Finite Group Databasebeta

← #35564729

Subgroups — #35564729

Order 21248.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581772C211yesmaximal, characteristic
2C2×#19128178C211yesmaximal, characteristic, complemented
3C2×#117×D83C211yesmaximal, characteristic
4#24910489C214yesmaximal, complemented
5#24910467C214yesmaximal, complemented
6#24910334C214yesmaximal, complemented
7#24910309C214yesmaximal
8#24910143C214yesmaximal, complemented
9#24910047C211yesmaximal, characteristic, complemented
10#24909993C214yesmaximal, complemented
11#24909842C211yesmaximal, characteristic
12#24909820C211yesmaximal, characteristic, complemented
13C83×#2316C211yesmaximal, characteristic, complemented
14C23×Dic166C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16C2×#14581772C4116yescomplemented
17C2×#14581772C2212yes—
18C2×#14581772C2212yes—
19C2×#14581772C2212yes—
20C2×#14581772C2212yes—
21#19128178—28no—
22C2×C4×Dic166C2214yescomplemented
23C2×#14581757C2214yescomplemented
24C2×#14581751C2211yescharacteristic
25C2×#14581751C2211yescharacteristic
26C2×#14581750C2212yes—
27C2×#14581749C2212yes—
28C2×#14581748C2212yes—
29C2×#14581748C2214yes—
30#117×D83—28no—
31C22×C4⋊Dic83C2212yes—
32C2×C4⋊C4×D83C2212yes—
33C22×(C2×D83)⋊C4C2212yes—
34C2×C22⋊C4×D83C2212yes—
35C2×C42×D83C2212yes—
36#19127607C2214yescomplemented
37#19127606C2214yescomplemented
38#19127597C2218yescomplemented
39#19127591C2218yes—
40#19127590C2218yes—
41#19127584C2218yescomplemented
42#19127584C2218yescomplemented
43#19127572C2218yes—
44#19127562C2212yes—
45#19127562C2214yes—
46#19127558C2212yes—
47#19127558C2212yes—
48#19127556C2212yes—
49C4⋊C4×Dic83C2214yescomplemented
50C4⋊C4×Dic83C2214yescomplemented
51C4⋊C4×Dic83C2218yescomplemented
52C4×(C2×D83)⋊C4C2212yes—
53C4×(C2×D83)⋊C4C2212yes—
54C4×C22⋊Dic83C2214yes—
55#19127548C2218yes—
56C2×#117×C83C2211yescharacteristic
57C2×#117×C83C2211yescharacteristic
58C22×C4⋊C4×C83C2211yescharacteristic
59#389×C83C2214yescomplemented
60#389×C83C2214yescomplemented
61#380×C83C2214yes—
62C24×D83C2311yescharacteristic
63C22×Dic166—24no—
64C22×Dic166—24no—
65C22×Dic166C2×C418yescomplemented
66C22×Dic166C2312yes—
67C22×Dic166C2312yes—
68C23×Dic83C2312yes—
69C23×Dic83C2311yescharacteristic
70C22×C4×D83—24no—
71C22×C4×D83—24no—
72C22×C4×D83C2×C418yes—
73C22×C4×D83C2314yes—
74C22×C4×D83C2311yescharacteristic
75C22×C4×D83C2311yescharacteristic
76C22×C4×D83C2311yescharacteristic
77C22×C4×D83C2311yescharacteristic
78#14581772C2×C4116yes—
79#14581772C2×C4116yes—
80#14581772C2×C4116yescomplemented
81#14581772C2×C4116yescomplemented
82C4×Dic166—216no—
83#14581757—216no—
84#14581751—24no—
85#14581751D418yes—
86#14581750—28no—
87#14581749—28no—
88#14581748—28no—
89#14581748—216no—
90C2×C4⋊Dic83C2×C4116yescomplemented
91C2×C4⋊Dic83C2312yes—
92C2×C4⋊Dic83C2314yes—
93C2×C4⋊Dic83C2314yes—
94C2×C4⋊Dic83C2312yes—
95C2×Dic83⋊C4C2318yes—
96C2×Dic83⋊C4C2314yes—
97C4⋊C4×D83—28no—
98C2×C22⋊Dic83C2312yes—
99C2×C22⋊Dic83C2312yes—
100C2×(C2×D83)⋊C4C2×C4116yes—
101C2×(C2×D83)⋊C4C2314yes—
102C2×(C2×D83)⋊C4C2314yes—
103C2×(C2×D83)⋊C4C2314yes—
104C2×(C2×D83)⋊C4C2314yes—
105C2×(C2×D83)⋊C4C2314yes—
106C22⋊C4×D83—28no—
107C2×C4×Dic83C2314yes—
108C2×C4×Dic83C2312yes—
109C2×C4×Dic83C2318yes—
110C2×C4×Dic83C2318yes—
111C2×C4×Dic83C2312yes—
112C42×D83—28no—
113C83⋊#95C2314yes—
114C83⋊#95C2318yes—
115C83⋊#95C2318yes—
116C83⋊#95C2314yes—
117C83⋊#95C2318yes—
118C23×C4×C83C2312yes—
119C23×C4×C83C2311yescharacteristic
120#117×C83—24no—
121#117×C83—24no—
122C2×C4⋊C4×C83C2×C418yescomplemented
123C2×C4⋊C4×C83C2312yes—
124C2×C4⋊C4×C83C2312yes—
125C2×C4⋊C4×C83C2318yes—
126C2×C4⋊C4×C83C2312yes—
127C2×C4⋊C4×C83C2312yes—
128C2×C22⋊C4×C83C2312yes—
129C2×C22⋊C4×C83C2312yes—
130C2×C42×C83C2312yes—
131C2×C42×C83C2312yes—
132#95×C83C2318yes—
133C23×D83—22no—
134C23×D83—22no—
135C23×D83—24no—
136C23×D83—42no—
137C23×D83C22×C414yes—
138C23×D83C2411yescharacteristic
139C23×D83C2411yescharacteristic
140C2×Dic166—232no—
141C2×Dic166—24no—
142C2×Dic166—24no—
143C2×Dic166—24no—
144C2×Dic166—24no—
145C2×Dic166C22×C418yes—
146C2×Dic166C22×C418yes—
147C22×Dic83—22no—
148C22×Dic83—24no—
149C22×Dic83—22no—
150C22×Dic83—24no—
151C22×Dic83C22×C414yes—
152C22×Dic83C22×C418yes—
153C22×Dic83C2414yes—
154C22×Dic83C2411yescharacteristic
155C22×Dic83C2411yescharacteristic
156C22×Dic83C2414yes—
157C22×Dic83C2414yes—
158C2×C4×D83—232no—
159C2×C4×D83—22no—
160C2×C4×D83—22no—
161C2×C4×D83—22no—
162C2×C4×D83—22no—
163C2×C4×D83—22no—
164C2×C4×D83—22no—
165C2×C4×D83—22no—
166C2×C4×D83—22no—
167C2×C4×D83—28no—
168C2×C4×D83—28no—
169C2×C4×D83—44no—
170C2×C4×D83C22×C414yes—
171C2×C4×D83C22×C414yes—
172C2×C4×D83C22×C414yes—
173C2×C4×D83C22×C414yes—
174C4⋊Dic83—24no—
175C4⋊Dic83C22×C4116yes—
176C4⋊Dic83C22×C4116yes—
177C4⋊Dic83C2×D418yes—
178Dic83⋊C4—28no—
179Dic83⋊C4—216no—
180C22⋊Dic83—24no—
181C22⋊Dic83C2×D418yes—
182(C2×D83)⋊C4—28no—
183(C2×D83)⋊C4—28no—
184(C2×D83)⋊C4—44no—
185(C2×D83)⋊C4C22×C4116yes—
186(C2×D83)⋊C4C22×C4116yes—
187C4×Dic83—24no—
188C4×Dic83—28no—
189C4×Dic83—216no—
190C4×Dic83—48no—
191C4×Dic83C2×D418yes—
192C24×C83C2411yescharacteristic
193C22×C4×C83—44no—
194C22×C4×C83C22×C418yes—
195C22×C4×C83C22×C414yes—
196C22×C4×C83C2×D414yes—
197C22×C4×C83#4114yes—
198C22×C4×C83C2414yes—
199C22×C4×C83C2414yes—
200C22×C4×C83C2414yes—
201C22×C4×C83C2411yescharacteristic
202C22×C4×C83C2411yescharacteristic
203C4⋊C4×C83—24no—
204C4⋊C4×C83—24no—
205C4⋊C4×C83—48no—
206C4⋊C4×C83C22×C418yes—
207C4⋊C4×C83C22×C418yes—
208C22⋊C4×C83—24no—
209C22⋊C4×C83—24no—
210C42×C83—24no—
211C42×C83—24no—
212C23×C83—22no—
213C23×C83—22no—
214C23×C83C23×C412yes—
215C23×C83C22×D412yes—
216C23×C83C2×#4112yes—
217C23×C83C2511yescharacteristic
218C22×D83—21no—
219C22×D83—21no—
220C22×D83—22no—
221C22×D83—28no—
222C22×D83—21no—
223C22×D83—21no—
224C22×D83—22no—
225C22×D83—22no—
226C22×D83—28no—
227C22×D83—44no—
228C22×D83—48no—
229C22×D83—48no—
230C22×D83C23×C412yes—
231C22×D83C23×C412yes—
232C22×D832−1+414yes—
233C2×C4×C83—28no—
234C2×C4×C83—24no—
235C2×C4×C83—24no—
236C2×C4×C83—24no—
237C2×C4×C83—24no—
238C2×C4×C83—24no—
239C2×C4×C83—24no—
240C2×C4×C83—48no—
241C2×C4×C83C4×D4116yes—
242C2×C4×C83C23×C418yes—
243C2×C4×C83C23×C412yes—
244C2×C4×C83C23×C412yes—
245C2×C4×C83C22×D412yes—
246C2×C4×C83C22×D412yes—
247C2×C4×C83C2×#4112yes—
248C2×C4×C83C2×#4112yes—
249C2×Dic83—28no—
250C2×Dic83—216no—
251C2×Dic83—21no—
252C2×Dic83—24no—
253C2×Dic83—21no—
254C2×Dic83—21no—
255C2×Dic83—21no—
256C2×Dic83—24no—
257C2×Dic83—24no—
258C2×Dic83—24no—
259C2×Dic83—24no—
260C2×Dic83—24no—
261C2×Dic83—24no—
262C2×Dic83C23×C418yes—
263C2×Dic83C23×C412yes—
264C2×Dic83C23×C412yes—
265C2×Dic83C22×D418yes—
266C4×D83—28no—
267C4×D83—28no—
268C4×D83—28no—
269C4×D83—28no—
270C4×D83—416no—
271Dic166—216no—
272Dic166—216no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—24no—
276C22×C83—24no—
277C22×C83C2×C4×D414yes—
278C22×C83C24×C411yescharacteristic
279C22×C83C23×D411yescharacteristic
280C22×C83C22×#4111yescharacteristic
281C22×C83C2×2−1+412yes—
282C22×C83#58512yes—
283C2×D83—22no—
284C2×D83—22no—
285C2×D83—24no—
286C2×D83—24no—
287C2×D83—22no—
288C2×D83—22no—
289C2×D83—22no—
290C2×D83—416no—
291C2×D83—42no—
292C2×D83—42no—
293C2×D83#51914yescomplemented
294C4×C83—28no—
295C4×C83—44no—
296C4×C83—44no—
297C4×C83C2×C4×D414yes—
298C4×C83C2×C4×D414yes—
299Dic83—28no—
300Dic83—22no—
301Dic83—22no—
302Dic83—28no—
303Dic83—44no—
304#34203—831nomaximal, Sylow
305C2×C83—24no—
306C2×C83C22×C4×D411yescharacteristic
307C2×C83C2×#51911yescharacteristic
308C2×C83#342811yescharacteristic
309C2×C83#348912yes—
310C2×C83#349512yes—
311D83—24no—
312D83—42no—
313C22×C22⋊Q8—831no—
314C2×#519—831no—
315C22×#117—831no—
316#2362—834no—
317#2360—834no—
318#2344—834no—
319#2342—834no—
320#2323—834no—
321#2316D8311yescharacteristic, complemented
322#2310—831no—
323C4×C22⋊Q8—834no—
324#2280—831no—
325#2274—831no—
326C83#3420311yescharacteristic, complemented, Sylow
327C23×Q8—831no—
328C24×C4—831no—
329C2×C22⋊Q8—8316no—
330C2×C22⋊Q8—832no—
331C2×C22⋊Q8—832no—
332C2×C22⋊Q8—832no—
333C2×C22⋊Q8—832no—
334#519—1668no—
335C2×C4×Q8—834no—
336C2×C4×Q8—834no—
337C2×#117—831no—
338C2×#117—831no—
339C2×#117—832no—
340C2×#117—834no—
341C2×#117—832no—
342C2×#117—832no—
343C2×#117—1668no—
344C2×#117C2×D8311yescharacteristic
345C2×#117C2×D8311yescharacteristic
346C22×C4⋊C4—832no—
347C22×C4⋊C4—832no—
348C22×C4⋊C4C2×D8311yescharacteristic
349C22×C22⋊C4—832no—
350C22×C22⋊C4—832no—
351C22×C42—832no—
352#391—834no—
353#391—834no—
354#389—838no—
355#389C2×D8314yescomplemented
356#389C2×D8314yescomplemented
357#388—838no—
358#388—838no—
359#387—838no—
360#387—838no—
361#385—838no—
362#381—834no—
363#381—832no—
364#380—832no—
365#380—832no—
366#380C2×D8314yes—
367#379—832no—
368C4×C4⋊C4—834no—
369C4×C4⋊C4—838no—
370C4×C4⋊C4—834no—
371C4×C22⋊C4—834no—
372C4×C22⋊C4—832no—
373C4×C22⋊C4—832no—
374C4×C22⋊C4—838no—
375C25—831no—
376C22×Q8—838no—
377C22×Q8—832no—
378C22×Q8—832no—
379C22×Q8—1664no—
380C22×Q8—1664no—
381C23×C4—834no—
382C23×C4—832no—
383C23×C4—831no—
384C23×C4—838no—
385C23×C4—831no—
386C23×C4—831no—
387C23×C4—831no—
388C23×C4—831no—
389C23×C4—1664no—
390C23×C4—1664no—
391C23×C4C22×D8312yes—
392C23×C4C22×D8311yescharacteristic
393C22⋊Q8—8316no—
394C22⋊Q8—8316no—
395C22⋊Q8—8316no—
396C22⋊Q8—8316no—
397C4×Q8—16616no—
398C4×Q8—16616no—
399#117—24no—
400#117—24no—
401#117—838no—
402#117—1668no—
403#117—1668no—
404#117—16616no—
405#117—1668no—
406#117—1664no—
407C2×C4⋊C4—8316no—
408C2×C4⋊C4—832no—
409C2×C4⋊C4—834no—
410C2×C4⋊C4—834no—
411C2×C4⋊C4—834no—
412C2×C4⋊C4—838no—
413C2×C4⋊C4—832no—
414C2×C4⋊C4—1668no—
415C2×C4⋊C4C4×D8318yescomplemented
416C2×C4⋊C4C22×D8312yes—
417C2×C4⋊C4C22×D8312yes—
418C2×C4⋊C4C22×D8318yes—
419C2×C4⋊C4C22×D8312yes—
420C2×C4⋊C4C22×D8312yes—
421C2×C22⋊C4—834no—
422C2×C22⋊C4—832no—
423C2×C22⋊C4—8316no—
424C2×C22⋊C4—834no—
425C2×C22⋊C4—834no—
426C2×C22⋊C4—832no—
427C2×C22⋊C4—834no—
428C2×C22⋊C4—834no—
429C2×C22⋊C4—1668no—
430C2×C22⋊C4C22×D8312yes—
431C2×C22⋊C4C22×D8312yes—
432C2×C42—838no—
433C2×C42—832no—
434C2×C42—834no—
435C2×C42—838no—
436C2×C42—832no—
437C2×C42—1668no—
438C2×C42C22×D8312yes—
439C2×C42C22×D8312yes—
440#95—834no—
441#95—838no—
442#95—834no—
443#95—838no—
444#95—838no—
445#95C22×D8318yes—
446C24—3322no—
447C24—834no—
448C24—831no—
449C24—831no—
450C24—1664no—
451C24—1662no—
452C24—1662no—
453C24C23×D8311yescharacteristic
454C2×Q8—838no—
455C2×Q8—838no—
456C2×Q8—1664no—
457C2×Q8—1664no—
458C2×Q8—1664no—
459C2×Q8—1664no—
460C2×Q8—16632no—
461C22×C4—44no—
462C22×C4—3324no—
463C22×C4—838no—
464C22×C4—834no—
465C22×C4—831no—
466C22×C4—834no—
467C22×C4—831no—
468C22×C4—834no—
469C22×C4—834no—
470C22×C4—834no—
471C22×C4—834no—
472C22×C4—834no—
473C22×C4—834no—
474C22×C4—1668no—
475C22×C4—1668no—
476C22×C4—1662no—
477C22×C4—1664no—
478C22×C4—1662no—
479C22×C4—1664no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1662no—
483C22×C4—1662no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—16632no—
489C22×C4C2×C4×D8318yes—
490C22×C4C2×C4×D8314yes—
491C22×C4D4×D8314yes—
492C22×C4#1305059114yes—
493C22×C4C23×D8314yes—
494C22×C4C23×D8314yes—
495C22×C4C23×D8314yes—
496C22×C4C23×D8311yescharacteristic
497C22×C4C23×D8311yescharacteristic
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—48no—
501C4⋊C4—8316no—
502C4⋊C4—8316no—
503C4⋊C4—838no—
504C4⋊C4—1668no—
505C4⋊C4—16616no—
506C4⋊C4—1664no—
507C4⋊C4C2×C4×D8318yes—
508C4⋊C4C2×C4×D8318yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—3324no—
512C22⋊C4—838no—
513C22⋊C4—8316no—
514C22⋊C4—8316no—
515C22⋊C4—1668no—
516C22⋊C4—1668no—
517C22⋊C4—1664no—
518C42—24no—
519C42—24no—
520C42—3328no—
521C42—838no—
522C42—1668no—
523C42—16616no—
524C42—1664no—
525C23—22no—
526C23—22no—
527C23—3328no—
528C23—3328no—
529C23—3324no—
530C23—832no—
531C23—834no—
532C23—832no—
533C23—1668no—
534C23—1668no—
535C23—1661no—
536C23—1661no—
537C23—1662no—
538C23—1662no—
539C23—1661no—
540C23—1661no—
541C23—1662no—
542C23C22×C4×D8312yes—
543C23C2×D4×D8312yes—
544C23C2×#1305059112yes—
545C23C24×D8311yescharacteristic
546Q8—16616no—
547Q8—16616no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—48no—
556C2×C4—33216no—
557C2×C4—838no—
558C2×C4—832no—
559C2×C4—832no—
560C2×C4—838no—
561C2×C4—1664no—
562C2×C4—1664no—
563C2×C4—1664no—
564C2×C4—1668no—
565C2×C4—16616no—
566C2×C4—1661no—
567C2×C4—1664no—
568C2×C4—1661no—
569C2×C4—1661no—
570C2×C4—1661no—
571C2×C4—1664no—
572C2×C4—1664no—
573C2×C4—1664no—
574C2×C4—1668no—
575C2×C4—1668no—
576C2×C4—1668no—
577C2×C4—1668no—
578C2×C4#14581754116yes—
579C2×C4C22×C4×D8318yes—
580C2×C4C22×C4×D8312yes—
581C2×C4C22×C4×D8312yes—
582C2×C4C2×D4×D8312yes—
583C2×C4C2×D4×D8312yes—
584C2×C4C2×#1305059112yes—
585C2×C4C2×#1305059112yes—
586C22—24no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—33216no—
591C22—3322no—
592C22—3322no—
593C22—834no—
594C22—1662no—
595C22—1662no—
596C22—1662no—
597C22—1664no—
598C22—1662no—
599C22—1662no—
600C22—1664no—
601C22C2×#1458175414yes—
602C22C23×C4×D8311yescharacteristic
603C22C22×D4×D8311yescharacteristic
604C22C22×#1305059111yescharacteristic
605C222−1+4×D8312yes—
606C22#1912853612yes—
607C4—28no—
608C4—44no—
609C4—44no—
610C4—3324no—
611C4—1668no—
612C4—1662no—
613C4—1662no—
614C4—1668no—
615C4C2×#1458175414yes—
616C4C2×#1458175414yes—
617C2—24no—
618C2—3322no—
619C2—1664no—
620C2C22×#1458175411yescharacteristic
621C2#519×D8311yescharacteristic
622C2#2492066111yescharacteristic
623C2#2492113812yes—
624C2#2492121812yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2316C2
socle, cosocleC23×C83C25
prime radicalC24×D83C23
hypercenter#2316D83
hypocenterC83#34203
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C2×#19128178C2semidirect
4. #24910489C2semidirect
5. #24910467C2semidirect
6. #24910334C2semidirect
8. #24910143C2semidirect
9. #24910047C2semidirect
10. #24909993C2semidirect
12. #24909820C2semidirect
13. C83×#2316C2semidirect
16. C2×#14581772C4semidirect
22. C2×C4×Dic166C22semidirect
23. C2×#14581757C22semidirect
36. #19127607C22semidirect
37. #19127606C22semidirect
38. #19127597C22semidirect
41. #19127584C22semidirect
42. #19127584C22semidirect
49. C4⋊C4×Dic83C22semidirect
50. C4⋊C4×Dic83C22semidirect
51. C4⋊C4×Dic83C22semidirect
59. #389×C83C22semidirect
60. #389×C83C22semidirect
65. C22×Dic166C2×C4semidirect
80. #14581772C2×C4semidirect
81. #14581772C2×C4semidirect
90. C2×C4⋊Dic83C2×C4semidirect
122. C2×C4⋊C4×C83C2×C4semidirect
293. C2×D83#519semidirect
321. #2316D83semidirect
326. C83#34203semidirect
355. #389C2×D83semidirect
356. #389C2×D83semidirect
415. C2×C4⋊C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C22×#145817722
C2×#191281782
C2×#117×D832
#249104892
#249104672
#249103342
#249103092
#249101432
#249100472
#249099932
#249098422
#249098202
C83×#23162
#3420383