611C8—41no—
612C8—834no—
613C8—1662no—
614C22—3322no—
615C22—3322no—
616C22—3322no—
617C22—3322no—
618C22—3322no—
619C22—3322no—
620C22—1664no—
621C22C2×(C22⋊D83)⋊C411yescharacteristic
622C22#1912833511yescharacteristic
623C22#1912836411yescharacteristic
624C22#137×D8311yescharacteristic
625C22#1912850011yescharacteristic
626C22#1912850311yescharacteristic
627C22#1912850811yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—3322no—
635C4—3321no—
636C4—3321no—
637C4—3321no—
638C4—6641no—
639C4—6641no—
640C4—1662no—
641C4C2×C4⋊C4×D8311yescharacteristic
642C4#1912818111yescharacteristic
643C4#1912835111yescharacteristic
644C4#1912840911yescharacteristic
645C2—3324no—
646C2#2491035911yescharacteristic
647C2#2491627811yescharacteristic
648C2#2491629811yescharacteristic
649C2#2491896811yescharacteristic
650C2#2491899811yescharacteristic
651C2#2492037611yescharacteristic
652C2#2492041111yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1805C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenter#1805D83
hypocenterC83#22678
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. Q8×C4⋊Dic83C2semidirect
6. #24908200C2semidirect
7. #24908195C2semidirect
8. #24907729C2semidirect
10. #24907715C2semidirect
12. #24907632C2semidirect
14. #24907340C2semidirect
15. C83×#1805C2semidirect
23. #19127799C4semidirect
103. #14581725C2×C4semidirect
106. #14581716C2×C4semidirect
110. #14581703C2×C4semidirect
123. C4×Q8×C83C2×C4semidirect
158. #13050577C4⋊C4semidirect
339. #1805D83semidirect
342. C83#22678semidirect
406. C4×Q8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191278042
C2×#191277992
C2×#191277862
Q8×C4⋊Dic832
#249103712
#249082002
#249081952
#249077292
#249077252
#249077152
#249077112
#249076322
#249074642
#249073402
C83×#18052
#2267883