Finite Group Databasebeta

← #35330608

Subgroups — #35330608

Order 21248.

1470 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127620C211yesmaximal, characteristic
2C2×C4×Dic83⋊C4C211yesmaximal, characteristic
3#24905734C211yesmaximal, characteristic
4#24905726C211yesmaximal, characteristic
5#24905713C211yesmaximal, characteristic
6#24905698C212yesmaximal
7#24905694C212yesmaximal
8#24905694C212yesmaximal
9Dic83⋊#95C212yesmaximal
10C4×C83⋊#95C211yesmaximal, characteristic
11C2×#391×C83C211yesmaximal, characteristic
12C22×Dic83⋊C4C2212yes—
13C22×Dic83⋊C4C2211yescharacteristic
14C22×Dic83⋊C4C2211yescharacteristic
15C22×Dic83⋊C4C2212yes—
16C22×C4×Dic83C2211yescharacteristic
17C22×C4×Dic83C2212yes—
18C22×C4×Dic83C2211yescharacteristic
19#19127620C418yescomplemented
20C4×Dic83⋊C4—24no—
21C2×C83⋊#95C2211yescharacteristic
22C2×C83⋊#95C2211yescharacteristic
23C2×C83⋊#95C2211yescharacteristic
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2211yescharacteristic
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C23×Q8×C83C2211yescharacteristic
33C22×C4⋊C4×C83C2211yescharacteristic
34C22×C42×C83C2211yescharacteristic
35#391×C83—24no—
36C2×#95×C83C2212yes—
37C2×#95×C83C2212yes—
38C23×Dic83C2311yescharacteristic
39C23×Dic83C2311yescharacteristic
40C23×Dic83C2312yes—
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2312yes—
44C2×Dic83⋊C4—24no—
45C2×Dic83⋊C4—24no—
46C2×Dic83⋊C4—22no—
47C2×Dic83⋊C4—22no—
48C2×Dic83⋊C4—22no—
49C2×Dic83⋊C4—22no—
50C2×Dic83⋊C4—42no—
51C2×Dic83⋊C4—42no—
52C2×Dic83⋊C4—42no—
53C2×Dic83⋊C4C2×C414yes—
54C2×Dic83⋊C4C2×C418yes—
55C2×Dic83⋊C4Q814yescomplemented
56C2×C4×Dic83—22no—
57C2×C4×Dic83—22no—
58C2×C4×Dic83—24no—
59C2×C4×Dic83—22no—
60C2×C4×Dic83—22no—
61C2×C4×Dic83—24no—
62C2×C4×Dic83—42no—
63C2×C4×Dic83—42no—
64C83⋊#95—24no—
65C83⋊#95—24no—
66C83⋊#95—22no—
67C83⋊#95—22no—
68C83⋊#95—22no—
69C83⋊#95—42no—
70C83⋊#95—42no—
71C83⋊#95—42no—
72C83⋊#95C2×C418yes—
73C83⋊#95C2×C414yes—
74C83⋊#95Q818yescomplemented
75C22×Q8×C83—22no—
76C22×Q8×C83—22no—
77C22×Q8×C83—42no—
78C22×Q8×C83—42no—
79C22×Q8×C83C2×C414yes—
80C23×C4×C83C2312yes—
81C23×C4×C83C2312yes—
82C23×C4×C83C2311yescharacteristic
83C23×C4×C83C2311yescharacteristic
84C23×C4×C83C2311yescharacteristic
85C2×C4⋊C4×C83—22no—
86C2×C4⋊C4×C83—42no—
87C2×C42×C83—42no—
88C2×C42×C83D414yes—
89#95×C83—24no—
90#95×C83—24no—
91C22×Dic83—21no—
92C22×Dic83—21no—
93C22×Dic83—22no—
94C22×Dic83—22no—
95C22×Dic83—21no—
96C22×Dic83—21no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—21no—
100C22×Dic83—21no—
101C22×Dic83—21no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—21no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—21no—
110C22×Dic83—21no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83C22×C412yes—
118C22×Dic83C22×C412yes—
119C22×Dic83C22×C414yes—
120C22×Dic83C2×Q814yes—
121C22×Dic83C2×Q812yes—
122C22×Dic83C2×Q814yes—
123C22×Dic83C2×Q812yes—
124C22×Dic83#4112yes—
125C22×Dic83#4112yes—
126C22×Dic83#4114yes—
127Dic83⋊C4—28no—
128Dic83⋊C4—28no—
129Dic83⋊C4—48no—
130Dic83⋊C4—48no—
131C4×Dic83—28no—
132C4×Dic83—48no—
133C4×Dic83—44no—
134C24×C83C2411yescharacteristic
135C2×Q8×C83—28no—
136C2×Q8×C83—48no—
137C2×Q8×C83—44no—
138C2×Q8×C83—44no—
139C2×Q8×C83—48no—
140C22×C4×C83—21no—
141C22×C4×C83—21no—
142C22×C4×C83—22no—
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83C22×C414yes—
161C22×C4×C83C22×C412yes—
162C22×C4×C83C2×D412yes—
163C22×C4×C83C2×D412yes—
164C22×C4×C83C2×D412yes—
165C22×C4×C83C2×Q812yes—
166C22×C4×C83C2×Q814yes—
167C22×C4×C83#4112yes—
168C22×C4×C83#4112yes—
169C4⋊C4×C83—44no—
170C42×C83—44no—
171C23×C83C23×C411yescharacteristic
172C23×C83C22×D411yescharacteristic
173C23×C83C22×Q811yescharacteristic
174C23×C83C22×Q812yes—
175C23×C83C2×#4111yescharacteristic
176C23×C83C2×#4112yes—
177C23×C83C2×#4111yescharacteristic
178C23×C83C2×#4111yescharacteristic
179C23×C83C2×#4111yescharacteristic
180C23×C83H(5,𝔽2)12yes—
181C23×C83H(5,𝔽2)12yes—
182Q8×C83—416no—
183C2×C4×C83—24no—
184C2×C4×C83—22no—
185C2×C4×C83—22no—
186C2×C4×C83—22no—
187C2×C4×C83—22no—
188C2×C4×C83—22no—
189C2×C4×C83—22no—
190C2×C4×C83—24no—
191C2×C4×C83—24no—
192C2×C4×C83—24no—
193C2×C4×C83—24no—
194C2×C4×C83—24no—
195C2×C4×C83—24no—
196C2×C4×C83—24no—
197C2×C4×C83—24no—
198C2×C4×C83—42no—
199C2×C4×C83—42no—
200C2×C4×C83—42no—
201C2×C4×C83—42no—
202C2×C4×C83—42no—
203C2×C4×C83—42no—
204C2×C4×C83—42no—
205C2×C4×C83—42no—
206C2×C4×C83—42no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83C4×D414yes—
210C2×C4×C83C22⋊Q814yes—
211C2×C4×C83C4⋊Q814yes—
212C2×Dic83—22no—
213C2×Dic83—24no—
214C2×Dic83—22no—
215C2×Dic83—24no—
216C2×Dic83—22no—
217C2×Dic83—22no—
218C2×Dic83—22no—
219C2×Dic83—22no—
220C2×Dic83—24no—
221C2×Dic83—24no—
222C2×Dic83—24no—
223C2×Dic83—24no—
224C2×Dic83—22no—
225C2×Dic83—24no—
226C2×Dic83—22no—
227C2×Dic83—22no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83—24no—
231C2×Dic83—24no—
232C2×Dic83—24no—
233C2×Dic83—42no—
234C2×Dic83—41no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—42no—
241C2×Dic83—42no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83C4×Q818yes—
247C2×Dic83C4×Q814yes—
248C22×C83C2×C4×D411yescharacteristic
249C22×C83C2×C4×Q812yes—
250C22×C83C2×C4×Q811yescharacteristic
251C22×C83C4×#4111yescharacteristic
252C22×C83#51812yes—
253C22×C83C2×C22⋊Q811yescharacteristic
254C22×C83#52511yescharacteristic
255C22×C83#52911yescharacteristic
256C22×C83C2×C4⋊Q811yescharacteristic
257C22×C83#53211yescharacteristic
258C22×C83#53312yes—
259C22×C83#53311yescharacteristic
260C22×C83#53411yescharacteristic
261C22×C83#53511yescharacteristic
262C22×C83#53812yes—
263C22×C83#55112yes—
264C22×C83#55112yes—
265C22×C83#55412yes—
266C22×C83#55412yes—
267C22×C83#55412yes—
268C22×C83#55412yes—
269C22×C83#55512yes—
270C22×C83Q8212yes—
271C4×C83—24no—
272C4×C83—24no—
273C4×C83—24no—
274C4×C83—44no—
275C4×C83—82no—
276Dic83—28no—
277Dic83—24no—
278Dic83—42no—
279Dic83—44no—
280Dic83—42no—
281Dic83—42no—
282#14241—831nomaximal, Sylow
283C2×C83C4×C22⋊Q811yescharacteristic
284C2×C83C4×C4⋊Q811yescharacteristic
285C2×C83#232111yescharacteristic
286C2×C83#236812yes—
287C2×C83#236912yes—
288C2×C83#256411yescharacteristic
289C2×C83#256711yescharacteristic
290C2×C83#257412yes—
291C2×C83#258311yescharacteristic
292C2×C83#258612yes—
293C2×C83#261011yescharacteristic
294C2×#395—831no—
295C2×#391D8311yescharacteristic
296C2×C4×C4⋊C4—831no—
297#1446—831no—
298#1444—831no—
299#1441—831no—
300#1436—832no—
301#1436—832no—
302#1436—832no—
303C4⋊#95—832no—
304C4×#95—831no—
305C83#1424111yescharacteristic, complemented, Sylow
306C23×Q8C2×D8311yescharacteristic
307C22×C4⋊C4—832no—
308C22×C4⋊C4—832no—
309C22×C4⋊C4—831no—
310C22×C4⋊C4—831no—
311C22×C4⋊C4C2×D8311yescharacteristic
312C22×C42—831no—
313C22×C42—832no—
314C22×C42—831no—
315C22×C42C2×D8311yescharacteristic
316#395—838no—
317#391—24no—
318C4×C4⋊C4—1664no—
319C2×#95—832no—
320C2×#95—832no—
321C2×#95—832no—
322C2×#95—831no—
323C2×#95—832no—
324C2×#95—832no—
325C2×#95—831no—
326C2×#95—831no—
327C2×#95—832no—
328C2×#95—832no—
329C2×#95—831no—
330C2×#95C2×D8312yes—
331C2×#95C2×D8312yes—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—42no—
335C22×Q8—42no—
336C22×Q8C4×D8314yes—
337C23×C4—832no—
338C23×C4—831no—
339C23×C4—831no—
340C23×C4—831no—
341C23×C4—831no—
342C23×C4—832no—
343C23×C4C22×D8312yes—
344C23×C4C22×D8312yes—
345C23×C4C22×D8311yescharacteristic
346C23×C4C22×D8311yescharacteristic
347C23×C4C22×D8311yescharacteristic
348C2×C4⋊C4—22no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—3322no—
351C2×C4⋊C4—3322no—
352C2×C4⋊C4—3322no—
353C2×C4⋊C4—838no—
354C2×C4⋊C4—834no—
355C2×C4⋊C4—834no—
356C2×C4⋊C4—1664no—
357C2×C4⋊C4—1664no—
358C2×C4⋊C4—1662no—
359C2×C4⋊C4—1662no—
360C2×C4⋊C4—1662no—
361C2×C4⋊C4—1662no—
362C2×C42—42no—
363C2×C42—3322no—
364C2×C42—3322no—
365C2×C42—1664no—
366C2×C42—1662no—
367C2×C42—1664no—
368C2×C42—1662no—
369C2×C42—1662no—
370C2×C42—1662no—
371C2×C42C22⋊D8314yes—
372#95—24no—
373#95—24no—
374#95—3322no—
375#95—3322no—
376#95—3322no—
377#95—838no—
378#95—834no—
379#95—838no—
380#95—1664no—
381#95—1664no—
382#95—1662no—
383#95—1662no—
384#95—1662no—
385C24C23×D8311yescharacteristic
386C2×Q8—28no—
387C2×Q8—48no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—48no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—832no—
412C22×C4—834no—
413C22×C4—834no—
414C22×C4—832no—
415C22×C4—832no—
416C22×C4—834no—
417C22×C4—832no—
418C22×C4—834no—
419C22×C4—832no—
420C22×C4—832no—
421C22×C4—1661no—
422C22×C4—1661no—
423C22×C4—1661no—
424C22×C4—1662no—
425C22×C4—1662no—
426C22×C4—1662no—
427C22×C4—1661no—
428C22×C4—1662no—
429C22×C4—1662no—
430C22×C4—1661no—
431C22×C4—1662no—
432C22×C4—1661no—
433C22×C4—1662no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4—1662no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4—1661no—
440C22×C4—1661no—
441C22×C4—1662no—
442C22×C4—1661no—
443C22×C4—1662no—
444C22×C4—1661no—
445C22×C4—1662no—
446C22×C4—1662no—
447C22×C4C2×C4×D8314yes—
448C22×C4C2×C4×D8312yes—
449C22×C4C2×C22⋊D8312yes—
450C22×C4C2×C22⋊D8312yes—
451C22×C4C2×C22⋊D8312yes—
452C22×C4Q8×D8312yes—
453C22×C4Q8×D8314yes—
454C22×C4#1305059012yes—
455C22×C4#1305059112yes—
456C4⋊C4—44no—
457C4⋊C4—3328no—
458C4⋊C4—3328no—
459C4⋊C4—1668no—
460C4⋊C4—1668no—
461C42—44no—
462C42—3324no—
463C42—3328no—
464C42—1668no—
465C23C22×C4×D8311yescharacteristic
466C23C22×C22⋊D8311yescharacteristic
467C23C2×Q8×D8312yes—
468C23C2×Q8×D8311yescharacteristic
469C23#41×D8312yes—
470C23#41×D8311yescharacteristic
471C23#41×D8311yescharacteristic
472C23C2×#1305059011yescharacteristic
473C23C2×#1305059111yescharacteristic
474C23#1458183712yes—
475C23#1458183712yes—
476Q8—416no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—3322no—
504C2×C4—3322no—
505C2×C4—3322no—
506C2×C4—3322no—
507C2×C4—3322no—
508C2×C4—3321no—
509C2×C4—3321no—
510C2×C4—3321no—
511C2×C4—3321no—
512C2×C4—3321no—
513C2×C4—3321no—
514C2×C4—3322no—
515C2×C4—3322no—
516C2×C4—834no—
517C2×C4—838no—
518C2×C4—1664no—
519C2×C4—1664no—
520C2×C4—1664no—
521C2×C4—1664no—
522C2×C4—1662no—
523C2×C4—1664no—
524C2×C4—1662no—
525C2×C4—1662no—
526C2×C4—1664no—
527C2×C4—1664no—
528C2×C4—1664no—
529C2×C4—1664no—
530C2×C4—1664no—
531C2×C4—1662no—
532C2×C4—1662no—
533C2×C4—1664no—
534C2×C4—1664no—
535C2×C4—1662no—
536C2×C4—1662no—
537C2×C4—1662no—
538C2×C4—1662no—
539C2×C4C4×C22⋊D8314yes—
540C2×C4#1458176814yes—
541C2×C4#1458179014yes—
542C22C2×C4×C22⋊D8311yescharacteristic
543C22C4×Q8×D8311yescharacteristic
544C22C4×Q8×D8312yes—
545C22C4×#1305059111yescharacteristic
546C22#1912817412yes—
547C22C2×#1458176811yescharacteristic
548C22#1912820811yescharacteristic
549C22#1912823811yescharacteristic
550C22C2×#1458179011yescharacteristic
551C22#1912824711yescharacteristic
552C22#1912825512yes—
553C22#1912825511yescharacteristic
554C22#1912826011yescharacteristic
555C22#1912826811yescharacteristic
556C22#1912828912yes—
557C22#1912837812yes—
558C22#1912838112yes—
559C22#1912840512yes—
560C22#1912840512yes—
561C22#1912840512yes—
562C22#1912841212yes—
563C22#1912841312yes—
564C22#1912843112yes—
565C4—24no—
566C4—24no—
567C4—24no—
568C4—44no—
569C4—82no—
570C4—3322no—
571C4—3322no—
572C4—3322no—
573C4—3324no—
574C4—1664no—
575C4—1668no—
576C2C4×#1458176811yescharacteristic
577C2C4×#1458179011yescharacteristic
578C2#2491012111yescharacteristic
579C2#2491054212yes—
580C2#2491055112yes—
581C2#2491264511yescharacteristic
582C2#2491267211yescharacteristic
583C2#2491273912yes—
584C2#2491284411yescharacteristic
585C2#2491287412yes—
586C2#2491315211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#391×C83C2
socle, prime radical, cosocleC24×C83C24
hypercenterC2×#391D83
hypocenterC83#14241
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
19. #19127620C4semidirect
55. C2×Dic83⋊C4Q8semidirect
74. C83⋊#95Q8semidirect
305. C83#14241semidirect

Maximal subgroups

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

subgroupindex
C2×#191276202
C2×C4×Dic83⋊C42
#249057342
#249057262
#249057132
#249056982
#249056942
Dic83⋊#952
C4×C83⋊#952
C2×#391×C832
#1424183