Finite Group Databasebeta

← #35564730

Subgroups — #35564730

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×#14581771C211yesmaximal, characteristic
2C2×#19128178C211yesmaximal, characteristic, complemented
3#24910490C214yesmaximal, complemented
4#24910468C214yesmaximal, complemented
5#24910333C214yesmaximal, complemented
6#24910308C214yesmaximal
7#24910145C214yesmaximal, complemented
8#24910079C211yesmaximal, characteristic, complemented
9#24910047C211yesmaximal, characteristic, complemented
10C4×#14581771C214yesmaximal, complemented
11#24909846C211yesmaximal, characteristic
12#24909819C211yesmaximal, characteristic, complemented
13C22×#117×C83C211yesmaximal, characteristic
14C23×Dic166C2211yescharacteristic
15C2×#14581771C4116yescomplemented
16C2×#14581771C2212yes—
17C2×#14581771C2212yes—
18C2×#14581771C2212yes—
19C2×#14581771C2212yes—
20#19128178—28no—
21C2×C4×Dic166C2214yescomplemented
22C2×#14581757C2214yescomplemented
23C2×#14581751C2211yescharacteristic
24C2×#14581751C2211yescharacteristic
25C2×#14581748C2214yes—
26C22×C4⋊Dic83C2211yescharacteristic
27C22×Dic83⋊C4C2212yes—
28C22×C22⋊Dic83C2212yes—
29#19127607C2214yescomplemented
30#19127606C2214yescomplemented
31#19127599C2214yescomplemented
32#19127597C2214yescomplemented
33#19127596C2218yescomplemented
34#19127595C2218yes—
35#19127589C2218yes—
36#19127585C2218yescomplemented
37#19127582C2218yescomplemented
38#19127574C2218yes—
39#19127564C2212yes—
40#19127562C2214yes—
41#19127561C2212yes—
42#19127561C2212yes—
43#19127559C2214yes—
44#19127557C2212yes—
45C4⋊C4×Dic83C2214yescomplemented
46C4×Dic83⋊C4C2218yescomplemented
47C4×C4⋊Dic83C2214yescomplemented
48C22⋊C4×Dic83C2214yes—
49C4×C22⋊Dic83C2212yes—
50C4×C22⋊Dic83C2212yes—
51C4×C22⋊Dic83C2218yes—
52C24×C4×C83C2211yescharacteristic
53C2×#117×C83—28no—
54C2×#117×C83C2212yes—
55C2×#117×C83C2212yes—
56C2×#117×C83C2211yescharacteristic
57C2×#117×C83C2212yes—
58C2×#117×C83C2211yescharacteristic
59C22×C4⋊C4×C83C2212yes—
60C22×C22⋊C4×C83C2212yes—
61C22×C42×C83C2212yes—
62C22×Dic166—24no—
63C22×Dic166—24no—
64C22×Dic166C2×C418yescomplemented
65C22×Dic166C2312yes—
66C22×Dic166C2312yes—
67C23×Dic83C2312yes—
68C23×Dic83C2312yes—
69#14581771C2×C4116yes—
70#14581771C2×C4116yes—
71#14581771C2×C4116yescomplemented
72#14581771C2×C4116yescomplemented
73C4×Dic166—216no—
74#14581757—216no—
75#14581751—24no—
76#14581751—24no—
77#14581748—216no—
78C2×C4⋊Dic83C2×C418yescomplemented
79C2×C4⋊Dic83C2312yes—
80C2×C4⋊Dic83C2312yes—
81C2×C4⋊Dic83C2312yes—
82C2×C4⋊Dic83C2312yes—
83C2×Dic83⋊C4C2×C4116yescomplemented
84C2×Dic83⋊C4C2318yes—
85C2×Dic83⋊C4C2318yes—
86C2×Dic83⋊C4C2314yes—
87C2×Dic83⋊C4C2314yes—
88C2×C22⋊Dic83C2×C4116yes—
89C2×C22⋊Dic83C2312yes—
90C2×C22⋊Dic83C2314yes—
91C2×C22⋊Dic83C2312yes—
92C2×C22⋊Dic83C2314yes—
93C2×C22⋊Dic83C2314yes—
94C2×C22⋊Dic83C2314yes—
95C2×C4×Dic83C2318yes—
96C2×C4×Dic83C2312yes—
97C2×C4×Dic83C2312yes—
98C2×C4×Dic83C2318yes—
99C83⋊#95C2318yes—
100C83⋊#95C2314yes—
101C83⋊#95C2318yes—
102C83⋊#95C2318yes—
103C83⋊#95C2314yes—
104C83⋊#95C2318yes—
105C25×C83C2311yescharacteristic
106C23×C4×C83—24no—
107C23×C4×C83—24no—
108C23×C4×C83C2×C418yes—
109C23×C4×C83C2314yes—
110C23×C4×C83C2311yescharacteristic
111C23×C4×C83C2311yescharacteristic
112C23×C4×C83C2311yescharacteristic
113C23×C4×C83C2311yescharacteristic
114C23×C4×C83C2311yescharacteristic
115C23×C4×C83C2311yescharacteristic
116#117×C83—24no—
117#117×C83—28no—
118#117×C83—28no—
119#117×C83—28no—
120#117×C83D418yes—
121C2×C4⋊C4×C83—28no—
122C2×C4⋊C4×C83C2314yes—
123C2×C4⋊C4×C83C2312yes—
124C2×C4⋊C4×C83C2312yes—
125C2×C22⋊C4×C83—28no—
126C2×C22⋊C4×C83C2312yes—
127C2×C22⋊C4×C83C2314yes—
128C2×C22⋊C4×C83C2312yes—
129C2×C42×C83—28no—
130C2×C42×C83C2314yes—
131C2×C42×C83C2312yes—
132C2×C42×C83C2312yes—
133C2×Dic166—232no—
134C2×Dic166—24no—
135C2×Dic166—24no—
136C2×Dic166—24no—
137C2×Dic166—24no—
138C2×Dic166C22×C418yes—
139C2×Dic166C22×C418yes—
140C22×Dic83—24no—
141C22×Dic83—24no—
142C22×Dic83—44no—
143C22×Dic83C22×C418yes—
144C22×Dic83C22×C418yes—
145C22×Dic83C2414yes—
146C22×Dic83C2414yes—
147C22×Dic83C2414yes—
148C22×Dic83C2414yes—
149C4⋊Dic83—24no—
150C4⋊Dic83—24no—
151C4⋊Dic83C22×C418yes—
152C4⋊Dic83C22×C418yes—
153Dic83⋊C4—216no—
154Dic83⋊C4—48no—
155Dic83⋊C4C22×C4116yes—
156Dic83⋊C4C22×C4116yes—
157C22⋊Dic83—24no—
158C22⋊Dic83—24no—
159C22⋊Dic83—28no—
160C22⋊Dic83—44no—
161C22⋊Dic83C22×C4116yes—
162C22⋊Dic83C22×C4116yes—
163C4×Dic83—216no—
164C4×Dic83—24no—
165C4×Dic83—24no—
166C4×Dic83—48no—
167C24×C83—22no—
168C24×C83—22no—
169C24×C83—24no—
170C24×C83—42no—
171C24×C83C22×C414yes—
172C24×C83C2411yescharacteristic
173C24×C83C2411yescharacteristic
174C24×C83C2411yescharacteristic
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83—232no—
178C22×C4×C83—22no—
179C22×C4×C83—22no—
180C22×C4×C83—22no—
181C22×C4×C83—22no—
182C22×C4×C83—22no—
183C22×C4×C83—22no—
184C22×C4×C83—22no—
185C22×C4×C83—22no—
186C22×C4×C83—28no—
187C22×C4×C83—28no—
188C22×C4×C83—44no—
189C22×C4×C83C22×C414yes—
190C22×C4×C83C22×C414yes—
191C22×C4×C83C22×C414yes—
192C22×C4×C83C22×C414yes—
193C22×C4×C83C22×C414yes—
194C22×C4×C83C22×C414yes—
195C22×C4×C83C2×D414yes—
196C22×C4×C83#4114yes—
197C22×C4×C83C2414yes—
198C22×C4×C83C2411yescharacteristic
199C22×C4×C83C2411yescharacteristic
200C22×C4×C83C2414yes—
201C22×C4×C83C2411yescharacteristic
202C22×C4×C83C2411yescharacteristic
203C4⋊C4×C83—28no—
204C4⋊C4×C83—24no—
205C4⋊C4×C83C2×D418yes—
206C22⋊C4×C83—24no—
207C22⋊C4×C83—28no—
208C22⋊C4×C83C2×D418yes—
209C42×C83—28no—
210C42×C83—24no—
211C42×C83C2×D418yes—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83—22no—
215C23×C83—22no—
216C23×C83—22no—
217C23×C83—28no—
218C23×C83—28no—
219C23×C83—21no—
220C23×C83—21no—
221C23×C83—22no—
222C23×C83—22no—
223C23×C83—44no—
224C23×C83—48no—
225C23×C83—48no—
226C23×C83C23×C412yes—
227C23×C83C23×C412yes—
228C23×C83C23×C412yes—
229C23×C83C22×D412yes—
230C23×C83C2×#4112yes—
231C23×C832−1+414yes—
232C23×C83C2511yescharacteristic
233C2×C4×C83—28no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—24no—
239C2×C4×C83—24no—
240C2×C4×C83—28no—
241C2×C4×C83—28no—
242C2×C4×C83—28no—
243C2×C4×C83—28no—
244C2×C4×C83—24no—
245C2×C4×C83—24no—
246C2×C4×C83—24no—
247C2×C4×C83—416no—
248C2×C4×C83C4×D4116yes—
249C2×C4×C83C23×C412yes—
250C2×C4×C83C23×C412yes—
251C2×C4×C83C23×C412yes—
252C2×C4×C83C23×C412yes—
253C2×C4×C83C22×D418yes—
254C2×C4×C83C22×D412yes—
255C2×C4×C83C22×D412yes—
256C2×C4×C83C2×#4112yes—
257C2×C4×C83C2×#4112yes—
258C2×Dic83—216no—
259C2×Dic83—24no—
260C2×Dic83—24no—
261C2×Dic83—28no—
262C2×Dic83—24no—
263C2×Dic83—24no—
264C2×Dic83—24no—
265C2×Dic83—24no—
266C2×Dic83—24no—
267C2×Dic83—24no—
268C2×Dic83—48no—
269C2×Dic83C23×C418yes—
270C2×Dic83C23×C418yes—
271Dic166—216no—
272Dic166—216no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—22no—
276C22×C83—22no—
277C22×C83—24no—
278C22×C83—24no—
279C22×C83—24no—
280C22×C83—24no—
281C22×C83—22no—
282C22×C83—22no—
283C22×C83—22no—
284C22×C83—416no—
285C22×C83—42no—
286C22×C83—42no—
287C22×C83C2×C4×D414yes—
288C22×C83#51914yescomplemented
289C22×C83C24×C411yescharacteristic
290C22×C83C23×D411yescharacteristic
291C22×C83C22×#4111yescharacteristic
292C22×C83C2×2−1+412yes—
293C22×C83#58512yes—
294C4×C83—22no—
295C4×C83—22no—
296C4×C83—28no—
297C4×C83—44no—
298C4×C83C2×C4×D414yes—
299C4×C83C2×C4×D414yes—
300Dic83—28no—
301Dic83—28no—
302Dic83—44no—
303Dic83—44no—
304#34203—831nomaximal, Sylow
305C2×C83—24no—
306C2×C83—24no—
307C2×C83—42no—
308C2×C83C22×C4×D411yescharacteristic
309C2×C83C2×#51911yescharacteristic
310C2×C83#342811yescharacteristic
311C2×C83#348912yes—
312C2×C83#349512yes—
313C22×C22⋊Q8—831no—
314C2×#519—831no—
315C22×#117D8311yescharacteristic
316#2362—834no—
317#2360—834no—
318#2344—834no—
319#2342—834no—
320#2323—834no—
321#2316—831no—
322#2310—831no—
323C4×C22⋊Q8—834no—
324#2280—831no—
325#2274—831no—
326C83#3420311yescharacteristic, complemented, Sylow
327C23×Q8—831no—
328C24×C4C2×D8311yescharacteristic
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—28no—
338C2×#117—831no—
339C2×#117—831no—
340C2×#117—834no—
341C2×#117C2×D8311yescharacteristic
342C2×#117C2×D8312yes—
343C2×#117C2×D8312yes—
344C2×#117C2×D8312yes—
345C2×#117C2×D8311yescharacteristic
346C22×C4⋊C4—831no—
347C22×C4⋊C4—832no—
348C22×C4⋊C4C2×D8312yes—
349C22×C22⋊C4—832no—
350C22×C22⋊C4C2×D8312yes—
351C22×C42C2×D8312yes—
352#391—834no—
353#391—834no—
354#389—834no—
355#389—838no—
356#389—834no—
357#388—838no—
358#388—838no—
359#387—838no—
360#387—838no—
361#385—838no—
362#381—832no—
363#381—834no—
364#380—832no—
365#380—832no—
366#380—834no—
367#379—832no—
368C4×C4⋊C4—834no—
369C4×C4⋊C4—834no—
370C4×C4⋊C4—838no—
371C4×C22⋊C4—838no—
372C4×C22⋊C4—832no—
373C4×C22⋊C4—832no—
374C4×C22⋊C4—834no—
375C25C22×D8311yescharacteristic
376C22×Q8—838no—
377C22×Q8—832no—
378C22×Q8—832no—
379C22×Q8—1664no—
380C22×Q8—1664no—
381C23×C4—24no—
382C23×C4—24no—
383C23×C4—832no—
384C23×C4—832no—
385C23×C4C4×D8318yes—
386C23×C4C22×D8314yes—
387C23×C4C22×D8311yescharacteristic
388C23×C4C22×D8311yescharacteristic
389C23×C4C22×D8311yescharacteristic
390C23×C4C22×D8311yescharacteristic
391C23×C4C22×D8311yescharacteristic
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—28no—
401#117—28no—
402#117—28no—
403#117—1664no—
404#117—1664no—
405#117—16616no—
406#117C22⋊D8318yes—
407C2×C4⋊C4—28no—
408C2×C4⋊C4—838no—
409C2×C4⋊C4—8316no—
410C2×C4⋊C4—838no—
411C2×C4⋊C4—834no—
412C2×C4⋊C4—832no—
413C2×C4⋊C4—832no—
414C2×C4⋊C4—834no—
415C2×C4⋊C4—832no—
416C2×C4⋊C4—838no—
417C2×C4⋊C4—832no—
418C2×C4⋊C4C22×D8314yes—
419C2×C4⋊C4C22×D8312yes—
420C2×C4⋊C4C22×D8312yes—
421C2×C22⋊C4—28no—
422C2×C22⋊C4—834no—
423C2×C22⋊C4—8316no—
424C2×C22⋊C4—834no—
425C2×C22⋊C4—834no—
426C2×C22⋊C4—832no—
427C2×C22⋊C4—832no—
428C2×C22⋊C4—834no—
429C2×C22⋊C4C22×D8312yes—
430C2×C22⋊C4C22×D8314yes—
431C2×C22⋊C4C22×D8312yes—
432C2×C42—28no—
433C2×C42—838no—
434C2×C42—832no—
435C2×C42—838no—
436C2×C42—832no—
437C2×C42C22×D8314yes—
438C2×C42C22×D8312yes—
439C2×C42C22×D8312yes—
440#95—834no—
441#95—838no—
442#95—834no—
443#95—838no—
444#95—838no—
445#95—838no—
446C24—22no—
447C24—22no—
448C24—24no—
449C24—42no—
450C24C2×C4×D8314yes—
451C24C23×D8311yescharacteristic
452C24C23×D8311yescharacteristic
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—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—232no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—44no—
475C22×C4—3324no—
476C22×C4—838no—
477C22×C4—838no—
478C22×C4—834no—
479C22×C4—834no—
480C22×C4—834no—
481C22×C4—834no—
482C22×C4—1664no—
483C22×C4—1664no—
484C22×C4C2×C4×D8314yes—
485C22×C4C2×C4×D8314yes—
486C22×C4C2×C4×D8314yes—
487C22×C4C2×C4×D8314yes—
488C22×C4C2×C4×D8314yes—
489C22×C4C2×C4×D8314yes—
490C22×C4C2×C22⋊D8314yes—
491C22×C4#1305059014yes—
492C22×C4C23×D8314yes—
493C22×C4C23×D8311yescharacteristic
494C22×C4C23×D8311yescharacteristic
495C22×C4C23×D8314yes—
496C22×C4C23×D8311yescharacteristic
497C22×C4C23×D8311yescharacteristic
498C4⋊C4—28no—
499C4⋊C4—24no—
500C4⋊C4—3328no—
501C4⋊C4—8316no—
502C4⋊C4—8316no—
503C4⋊C4—838no—
504C4⋊C4—838no—
505C4⋊C4—16616no—
506C4⋊C4—1664no—
507C4⋊C4—1664no—
508C4⋊C4C2×C22⋊D8318yes—
509C22⋊C4—24no—
510C22⋊C4—28no—
511C22⋊C4—3324no—
512C22⋊C4—8316no—
513C22⋊C4—8316no—
514C22⋊C4—1664no—
515C22⋊C4—1664no—
516C22⋊C4—1668no—
517C22⋊C4C2×C22⋊D8318yes—
518C42—28no—
519C42—24no—
520C42—3328no—
521C42—1664no—
522C42—16616no—
523C42—1664no—
524C42C2×C22⋊D8318yes—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—28no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—48no—
538C23—48no—
539C23C22×C4×D8312yes—
540C23C22×C4×D8312yes—
541C23C22×C4×D8312yes—
542C23C22×C22⋊D8312yes—
543C23C2×#1305059012yes—
544C23#1458183914yes—
545C23C24×D8311yescharacteristic
546Q8—16616no—
547Q8—16616no—
548C2×C4—28no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—416no—
563C2×C4—3328no—
564C2×C4—838no—
565C2×C4—838no—
566C2×C4—1664no—
567C2×C4—1664no—
568C2×C4—1664no—
569C2×C4—1664no—
570C2×C4—1664no—
571C2×C4—1664no—
572C2×C4—16616no—
573C2×C4—1664no—
574C2×C4—1664no—
575C2×C4—1668no—
576C2×C4C4×C22⋊D83116yes—
577C2×C4C22×C4×D8312yes—
578C2×C4C22×C4×D8312yes—
579C2×C4C22×C4×D8312yes—
580C2×C4C22×C4×D8312yes—
581C2×C4C22×C22⋊D8318yes—
582C2×C4C22×C22⋊D8312yes—
583C2×C4C22×C22⋊D8312yes—
584C2×C4C2×#1305059012yes—
585C2×C4C2×#1305059012yes—
586C22—24no—
587C22—24no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—24no—
593C22—24no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—416no—
598C22—42no—
599C22—42no—
600C22C2×C4×C22⋊D8314yes—
601C22#1912817814yescomplemented
602C22C23×C4×D8311yescharacteristic
603C22C23×C22⋊D8311yescharacteristic
604C22C22×#1305059011yescharacteristic
605C22C2×#1458183912yes—
606C22#1912853712yes—
607C4—28no—
608C4—22no—
609C4—22no—
610C4—44no—
611C4—3324no—
612C4—3324no—
613C4—1668no—
614C4—1668no—
615C4C2×C4×C22⋊D8314yes—
616C4C2×C4×C22⋊D8314yes—
617C2—24no—
618C2—24no—
619C2—42no—
620C2C22×C4×C22⋊D8311yescharacteristic
621C2C2×#1912817811yescharacteristic
622C2#2492066411yescharacteristic
623C2#2492114312yes—
624C2#2492121512yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC22×#117×C83C2
socle, cosocleC23×C83C25
prime radicalC25×C83C23
hypercenterC22×#117D83
hypocenterC83#34203
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#19128178C2semidirect
3. #24910490C2semidirect
4. #24910468C2semidirect
5. #24910333C2semidirect
7. #24910145C2semidirect
8. #24910079C2semidirect
9. #24910047C2semidirect
10. C4×#14581771C2semidirect
12. #24909819C2semidirect
15. C2×#14581771C4semidirect
21. C2×C4×Dic166C22semidirect
22. C2×#14581757C22semidirect
29. #19127607C22semidirect
30. #19127606C22semidirect
31. #19127599C22semidirect
32. #19127597C22semidirect
33. #19127596C22semidirect
36. #19127585C22semidirect
37. #19127582C22semidirect
45. C4⋊C4×Dic83C22semidirect
46. C4×Dic83⋊C4C22semidirect
47. C4×C4⋊Dic83C22semidirect
64. C22×Dic166C2×C4semidirect
71. #14581771C2×C4semidirect
72. #14581771C2×C4semidirect
78. C2×C4⋊Dic83C2×C4semidirect
83. C2×Dic83⋊C4C2×C4semidirect
288. C22×C83#519semidirect
326. C83#34203semidirect
601. C22#19128178semidirect

Maximal subgroups

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

subgroupindex
C22×#145817712
C2×#191281782
#249104902
#249104682
#249103332
#249103082
#249101452
#249100792
#249100472
C4×#145817712
#249098462
#249098192
C22×#117×C832
#3420383