Finite Group Databasebeta

← #35564505

Subgroups — #35564505

Order 21248.

3206 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128182C211yesmaximal, characteristic
2C2×#19128181C211yesmaximal, characteristic
3C2×#19128169C211yesmaximal, characteristic
4#24910377C214yesmaximal, complemented
5#24910364C214yesmaximal, complemented
6#24910361C218yesmaximal, complemented
7#24910088C212yesmaximal, complemented
8#24910086C211yesmaximal, characteristic, complemented
9#24910084C212yesmaximal, complemented
10#24909869C212yesmaximal, complemented
11#24909866C211yesmaximal, characteristic, complemented
12C2×#19127570C211yesmaximal, characteristic, complemented
13#384×D83C212yesmaximal, complemented
14C83×#2317C211yesmaximal, characteristic, complemented
15C22×#13050590C2211yescharacteristic
16#19128182—28no—
17#19128181—28no—
18#19128169C4116yescomplemented
19C2×C4×Dic166C2212yes—
20C2×#14581757C2212yes—
21C2×#14581757C2212yes—
22C2×C4×D332C2212yes—
23C2×#14581754C2212yes—
24C2×#14581754C2212yes—
25C2×C4×C22⋊D83C2214yes—
26C2×(C22⋊D83)⋊C4C2214yes—
27C2×(C22⋊D83)⋊C4C2214yes—
28C2×#14581751C2211yescharacteristic
29C2×#14581750C2212yes—
30C2×#14581750C2212yes—
31C2×#14581749C2212yes—
32C22×C4⋊Dic83C2211yescharacteristic
33C22×Dic83⋊C4C2214yes—
34C2×C4⋊C4×D83C2212yes—
35C2×C4⋊C4×D83C2212yes—
36C2×C4⋊C4×D83C2218yes—
37C22×C4×Dic83C2211yescharacteristic
38C2×C42×D83C2212yes—
39#19127600C2214yes—
40#19127598C2218yes—
41#19127596C2218yes—
42#19127576C2214yes—
43#19127575C2218yes—
44#19127572C2218yes—
45#19127571C2212yes—
46#19127570C2212yes—
47#19127570C2218yes—
48#19127570C2214yes—
49#19127570C2211yescharacteristic
50#19127570C2212yes—
51#19127570C2211yescharacteristic
52#19127567C2212yes—
53#19127566C2211yescharacteristic
54#19127566C2211yescharacteristic
55#19127560C2218yes—
56#19127559C2214yes—
57C4×Dic83⋊C4C2214yes—
58#19127552C2218yes—
59C2×#117×C83C2211yescharacteristic
60C2×#117×C83C2211yescharacteristic
61C22×C4⋊C4×C83C2211yescharacteristic
62#389×C83C2214yes—
63#385×C83C2214yes—
64#384×C83C2212yes—
65#382×C83C2212yes—
66C2×#13050590C2×C418yes—
67C2×#13050590D418yes—
68C2×#13050590Q818yes—
69C22×Dic166C2311yescharacteristic
70C22×D332C2311yescharacteristic
71C22×C22⋊D83C2312yes—
72C23×Dic83C2312yes—
73C23×Dic83C2312yes—
74C22×C4×D83C2314yes—
75C22×C4×D83C2314yes—
76C22×C4×D83C2314yes—
77C22×C4×D83C2312yes—
78C4×Dic166—28no—
79#14581757—28no—
80#14581757C2×C4116yescomplemented
81C4×D332—28no—
82#14581754—28no—
83#14581754C2×C4116yescomplemented
84C4×C22⋊D83—216no—
85(C22⋊D83)⋊C4—216no—
86(C22⋊D83)⋊C4C2×C4132yescomplemented
87#14581751—24no—
88#14581750—28no—
89#14581750—28no—
90#14581749C2×C4116yescomplemented
91C2×C4⋊Dic83—24no—
92C2×C4⋊Dic83C2311yescharacteristic
93C2×C4⋊Dic83C2311yescharacteristic
94C2×C4⋊Dic83C2312yes—
95C2×C4⋊Dic83C2312yes—
96C2×C4⋊Dic83C2318yes—
97C2×Dic83⋊C4—48no—
98C2×Dic83⋊C4C2314yes—
99C2×Dic83⋊C4C2318yes—
100C2×Dic83⋊C4C2314yes—
101C2×Dic83⋊C4C2314yes—
102C2×Dic83⋊C4C2314yes—
103C2×Dic83⋊C4C2318yes—
104C2×Dic83⋊C4C2314yes—
105C4⋊C4×D83—28no—
106C4⋊C4×D83—28no—
107C4⋊C4×D83—416no—
108C2×C22⋊Dic83C2312yes—
109C2×(C2×D83)⋊C4C2314yes—
110C2×(C2×D83)⋊C4C2314yes—
111C2×(C2×D83)⋊C4C2314yes—
112C2×C4×Dic83C2×C418yescomplemented
113C2×C4×Dic83C2314yes—
114C2×C4×Dic83C2314yes—
115C2×C4×Dic83C2312yes—
116C2×C4×Dic83C2314yes—
117C2×C4×Dic83C2312yes—
118C2×C4×Dic83C2311yescharacteristic
119C2×C4×Dic83C2311yescharacteristic
120C42×D83C2×C4116yescomplemented
121C83⋊#95C2318yes—
122C83⋊#95C2314yes—
123C83⋊#95C2314yes—
124C23×C4×C83C2312yes—
125C23×C4×C83C2311yescharacteristic
126#117×C83—24no—
127#117×C83C2×C418yescomplemented
128C2×C4⋊C4×C83—24no—
129C2×C4⋊C4×C83C2311yescharacteristic
130C2×C4⋊C4×C83C2311yescharacteristic
131C2×C4⋊C4×C83C2312yes—
132C2×C4⋊C4×C83C2312yes—
133C2×C4⋊C4×C83C2318yes—
134C2×C4⋊C4×C83C2312yes—
135C2×C22⋊C4×C83C2312yes—
136C2×C22⋊C4×C83C2312yes—
137C2×C42×C83C2312yes—
138C2×C42×C83C2312yes—
139#95×C83C2318yes—
140C23×D83C2412yes—
141#13050590C4⋊C4164yescomplemented
142C2×Dic166C22×C414yes—
143C2×Dic166C2×D414yes—
144C2×Dic166C2×Q814yes—
145C2×D332C22×C414yes—
146C2×D332C2×D414yes—
147C2×D332C2×Q814yes—
148C2×C22⋊D83C22×C418yes—
149C2×C22⋊D83C2×D418yes—
150C2×C22⋊D83C2×Q818yes—
151C22×Dic83—24no—
152C22×Dic83—24no—
153C22×Dic83—24no—
154C22×Dic83—44no—
155C22×Dic83C22×C418yes—
156C22×Dic83C2412yes—
157C22×Dic83C2414yes—
158C22×Dic83C2412yes—
159C22×Dic83C2412yes—
160C22×Dic83C2412yes—
161C22×Dic83C2412yes—
162C2×C4×D83—28no—
163C2×C4×D83—28no—
164C2×C4×D83—28no—
165C2×C4×D83—28no—
166C2×C4×D83—48no—
167C2×C4×D83—48no—
168C2×C4×D83C22×C4116yes—
169C2×C4×D83C22×C418yes—
170C2×C4×D83C2×D418yes—
171C2×C4×D83C2×Q818yes—
172C4⋊Dic83—22no—
173C4⋊Dic83—24no—
174C4⋊Dic83—24no—
175C4⋊Dic83—22no—
176C4⋊Dic83—48no—
177Dic83⋊C4—28no—
178Dic83⋊C4—28no—
179Dic83⋊C4—28no—
180Dic83⋊C4—216no—
181Dic83⋊C4—28no—
182Dic83⋊C4—48no—
183Dic83⋊C4C22×C4116yes—
184C22⋊Dic83—24no—
185(C2×D83)⋊C4—28no—
186(C2×D83)⋊C4—28no—
187(C2×D83)⋊C4C22×C4116yes—
188C4×Dic83—24no—
189C4×Dic83—28no—
190C4×Dic83—28no—
191C4×Dic83C22×C414yes—
192C4×Dic83C22×C4116yes—
193C4×Dic83C22×C414yes—
194C4×Dic83C22×C418yes—
195C24×C83C2411yescharacteristic
196C22×C4×C83—24no—
197C22×C4×C83—44no—
198C22×C4×C83C22×C414yes—
199C22×C4×C83C2×D414yes—
200C22×C4×C83C2×Q814yes—
201C22×C4×C83C2412yes—
202C22×C4×C83C2414yes—
203C22×C4×C83C2412yes—
204C22×C4×C83C2414yes—
205C22×C4×C83C2411yescharacteristic
206C22×C4×C83C2411yescharacteristic
207C4⋊C4×C83—24no—
208C4⋊C4×C83—24no—
209C4⋊C4×C83—22no—
210C4⋊C4×C83—22no—
211C4⋊C4×C83—48no—
212C4⋊C4×C83C22×C418yes—
213C22⋊C4×C83—24no—
214C22⋊C4×C83C22×C418yes—
215C42×C83—24no—
216C42×C83C22×C418yes—
217C23×C83—22no—
218C23×C83—22no—
219C23×C83C23×C412yes—
220C23×C83C22×D412yes—
221C23×C83C22×Q812yes—
222C23×C83C2511yescharacteristic
223C22×D83—24no—
224C22×D83—24no—
225C22×D83C23×C414yes—
226C22×D83C22×D414yes—
227C22×D83C22×Q814yes—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—28no—
235C2×C4×C83—24no—
236C2×C4×C83—84no—
237C2×C4×C83C2×C4⋊C4116yescomplemented
238C2×C4×C83C23×C418yes—
239C2×C4×C83C23×C412yes—
240C2×C4×C83C23×C412yes—
241C2×C4×C83C22×D412yes—
242C2×C4×C83C22×D412yes—
243C2×C4×C83C22×Q812yes—
244C2×C4×C83C22×Q812yes—
245C22⋊D83C2×C4⋊C4132yescomplemented
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—216no—
249C2×Dic83—22no—
250C2×Dic83—24no—
251C2×Dic83—22no—
252C2×Dic83—28no—
253C2×Dic83—24no—
254C2×Dic83—24no—
255C2×Dic83—84no—
256C2×Dic83C23×C414yes—
257C2×Dic83C23×C414yes—
258C2×Dic83C23×C414yes—
259C2×Dic83C22×D414yes—
260C2×Dic83C22×Q814yes—
261C2×Dic83H(5,𝔽2)14yes—
262C2×Dic832−1+414yes—
263D332C2×C4⋊C4116yescomplemented
264C4×D83—232no—
265C4×D83—88no—
266C4×D83—88no—
267C4×D83C2×C4⋊C4132yescomplemented
268Dic166C2×C4⋊C4116yescomplemented
269C22×C83—24no—
270C22×C83—22no—
271C22×C83—22no—
272C22×C83—24no—
273C22×C83C22×C4⋊C414yes—
274C22×C83C24×C411yescharacteristic
275C22×C83C23×D411yescharacteristic
276C22×C83C23×Q811yescharacteristic
277C22×C83C2×H(5,𝔽2)12yes—
278C22×C83C2×2−1+412yes—
279C2×D83—28no—
280C2×D83—24no—
281C2×D83—24no—
282C2×D83—28no—
283C2×D83C22×C4⋊C418yes—
284C4×C83—28no—
285C4×C83—42no—
286C4×C83—44no—
287C4×C83—42no—
288C4×C83C22×C4⋊C414yes—
289C4×C83C22×C4⋊C414yes—
290Dic83—24no—
291Dic83—42no—
292Dic83—44no—
293Dic83—42no—
294Dic83C22×C4⋊C418yes—
295Dic83#52018yescomplemented
296#34183—831nomaximal, Sylow
297C2×C83—24no—
298C2×C83C23×C4⋊C411yescharacteristic
299C2×C83C2×#52012yes—
300C2×C83#349811yescharacteristic
301C2×C83#349911yescharacteristic
302C2×C83#350312yes—
303D83—28no—
304C2×#520—831no—
305C2×#520—831no—
306C2×C4×#41—831no—
307Q8×C4⋊C4—834no—
308D4×C4⋊C4—834no—
309D4×C4⋊C4—838no—
310#2317—832no—
311#2317—832no—
312#2317—831no—
313#2317D8311yescharacteristic, complemented
314#2285—832no—
315#2285—831no—
316C2×#384—832no—
317C2×#384—831no—
318C83#3418311yescharacteristic, complemented, Sylow
319C22×#41—831no—
320#520—1668no—
321#520—1668no—
322C4×#41—8316no—
323C2×C4×Q8—832no—
324C2×C4×Q8—832no—
325C2×C4×Q8—832no—
326C2×C4×D4—834no—
327C2×C4×D4—832no—
328C2×C4×D4—832no—
329C2×C4×D4—834no—
330C2×C4×D4—834no—
331C2×C4×D4—832no—
332C2×#117—831no—
333C2×#117—832no—
334C2×#117—832no—
335C2×#117—832no—
336C2×#117C2×D8311yescharacteristic
337C2×#117C2×D8311yescharacteristic
338C22×C4⋊C4—834no—
339C22×C4⋊C4—838no—
340C22×C4⋊C4—831no—
341C22×C4⋊C4—832no—
342C22×C4⋊C4—832no—
343C22×C4⋊C4C2×D8311yescharacteristic
344C22×C42—831no—
345C22×C42—832no—
346#389—838no—
347#389—834no—
348#389—838no—
349#389C2×D8314yes—
350#385—838no—
351#385—834no—
352#385—838no—
353#385C2×D8314yes—
354#384—832no—
355#384—834no—
356#384—838no—
357#384—831no—
358#384—832no—
359#384—831no—
360#384—832no—
361#384C2×D8312yes—
362#382—831no—
363#382—831no—
364#382—832no—
365#382C2×D8312yes—
366#380—834no—
367#380—838no—
368C4×C4⋊C4—834no—
369C4×C4⋊C4—838no—
370C2×#41—838no—
371C2×#41—838no—
372C2×#41—838no—
373C22×Q8—831no—
374C22×D4—832no—
375C22×D4—831no—
376C23×C4—832no—
377C23×C4—834no—
378C23×C4—834no—
379C23×C4—832no—
380C23×C4—832no—
381C23×C4—834no—
382C23×C4C22×D8312yes—
383C23×C4C22×D8311yescharacteristic
384C4×Q8—8316no—
385C4×Q8—1668no—
386C4×Q8—1668no—
387C4×D4—8316no—
388C4×D4—8332no—
389C4×D4—1668no—
390C4×D4—16616no—
391C4×D4—1668no—
392C4×D4—16616no—
393#117—24no—
394#117—8316no—
395#117—1668no—
396#117—1668no—
397#117—1664no—
398#117C4×D8318yescomplemented
399C2×C4⋊C4—24no—
400C2×C4⋊C4—33216no—
401C2×C4⋊C4—3328no—
402C2×C4⋊C4—838no—
403C2×C4⋊C4—838no—
404C2×C4⋊C4—834no—
405C2×C4⋊C4—832no—
406C2×C4⋊C4—831no—
407C2×C4⋊C4—834no—
408C2×C4⋊C4—838no—
409C2×C4⋊C4—834no—
410C2×C4⋊C4—834no—
411C2×C4⋊C4—832no—
412C2×C4⋊C4—831no—
413C2×C4⋊C4—834no—
414C2×C4⋊C4—1668no—
415C2×C4⋊C4—1668no—
416C2×C4⋊C4—1664no—
417C2×C4⋊C4C22×D8311yescharacteristic
418C2×C4⋊C4C22×D8312yes—
419C2×C4⋊C4C22×D8312yes—
420C2×C4⋊C4C22×D8311yescharacteristic
421C2×C4⋊C4C22×D8318yes—
422C2×C4⋊C4C22×D8312yes—
423C2×C22⋊C4—834no—
424C2×C22⋊C4—832no—
425C2×C22⋊C4—834no—
426C2×C22⋊C4—834no—
427C2×C22⋊C4C22×D8312yes—
428C2×C22⋊C4C22×D8312yes—
429C2×C42—838no—
430C2×C42—831no—
431C2×C42—834no—
432C2×C42—832no—
433C2×C42—831no—
434C2×C42—8316no—
435C2×C42—834no—
436C2×C42—834no—
437C2×C42—832no—
438C2×C42C22×D8312yes—
439C2×C42C22×D8312yes—
440#95—834no—
441#95—834no—
442#95—838no—
443#95C22×D8318yes—
444C24—832no—
445C24C23×D8311yescharacteristic
446#41—8364no—
447C2×Q8—834no—
448C2×Q8—834no—
449C2×Q8—834no—
450C2×D4—838no—
451C2×D4—838no—
452C2×D4—838no—
453C2×D4—834no—
454C2×D4—834no—
455C2×D4—834no—
456C22×C4—24no—
457C22×C4—44no—
458C22×C4—3328no—
459C22×C4—3328no—
460C22×C4—3324no—
461C22×C4—832no—
462C22×C4—832no—
463C22×C4—838no—
464C22×C4—838no—
465C22×C4—838no—
466C22×C4—838no—
467C22×C4—832no—
468C22×C4—8316no—
469C22×C4—832no—
470C22×C4—832no—
471C22×C4—834no—
472C22×C4—1668no—
473C22×C4—1668no—
474C22×C4—1668no—
475C22×C4—1668no—
476C22×C4—1664no—
477C22×C4—1664no—
478C22×C4—1664no—
479C22×C4C2×C4×D8314yes—
480C22×C4D4×D8314yes—
481C22×C4Q8×D8314yes—
482C22×C4C23×D8312yes—
483C22×C4C23×D8312yes—
484C22×C4C23×D8314yes—
485C22×C4C23×D8314yes—
486C22×C4C23×D8311yescharacteristic
487C22×C4C23×D8311yescharacteristic
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—48no—
493C4⋊C4—3328no—
494C4⋊C4—3328no—
495C4⋊C4—8316no—
496C4⋊C4—1668no—
497C4⋊C4—16616no—
498C4⋊C4—1668no—
499C4⋊C4—1668no—
500C4⋊C4—1668no—
501C4⋊C4—1662no—
502C4⋊C4—1664no—
503C4⋊C4—1664no—
504C4⋊C4—1662no—
505C4⋊C4C2×C4×D8318yes—
506C22⋊C4—24no—
507C22⋊C4—8316no—
508C22⋊C4—1668no—
509C22⋊C4—1668no—
510C22⋊C4—1664no—
511C22⋊C4C2×C4×D8318yes—
512C42—24no—
513C42—834no—
514C42—834no—
515C42—8316no—
516C42—838no—
517C42—1668no—
518C42—1668no—
519C42—1664no—
520C42C2×C4×D8318yes—
521C23—22no—
522C23—22no—
523C23—834no—
524C23—834no—
525C23—834no—
526C23—1664no—
527C23—1664no—
528C23C22×C4×D8312yes—
529C23C2×D4×D8312yes—
530C23C2×Q8×D8312yes—
531C23C24×D8311yescharacteristic
532Q8—8316no—
533D4—8332no—
534D4—8316no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—84no—
544C2×C4—834no—
545C2×C4—834no—
546C2×C4—834no—
547C2×C4—834no—
548C2×C4—834no—
549C2×C4—834no—
550C2×C4—8332no—
551C2×C4—834no—
552C2×C4—6648no—
553C2×C4—6648no—
554C2×C4—6644no—
555C2×C4—16632no—
556C2×C4—1662no—
557C2×C4—1664no—
558C2×C4—1662no—
559C2×C4—1668no—
560C2×C4—1664no—
561C2×C4—1664no—
562C2×C4—16616no—
563C2×C4—1662no—
564C2×C4—1662no—
565C2×C4C4⋊C4×D83116yescomplemented
566C2×C4C22×C4×D8318yes—
567C2×C4C22×C4×D8312yes—
568C2×C4C22×C4×D8312yes—
569C2×C4C2×D4×D8312yes—
570C2×C4C2×D4×D8312yes—
571C2×C4C2×Q8×D8312yes—
572C2×C4C2×Q8×D8312yes—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—24no—
577C22—838no—
578C22—1668no—
579C22—1664no—
580C22—1664no—
581C22—1668no—
582C22C2×C4⋊C4×D8314yes—
583C22C23×C4×D8311yescharacteristic
584C22C22×D4×D8311yescharacteristic
585C22C22×Q8×D8311yescharacteristic
586C22H(5,𝔽2)×D8312yes—
587C222−1+4×D8312yes—
588C4—28no—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—3322no—
593C4—3324no—
594C4—3322no—
595C4—838no—
596C4—838no—
597C4—1664no—
598C4C2×C4⋊C4×D8314yes—
599C4C2×C4⋊C4×D8314yes—
600C2—24no—
601C2—1668no—
602C2C22×C4⋊C4×D8311yescharacteristic
603C2#520×D8312yes—
604C2#2492123811yescharacteristic
605C2#2492124511yescharacteristic
606C2#2492127812yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2317C2
socle, cosocleC23×C83C25
prime radicalC22×#13050590C22
hypercenter#2317D83
hypocenterC83#34183
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2317

Complements

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

NH ≅ G/Ntype
4. #24910377C2semidirect
5. #24910364C2semidirect
6. #24910361C2semidirect
7. #24910088C2semidirect
8. #24910086C2semidirect
9. #24910084C2semidirect
10. #24909869C2semidirect
11. #24909866C2semidirect
12. C2×#19127570C2semidirect
13. #384×D83C2semidirect
14. C83×#2317C2semidirect
18. #19128169C4semidirect
80. #14581757C2×C4semidirect
83. #14581754C2×C4semidirect
86. (C22⋊D83)⋊C4C2×C4semidirect
90. #14581749C2×C4semidirect
112. C2×C4×Dic83C2×C4semidirect
120. C42×D83C2×C4semidirect
127. #117×C83C2×C4semidirect
141. #13050590C4⋊C4semidirect
237. C2×C4×C83C2×C4⋊C4semidirect
245. C22⋊D83C2×C4⋊C4semidirect
263. D332C2×C4⋊C4semidirect
267. C4×D83C2×C4⋊C4semidirect
268. Dic166C2×C4⋊C4semidirect
295. Dic83#520semidirect
313. #2317D83semidirect
318. C83#34183semidirect
398. #117C4×D83semidirect
565. C2×C4C4⋊C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281822
C2×#191281812
C2×#191281692
#249103772
#249103642
#249103612
#249100882
#249100862
#249100842
#249098692
#249098662
C2×#191275702
#384×D832
C83×#23172
#3418383