Finite Group Databasebeta

← #35567587

Subgroups — #35567587

Order 21248.

1998 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128256C211yesmaximal, characteristic
2#24910627C214yesmaximal
3#24910552C211yesmaximal, characteristic, complemented
4#24910552C211yesmaximal, characteristic, complemented
5#24910523C211yesmaximal, characteristic, complemented
6#24910427C212yesmaximal, complemented
7#24910381C212yesmaximal, complemented
8#24910352C212yesmaximal
9#24910347C212yesmaximal
10#24910203C212yesmaximal, complemented
11#24910200C212yesmaximal, complemented
12#24910148C212yesmaximal
13#24910139C212yesmaximal
14#24910112C214yesmaximal, complemented
15#24909867C211yesmaximal, characteristic
16#24909859C211yesmaximal, characteristic
17C83×#2365C211yesmaximal, characteristic, complemented
18#19128256C4116yescomplemented
19C2×#14581792C2211yescharacteristic
20C2×#14581783C2212yes—
21C2×#14581772C2212yes—
22C2×#14581768C2212yes—
23C2×Q8×Dic83C2212yes—
24C2×Q8×Dic83C2212yes—
25C2×#14581757C2212yes—
26C2×#14581750C2212yes—
27C2×#14581749C2212yes—
28C2×C42×D83C2211yescharacteristic
29#19127608C2212yes—
30#19127605C2214yes—
31#19127605C2214yes—
32#19127600C2212yes—
33#19127599C2212yes—
34#19127599C2214yes—
35#19127599C2212yes—
36#19127598C2214yes—
37#19127597C2212yes—
38#19127597C2214yes—
39#19127597C2212yes—
40#19127596C2214yes—
41#19127593C2214yes—
42#19127592C2214yes—
43#19127591C2214yes—
44#19127590C2214yes—
45#19127587C2214yes—
46#19127586C2214yes—
47#19127585C2214yes—
48#19127584C2214yes—
49#19127583C2214yes—
50#19127570C2211yescharacteristic
51#19127570C2211yescharacteristic
52#19127568C2214yes—
53#19127568C2212yes—
54#19127566C2214yes—
55#19127566C2211yescharacteristic
56#19127566C2211yescharacteristic
57#19127565C2212yes—
58#19127563C2212yes—
59#19127560C2212yes—
60#19127558C2211yescharacteristic
61#19127558C2211yescharacteristic
62#19127558C2211yescharacteristic
63#19127558C2211yescharacteristic
64C4⋊C4×Dic83C2212yes—
65C4⋊C4×Dic83C2212yes—
66C4⋊C4×Dic83C2212yes—
67C4×Dic83⋊C4C2214yes—
68C4×Dic83⋊C4C2212yes—
69C4×C4⋊Dic83C2212yes—
70C4×C4⋊Dic83C2211yescharacteristic
71C4×C4⋊Dic83C2211yescharacteristic
72C4×C4⋊Dic83C2212yes—
73C4×(C2×D83)⋊C4C2212yes—
74#19127547C2212yes—
75#19127546C2214yes—
76C2×C4⋊Q8×C83C2211yescharacteristic
77C2×C4×Q8×C83C2212yes—
78#391×C83C2212yes—
79#389×C83C2212yes—
80#387×C83C2214yes—
81C4×C4⋊C4×C83C2212yes—
82C4×C4⋊C4×C83C2211yescharacteristic
83C4×C4⋊C4×C83C2211yescharacteristic
84C22×Dic166C2312yes—
85C22×C4×D83C2311yescharacteristic
86C22×C4×D83C2311yescharacteristic
87C22×C4×D83C2311yescharacteristic
88#14581792C2×C418yes—
89#14581783C2×C4116yes—
90#14581772C2×C4116yes—
91#14581768C2×C4116yes—
92Q8×Dic83—44no—
93Q8×Dic83C2×C4116yes—
94#14581757C2×C4116yes—
95#14581750C2×C4116yes—
96#14581749—44no—
97C2×C4⋊Dic83C2311yescharacteristic
98C2×C4⋊Dic83C2311yescharacteristic
99C2×C4⋊Dic83C2311yescharacteristic
100C2×C4⋊Dic83C2311yescharacteristic
101C2×C4⋊Dic83C2314yes—
102C2×C4⋊Dic83C2312yes—
103C2×C4⋊Dic83C2312yes—
104C2×C4⋊Dic83C2314yes—
105C2×Dic83⋊C4C2312yes—
106C2×Dic83⋊C4C2312yes—
107C2×Dic83⋊C4C2314yes—
108C2×Dic83⋊C4C2314yes—
109C2×Dic83⋊C4C2312yes—
110C2×Dic83⋊C4C2312yes—
111C2×Dic83⋊C4C2314yes—
112C2×(C2×D83)⋊C4C2312yes—
113C2×(C2×D83)⋊C4C2312yes—
114C2×(C2×D83)⋊C4C2312yes—
115C2×(C2×D83)⋊C4C2312yes—
116C2×(C2×D83)⋊C4C2314yes—
117C2×C4×Dic83C2312yes—
118C2×C4×Dic83C2312yes—
119C2×C4×Dic83C2314yes—
120C2×C4×Dic83C2314yes—
121C2×C4×Dic83C2314yes—
122C2×C4×Dic83C2311yescharacteristic
123C2×C4×Dic83C2314yes—
124C2×C4×Dic83C2311yescharacteristic
125C2×C4×Dic83C2311yescharacteristic
126C42×D83C2×C418yes—
127C83⋊#95C2312yes—
128C83⋊#95C2312yes—
129C83⋊#95C2314yes—
130C83⋊#95C2314yes—
131C83⋊#95C2312yes—
132C83⋊#95C2312yes—
133C83⋊#95C2312yes—
134C83⋊#95C2314yes—
135C83⋊#95C2314yes—
136C83⋊#95C2314yes—
137C83⋊#95C2312yes—
138C83⋊#95C2312yes—
139C83⋊#95C2312yes—
140C83⋊#95C2312yes—
141C83⋊#95C2312yes—
142C83⋊#95C2314yes—
143C22×Q8×C83C2312yes—
144C4⋊Q8×C83C2×C418yes—
145C4×Q8×C83—44no—
146C2×C4⋊C4×C83C2311yescharacteristic
147C2×C4⋊C4×C83C2311yescharacteristic
148C2×C4⋊C4×C83C2311yescharacteristic
149C2×C4⋊C4×C83C2311yescharacteristic
150C2×C4⋊C4×C83C2314yes—
151C2×C4⋊C4×C83C2312yes—
152C2×C4⋊C4×C83C2312yes—
153C2×C42×C83C2312yes—
154C2×C42×C83C2312yes—
155C2×C42×C83C2314yes—
156C2×C42×C83C2311yescharacteristic
157#95×C83C2312yes—
158#95×C83C2312yes—
159#95×C83C2314yes—
160#95×C83C2314yes—
161C23×D83C2411yescharacteristic
162C2×Dic166—42no—
163C2×Dic166—42no—
164C2×Dic166C22×C418yes—
165C22×Dic83C2412yes—
166C22×Dic83C2412yes—
167C22×Dic83C2412yes—
168C22×Dic83C2412yes—
169C22×Dic83C2411yescharacteristic
170C22×Dic83C2411yescharacteristic
171C22×Dic83C2411yescharacteristic
172C22×Dic83C2414yes—
173C2×C4×D83—22no—
174C2×C4×D83—22no—
175C2×C4×D83—22no—
176C2×C4×D83—22no—
177C2×C4×D83—41no—
178C2×C4×D83—41no—
179C2×C4×D83C22×C414yes—
180C2×C4×D83C22×C414yes—
181C2×C4×D83C22×C414yes—
182C4⋊Dic83—22no—
183C4⋊Dic83—22no—
184C4⋊Dic83—22no—
185C4⋊Dic83—22no—
186C4⋊Dic83—44no—
187C4⋊Dic83C22×C418yes—
188C4⋊Dic83C22×C4116yes—
189C4⋊Dic83C22×C418yes—
190Dic83⋊C4—44no—
191Dic83⋊C4—44no—
192Dic83⋊C4—42no—
193Dic83⋊C4—42no—
194Dic83⋊C4—42no—
195Dic83⋊C4—42no—
196Dic83⋊C4C22×C4116yes—
197(C2×D83)⋊C4—42no—
198(C2×D83)⋊C4—42no—
199(C2×D83)⋊C4—42no—
200(C2×D83)⋊C4—42no—
201(C2×D83)⋊C4C22×C4116yes—
202C4×Dic83—24no—
203C4×Dic83—24no—
204C4×Dic83—44no—
205C4×Dic83—44no—
206C4×Dic83—44no—
207C4×Dic83C22×C414yes—
208C4×Dic83C22×C414yes—
209C4×Dic83C22×C414yes—
210C4×Dic83C22×C4116yes—
211C2×Q8×C83—24no—
212C2×Q8×C83—24no—
213C2×Q8×C83C22×C418yes—
214C22×C4×C83C2412yes—
215C22×C4×C83C2412yes—
216C22×C4×C83C2412yes—
217C22×C4×C83C2412yes—
218C22×C4×C83C2411yescharacteristic
219C22×C4×C83C2414yes—
220C22×C4×C83C2411yescharacteristic
221C22×C4×C83C2411yescharacteristic
222C4⋊C4×C83—22no—
223C4⋊C4×C83—22no—
224C4⋊C4×C83—22no—
225C4⋊C4×C83—22no—
226C4⋊C4×C83—44no—
227C4⋊C4×C83C22×C418yes—
228C4⋊C4×C83C22×C418yes—
229C42×C83—24no—
230C42×C83—24no—
231C42×C83—44no—
232C42×C83C22×C414yes—
233C23×C83C2511yescharacteristic
234C22×D83—21no—
235C22×D83—21no—
236C22×D83—21no—
237C22×D83—21no—
238C22×D83—21no—
239C22×D83—21no—
240C22×D83C23×C412yes—
241Q8×C83—216no—
242C2×C4×C83—24no—
243C2×C4×C83—24no—
244C2×C4×C83—21no—
245C2×C4×C83—21no—
246C2×C4×C83—22no—
247C2×C4×C83—22no—
248C2×C4×C83—22no—
249C2×C4×C83—22no—
250C2×C4×C83—22no—
251C2×C4×C83—22no—
252C2×C4×C83—22no—
253C2×C4×C83—22no—
254C2×C4×C83—22no—
255C2×C4×C83—22no—
256C2×C4×C83—22no—
257C2×C4×C83—22no—
258C2×C4×C83C23×C412yes—
259C2×C4×C83C23×C418yes—
260C2×C4×C83C23×C412yes—
261C2×C4×C83C23×C412yes—
262C2×C4×C83H(5,𝔽2)12yes—
263C2×C4×C832−1+412yes—
264C2×C4×C832−1+412yes—
265C2×C4×C832−1+412yes—
266C2×Dic83—21no—
267C2×Dic83—21no—
268C2×Dic83—21no—
269C2×Dic83—21no—
270C2×Dic83—24no—
271C2×Dic83—24no—
272C2×Dic83—21no—
273C2×Dic83—21no—
274C2×Dic83—22no—
275C2×Dic83—22no—
276C2×Dic83—22no—
277C2×Dic83—22no—
278C2×Dic83—22no—
279C2×Dic83—22no—
280C2×Dic83—22no—
281C2×Dic83—22no—
282C2×Dic83—22no—
283C2×Dic83—22no—
284C2×Dic83—22no—
285C2×Dic83—22no—
286C2×Dic83C23×C412yes—
287C2×Dic83C23×C412yes—
288C2×Dic83C23×C412yes—
289C2×Dic83C23×C418yes—
290C4×D83—28no—
291C4×D83—28no—
292C4×D83—44no—
293Dic166—48no—
294C22×C83C24×C411yescharacteristic
295C22×C83C2×H(5,𝔽2)11yescharacteristic
296C22×C83C2×2−1+411yescharacteristic
297C22×C83C2×2−1+411yescharacteristic
298C22×C83C2×2−1+411yescharacteristic
299C22×C83#58511yescharacteristic
300C22×C83#58511yescharacteristic
301C2×D83—22no—
302C2×D83—22no—
303C2×D83—22no—
304C2×D83—41no—
305C2×D83—41no—
306C2×D83—41no—
307C2×D83—41no—
308C4×C83—28no—
309C4×C83—22no—
310C4×C83—42no—
311C4×C83—42no—
312C4×C83—42no—
313C4×C83—42no—
314C4×C83#51914yes—
315C4×C83#52014yes—
316Dic83—22no—
317Dic83—22no—
318Dic83—28no—
319Dic83—22no—
320Dic83—42no—
321Dic83—42no—
322Dic83—42no—
323Dic83—42no—
324#34394—831nomaximal, Sylow
325C2×C83C2×#51911yescharacteristic
326C2×C83C2×#52011yescharacteristic
327C2×C83#342811yescharacteristic
328C2×C83#354711yescharacteristic
329C2×C83#354711yescharacteristic
330C2×C83#355511yescharacteristic
331C2×C83#355511yescharacteristic
332D83—42no—
333C2×#533—831no—
334#2378—834no—
335#2369—831no—
336#2369—831no—
337#2365—831no—
338#2365D8311yescharacteristic, complemented
339#2356—832no—
340#2351—832no—
341#2347—832no—
342#2346—832no—
343#2330—832no—
344#2330—832no—
345#2323—832no—
346#2323—832no—
347#2320—834no—
348#2285—831no—
349#2283—831no—
350C83#3439411yescharacteristic, complemented, Sylow
351#533—8316no—
352C2×C4⋊Q8—831no—
353C2×C4⋊Q8C2×D8311yescharacteristic
354C2×#125—832no—
355C2×C22⋊Q8—832no—
356C2×C22⋊Q8—832no—
357C2×C4×Q8—832no—
358C2×C4×Q8—832no—
359C2×C4×Q8—832no—
360C2×C4×Q8C2×D8312yes—
361C2×#117—832no—
362C2×#117—832no—
363C22×C42—831no—
364#391—832no—
365#391—834no—
366#391—834no—
367#391C2×D8312yes—
368#389—834no—
369#389—832no—
370#389—832no—
371#389—832no—
372#389—834no—
373#389—832no—
374#389—832no—
375#389—834no—
376#389—834no—
377#389C2×D8312yes—
378#388—834no—
379#388—834no—
380#388—834no—
381#388—834no—
382#387—834no—
383#387—834no—
384#387—834no—
385#387—834no—
386#387—834no—
387#387C2×D8314yes—
388#384—831no—
389#384—831no—
390#383—832no—
391#383—834no—
392#382—834no—
393#382—831no—
394#382—831no—
395#382—832no—
396#381—832no—
397#380—831no—
398#380—831no—
399#380—831no—
400#380—831no—
401#380—832no—
402C4×C4⋊C4—832no—
403C4×C4⋊C4—832no—
404C4×C4⋊C4—832no—
405C4×C4⋊C4—832no—
406C4×C4⋊C4—832no—
407C4×C4⋊C4—831no—
408C4×C4⋊C4—831no—
409C4×C4⋊C4—834no—
410C4×C4⋊C4—832no—
411C4×C4⋊C4C2×D8312yes—
412C4×C4⋊C4C2×D8311yescharacteristic
413C4×C4⋊C4C2×D8311yescharacteristic
414C4×C22⋊C4—832no—
415#376—832no—
416#376—834no—
417C22×Q8—832no—
418C22×Q8C22×D8312yes—
419C23×C4—831no—
420C23×C4—831no—
421C23×C4—831no—
422C4⋊Q8—838no—
423C4⋊Q8C4×D8318yes—
424#125—8316no—
425C22⋊Q8—8316no—
426C22⋊Q8—8316no—
427C4×Q8—44no—
428C4×Q8—3324no—
429C4×Q8—8316no—
430C4×Q8—8316no—
431#117—3324no—
432#117—8316no—
433C2×C4⋊C4—832no—
434C2×C4⋊C4—832no—
435C2×C4⋊C4—834no—
436C2×C4⋊C4—832no—
437C2×C4⋊C4—832no—
438C2×C4⋊C4—834no—
439C2×C4⋊C4—834no—
440C2×C4⋊C4—832no—
441C2×C4⋊C4—834no—
442C2×C4⋊C4—834no—
443C2×C4⋊C4—832no—
444C2×C4⋊C4—831no—
445C2×C4⋊C4—831no—
446C2×C4⋊C4—831no—
447C2×C4⋊C4—831no—
448C2×C4⋊C4C22×D8311yescharacteristic
449C2×C4⋊C4C22×D8311yescharacteristic
450C2×C4⋊C4C22×D8311yescharacteristic
451C2×C4⋊C4C22×D8311yescharacteristic
452C2×C4⋊C4C22×D8314yes—
453C2×C4⋊C4C22×D8312yes—
454C2×C4⋊C4C22×D8312yes—
455C2×C22⋊C4—832no—
456C2×C22⋊C4—832no—
457C2×C22⋊C4—832no—
458C2×C22⋊C4—832no—
459C2×C22⋊C4—834no—
460C2×C42—834no—
461C2×C42—834no—
462C2×C42—834no—
463C2×C42—834no—
464C2×C42—831no—
465C2×C42—831no—
466C2×C42—831no—
467C2×C42—838no—
468C2×C42—832no—
469C2×C42—832no—
470C2×C42C22×D8312yes—
471C2×C42C22×D8312yes—
472C2×C42C22×D8314yes—
473C2×C42C22×D8311yescharacteristic
474#95—832no—
475#95—832no—
476#95—832no—
477#95—832no—
478#95—832no—
479#95—832no—
480#95—834no—
481#95—832no—
482#95—832no—
483#95—832no—
484#95—832no—
485#95—834no—
486#95—834no—
487#95—834no—
488#95—834no—
489#95—834no—
490#95C22×D8314yes—
491#95C22×D8312yes—
492#95C22×D8312yes—
493#95C22×D8314yes—
494C24—831no—
495C2×Q8—24no—
496C2×Q8—24no—
497C2×Q8—3322no—
498C2×Q8—3322no—
499C2×Q8—838no—
500C2×Q8C2×C4×D8318yes—
501C22×C4—3321no—
502C22×C4—3321no—
503C22×C4—834no—
504C22×C4—831no—
505C22×C4—834no—
506C22×C4—831no—
507C22×C4—831no—
508C22×C4—834no—
509C22×C4—834no—
510C22×C4—832no—
511C22×C4—832no—
512C22×C4—832no—
513C22×C4—832no—
514C22×C4—1662no—
515C22×C4—1662no—
516C22×C4—1662no—
517C22×C4—1662no—
518C22×C4C23×D8312yes—
519C22×C4C23×D8312yes—
520C22×C4C23×D8312yes—
521C22×C4C23×D8312yes—
522C22×C4C23×D8311yescharacteristic
523C22×C4C23×D8314yes—
524C22×C4C23×D8311yescharacteristic
525C22×C4C23×D8311yescharacteristic
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—44no—
531C4⋊C4—3322no—
532C4⋊C4—3322no—
533C4⋊C4—3324no—
534C4⋊C4—3322no—
535C4⋊C4—3324no—
536C4⋊C4—3322no—
537C4⋊C4—3324no—
538C4⋊C4—838no—
539C4⋊C4—8316no—
540C4⋊C4—8316no—
541C4⋊C4—838no—
542C4⋊C4—1662no—
543C4⋊C4—1662no—
544C4⋊C4—1662no—
545C4⋊C4—1662no—
546C4⋊C4C2×C4×D8318yes—
547C4⋊C4C2×C4×D8318yes—
548C22⋊C4—3322no—
549C22⋊C4—3322no—
550C22⋊C4—3322no—
551C22⋊C4—3322no—
552C22⋊C4—8316no—
553C42—24no—
554C42—24no—
555C42—44no—
556C42—3324no—
557C42—3324no—
558C42—3324no—
559C42—834no—
560C42—834no—
561C42—834no—
562C42—8316no—
563C42—1664no—
564C42—1664no—
565C42C2×C4×D8314yes—
566C23—832no—
567C23—1661no—
568C23—1661no—
569C23—1661no—
570C23—1661no—
571C23—1661no—
572C23—1661no—
573C23C24×D8311yescharacteristic
574Q8—216no—
575Q8—3328no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—3324no—
593C2×C4—832no—
594C2×C4—832no—
595C2×C4—832no—
596C2×C4—838no—
597C2×C4—1662no—
598C2×C4—1662no—
599C2×C4—1662no—
600C2×C4—1662no—
601C2×C4—1662no—
602C2×C4—1662no—
603C2×C4—1662no—
604C2×C4—1662no—
605C2×C4—1662no—
606C2×C4—1662no—
607C2×C4—1662no—
608C2×C4—1662no—
609C2×C4—1661no—
610C2×C4—1661no—
611C2×C4—1661no—
612C2×C4—1661no—
613C2×C4—1664no—
614C2×C4—1664no—
615C2×C4—1661no—
616C2×C4—1661no—
617C2×C4—1668no—
618C2×C4—1668no—
619C2×C4C22×C4×D8312yes—
620C2×C4C22×C4×D8318yes—
621C2×C4C22×C4×D8312yes—
622C2×C4C22×C4×D8312yes—
623C2×C4#1458183712yes—
624C2×C4#1458183912yes—
625C2×C4#1458183912yes—
626C2×C4#1458184012yes—
627C22—3321no—
628C22—3321no—
629C22—3321no—
630C22—3321no—
631C22—1662no—
632C22—1662no—
633C22—1662no—
634C22C23×C4×D8311yescharacteristic
635C22C2×#1458183711yescharacteristic
636C22C2×#1458183911yescharacteristic
637C22C2×#1458183911yescharacteristic
638C22C2×#1458184011yescharacteristic
639C22#1912853811yescharacteristic
640C22#1912853811yescharacteristic
641C4—28no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—3322no—
648C4—3322no—
649C4—3322no—
650C4—3322no—
651C4—1662no—
652C4—1662no—
653C4—1662no—
654C4—1668no—
655C4#1912817714yes—
656C4#1912818014yes—
657C2—3322no—
658C2C2×#1912817711yescharacteristic
659C2C2×#1912818011yescharacteristic
660C2#2492066511yescharacteristic
661C2#2492184011yescharacteristic
662C2#2492184811yescharacteristic
663C2#2492193911yescharacteristic
664C2#2492194711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2365C2
socle, cosocleC23×C83C25
prime radicalC23×D83C24
hypercenter#2365D83
hypocenterC83#34394
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2365

Complements

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

NH ≅ G/Ntype
3. #24910552C2semidirect
4. #24910552C2semidirect
5. #24910523C2semidirect
6. #24910427C2semidirect
7. #24910381C2semidirect
10. #24910203C2semidirect
11. #24910200C2semidirect
14. #24910112C2semidirect
17. C83×#2365C2semidirect
18. #19128256C4semidirect
338. #2365D83semidirect
350. C83#34394semidirect

Maximal subgroups

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

subgroupindex
C2×#191282562
#249106272
#249105522
#249105232
#249104272
#249103812
#249103522
#249103472
#249102032
#249102002
#249101482
#249101392
#249101122
#249098672
#249098592
C83×#23652
#3439483