Finite Group Databasebeta

← #35563827

Subgroups — #35563827

Order 21248.

2434 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581757C211yesmaximal, characteristic
2#24910417C214yesmaximal, complemented
3#24910416C212yesmaximal, complemented
4#24910377C212yesmaximal, complemented
5#24910341C212yesmaximal
6#24910340C211yesmaximal, characteristic
7#24910335C212yesmaximal
8C22⋊C4×Dic166C211yesmaximal, characteristic
9#24910295C211yesmaximal, characteristic
10#24909936C214yesmaximal, complemented
11#24909934C212yesmaximal, complemented
12#24909865C211yesmaximal, characteristic
13#24909865C212yesmaximal
14C4×#14581757C212yesmaximal, complemented
15C4×#14581748C212yesmaximal
16C4×#14581748C211yesmaximal, characteristic
17C83×#2345C211yesmaximal, characteristic
18C23×Dic166C2211yescharacteristic
19C2×C4×Dic166C2212yes—
20C2×#14581757C4116yescomplemented
21C2×#14581757C2212yes—
22C2×#14581757C2211yescharacteristic
23C2×#14581757C2211yescharacteristic
24C2×#14581757C2212yes—
25C2×#14581757C2211yescharacteristic
26C2×#14581757C2212yes—
27C2×#14581757C2211yescharacteristic
28C2×#14581748C2212yes—
29C2×#14581748C2211yescharacteristic
30C2×#14581748C2211yescharacteristic
31C22×Dic83⋊C4C2212yes—
32C22×C4×Dic83C2212yes—
33C22×C4×Dic83C2211yescharacteristic
34#19127607C2212yes—
35#19127606C2212yes—
36#19127605C2214yes—
37#19127605C2214yes—
38#19127600C2212yes—
39#19127598C2214yes—
40#19127596C2214yes—
41#19127588C2214yes—
42#19127587C2214yes—
43#19127585C2214yes—
44#19127583C2214yes—
45#19127582C2214yes—
46#19127570C2212yes—
47#19127570C2214yes—
48#19127565C2214yes—
49#19127565C2212yes—
50#19127561C2212yes—
51#19127561C2211yescharacteristic
52#19127559C2212yes—
53#19127559C2212yes—
54#19127559C2212yes—
55C4⋊C4×Dic83C2212yes—
56C4⋊C4×Dic83C2214yes—
57C4⋊C4×Dic83C2212yes—
58C4×Dic83⋊C4C2214yes—
59C4×Dic83⋊C4C2212yes—
60C4×Dic83⋊C4C2214yes—
61#19127552C2212yes—
62#19127552C2214yes—
63#19127552C2214yes—
64#19127552C2212yes—
65#19127552C2214yes—
66C22⋊C4×Dic83C2211yescharacteristic
67C22⋊C4×Dic83C2212yes—
68C22⋊C4×Dic83C2212yes—
69C22⋊C4×Dic83C2211yescharacteristic
70C4×C22⋊Dic83C2211yescharacteristic
71C4×C22⋊Dic83C2212yes—
72#19127546C2214yes—
73#19127546C2212yes—
74C42×Dic83C2212yes—
75C42×Dic83C2214yes—
76C22×C4⋊C4×C83C2211yescharacteristic
77#389×C83C2212yes—
78#387×C83C2214yes—
79#380×C83C2211yescharacteristic
80#380×C83C2212yes—
81C4×C4⋊C4×C83C2212yes—
82C4×C22⋊C4×C83C2212yes—
83C4×C22⋊C4×C83C2211yescharacteristic
84C22×Dic166—28no—
85C22×Dic166C2×C418yes—
86C22×Dic166C2312yes—
87C22×Dic166C2311yescharacteristic
88C22×Dic166C2311yescharacteristic
89C23×Dic83C2312yes—
90C23×Dic83C2311yescharacteristic
91C23×Dic83C2311yescharacteristic
92C4×Dic166—28no—
93#14581757—28no—
94#14581757C2×C418yes—
95#14581757C2×C4116yes—
96#14581757C2×C418yes—
97#14581757C2×C4116yes—
98#14581757C2×C418yes—
99#14581757C2×C418yes—
100#14581748—28no—
101#14581748—24no—
102#14581748Q818yescomplemented
103C2×C4⋊Dic83C2312yes—
104C2×C4⋊Dic83C2314yes—
105C2×Dic83⋊C4C2×C4116yes—
106C2×Dic83⋊C4C2314yes—
107C2×Dic83⋊C4C2314yes—
108C2×Dic83⋊C4C2314yes—
109C2×Dic83⋊C4C2312yes—
110C2×Dic83⋊C4C2314yes—
111C2×Dic83⋊C4C2312yes—
112C2×Dic83⋊C4C2312yes—
113C2×Dic83⋊C4C2312yes—
114C2×Dic83⋊C4C2312yes—
115C2×Dic83⋊C4C2312yes—
116C2×C22⋊Dic83C2311yescharacteristic
117C2×C22⋊Dic83C2312yes—
118C2×C22⋊Dic83C2311yescharacteristic
119C2×C4×Dic83C2×C418yes—
120C2×C4×Dic83C2×C4116yes—
121C2×C4×Dic83C2314yes—
122C2×C4×Dic83C2312yes—
123C2×C4×Dic83C2314yes—
124C2×C4×Dic83C2314yes—
125C2×C4×Dic83C2312yes—
126C2×C4×Dic83C2312yes—
127C2×C4×Dic83C2312yes—
128C2×C4×Dic83C2314yes—
129C2×C4×Dic83C2311yescharacteristic
130C2×C4×Dic83C2312yes—
131C2×C4×Dic83C2312yes—
132C2×C4×Dic83C2311yescharacteristic
133C2×C4×Dic83C2312yes—
134C2×C4×Dic83C2311yescharacteristic
135C2×C4×Dic83C2311yescharacteristic
136C2×C4×Dic83C2312yes—
137C2×C4×Dic83C2312yes—
138C83⋊#95C2314yes—
139C83⋊#95C2314yes—
140C83⋊#95C2312yes—
141C83⋊#95C2314yes—
142C83⋊#95C2312yes—
143C83⋊#95C2314yes—
144C83⋊#95C2312yes—
145C83⋊#95C2312yes—
146C83⋊#95C2314yes—
147C83⋊#95C2314yes—
148C83⋊#95C2314yes—
149C23×C4×C83C2311yescharacteristic
150C23×C4×C83C2312yes—
151C2×C4⋊C4×C83C2×C418yes—
152C2×C4⋊C4×C83C2312yes—
153C2×C4⋊C4×C83C2314yes—
154C2×C4⋊C4×C83C2311yescharacteristic
155C2×C4⋊C4×C83C2312yes—
156C2×C4⋊C4×C83C2311yescharacteristic
157C2×C22⋊C4×C83C2312yes—
158C2×C22⋊C4×C83C2311yescharacteristic
159C2×C22⋊C4×C83C2311yescharacteristic
160C2×C42×C83C2312yes—
161C2×C42×C83C2314yes—
162#95×C83C2314yes—
163#95×C83C2312yes—
164#95×C83C2314yes—
165#95×C83C2312yes—
166C2×Dic166—24no—
167C2×Dic166—28no—
168C2×Dic166—232no—
169C2×Dic166C22×C414yes—
170C2×Dic166C22×C418yes—
171C2×Dic166C22×C414yes—
172C2×Dic166#4118yes—
173C22×Dic83—28no—
174C22×Dic83—42no—
175C22×Dic83C22×C418yes—
176C22×Dic83C22×C414yes—
177C22×Dic83C22×C414yes—
178C22×Dic83C2×Q814yes—
179C22×Dic83#4114yes—
180C22×Dic83C2412yes—
181C22×Dic83C2414yes—
182C22×Dic83C2412yes—
183C22×Dic83C2411yescharacteristic
184C22×Dic83C2411yescharacteristic
185C22×Dic83C2412yes—
186C22×Dic83C2411yescharacteristic
187C22×Dic83C2411yescharacteristic
188C22×Dic83C2412yes—
189C4⋊Dic83—24no—
190C4⋊Dic83—44no—
191Dic83⋊C4—28no—
192Dic83⋊C4—24no—
193Dic83⋊C4—28no—
194Dic83⋊C4—28no—
195Dic83⋊C4—44no—
196Dic83⋊C4C22×C418yes—
197Dic83⋊C4C22×C418yes—
198Dic83⋊C4C22×C418yes—
199Dic83⋊C4C22×C418yes—
200Dic83⋊C4C2×Q818yescomplemented
201C22⋊Dic83—22no—
202C22⋊Dic83—24no—
203C22⋊Dic83C2×Q814yes—
204C4×Dic83—24no—
205C4×Dic83—28no—
206C4×Dic83—24no—
207C4×Dic83—28no—
208C4×Dic83—28no—
209C4×Dic83—42no—
210C4×Dic83—44no—
211C4×Dic83—42no—
212C4×Dic83C22×C414yes—
213C4×Dic83C22×C414yes—
214C4×Dic83C22×C414yes—
215C4×Dic83C22×C414yes—
216C4×Dic83C22×C418yes—
217C4×Dic83C22×C418yes—
218C4×Dic83C22×C418yes—
219C4×Dic83C22×C418yes—
220C4×Dic83C2×Q818yescomplemented
221C24×C83C2411yescharacteristic
222C22×C4×C83—24no—
223C22×C4×C83—44no—
224C22×C4×C83C22×C414yes—
225C22×C4×C83C22×C418yes—
226C22×C4×C83C2412yes—
227C22×C4×C83C2414yes—
228C22×C4×C83C2411yescharacteristic
229C22×C4×C83C2412yes—
230C22×C4×C83C2412yes—
231C22×C4×C83C2412yes—
232C22×C4×C83C2411yescharacteristic
233C4⋊C4×C83—24no—
234C4⋊C4×C83—44no—
235C4⋊C4×C83C22×C418yes—
236C4⋊C4×C83C22×C414yes—
237C4⋊C4×C83C22×C414yes—
238C22⋊C4×C83—24no—
239C22⋊C4×C83—22no—
240C22⋊C4×C83C2×Q814yes—
241C42×C83—24no—
242C42×C83—44no—
243C23×C83—22no—
244C23×C83—22no—
245C23×C83C23×C412yes—
246C23×C83C22×Q812yes—
247C23×C83C2×#4112yes—
248C23×C83C2511yescharacteristic
249C2×C4×C83—22no—
250C2×C4×C83—22no—
251C2×C4×C83—22no—
252C2×C4×C83—24no—
253C2×C4×C83—24no—
254C2×C4×C83—28no—
255C2×C4×C83—22no—
256C2×C4×C83—22no—
257C2×C4×C83—24no—
258C2×C4×C83—24no—
259C2×C4×C83—24no—
260C2×C4×C83—22no—
261C2×C4×C83—48no—
262C2×C4×C83C23×C414yes—
263C2×C4×C83C23×C414yes—
264C2×C4×C83C23×C412yes—
265C2×C4×C83C23×C412yes—
266C2×C4×C83C22×Q814yes—
267C2×C4×C83C2×#4114yes—
268C2×Dic83—22no—
269C2×Dic83—24no—
270C2×Dic83—22no—
271C2×Dic83—22no—
272C2×Dic83—22no—
273C2×Dic83—216no—
274C2×Dic83—22no—
275C2×Dic83—24no—
276C2×Dic83—22no—
277C2×Dic83—24no—
278C2×Dic83—24no—
279C2×Dic83—22no—
280C2×Dic83—44no—
281C2×Dic83C4×Q8116yescomplemented
282C2×Dic83C23×C412yes—
283C2×Dic83C23×C412yes—
284C2×Dic83C23×C414yes—
285C2×Dic83C23×C414yes—
286C2×Dic83C23×C412yes—
287C2×Dic83C23×C412yes—
288C2×Dic83C22×Q814yes—
289C2×Dic83C22×Q812yes—
290C2×Dic83C22×Q812yes—
291C2×Dic83C2×#4118yes—
292C2×Dic83C2×#4112yes—
293C2×Dic83C2×#4112yes—
294Dic166—28no—
295Dic166—216no—
296Dic166#117116yescomplemented
297C22×C83—22no—
298C22×C83—22no—
299C22×C83—24no—
300C22×C83—24no—
301C22×C83C2×C4×Q814yes—
302C22×C83C24×C411yescharacteristic
303C22×C83C23×Q811yescharacteristic
304C22×C83C22×#4112yes—
305C22×C83C22×#4111yescharacteristic
306C22×C83#58512yes—
307C4×C83—22no—
308C4×C83—24no—
309C4×C83—24no—
310C4×C83—24no—
311C4×C83—42no—
312C4×C83—44no—
313C4×C83C2×#11714yes—
314Dic83—24no—
315Dic83—22no—
316Dic83—22no—
317Dic83—24no—
318Dic83—42no—
319Dic83—44no—
320Dic83C2×#11718yes—
321Dic83C2×C4×Q814yes—
322Dic83C2×C4×Q814yes—
323Q8×#117—831nomaximal, Sylow
324C2×C83—24no—
325C2×C83C22×#11711yescharacteristic
326C2×C83C22×C4×Q811yescharacteristic
327C2×C83#342811yescharacteristic
328C2×C83Q8×#4112yes—
329C2×C83#348012yes—
330C22×C4×Q8—831no—
331#2355—834no—
332#2355—832no—
333Q8×C4⋊C4—832no—
334#2345—832no—
335#2345—832no—
336#2345—831no—
337#2345D8311yescharacteristic
338Q8×C22⋊C4—831no—
339Q8×C22⋊C4—831no—
340#2295—832no—
341#2295—834no—
342#2285—831no—
343#2285—832no—
344C42×Q8—832no—
345C4×#117—831no—
346C4×#117—832no—
347C83Q8×#11711yescharacteristic, complemented, Sylow
348C23×Q8—831no—
349C2×C4×Q8—8316no—
350C2×C4×Q8—831no—
351C2×C4×Q8—832no—
352C2×C4×Q8—831no—
353C2×C4×Q8—831no—
354C2×C4×Q8—831no—
355C2×C4×Q8—832no—
356C2×C4×Q8—832no—
357C2×C4×Q8—832no—
358C2×#117—831no—
359C2×#117—832no—
360C2×#117—831no—
361C22×C4⋊C4—832no—
362C22×C4⋊C4C2×D8311yescharacteristic
363C22×C42—831no—
364C22×C42—832no—
365#391—832no—
366#391—832no—
367#391—834no—
368#391—834no—
369#389—834no—
370#389—832no—
371#389—834no—
372#389C2×D8312yes—
373#387—834no—
374#387—834no—
375#387—834no—
376#387—834no—
377#387—834no—
378#387C2×D8314yes—
379#384—832no—
380#384—834no—
381#382—832no—
382#382—834no—
383#380—831no—
384#380—832no—
385#380—832no—
386#380—832no—
387#380—832no—
388#380C2×D8311yescharacteristic
389#380C2×D8312yes—
390C4×C4⋊C4—832no—
391C4×C4⋊C4—832no—
392C4×C4⋊C4—834no—
393C4×C4⋊C4—834no—
394C4×C4⋊C4—832no—
395C4×C4⋊C4—834no—
396C4×C4⋊C4—834no—
397C4×C4⋊C4—834no—
398C4×C4⋊C4—832no—
399C4×C4⋊C4—834no—
400C4×C4⋊C4—832no—
401C4×C4⋊C4C2×D8312yes—
402C4×C22⋊C4—831no—
403C4×C22⋊C4—832no—
404C4×C22⋊C4—832no—
405C4×C22⋊C4—832no—
406C4×C22⋊C4—831no—
407C4×C22⋊C4—831no—
408C4×C22⋊C4C2×D8311yescharacteristic
409C4×C22⋊C4C2×D8312yes—
410#376—832no—
411#376—834no—
412C43—832no—
413C43—834no—
414C22×Q8—838no—
415C22×Q8—831no—
416C22×Q8—832no—
417C22×Q8—831no—
418C22×Q8—1668no—
419C23×C4—831no—
420C23×C4—831no—
421C23×C4—832no—
422C23×C4C22×D8311yescharacteristic
423C23×C4C22×D8312yes—
424C4×Q8—838no—
425C4×Q8—838no—
426C4×Q8—8316no—
427C4×Q8—838no—
428C4×Q8—8316no—
429C4×Q8—838no—
430C4×Q8—1668no—
431C4×Q8—1668no—
432#117—838no—
433#117—1668no—
434#117—1664no—
435C2×C4⋊C4—834no—
436C2×C4⋊C4—834no—
437C2×C4⋊C4—8316no—
438C2×C4⋊C4—832no—
439C2×C4⋊C4—832no—
440C2×C4⋊C4—832no—
441C2×C4⋊C4—832no—
442C2×C4⋊C4—832no—
443C2×C4⋊C4—832no—
444C2×C4⋊C4—834no—
445C2×C4⋊C4—834no—
446C2×C4⋊C4—834no—
447C2×C4⋊C4—832no—
448C2×C4⋊C4C4×D8318yes—
449C2×C4⋊C4C22×D8312yes—
450C2×C4⋊C4C22×D8314yes—
451C2×C4⋊C4C22×D8311yescharacteristic
452C2×C4⋊C4C22×D8312yes—
453C2×C4⋊C4C22×D8311yescharacteristic
454C2×C22⋊C4—831no—
455C2×C22⋊C4—831no—
456C2×C22⋊C4—832no—
457C2×C22⋊C4C22×D8312yes—
458C2×C22⋊C4C22×D8311yescharacteristic
459C2×C22⋊C4C22×D8311yescharacteristic
460C2×C42—832no—
461C2×C42—834no—
462C2×C42—832no—
463C2×C42—838no—
464C2×C42—8316no—
465C2×C42—832no—
466C2×C42—832no—
467C2×C42—831no—
468C2×C42—832no—
469C2×C42—831no—
470C2×C42—832no—
471C2×C42—831no—
472C2×C42—832no—
473C2×C42—831no—
474C2×C42—834no—
475C2×C42—832no—
476C2×C42—834no—
477C2×C42—834no—
478C2×C42—832no—
479C2×C42C22×D8312yes—
480C2×C42C22×D8314yes—
481#95—832no—
482#95—834no—
483#95—832no—
484#95—834no—
485#95—834no—
486#95—832no—
487#95—834no—
488#95—834no—
489#95—834no—
490#95—834no—
491#95—832no—
492#95C22×D8314yes—
493#95C22×D8312yes—
494#95C22×D8314yes—
495#95C22×D8312yes—
496C24C23×D8311yescharacteristic
497C2×Q8—834no—
498C2×Q8—838no—
499C2×Q8—834no—
500C2×Q8—838no—
501C2×Q8—1668no—
502C2×Q8—16632no—
503C2×Q8—1664no—
504C22×C4—24no—
505C22×C4—44no—
506C22×C4—3322no—
507C22×C4—834no—
508C22×C4—834no—
509C22×C4—834no—
510C22×C4—834no—
511C22×C4—831no—
512C22×C4—831no—
513C22×C4—832no—
514C22×C4—838no—
515C22×C4—832no—
516C22×C4—832no—
517C22×C4—831no—
518C22×C4—831no—
519C22×C4—834no—
520C22×C4—832no—
521C22×C4—1668no—
522C22×C4C2×C4×D8314yes—
523C22×C4C2×C4×D8318yes—
524C22×C4C23×D8312yes—
525C22×C4C23×D8314yes—
526C22×C4C23×D8311yescharacteristic
527C22×C4C23×D8312yes—
528C22×C4C23×D8312yes—
529C22×C4C23×D8312yes—
530C22×C4C23×D8311yescharacteristic
531C4⋊C4—24no—
532C4⋊C4—44no—
533C4⋊C4—3324no—
534C4⋊C4—3324no—
535C4⋊C4—838no—
536C4⋊C4—838no—
537C4⋊C4—838no—
538C4⋊C4—838no—
539C4⋊C4—838no—
540C4⋊C4—1664no—
541C4⋊C4—1668no—
542C4⋊C4—1668no—
543C4⋊C4—1668no—
544C4⋊C4—1664no—
545C4⋊C4C2×C4×D8318yes—
546C4⋊C4C2×C4×D8314yes—
547C4⋊C4C2×C4×D8314yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—834no—
551C22⋊C4—1662no—
552C22⋊C4—1664no—
553C22⋊C4Q8×D8314yes—
554C42—24no—
555C42—44no—
556C42—3322no—
557C42—3324no—
558C42—3322no—
559C42—838no—
560C42—838no—
561C42—834no—
562C42—834no—
563C42—834no—
564C42—834no—
565C42—838no—
566C42—838no—
567C42—838no—
568C42—1668no—
569C42—1664no—
570C42—1668no—
571C42—1668no—
572C42—1664no—
573C23—22no—
574C23—22no—
575C23C22×C4×D8312yes—
576C23C2×Q8×D8312yes—
577C23#41×D8312yes—
578C23C24×D8311yescharacteristic
579Q8—8316no—
580Q8—16616no—
581Q8—1668no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—3324no—
596C2×C4—8316no—
597C2×C4—832no—
598C2×C4—832no—
599C2×C4—834no—
600C2×C4—832no—
601C2×C4—832no—
602C2×C4—832no—
603C2×C4—838no—
604C2×C4—832no—
605C2×C4—832no—
606C2×C4—832no—
607C2×C4—834no—
608C2×C4—834no—
609C2×C4—1664no—
610C2×C4—1662no—
611C2×C4—1664no—
612C2×C4—1662no—
613C2×C4—1664no—
614C2×C4—1662no—
615C2×C4—16616no—
616C2×C4—1662no—
617C2×C4—1664no—
618C2×C4—1662no—
619C2×C4—1662no—
620C2×C4—1662no—
621C2×C4C22×C4×D8314yes—
622C2×C4C22×C4×D8314yes—
623C2×C4C22×C4×D8312yes—
624C2×C4C22×C4×D8312yes—
625C2×C4C2×Q8×D8314yes—
626C2×C4#41×D8314yes—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—24no—
631C22C4×Q8×D8314yes—
632C22C23×C4×D8311yescharacteristic
633C22C22×Q8×D8311yescharacteristic
634C22C2×#41×D8312yes—
635C22C2×#41×D8311yescharacteristic
636C22#1912853612yes—
637C4—22no—
638C4—24no—
639C4—24no—
640C4—24no—
641C4—42no—
642C4—44no—
643C4—3324no—
644C4—3322no—
645C4—838no—
646C4—834no—
647C4—834no—
648C4—1664no—
649C4—1662no—
650C4—1662no—
651C4—1664no—
652C4#117×D8314yes—
653C2—24no—
654C2C2×#117×D8311yescharacteristic
655C2C2×C4×Q8×D8311yescharacteristic
656C2#2492066111yescharacteristic
657C2#2492104212yes—
658C2#2492105712yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2345C2
socle, cosocleC23×C83C25
prime radicalC24×C83C24
hypercenter#2345D83
hypocenterC83Q8×#117
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2345

Complements

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

NH ≅ G/Ntype
2. #24910417C2semidirect
3. #24910416C2semidirect
4. #24910377C2semidirect
10. #24909936C2semidirect
11. #24909934C2semidirect
14. C4×#14581757C2semidirect
20. C2×#14581757C4semidirect
102. #14581748Q8semidirect
200. Dic83⋊C4C2×Q8semidirect
220. C4×Dic83C2×Q8semidirect
281. C2×Dic83C4×Q8semidirect
296. Dic166#117semidirect
347. C83Q8×#117semidirect

Maximal subgroups

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

subgroupindex
C22×#145817572
#249104172
#249104162
#249103772
#249103412
#249103402
#249103352
C22⋊C4×Dic1662
#249102952
#249099362
#249099342
#249098652
C4×#145817572
C4×#145817482
C83×#23452
Q8×#11783