Finite Group Databasebeta

← #35326214

Subgroups — #35326214

Order 21248.

1942 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581757C211yesmaximal, characteristic
2C2×#19127622C211yesmaximal, characteristic
3C2×C4⋊C4×Dic83C212yesmaximal
4C2×#19127552C211yesmaximal, characteristic
5#24905733C211yesmaximal, characteristic
6#24905726C212yesmaximal
7#24905715C212yesmaximal
8#24905713C211yesmaximal, characteristic
9#95×Dic83C211yesmaximal, characteristic
10#95×Dic83C212yesmaximal
11C83×#1444C211yesmaximal, characteristic
12C23×Dic166C2211yescharacteristic
13C2×#14581757—28no—
14C2×#14581757—24no—
15C22×C4⋊Dic83C2212yes—
16C22×Dic83⋊C4C2212yes—
17C22×Dic83⋊C4C2211yescharacteristic
18C22×Dic83⋊C4C2211yescharacteristic
19C22×C4×Dic83C2211yescharacteristic
20C22×C4×Dic83C2211yescharacteristic
21C22×C4×Dic83C2212yes—
22C22×C4×Dic83C2211yescharacteristic
23C22×C4×Dic83C2212yes—
24#19127622C418yescomplemented
25C4⋊C4×Dic83—28no—
26#19127552—24no—
27C2×C83⋊#95C2211yescharacteristic
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2211yescharacteristic
33C2×C83⋊#95C2212yes—
34C2×C83⋊#95C2212yes—
35C22×C4⋊C4×C83C2211yescharacteristic
36C22×C4⋊C4×C83C2212yes—
37C2×#95×C83C2211yescharacteristic
38C2×#95×C83C2212yes—
39C2×#95×C83C2211yescharacteristic
40C22×Dic166—24no—
41C22×Dic166—24no—
42C22×Dic166—22no—
43C22×Dic166—22no—
44C22×Dic166C2×C414yes—
45C23×Dic83C2311yescharacteristic
46C23×Dic83C2311yescharacteristic
47C23×Dic83C2312yes—
48C23×Dic83C2312yes—
49C23×Dic83C2311yescharacteristic
50C23×Dic83C2311yescharacteristic
51#14581757—232no—
52C2×C4⋊Dic83—24no—
53C2×C4⋊Dic83—24no—
54C2×C4⋊Dic83C2×C418yes—
55C2×Dic83⋊C4—22no—
56C2×Dic83⋊C4—24no—
57C2×Dic83⋊C4—24no—
58C2×Dic83⋊C4—22no—
59C2×Dic83⋊C4—22no—
60C2×Dic83⋊C4—24no—
61C2×Dic83⋊C4—22no—
62C2×Dic83⋊C4—22no—
63C2×Dic83⋊C4C2×C414yes—
64C2×C4×Dic83—22no—
65C2×C4×Dic83—24no—
66C2×C4×Dic83—24no—
67C2×C4×Dic83—24no—
68C2×C4×Dic83—24no—
69C2×C4×Dic83—24no—
70C2×C4×Dic83—22no—
71C2×C4×Dic83—22no—
72C2×C4×Dic83—24no—
73C2×C4×Dic83D414yes—
74C2×C4×Dic83D418yes—
75C2×C4×Dic83Q814yescomplemented
76C2×C4×Dic83Q818yescomplemented
77C83⋊#95—22no—
78C83⋊#95—24no—
79C83⋊#95—24no—
80C83⋊#95—24no—
81C83⋊#95—24no—
82C83⋊#95—24no—
83C83⋊#95—22no—
84C83⋊#95C2×C418yes—
85C23×C4×C83C2312yes—
86C23×C4×C83C2311yescharacteristic
87C23×C4×C83C2311yescharacteristic
88C23×C4×C83C2311yescharacteristic
89C23×C4×C83C2312yes—
90C2×C4⋊C4×C83—24no—
91C2×C4⋊C4×C83—24no—
92C2×C4⋊C4×C83—22no—
93C2×C4⋊C4×C83—24no—
94C2×C4⋊C4×C83—24no—
95#95×C83—22no—
96#95×C83—24no—
97#95×C83C2×C414yes—
98C2×Dic166—216no—
99C2×Dic166—28no—
100C2×Dic166—28no—
101C2×Dic166—28no—
102C2×Dic166—216no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—21no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—21no—
109C22×Dic83—21no—
110C22×Dic83—21no—
111C22×Dic83—22no—
112C22×Dic83—21no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—22no—
125C22×Dic83C22×C412yes—
126C22×Dic83C22×C414yes—
127C22×Dic83C22×C412yes—
128C22×Dic83C2×D412yes—
129C22×Dic83C2×D414yes—
130C22×Dic83C2×D412yes—
131C22×Dic83C2×D414yes—
132C22×Dic83C2×Q814yes—
133C22×Dic83C2×Q812yes—
134C22×Dic83C2×Q812yes—
135C22×Dic83C2×Q814yes—
136C22×Dic83#4114yes—
137C22×Dic83#4112yes—
138C22×Dic83#4112yes—
139C4⋊Dic83—216no—
140Dic83⋊C4—28no—
141Dic83⋊C4—216no—
142Dic83⋊C4—28no—
143C4×Dic83—216no—
144C4×Dic83—28no—
145C4×Dic83—28no—
146C4×Dic83—216no—
147C4×Dic83—44no—
148C24×C83C2411yescharacteristic
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—21no—
156C22×C4×C83—21no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83C22×C412yes—
170C22×C4×C83C22×C414yes—
171C22×C4×C83C2×D414yes—
172C22×C4×C83C2×D412yes—
173C22×C4×C83C2×Q812yes—
174C22×C4×C83C2×Q814yes—
175C22×C4×C83#4112yes—
176C22×C4×C83#4114yes—
177C22×C4×C83#4112yes—
178C4⋊C4×C83—216no—
179C4⋊C4×C83—28no—
180C23×C83C23×C411yescharacteristic
181C23×C83C22×D412yes—
182C23×C83C22×D411yescharacteristic
183C23×C83C22×Q811yescharacteristic
184C23×C83C22×Q812yes—
185C23×C83C2×#4112yes—
186C23×C83C2×#4111yescharacteristic
187C23×C83C2×#4111yescharacteristic
188C23×C83C2×#4112yes—
189C23×C83C2×#4111yescharacteristic
190C23×C83C2×#4111yescharacteristic
191C2×C4×C83—24no—
192C2×C4×C83—24no—
193C2×C4×C83—24no—
194C2×C4×C83—22no—
195C2×C4×C83—24no—
196C2×C4×C83—24no—
197C2×C4×C83—22no—
198C2×C4×C83—24no—
199C2×C4×C83—24no—
200C2×C4×C83—24no—
201C2×C4×C83—24no—
202C2×C4×C83—22no—
203C2×C4×C83—24no—
204C2×C4×C83—24no—
205C2×C4×C83—22no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83—41no—
209C2×C4×C83—42no—
210C2×C4×C83—41no—
211C2×C4×C83—42no—
212C2×C4×C83—42no—
213C2×C4×C83—42no—
214C2×C4×C83—42no—
215C2×C4×C83—41no—
216C2×C4×C83—42no—
217C2×C4×C83—41no—
218C2×C4×C83C4×D414yes—
219C2×C4×C83C4×Q818yescomplemented
220C2×Dic83—22no—
221C2×Dic83—24no—
222C2×Dic83—24no—
223C2×Dic83—22no—
224C2×Dic83—24no—
225C2×Dic83—22no—
226C2×Dic83—22no—
227C2×Dic83—24no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83—22no—
231C2×Dic83—22no—
232C2×Dic83—22no—
233C2×Dic83—24no—
234C2×Dic83—22no—
235C2×Dic83—24no—
236C2×Dic83—22no—
237C2×Dic83—22no—
238C2×Dic83—24no—
239C2×Dic83—24no—
240C2×Dic83—41no—
241C2×Dic83—42no—
242C2×Dic83—41no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249C2×Dic83C4×D418yes—
250C2×Dic83C4×Q814yescomplemented
251C2×Dic83#12014yes—
252C2×Dic83C22⋊Q814yes—
253C2×Dic83C22⋊Q818yes—
254C2×Dic83C4⋊Q818yes—
255C2×Dic83C4⋊Q814yes—
256Dic166—232no—
257C22×C83C2×C4×D412yes—
258C22×C83C2×C4×D411yescharacteristic
259C22×C83C2×C4×Q811yescharacteristic
260C22×C83C2×C4×Q812yes—
261C22×C83C4×#4111yescharacteristic
262C22×C83C2×#12011yescharacteristic
263C22×C83C2×C22⋊Q811yescharacteristic
264C22×C83C2×C22⋊Q812yes—
265C22×C83#52512yes—
266C22×C83#52511yescharacteristic
267C22×C83#52511yescharacteristic
268C22×C83#52512yes—
269C22×C83#52912yes—
270C22×C83#52911yescharacteristic
271C22×C83C2×C4⋊Q812yes—
272C22×C83C2×C4⋊Q811yescharacteristic
273C22×C83#53212yes—
274C22×C83#53211yescharacteristic
275C22×C83#53312yes—
276C22×C83#53312yes—
277C22×C83#53312yes—
278C22×C83#53311yescharacteristic
279C22×C83#53312yes—
280C4×C83—24no—
281C4×C83—28no—
282C4×C83—44no—
283C4×C83—42no—
284C4×C83—42no—
285Dic83—28no—
286Dic83—24no—
287Dic83—42no—
288Dic83—44no—
289Dic83—42no—
290Dic83#39518yescomplemented
291C4×#395—831nomaximal, Sylow
292C2×C83C4×#12011yescharacteristic
293C2×C83C4×C22⋊Q811yescharacteristic
294C2×C83C4×C22⋊Q812yes—
295C2×C83C4×C4⋊Q811yescharacteristic
296C2×C83C4×C4⋊Q812yes—
297C2×C83C2×#39511yescharacteristic
298C2×C83#238811yescharacteristic
299C2×C83#239611yescharacteristic
300C2×C83#239612yes—
301C2×C83#239912yes—
302C2×C83#239911yescharacteristic
303C22×C4×Q8—831no—
304C2×#395—831no—
305C2×C4×C4⋊C4—831no—
306C2×C4×C4⋊C4—832no—
307#1446—831no—
308#1444—832no—
309#1444D8311yescharacteristic
310#1441—832no—
311#1441—831no—
312C4×#95—832no—
313C4×#95—831no—
314C83C4×#39511yescharacteristic, complemented, Sylow
315C23×Q8—831no—
316C2×C4×Q8—1664no—
317C2×C4×Q8—1668no—
318C22×C4⋊C4—831no—
319C22×C4⋊C4—832no—
320C22×C4⋊C4—832no—
321C22×C4⋊C4—831no—
322C22×C4⋊C4C2×D8311yescharacteristic
323C22×C4⋊C4C2×D8312yes—
324C22×C42—832no—
325C22×C42—831no—
326C22×C42—831no—
327C22×C42—832no—
328C22×C42—831no—
329#395—838no—
330C4×C4⋊C4—1664no—
331C4×C4⋊C4—1668no—
332C2×#95—832no—
333C2×#95—832no—
334C2×#95—832no—
335C2×#95—832no—
336C2×#95—831no—
337C2×#95—831no—
338C2×#95—832no—
339C2×#95—832no—
340C2×#95C2×D8312yes—
341C2×#95C2×D8311yescharacteristic
342C2×#95C2×D8311yescharacteristic
343C22×Q8—834no—
344C22×Q8—1664no—
345C22×Q8—1664no—
346C22×Q8—1662no—
347C22×Q8—1662no—
348C23×C4—831no—
349C23×C4—832no—
350C23×C4—831no—
351C23×C4—831no—
352C23×C4—831no—
353C23×C4—832no—
354C23×C4C22×D8312yes—
355C23×C4C22×D8311yescharacteristic
356C23×C4C22×D8311yescharacteristic
357C23×C4C22×D8311yescharacteristic
358C23×C4C22×D8312yes—
359C4×Q8—16632no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—834no—
366C2×C4⋊C4—838no—
367C2×C4⋊C4—1664no—
368C2×C4⋊C4—1664no—
369C2×C4⋊C4—1662no—
370C2×C4⋊C4—1664no—
371C2×C4⋊C4—1662no—
372C2×C4⋊C4—1664no—
373C2×C4⋊C4—1664no—
374C2×C4⋊C4—1662no—
375C2×C4⋊C4—1662no—
376C2×C4⋊C4—1662no—
377C2×C42—838no—
378C2×C42—834no—
379C2×C42—834no—
380C2×C42—838no—
381C2×C42—1662no—
382C2×C42—1664no—
383C2×C42—1664no—
384C2×C42—1664no—
385C2×C42—1664no—
386C2×C42—1664no—
387C2×C42—1662no—
388C2×C42—1664no—
389C2×C42—1662no—
390#95—22no—
391#95—24no—
392#95—838no—
393#95—1662no—
394#95—1664no—
395#95—1664no—
396#95—1664no—
397#95—1664no—
398#95—1664no—
399#95—1662no—
400#95C4×D8314yes—
401C24C23×D8311yescharacteristic
402C2×Q8—16616no—
403C2×Q8—1668no—
404C2×Q8—1668no—
405C2×Q8—16616no—
406C2×Q8—1668no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—834no—
428C22×C4—832no—
429C22×C4—832no—
430C22×C4—834no—
431C22×C4—832no—
432C22×C4—832no—
433C22×C4—834no—
434C22×C4—834no—
435C22×C4—832no—
436C22×C4—832no—
437C22×C4—834no—
438C22×C4—832no—
439C22×C4—832no—
440C22×C4—834no—
441C22×C4—1662no—
442C22×C4—1661no—
443C22×C4—1662no—
444C22×C4—1662no—
445C22×C4—1662no—
446C22×C4—1662no—
447C22×C4—1662no—
448C22×C4—1662no—
449C22×C4—1661no—
450C22×C4—1662no—
451C22×C4—1662no—
452C22×C4—1661no—
453C22×C4—1662no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1662no—
457C22×C4—1662no—
458C22×C4—1661no—
459C22×C4—1662no—
460C22×C4—1661no—
461C22×C4—1661no—
462C22×C4—1662no—
463C22×C4C2×C4×D8312yes—
464C22×C4C2×C4×D8314yes—
465C22×C4D4×D8314yes—
466C22×C4D4×D8312yes—
467C22×C4Q8×D8312yes—
468C22×C4Q8×D8314yes—
469C22×C4#1305058912yes—
470C22×C4#1305058914yes—
471C22×C4#1305059112yes—
472C4⋊C4—28no—
473C4⋊C4—216no—
474C4⋊C4—16616no—
475C4⋊C4—16616no—
476C4⋊C4—1668no—
477C4⋊C4—1668no—
478C42—3324no—
479C42—1668no—
480C42—16616no—
481C42—16616no—
482C42—1668no—
483C23C22×C4×D8311yescharacteristic
484C23C2×D4×D8312yes—
485C23C2×D4×D8311yescharacteristic
486C23C2×Q8×D8311yescharacteristic
487C23C2×Q8×D8312yes—
488C23#41×D8312yes—
489C23#41×D8311yescharacteristic
490C23#41×D8311yescharacteristic
491C23C2×#1305058912yes—
492C23C2×#1305058911yescharacteristic
493C23C2×#1305059111yescharacteristic
494Q8—16632no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—3321no—
523C2×C4—3322no—
524C2×C4—3321no—
525C2×C4—3322no—
526C2×C4—3322no—
527C2×C4—3322no—
528C2×C4—3322no—
529C2×C4—3322no—
530C2×C4—3322no—
531C2×C4—834no—
532C2×C4—834no—
533C2×C4—838no—
534C2×C4—838no—
535C2×C4—834no—
536C2×C4—838no—
537C2×C4—834no—
538C2×C4—1664no—
539C2×C4—1662no—
540C2×C4—1662no—
541C2×C4—1664no—
542C2×C4—1662no—
543C2×C4—1662no—
544C2×C4—1664no—
545C2×C4—1664no—
546C2×C4—1664no—
547C2×C4—1662no—
548C2×C4—1664no—
549C2×C4—1662no—
550C2×C4—1664no—
551C2×C4—1662no—
552C2×C4—1664no—
553C2×C4—1662no—
554C2×C4—1664no—
555C2×C4—1664no—
556C2×C4—1662no—
557C2×C4—1662no—
558C2×C4#1458175414yes—
559C2×C4#1458175718yescomplemented
560C22C4×D4×D8312yes—
561C22C2×#1458175411yescharacteristic
562C22C4×Q8×D8311yescharacteristic
563C22C2×#1458175712yes—
564C22C4×#1305058911yescharacteristic
565C22#120×D8311yescharacteristic
566C22C22⋊Q8×D8311yescharacteristic
567C22C22⋊Q8×D8312yes—
568C22#1912820712yes—
569C22#1912820911yescharacteristic
570C22#1912821111yescharacteristic
571C22#1912821412yes—
572C22#1912822812yes—
573C22#1912823611yescharacteristic
574C22C4⋊Q8×D8312yes—
575C22C4⋊Q8×D8311yescharacteristic
576C22#1912824612yes—
577C22#1912824911yescharacteristic
578C22#1912825312yes—
579C22#1912825312yes—
580C22#1912825612yes—
581C22#1912825611yescharacteristic
582C22#1912825712yes—
583C4—24no—
584C4—28no—
585C4—44no—
586C4—42no—
587C4—42no—
588C4—3322no—
589C4—3322no—
590C4—3324no—
591C4—838no—
592C4—1668no—
593C4—1664no—
594C2#2490996411yescharacteristic
595C2#2490999311yescharacteristic
596C2#2490999512yes—
597C2#2491003011yescharacteristic
598C2#2491003412yes—
599C2#395×D8311yescharacteristic
600C2#2491067511yescharacteristic
601C2#2491073211yescharacteristic
602C2#2491074012yes—
603C2#2491075312yes—
604C2#2491075511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1444C2
socle, prime radical, cosocleC24×C83C24
hypercenter#1444D83
hypocenterC83C4×#395
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
24. #19127622C4semidirect
75. C2×C4×Dic83Q8semidirect
76. C2×C4×Dic83Q8semidirect
219. C2×C4×C83C4×Q8semidirect
250. C2×Dic83C4×Q8semidirect
290. Dic83#395semidirect
314. C83C4×#395semidirect
559. C2×C4#14581757semidirect

Maximal subgroups

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

subgroupindex
C22×#145817572
C2×#191276222
C2×C4⋊C4×Dic832
C2×#191275522
#249057332
#249057262
#249057152
#249057132
#95×Dic832
C83×#14442
C4×#39583