Finite Group Databasebeta

← #35330293

Subgroups — #35330293

Order 21248.

1738 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127587C212yesmaximal, complemented
2C2×C22⋊C4×Dic83C212yesmaximal
3#24905708C212yesmaximal
4#24905708C211yesmaximal, characteristic
5#24905697C212yesmaximal, complemented
6#24905695C212yesmaximal, complemented
7#24905692C212yesmaximal, complemented
8#24905685C211yesmaximal, characteristic
9C2×#399×C83C211yesmaximal, characteristic
10C24×Dic83C2211yescharacteristic
11C22×C4⋊Dic83C2211yescharacteristic
12C22×C4⋊Dic83C2212yes—
13C22×C22⋊Dic83C2212yes—
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C4×Dic83C2212yes—
16C22×C4×Dic83C2212yes—
17#19127587—28no—
18C22⋊C4×Dic83—28no—
19C2×C83⋊#95C2212yes—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C22×C4⋊C4×C83C2212yes—
29C22×C4⋊C4×C83C2211yescharacteristic
30C22×C22⋊C4×C83C2212yes—
31C22×C22⋊C4×C83C2211yescharacteristic
32#399×C83C418yescomplemented
33C2×#95×C83C2211yescharacteristic
34C23×Dic83—24no—
35C23×Dic83—24no—
36C23×Dic83—41no—
37C23×Dic83—41no—
38C23×Dic83Q814yes—
39C23×Dic83C2312yes—
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43C2×C4⋊Dic83—24no—
44C2×C4⋊Dic83—24no—
45C2×C4⋊Dic83—24no—
46C2×C4⋊Dic83—42no—
47C2×C4⋊Dic83—41no—
48C2×C22⋊Dic83—24no—
49C2×C22⋊Dic83—24no—
50C2×C22⋊Dic83—24no—
51C2×C22⋊Dic83—22no—
52C2×C22⋊Dic83—41no—
53C2×C22⋊Dic83—41no—
54C2×C4×Dic83—24no—
55C2×C4×Dic83—24no—
56C2×C4×Dic83—24no—
57C2×C4×Dic83—24no—
58C2×C4×Dic83—42no—
59C2×C4×Dic83—42no—
60C83⋊#95—24no—
61C83⋊#95—24no—
62C83⋊#95—24no—
63C83⋊#95—24no—
64C83⋊#95—24no—
65C83⋊#95—24no—
66C83⋊#95—41no—
67C83⋊#95—42no—
68C83⋊#95D418yes—
69C25×C83C2311yescharacteristic
70C23×C4×C83C2312yes—
71C23×C4×C83C2311yescharacteristic
72C23×C4×C83C2311yescharacteristic
73C23×C4×C83C2312yes—
74C2×C4⋊C4×C83—24no—
75C2×C4⋊C4×C83—24no—
76C2×C4⋊C4×C83—41no—
77C2×C4⋊C4×C83—41no—
78C2×C4⋊C4×C83C2×C418yescomplemented
79C2×C4⋊C4×C83C2×C414yescomplemented
80C2×C22⋊C4×C83—24no—
81C2×C22⋊C4×C83—24no—
82C2×C22⋊C4×C83—41no—
83C2×C22⋊C4×C83—41no—
84C2×C22⋊C4×C83C2×C418yes—
85C2×C22⋊C4×C83C2×C414yes—
86#95×C83C2×C414yescomplemented
87C22×Dic83—22no—
88C22×Dic83—22no—
89C22×Dic83—22no—
90C22×Dic83—22no—
91C22×Dic83—22no—
92C22×Dic83—22no—
93C22×Dic83—22no—
94C22×Dic83—22no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—216no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—44no—
112C22×Dic83—44no—
113C22×Dic83—48no—
114C22×Dic83—44no—
115C22×Dic83C2×D414yes—
116C22×Dic83C2×D414yes—
117C22×Dic83C2×Q814yes—
118C22×Dic83#4114yes—
119C22×Dic83#4114yes—
120C4⋊Dic83—44no—
121C4⋊Dic83—48no—
122C22⋊Dic83—216no—
123C22⋊Dic83—44no—
124C4×Dic83—48no—
125C4×Dic83—48no—
126C24×C83—22no—
127C24×C83—22no—
128C24×C83—22no—
129C24×C83—22no—
130C24×C83—21no—
131C24×C83—21no—
132C24×C83—21no—
133C24×C83—21no—
134C24×C83C22×C412yes—
135C24×C83C2×Q812yes—
136C24×C83#4112yes—
137C24×C83C2411yescharacteristic
138C22×C4×C83—22no—
139C22×C4×C83—22no—
140C22×C4×C83—22no—
141C22×C4×C83—22no—
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—21no—
151C22×C4×C83—21no—
152C22×C4×C83—21no—
153C22×C4×C83—21no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83C22×C412yes—
157C22×C4×C83C22×C414yes—
158C22×C4×C83C22×C412yes—
159C22×C4×C83C22×C414yes—
160C22×C4×C83C2×D414yes—
161C22×C4×C83#4114yes—
162C4⋊C4×C83—216no—
163C4⋊C4×C83—44no—
164C22⋊C4×C83—216no—
165C22⋊C4×C83—44no—
166C23×C83—24no—
167C23×C83—24no—
168C23×C83—22no—
169C23×C83—22no—
170C23×C83—24no—
171C23×C83—24no—
172C23×C83—24no—
173C23×C83—24no—
174C23×C83—22no—
175C23×C83—22no—
176C23×C83—22no—
177C23×C83—22no—
178C23×C83—41no—
179C23×C83—42no—
180C23×C83—42no—
181C23×C83—42no—
182C23×C83—42no—
183C23×C83—41no—
184C23×C83—42no—
185C23×C83—42no—
186C23×C83—41no—
187C23×C83—41no—
188C23×C83C4×Q814yes—
189C23×C83C23×C411yescharacteristic
190C23×C83C22×D412yes—
191C23×C83C22×Q811yescharacteristic
192C23×C83C2×#4112yes—
193C23×C83C2×#4112yes—
194C23×C83C2×#4112yes—
195C23×C83C2×#4111yescharacteristic
196C23×C83H(5,𝔽2)11yescharacteristic
197C23×C83H(5,𝔽2)11yescharacteristic
198C23×C83H(5,𝔽2)11yescharacteristic
199C23×C832−1+411yescharacteristic
200C2×C4×C83—24no—
201C2×C4×C83—24no—
202C2×C4×C83—24no—
203C2×C4×C83—24no—
204C2×C4×C83—24no—
205C2×C4×C83—24no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83—24no—
209C2×C4×C83—24no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—24no—
213C2×C4×C83—42no—
214C2×C4×C83—41no—
215C2×C4×C83—42no—
216C2×C4×C83—42no—
217C2×C4×C83—42no—
218C2×C4×C83—41no—
219C2×C4×C83—42no—
220C2×C4×C83—41no—
221C2×C4×C83—42no—
222C2×C4×C83—41no—
223C2×C4×C83C4×D418yes—
224C2×Dic83—24no—
225C2×Dic83—24no—
226C2×Dic83—24no—
227C2×Dic83—24no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83—24no—
231C2×Dic83—24no—
232C2×Dic83—24no—
233C2×Dic83—24no—
234C2×Dic83—24no—
235C2×Dic83—24no—
236C2×Dic83—416no—
237C2×Dic83—42no—
238C2×Dic83—42no—
239C2×Dic83—42no—
240C2×Dic83—42no—
241C2×Dic83—42no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249C2×Dic83—42no—
250C2×Dic83—42no—
251C2×Dic83—42no—
252C2×Dic83C22⋊Q818yes—
253C22×C83—24no—
254C22×C83—28no—
255C22×C83—42no—
256C22×C83—44no—
257C22×C83—42no—
258C22×C83—42no—
259C22×C83—42no—
260C22×C83—44no—
261C22×C83—42no—
262C22×C83—44no—
263C22×C83—42no—
264C22×C83C2×C4×D412yes—
265C22×C83C2×C4×Q811yescharacteristic
266C22×C83C4×#4112yes—
267C22×C83#51811yescharacteristic
268C22×C83#52011yescharacteristic
269C22×C83C2×C22⋊Q812yes—
270C22×C83#52512yes—
271C22×C83#52912yes—
272C22×C83#53312yes—
273C22×C83#53811yescharacteristic
274C22×C83#53911yescharacteristic
275C22×C83#54311yescharacteristic
276C22×C83#54411yescharacteristic
277C22×C83D4211yescharacteristic
278C22×C83#54612yes—
279C22×C83#54712yes—
280C22×C83#55012yes—
281C22×C83#55112yes—
282C22×C83#55111yescharacteristic
283C22×C83#55212yes—
284C22×C83#55312yes—
285C22×C83#55911yescharacteristic
286C22×C83#55911yescharacteristic
287C4×C83—28no—
288C4×C83—44no—
289C4×C83—42no—
290C4×C83—42no—
291Dic83—28no—
292Dic83—44no—
293Dic83—44no—
294Dic83—82no—
295#14215—831nomaximal, Sylow
296C2×C83—44no—
297C2×C83C4×C22⋊Q812yes—
298C2×C83#232911yescharacteristic
299C2×C83#235912yes—
300C2×C83#236611yescharacteristic
301C2×C83#237411yescharacteristic
302C2×C83#243711yescharacteristic
303C2×C83#244112yes—
304C2×C83#245511yescharacteristic
305C2×C83#246712yes—
306C2×C83#251711yescharacteristic
307C2×C83#260711yescharacteristic
308C2×#399D8311yescharacteristic
309C2×#387—832no—
310C2×C4×C22⋊C4—832no—
311#1439—832no—
312#1439—831no—
313#1436—832no—
314#1436—832no—
315C4⋊#95—832no—
316C22⋊#95—831no—
317C83#1421511yescharacteristic, complemented, Sylow
318C24×C4—831no—
319C22×C4⋊C4—832no—
320C22×C4⋊C4—831no—
321C22×C4⋊C4C2×D8312yes—
322C22×C4⋊C4C2×D8311yescharacteristic
323C22×C22⋊C4—831no—
324C22×C22⋊C4—832no—
325C22×C22⋊C4C2×D8312yes—
326C22×C22⋊C4C2×D8311yescharacteristic
327C22×C42—832no—
328C22×C42—832no—
329#399Dic8318yescomplemented
330#387—1668no—
331C4×C22⋊C4—1668no—
332C2×#95—831no—
333C2×#95—832no—
334C2×#95—832no—
335C2×#95—832no—
336C2×#95—832no—
337C2×#95—832no—
338C2×#95—832no—
339C2×#95—832no—
340C2×#95—832no—
341C2×#95C2×D8311yescharacteristic
342C25C22×D8311yescharacteristic
343C23×C4—3321no—
344C23×C4—3321no—
345C23×C4—834no—
346C23×C4—832no—
347C23×C4—832no—
348C23×C4—832no—
349C23×C4—832no—
350C23×C4—1664no—
351C23×C4—1664no—
352C23×C4C22×D8311yescharacteristic
353C23×C4C22×D8312yes—
354C23×C4C22×D8311yescharacteristic
355C23×C4C22×D8312yes—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—41no—
359C2×C4⋊C4—41no—
360C2×C4⋊C4—3322no—
361C2×C4⋊C4—3321no—
362C2×C4⋊C4—1664no—
363C2×C4⋊C4—1664no—
364C2×C4⋊C4—1664no—
365C2×C4⋊C4C2×Dic8318yescomplemented
366C2×C4⋊C4C2×Dic8314yescomplemented
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—41no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—3321no—
372C2×C22⋊C4—3321no—
373C2×C22⋊C4—1662no—
374C2×C22⋊C4—1664no—
375C2×C22⋊C4—1664no—
376C2×C22⋊C4—1664no—
377C2×C22⋊C4C2×Dic8318yes—
378C2×C22⋊C4C2×Dic8314yes—
379C2×C42—3322no—
380C2×C42—3322no—
381C2×C42—1664no—
382C2×C42—1664no—
383C2×C42—1664no—
384C2×C42—1664no—
385#95—3321no—
386#95—3322no—
387#95—838no—
388#95—1664no—
389#95—1664no—
390#95—1664no—
391#95—1664no—
392#95—1664no—
393#95—1664no—
394#95C2×Dic8314yescomplemented
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—21no—
400C24—21no—
401C24—21no—
402C24—21no—
403C24C22×Dic8312yes—
404C24Q8×D8312yes—
405C24#1305059112yes—
406C24C23×D8311yescharacteristic
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—3324no—
426C22×C4—3324no—
427C22×C4—3328no—
428C22×C4—3324no—
429C22×C4—834no—
430C22×C4—834no—
431C22×C4—834no—
432C22×C4—834no—
433C22×C4—834no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4—1662no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4—1662no—
440C22×C4—1662no—
441C22×C4—16616no—
442C22×C4—1662no—
443C22×C4—1662no—
444C22×C4—1662no—
445C22×C4—1662no—
446C22×C4—1662no—
447C22×C4—1662no—
448C22×C4—1662no—
449C22×C4—1662no—
450C22×C4—1662no—
451C22×C4—1662no—
452C22×C4—1662no—
453C22×C4—1662no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4—1662no—
457C22×C4—1662no—
458C22×C4C22×Dic8312yes—
459C22×C4C22×Dic8314yes—
460C22×C4C22×Dic8312yes—
461C22×C4C22×Dic8314yes—
462C22×C4D4×D8314yes—
463C22×C4#1305058914yes—
464C4⋊C4—216no—
465C4⋊C4—44no—
466C4⋊C4—3328no—
467C4⋊C4—3324no—
468C22⋊C4—216no—
469C22⋊C4—44no—
470C22⋊C4—3324no—
471C22⋊C4—16616no—
472C42—3328no—
473C42—3328no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23Q8×Dic8314yes—
497C23C23×Dic8311yescharacteristic
498C23C2×D4×D8312yes—
499C23C2×Q8×D8311yescharacteristic
500C23#41×D8312yes—
501C23#41×D8312yes—
502C23C2×#1305058912yes—
503C23C2×#1305059111yescharacteristic
504C23#1458183711yescharacteristic
505C23#1458183711yescharacteristic
506C23#1458183811yescharacteristic
507C23#1458183911yescharacteristic
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—3322no—
532C2×C4—3322no—
533C2×C4—3322no—
534C2×C4—3322no—
535C2×C4—3322no—
536C2×C4—3322no—
537C2×C4—3322no—
538C2×C4—3322no—
539C2×C4—3322no—
540C2×C4—3322no—
541C2×C4—3322no—
542C2×C4—3322no—
543C2×C4—3322no—
544C2×C4—3322no—
545C2×C4—3322no—
546C2×C4—33216no—
547C2×C4—838no—
548C2×C4—1664no—
549C2×C4—1664no—
550C2×C4—1664no—
551C2×C4—1664no—
552C2×C4—1664no—
553C2×C4—1664no—
554C2×C4—1664no—
555C2×C4—1664no—
556C2×C4—1664no—
557C2×C4—1664no—
558C2×C4—1664no—
559C2×C4—1664no—
560C2×C4D4×Dic8318yes—
561C22—24no—
562C22—28no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22C2×D4×Dic8312yes—
573C22C2×Q8×Dic8311yescharacteristic
574C22#41×Dic8312yes—
575C22#1912817611yescharacteristic
576C22#1912818411yescharacteristic
577C22C22⋊Q8×D8312yes—
578C22#1912821412yes—
579C22#1912823712yes—
580C22#1912825312yes—
581C22#1912829111yescharacteristic
582C22#1912830011yescharacteristic
583C22#1912832611yescharacteristic
584C22#1912833011yescharacteristic
585C22#1912833611yescharacteristic
586C22#1912834312yes—
587C22#1912834912yes—
588C22#1912837112yes—
589C22#1912837412yes—
590C22#1912837911yescharacteristic
591C22#1912838812yes—
592C22#1912839612yes—
593C22#1912843511yescharacteristic
594C22#1912843711yescharacteristic
595C4—28no—
596C4—42no—
597C4—44no—
598C4—42no—
599C4—3324no—
600C4—3324no—
601C4—6642no—
602C4—1668no—
603C2—44no—
604C2C22⋊Q8×Dic8312yes—
605C2#2491019811yescharacteristic
606C2#2491045912yes—
607C2#2491052811yescharacteristic
608C2#2491059911yescharacteristic
609C2#2491113011yescharacteristic
610C2#2491116912yes—
611C2#2491134611yescharacteristic
612C2#2491151712yes—
613C2#2491211211yescharacteristic
614C2#2491312511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#399×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#399D83
hypocenterC83#14215
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C2×#19127587C2semidirect
5. #24905697C2semidirect
6. #24905695C2semidirect
7. #24905692C2semidirect
32. #399×C83C4semidirect
78. C2×C4⋊C4×C83C2×C4semidirect
79. C2×C4⋊C4×C83C2×C4semidirect
86. #95×C83C2×C4semidirect
317. C83#14215semidirect
329. #399Dic83semidirect
365. C2×C4⋊C4C2×Dic83semidirect
366. C2×C4⋊C4C2×Dic83semidirect
394. #95C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275872
C2×C22⋊C4×Dic832
#249057082
#249056972
#249056952
#249056922
#249056852
C2×#399×C832
#1421583