Finite Group Databasebeta

← #35405773

Subgroups — #35405773

Order 21248.

1802 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581823C211yesmaximal, characteristic
2C2×#19127700C211yesmaximal, characteristic
3C2×#19127710C211yesmaximal, characteristic, complemented
4#24909953C211yesmaximal, characteristic
5#24907818C214yesmaximal, complemented
6#24907804C214yesmaximal, complemented
7#24907182C211yesmaximal, characteristic, complemented
8#24907157C211yesmaximal, characteristic, complemented
9C22×C4×Q8×C83C211yesmaximal, characteristic, complemented
10C22×#13050590C2211yescharacteristic
11C2×#14581823—24no—
12C2×#14581823—24no—
13C2×#14581823C418yescomplemented
14C22×#13050580C2212yes—
15C22×#13050577C2212yes—
16C22×#13050574C2211yescharacteristic
17C22×#13050574C2211yescharacteristic
18C2×#14581751C2211yescharacteristic
19#19127710—24no—
20#19127700—24no—
21C2×#14581716C2214yes—
22C2×#14581715C2212yes—
23C2×#14581712C2212yes—
24#19127607C2212yes—
25#19127603C2212yes—
26#19127580—21no—
27#19127580—21no—
28C4×(C2×D83)⋊C4—22no—
29#19127463C2214yes—
30#19127456C2214yes—
31C23×Q8×C83C2211yescharacteristic
32C2×C4×Q8×C83—24no—
33C2×C4×Q8×C83—24no—
34C2×C4×Q8×C83—22no—
35C2×C4×Q8×C83C2214yes—
36C22×C4⋊C4×C83—21no—
37C22×C4⋊C4×C83C2211yescharacteristic
38C22×C42×C83—21no—
39C22×C42×C83C2211yescharacteristic
40C2×#13050590—22no—
41C2×#13050590—22no—
42C2×#13050590—22no—
43C2×#13050590—22no—
44C2×#13050590—22no—
45C2×#13050590C2×C414yes—
46C22×Dic166C2311yescharacteristic
47C22×D332C2311yescharacteristic
48C22×C22⋊D83—21no—
49C22×C4×D83—21no—
50#14581823—232no—
51C2×#13050580—24no—
52C2×#13050580—24no—
53C2×#13050580C2×C418yescomplemented
54C2×#13050577—24no—
55C2×#13050577—24no—
56C2×#13050577C2×C418yescomplemented
57C2×#13050574—22no—
58C2×#13050574—22no—
59C2×#13050574—22no—
60C2×#13050574—22no—
61C2×#13050574—22no—
62C2×#13050574C2×C414yes—
63C22×C83⋊C8C2312yes—
64C22×C83⋊C8C2312yes—
65#14581751—22no—
66#14581751—22no—
67C2×C4⋊Dic83C2312yes—
68C2×C22⋊Dic83—21no—
69C2×(C2×D83)⋊C4—22no—
70C2×(C2×D83)⋊C4—22no—
71C2×C4×Dic83—21no—
72#14581716—28no—
73#14581715—24no—
74#14581712—24no—
75C83⋊#95—22no—
76C83⋊#95—22no—
77C22×Q8×C83—22no—
78C22×Q8×C83—21no—
79C22×Q8×C83—22no—
80C22×Q8×C83—22no—
81C22×Q8×C83C2×C414yes—
82C22×Q8×C83D414yes—
83C22×Q8×C83D414yes—
84C22×Q8×C83C2312yes—
85C23×C4×C83—21no—
86C23×C4×C83—21no—
87C23×C4×C83D411yescharacteristic
88C23×C4×C83D411yescharacteristic
89C23×C4×C83C2311yescharacteristic
90C4×Q8×C83—216no—
91C4×Q8×C83—28no—
92C4×Q8×C83—28no—
93C2×C4⋊C4×C83—28no—
94C2×C4⋊C4×C83—24no—
95C2×C4⋊C4×C83—21no—
96C2×C4⋊C4×C83—22no—
97C2×C4⋊C4×C83D414yes—
98C2×C4⋊C4×C83C2312yes—
99C2×C42×C83—28no—
100C2×C42×C83—24no—
101C2×C42×C83—22no—
102C2×C42×C83—22no—
103C2×C42×C83D412yes—
104C2×C42×C83C2312yes—
105C23×D83—21no—
106#13050590—28no—
107#13050590—28no—
108#13050590—44no—
109#13050590—44no—
110C2×Dic166—21no—
111C2×Dic166—21no—
112C2×Dic166—21no—
113C2×Dic166—21no—
114C2×Dic166—21no—
115C2×Dic166C22×C412yes—
116C2×D332—21no—
117C2×D332—21no—
118C2×D332—21no—
119C2×D332—21no—
120C2×D332—21no—
121C2×D332C22×C412yes—
122C2×C22⋊D83—24no—
123C2×C22⋊D83—42no—
124C2×C22⋊D83—42no—
125C22×Dic83—21no—
126C22×Dic83—22no—
127C2×C4×D83—24no—
128C2×C4×D83—42no—
129C2×C4×D83—42no—
130#13050580—216no—
131#13050577—216no—
132#13050574—28no—
133#13050574—44no—
134C2×C83⋊C8—22no—
135C2×C83⋊C8—22no—
136C2×C83⋊C8—22no—
137C2×C83⋊C8—22no—
138C2×C83⋊C8—22no—
139C2×C83⋊C8C22×C414yes—
140C4⋊Dic83—22no—
141C4⋊Dic83—22no—
142C22⋊Dic83—42no—
143(C2×D83)⋊C4—44no—
144(C2×D83)⋊C4—44no—
145C4×Dic83—42no—
146C24×C83C2×D411yescharacteristic
147C2×Q8×C83—24no—
148C2×Q8×C83—28no—
149C2×Q8×C83—24no—
150C2×Q8×C83—24no—
151C2×Q8×C83—28no—
152C2×Q8×C83—28no—
153C2×Q8×C83—22no—
154C2×Q8×C83—22no—
155C2×Q8×C83—22no—
156C2×Q8×C83C22⋊C4116yescomplemented
157C2×Q8×C83C22×C414yes—
158C2×Q8×C83C2×D414yes—
159C2×Q8×C83C2×D414yes—
160C22×C4×C83—24no—
161C22×C4×C83—22no—
162C22×C4×C83—24no—
163C22×C4×C83—24no—
164C22×C4×C83—22no—
165C22×C4×C83—24no—
166C22×C4×C83—22no—
167C22×C4×C83—24no—
168C22×C4×C83—22no—
169C22×C4×C83—42no—
170C22×C4×C83—44no—
171C22×C4×C83C22×C412yes—
172C22×C4×C83C2×D412yes—
173C22×C4×C83C2×D412yes—
174C22×C4×C83C2×D412yes—
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D412yes—
177C22×C4×C83C2×D412yes—
178C22×C4×C83#4112yes—
179C22×C4×C83#4112yes—
180C22×C4×C83C2411yescharacteristic
181C4⋊C4×C83—216no—
182C4⋊C4×C83—24no—
183C4⋊C4×C83—22no—
184C4⋊C4×C83C2×D414yes—
185C42×C83—216no—
186C42×C83—24no—
187C42×C83—22no—
188C42×C83—22no—
189C23×C83—22no—
190C23×C83—22no—
191C23×C83C4×D412yes—
192C23×C83#12012yes—
193C23×C83#12112yes—
194C23×C83C22×D411yescharacteristic
195C22×D83—22no—
196C22×D83—41no—
197C22×D83—41no—
198C22×D83—42no—
199C22×D83—42no—
200Q8×C83—24no—
201Q8×C83—24no—
202Q8×C83—24no—
203Q8×C83—216no—
204Q8×C83C2×C22⋊C418yes—
205C2×C4×C83—28no—
206C2×C4×C83—24no—
207C2×C4×C83—22no—
208C2×C4×C83—24no—
209C2×C4×C83—24no—
210C2×C4×C83—22no—
211C2×C4×C83—216no—
212C2×C4×C83—22no—
213C2×C4×C83—24no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—44no—
217C2×C4×C83—48no—
218C2×C4×C83—44no—
219C2×C4×C83C2×C22⋊C414yes—
220C2×C4×C83C4×D414yes—
221C2×C4×C83C4×D412yes—
222C2×C4×C83#12012yes—
223C2×C4×C83#12114yes—
224C2×C4×C83#12112yes—
225C2×C4×C83#12314yes—
226C2×C4×C83#13714yescomplemented
227C2×C4×C83C23×C411yescharacteristic
228C2×C4×C83C22×D411yescharacteristic
229C2×C4×C83C22×D411yescharacteristic
230C2×C4×C83C22×D411yescharacteristic
231C2×C4×C83C2×#4111yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C22⋊D83—48no—
234C2×Dic83—22no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—42no—
238C2×Dic83—42no—
239C2×Dic83—42no—
240D332—22no—
241D332—22no—
242D332—41no—
243D332—41no—
244C4×D83—48no—
245Dic166—22no—
246Dic166—22no—
247Dic166—41no—
248Dic166—41no—
249C83⋊C8—24no—
250C83⋊C8—42no—
251C22×C83—24no—
252C22×C83—22no—
253C22×C83—22no—
254C22×C83—24no—
255C22×C83#38614yes—
256C22×C83C2×C4×D411yescharacteristic
257C22×C83C2×#12011yescharacteristic
258C22×C83C2×#12111yescharacteristic
259C22×C83C2×#13712yes—
260C22×C83#57512yes—
261C2×D83—44no—
262C2×D83—42no—
263C2×D83—81no—
264C2×D83—81no—
265C2×D83—82no—
266C4×C83—24no—
267C4×C83—28no—
268C4×C83—42no—
269C4×C83—44no—
270C4×C83#38614yes—
271C4×C83C22×C22⋊C411yescharacteristic
272C4×C83C2×C4×D411yescharacteristic
273C4×C83C2×#12111yescharacteristic
274C4×C83C2×#12311yescharacteristic
275Dic83—42no—
276Dic83—82no—
277#20926—831nomaximal, Sylow
278C2×C83—24no—
279C2×C83C2×#38611yescharacteristic
280C2×C83C4×#13712yes—
281C2×C83#300411yescharacteristic
282C2×C83#300511yescharacteristic
283C2×C83#303712yes—
284D83—82no—
285C22×#137—831no—
286C22×C4×Q8D8311yescharacteristic, complemented
287C2×#417—831no—
288C2×#419—831no—
289#2297—831no—
290#1880—834no—
291#1878—834no—
292#1736—831no—
293#1728—831no—
294C83#2092611yescharacteristic, complemented, Sylow
295C22×#41—831no—
296C23×Q8C2×D8311yescharacteristic
297C2×#137—838no—
298C2×#137—1664no—
299C2×#137—1664no—
300C22×Q16—832no—
301C22×SD16—832no—
302C22×M16—831no—
303C22×M16—831no—
304C2×C4×Q8—24no—
305C2×C4×Q8—24no—
306C2×C4×Q8—22no—
307C2×C4×Q8C2×D8314yes—
308C2×#117—831no—
309C22×C4⋊C4—21no—
310C22×C4⋊C4C2×D8311yescharacteristic
311C22×C42—21no—
312C22×C42C2×D8311yescharacteristic
313#419—1664no—
314#417—1664no—
315C2×#103—834no—
316C2×#103—832no—
317C2×#102—832no—
318#391—832no—
319#390—832no—
320#386—1661no—
321#386—1661no—
322C4×C22⋊C4—1662no—
323#340—834no—
324#336—834no—
325C2×#41—834no—
326C2×#41—1662no—
327C2×#41—1662no—
328C2×#41—1662no—
329C2×#41—1662no—
330C2×#41—1662no—
331C22×Q8—22no—
332C22×Q8—21no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—831no—
336C22×Q8C4×D8314yes—
337C22×Q8D33214yes—
338C22×Q8C22⋊D8314yes—
339C22×Q8C22×D8312yes—
340C22×D4—831no—
341C22×D4—1661no—
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—1661no—
345C23×C4C22⋊D8311yescharacteristic
346C23×C4C22⋊D8311yescharacteristic
347C23×C4C22×D8311yescharacteristic
348#137—16632no—
349C2×Q16—838no—
350C2×Q16—1664no—
351C2×Q16—1664no—
352C2×SD16—838no—
353C2×SD16—1664no—
354C2×SD16—1664no—
355C2×M16—834no—
356C2×M16—1662no—
357C2×M16—1662no—
358C2×M16—1662no—
359C2×M16—1662no—
360C2×M16—1662no—
361C22×C8—832no—
362C22×C8—832no—
363C4×Q8—216no—
364C4×Q8—28no—
365C4×Q8—28no—
366#117—1662no—
367#117—1662no—
368C2×C4⋊C4—28no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—832no—
373C2×C4⋊C4C22⋊D8314yes—
374C2×C4⋊C4C22×D8312yes—
375C2×C22⋊C4—1661no—
376C2×C22⋊C4—1662no—
377C2×C22⋊C4—1662no—
378C2×C42—28no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—24no—
382C2×C42—1661no—
383C2×C42C22⋊D8312yes—
384C2×C42C22×D8312yes—
385#103—1668no—
386#103—1664no—
387#102—1664no—
388#95—1662no—
389#95—1662no—
390C24—1661no—
391C24C2×C22⋊D8311yescharacteristic
392#41—3324no—
393#41—3324no—
394#41—1668no—
395#41—1668no—
396C2×Q8—24no—
397C2×Q8—28no—
398C2×Q8—24no—
399C2×Q8—28no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—28no—
404C2×Q8—24no—
405C2×Q8—832no—
406C2×Q8—1661no—
407C2×Q8—1661no—
408C2×Q8—1661no—
409C2×Q8—1661no—
410C2×Q8—1661no—
411C2×Q8(C2×D83)⋊C4116yescomplemented
412C2×Q8C2×C4×D8314yes—
413C2×Q8C2×C22⋊D8314yes—
414C2×Q8C2×D33214yes—
415C2×D4—3322no—
416C2×D4—3322no—
417C2×D4—832no—
418C2×D4—1664no—
419C2×D4—1661no—
420C2×D4—1661no—
421C2×D4—1661no—
422C2×D4—1661no—
423C2×D4—1661no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—3322no—
436C22×C4—3322no—
437C22×C4—1662no—
438C22×C4—1661no—
439C22×C4—1664no—
440C22×C4C2×C4×D8312yes—
441C22×C4C2×C22⋊D8312yes—
442C22×C4C2×C22⋊D8312yes—
443C22×C4C2×C22⋊D8311yescharacteristic
444C22×C4C2×C22⋊D8312yes—
445C22×C4C2×C22⋊D8312yes—
446C22×C4C2×D33212yes—
447C22×C4#1305059012yes—
448C22×C4#1305059012yes—
449C22×C4C23×D8311yescharacteristic
450Q16—16616no—
451SD16—16616no—
452M16—3324no—
453M16—1668no—
454C2×C8—834no—
455C2×C8—1662no—
456C2×C8—1662no—
457C2×C8—1662no—
458C2×C8—1662no—
459C2×C8—1662no—
460C4⋊C4—216no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—1662no—
464C4⋊C4—1662no—
465C4⋊C4C2×C22⋊D8314yes—
466C22⋊C4—3324no—
467C22⋊C4—3322no—
468C22⋊C4—3324no—
469C42—216no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—3322no—
474C23—22no—
475C23—22no—
476C23—3321no—
477C23—3321no—
478C23—3322no—
479C23—3322no—
480C23—1662no—
481C23C4×C22⋊D8312yes—
482C23#1458176112yes—
483C23#1458176612yes—
484C23C22×C22⋊D8311yescharacteristic
485Q8—216no—
486Q8—24no—
487Q8—24no—
488Q8—24no—
489Q8—3321no—
490Q8—3321no—
491Q8—1662no—
492Q8—1662no—
493Q8C2×(C2×D83)⋊C418yes—
494D4—3328no—
495D4—3321no—
496D4—3321no—
497D4—1662no—
498D4—1662no—
499C2×C4—28no—
500C2×C4—216no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—3321no—
514C2×C4—3321no—
515C2×C4—3328no—
516C2×C4—3322no—
517C2×C4—3322no—
518C2×C4—3322no—
519C2×C4—1662no—
520C2×C4C2×(C2×D83)⋊C414yes—
521C2×C4C4×C22⋊D8312yes—
522C2×C4C4×C22⋊D8314yes—
523C2×C4#1458176112yes—
524C2×C4#1458176612yes—
525C2×C4#1458176614yes—
526C2×C4#1458177514yes—
527C2×C4#1458182314yescomplemented
528C2×C4C22×C4×D8311yescharacteristic
529C2×C4C22×C22⋊D8311yescharacteristic
530C2×C4C22×C22⋊D8311yescharacteristic
531C2×C4C22×D33211yescharacteristic
532C2×C4C2×#1305059011yescharacteristic
533C2×C4C2×#1305059011yescharacteristic
534C8—3322no—
535C8—1664no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—3322no—
541C22—3324no—
542C22—6642no—
543C22—6641no—
544C22—6641no—
545C22#1912758014yes—
546C22C2×C4×C22⋊D8311yescharacteristic
547C22C2×#1458176111yescharacteristic
548C22C2×#1458176611yescharacteristic
549C22C2×#1458182312yes—
550C22#1912849912yes—
551C4—28no—
552C4—24no—
553C4—42no—
554C4—44no—
555C4—3322no—
556C4—6642no—
557C4#1912758014yes—
558C4C22×(C2×D83)⋊C411yescharacteristic
559C4C2×C4×C22⋊D8311yescharacteristic
560C4C2×#1458176611yescharacteristic
561C4C2×#1458177511yescharacteristic
562C2—24no—
563C2—6642no—
564C2C2×#1912758011yescharacteristic
565C2C4×#1458182312yes—
566C2#2491669711yescharacteristic
567C2#2491670511yescharacteristic
568C2#2491693512yes—
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC22×C4×Q8×C83C2
socleC23×C83C22×D4
prime radicalC22×#13050590C22
cosocleC22×C4×C83C24
hypercenterC22×C4×Q8D83
hypocenterC83#20926
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
3. C2×#19127710C2semidirect
5. #24907818C2semidirect
6. #24907804C2semidirect
7. #24907182C2semidirect
8. #24907157C2semidirect
9. C22×C4×Q8×C83C2semidirect
13. C2×#14581823C4semidirect
53. C2×#13050580C2×C4semidirect
56. C2×#13050577C2×C4semidirect
156. C2×Q8×C83C22⋊C4semidirect
226. C2×C4×C83#137semidirect
286. C22×C4×Q8D83semidirect
294. C83#20926semidirect
411. C2×Q8(C2×D83)⋊C4semidirect
527. C2×C4#14581823semidirect

Maximal subgroups

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

subgroupindex
C22×#145818232
C2×#191277002
C2×#191277102
#249099532
#249078182
#249078042
#249071822
#249071572
C22×C4×Q8×C832
#2092683