Finite Group Databasebeta

← #35333128

Subgroups — #35333128

Order 21248.

1398 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127607C211yesmaximal, characteristic
2C2×#19127605C212yesmaximal
3#24905727C212yesmaximal
4#24905726C212yesmaximal
5#24905715C212yesmaximal
6#24905697C212yesmaximal
7#24905695C212yesmaximal
8#24905693C211yesmaximal, characteristic
9C83×C4⋊#95C211yesmaximal, characteristic
10C23×Dic166C2211yescharacteristic
11C22×C4⋊Dic83C2211yescharacteristic
12C22×C4⋊Dic83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×Dic83⋊C4C2212yes—
15C22×Dic83⋊C4C2212yes—
16C22×C4×Dic83C2212yes—
17#19127607C418yescomplemented
18#19127605—28no—
19C2×C83⋊#95C2212yes—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C22×C4⋊C4×C83C2211yescharacteristic
29C22×C4⋊C4×C83C2211yescharacteristic
30C22×C42×C83C2211yescharacteristic
31C2×#95×C83C2212yes—
32C2×#95×C83C2212yes—
33C22×Dic166—41no—
34C22×Dic166—41no—
35C22×Dic166—41no—
36C22×Dic166—41no—
37C22×Dic166C2×C414yes—
38C22×Dic166D414yes—
39C22×Dic166Q814yes—
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43C23×Dic83C2312yes—
44C2×C4⋊Dic83—22no—
45C2×C4⋊Dic83—41no—
46C2×C4⋊Dic83—41no—
47C2×C4⋊Dic83—41no—
48C2×C4⋊Dic83—41no—
49C2×C4⋊Dic83—41no—
50C2×C4⋊Dic83—41no—
51C2×C4⋊Dic83C2×C414yes—
52C2×C4⋊Dic83Q814yescomplemented
53C2×Dic83⋊C4—24no—
54C2×Dic83⋊C4—42no—
55C2×Dic83⋊C4—42no—
56C2×Dic83⋊C4—42no—
57C2×Dic83⋊C4—42no—
58C2×Dic83⋊C4—42no—
59C2×C4×Dic83—24no—
60C2×C4×Dic83—42no—
61C2×C4×Dic83—42no—
62C83⋊#95—24no—
63C83⋊#95—24no—
64C83⋊#95—24no—
65C83⋊#95—24no—
66C83⋊#95—42no—
67C83⋊#95—42no—
68C83⋊#95—42no—
69C83⋊#95C2×C418yes—
70C83⋊#95C2×C418yes—
71C23×C4×C83C2312yes—
72C23×C4×C83C2312yes—
73C23×C4×C83C2311yescharacteristic
74C23×C4×C83C2311yescharacteristic
75C23×C4×C83C2311yescharacteristic
76C2×C4⋊C4×C83—22no—
77C2×C4⋊C4×C83—41no—
78C2×C4⋊C4×C83—41no—
79C2×C4⋊C4×C83—41no—
80C2×C4⋊C4×C83—41no—
81C2×C4⋊C4×C83Q814yescomplemented
82C2×C42×C83—41no—
83C2×C42×C83—41no—
84C2×C42×C83C2×C414yes—
85#95×C83—24no—
86#95×C83—24no—
87C2×Dic166—44no—
88C2×Dic166—44no—
89C2×Dic166—44no—
90C2×Dic166—44no—
91C2×Dic166—44no—
92C2×Dic166—44no—
93C2×Dic166C4⋊C4116yescomplemented
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—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83C22×C414yes—
118C22×Dic83C22×C414yes—
119C22×Dic83C2×D414yes—
120C22×Dic83C2×Q814yes—
121C22×Dic83C2×Q814yes—
122C4⋊Dic83—44no—
123C4⋊Dic83—44no—
124C4⋊Dic83—44no—
125Dic83⋊C4—48no—
126Dic83⋊C4—48no—
127C4×Dic83—48no—
128C24×C83C2411yescharacteristic
129C22×C4×C83—21no—
130C22×C4×C83—21no—
131C22×C4×C83—21no—
132C22×C4×C83—21no—
133C22×C4×C83—21no—
134C22×C4×C83—21no—
135C22×C4×C83—21no—
136C22×C4×C83—21no—
137C22×C4×C83—21no—
138C22×C4×C83—21no—
139C22×C4×C83—21no—
140C22×C4×C83—21no—
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—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83C22×C412yes—
155C22×C4×C83C22×C412yes—
156C22×C4×C83C22×C412yes—
157C22×C4×C83C2×D412yes—
158C22×C4×C83C2×Q812yes—
159C22×C4×C83C2×Q812yes—
160C22×C4×C83C2×Q814yes—
161C22×C4×C83C2×Q812yes—
162C22×C4×C83#4112yes—
163C22×C4×C83#4112yes—
164C4⋊C4×C83—44no—
165C4⋊C4×C83—44no—
166C42×C83—44no—
167C23×C83C23×C411yescharacteristic
168C23×C83C22×D411yescharacteristic
169C23×C83C22×Q811yescharacteristic
170C23×C83C22×Q811yescharacteristic
171C23×C83C22×Q811yescharacteristic
172C23×C83C2×#4111yescharacteristic
173C23×C83C2×#4111yescharacteristic
174C23×C83H(5,𝔽2)11yescharacteristic
175C23×C83H(5,𝔽2)11yescharacteristic
176C23×C83H(5,𝔽2)11yescharacteristic
177C23×C83H(5,𝔽2)11yescharacteristic
178C23×C83H(5,𝔽2)11yescharacteristic
179C23×C83H(5,𝔽2)11yescharacteristic
180C23×C832−1+411yescharacteristic
181C23×C832−1+411yescharacteristic
182C2×C4×C83—22no—
183C2×C4×C83—22no—
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—22no—
191C2×C4×C83—22no—
192C2×C4×C83—22no—
193C2×C4×C83—22no—
194C2×C4×C83—22no—
195C2×C4×C83—22no—
196C2×C4×C83—22no—
197C2×C4×C83—22no—
198C2×C4×C83—22no—
199C2×C4×C83—22no—
200C2×C4×C83—24no—
201C2×C4×C83—24no—
202C2×C4×C83—24no—
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×C83—42no—
210C2×C4×C83—42no—
211C2×C4×C83—42no—
212C2×C4×C83—42no—
213C2×C4×C83—42no—
214C2×C4×C83C2×C4⋊C414yes—
215C2×C4×C83C4×Q814yes—
216C2×C4×C83C4×Q814yes—
217C2×Dic83—24no—
218C2×Dic83—24no—
219C2×Dic83—24no—
220C2×Dic83—24no—
221C2×Dic83—24no—
222C2×Dic83—24no—
223C2×Dic83—24no—
224C2×Dic83—24no—
225C2×Dic83—24no—
226C2×Dic83—24no—
227C2×Dic83—24no—
228C2×Dic83—24no—
229C2×Dic83—42no—
230C2×Dic83—42no—
231C2×Dic83—42no—
232C2×Dic83—42no—
233C2×Dic83—42no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83—42no—
237C2×Dic83—42no—
238C2×Dic83—42no—
239C2×Dic83—42no—
240C2×Dic83—42no—
241C2×Dic83—42no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83C2×C4⋊C418yes—
245Dic166—416no—
246C22×C83C22×C4⋊C411yescharacteristic
247C22×C83C2×C4×Q811yescharacteristic
248C22×C83C2×C4×Q811yescharacteristic
249C22×C83#51811yescharacteristic
250C22×C83#51811yescharacteristic
251C22×C83#52011yescharacteristic
252C22×C83#52011yescharacteristic
253C22×C83#53411yescharacteristic
254C22×C83#53411yescharacteristic
255C22×C83#53811yescharacteristic
256C22×C83#53811yescharacteristic
257C22×C83#53911yescharacteristic
258C22×C83#53911yescharacteristic
259C22×C83#54311yescharacteristic
260C22×C83#54311yescharacteristic
261C22×C83#54311yescharacteristic
262C22×C83#54311yescharacteristic
263C22×C83#54411yescharacteristic
264C22×C83#54411yescharacteristic
265C22×C83D4×Q811yescharacteristic
266C22×C83D4×Q811yescharacteristic
267C22×C83#55011yescharacteristic
268C22×C83#55011yescharacteristic
269C22×C83#55411yescharacteristic
270C22×C83#55411yescharacteristic
271C22×C83#55511yescharacteristic
272C22×C83#55711yescharacteristic
273C22×C83#55711yescharacteristic
274C22×C83Q8211yescharacteristic
275C22×C83Q8211yescharacteristic
276C22×C83Q8211yescharacteristic
277C22×C83H(3,𝔽4)11yescharacteristic
278C22×C83#56211yescharacteristic
279C22×C83#56211yescharacteristic
280C22×C83#56311yescharacteristic
281C4×C83—24no—
282C4×C83—24no—
283C4×C83—24no—
284C4×C83—44no—
285C4×C83—82no—
286Dic83—28no—
287Dic83—44no—
288Dic83—44no—
289Dic83—82no—
290#14467—831nomaximal, Sylow
291C2×C83#231511yescharacteristic
292C2×C83#232911yescharacteristic
293C2×C83#232911yescharacteristic
294C2×C83Q8×C4⋊C411yescharacteristic
295C2×C83Q8×C4⋊C411yescharacteristic
296C2×C83#236911yescharacteristic
297C2×C83#237911yescharacteristic
298C2×C83#273211yescharacteristic
299C2×C83#273211yescharacteristic
300C2×C83#273311yescharacteristic
301C2×C83#273411yescharacteristic
302C2×C83#280611yescharacteristic
303C2×C83#280611yescharacteristic
304C2×C83#280911yescharacteristic
305C2×C83#281111yescharacteristic
306C2×#391—831no—
307C2×#391—832no—
308#1444—832no—
309#1444—832no—
310#1441—832no—
311#1436—832no—
312#1436—832no—
313C4⋊#95—831no—
314C4⋊#95D8311yescharacteristic
315C83#1446711yescharacteristic, complemented, Sylow
316C23×Q8—831no—
317C22×C4⋊C4—832no—
318C22×C4⋊C4—831no—
319C22×C4⋊C4—831no—
320C22×C4⋊C4—832no—
321C22×C4⋊C4—831no—
322C22×C4⋊C4C2×D8311yescharacteristic
323C22×C4⋊C4C2×D8311yescharacteristic
324C22×C42—832no—
325C22×C42C2×D8311yescharacteristic
326#391—838no—
327#391—1668no—
328C2×#95—832no—
329C2×#95—832no—
330C2×#95—832no—
331C2×#95—832no—
332C2×#95—832no—
333C2×#95—832no—
334C2×#95—832no—
335C2×#95—832no—
336C2×#95—832no—
337C2×#95C2×D8312yes—
338C2×#95C2×D8312yes—
339C22×Q8—3321no—
340C22×Q8—3321no—
341C22×Q8—3321no—
342C22×Q8—3321no—
343C22×Q8—834no—
344C22×Q8—834no—
345C22×Q8—834no—
346C23×C4—832no—
347C23×C4—832no—
348C23×C4—832no—
349C23×C4—832no—
350C23×C4C22×D8312yes—
351C23×C4C22×D8312yes—
352C23×C4C22×D8311yescharacteristic
353C23×C4C22×D8311yescharacteristic
354C23×C4C22×D8311yescharacteristic
355C2×C4⋊C4—22no—
356C2×C4⋊C4—41no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—41no—
359C2×C4⋊C4—41no—
360C2×C4⋊C4—3322no—
361C2×C4⋊C4—3322no—
362C2×C4⋊C4—3322no—
363C2×C4⋊C4—3322no—
364C2×C4⋊C4—3322no—
365C2×C4⋊C4—3321no—
366C2×C4⋊C4—3321no—
367C2×C4⋊C4—3321no—
368C2×C4⋊C4—3321no—
369C2×C4⋊C4—3321no—
370C2×C4⋊C4—3321no—
371C2×C4⋊C4—834no—
372C2×C4⋊C4—834no—
373C2×C4⋊C4—1664no—
374C2×C4⋊C4—1662no—
375C2×C4⋊C4Dic16614yescomplemented
376C2×C42—41no—
377C2×C42—41no—
378C2×C42—3322no—
379C2×C42—3322no—
380C2×C42—1664no—
381C2×C42C4×D8314yes—
382#95—24no—
383#95—24no—
384#95—3322no—
385#95—3322no—
386#95—3322no—
387#95—838no—
388#95—838no—
389#95—1664no—
390#95—1664no—
391#95—1664no—
392#95—1664no—
393C24C23×D8311yescharacteristic
394C2×Q8—3324no—
395C2×Q8—3324no—
396C2×Q8—3324no—
397C2×Q8—3324no—
398C2×Q8—3324no—
399C2×Q8—3324no—
400C2×Q8—8316no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—834no—
427C22×C4—834no—
428C22×C4—834no—
429C22×C4—834no—
430C22×C4—834no—
431C22×C4—1662no—
432C22×C4—1662no—
433C22×C4—1662no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4—1662no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4—1662no—
440C22×C4—1662no—
441C22×C4—1662no—
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×C4C2×C4×D8312yes—
455C22×C4C2×C4×D8312yes—
456C22×C4C2×C4×D8312yes—
457C22×C4D4×D8312yes—
458C22×C4Q8×D8312yes—
459C22×C4Q8×D8312yes—
460C22×C4C2×Dic16614yes—
461C22×C4C2×Dic16612yes—
462C22×C4#1305058912yes—
463C22×C4#1305059012yes—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—3324no—
467C4⋊C4—3324no—
468C4⋊C4—3324no—
469C4⋊C4—3328no—
470C4⋊C4—3328no—
471C42—44no—
472C42—3328no—
473C23C22×C4×D8311yescharacteristic
474C23C2×D4×D8311yescharacteristic
475C23C2×Q8×D8311yescharacteristic
476C23C2×Q8×D8311yescharacteristic
477C23C22×Dic16611yescharacteristic
478C23C2×#1305058911yescharacteristic
479C23C2×#1305059011yescharacteristic
480C23#1458183711yescharacteristic
481C23#1458183711yescharacteristic
482C23#1458183711yescharacteristic
483C23#1458183811yescharacteristic
484C23#1458183811yescharacteristic
485C23#1458183811yescharacteristic
486C23#1458183911yescharacteristic
487C23#1458184011yescharacteristic
488Q8—33216no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—3322no—
522C2×C4—3322no—
523C2×C4—3322no—
524C2×C4—3322no—
525C2×C4—3322no—
526C2×C4—3322no—
527C2×C4—3322no—
528C2×C4—3322no—
529C2×C4—3322no—
530C2×C4—3322no—
531C2×C4—3322no—
532C2×C4—3322no—
533C2×C4—3322no—
534C2×C4—3322no—
535C2×C4—3322no—
536C2×C4—838no—
537C2×C4—1664no—
538C2×C4—1664no—
539C2×C4—1664no—
540C2×C4—1664no—
541C2×C4—1664no—
542C2×C4—1664no—
543C2×C4—1664no—
544C2×C4—1664no—
545C2×C4—1664no—
546C2×C4—1664no—
547C2×C4—1664no—
548C2×C4—1664no—
549C2×C4C4⋊C4×D8314yes—
550C2×C4#1458175714yes—
551C2×C4C4×Dic16614yes—
552C22C2×C4⋊C4×D8311yescharacteristic
553C22C2×#1458175711yescharacteristic
554C22C2×C4×Dic16611yescharacteristic
555C22#1912817411yescharacteristic
556C22#1912817511yescharacteristic
557C22#1912818111yescharacteristic
558C22#1912818211yescharacteristic
559C22#1912825911yescharacteristic
560C22#1912825911yescharacteristic
561C22#1912828811yescharacteristic
562C22#1912828911yescharacteristic
563C22#1912829611yescharacteristic
564C22#1912829811yescharacteristic
565C22#1912832611yescharacteristic
566C22#1912832611yescharacteristic
567C22#1912832611yescharacteristic
568C22#1912832711yescharacteristic
569C22#1912833011yescharacteristic
570C22#1912833211yescharacteristic
571C22#1912836611yescharacteristic
572C22D4×Dic16611yescharacteristic
573C22#1912837111yescharacteristic
574C22#1912837211yescharacteristic
575C22#1912840811yescharacteristic
576C22#1912841111yescharacteristic
577C22#1912841611yescharacteristic
578C22#1912842611yescharacteristic
579C22#1912842911yescharacteristic
580C22#1912843111yescharacteristic
581C22Q8×Dic16611yescharacteristic
582C22Q8×Dic16611yescharacteristic
583C22#1912844311yescharacteristic
584C22#1912844511yescharacteristic
585C22#1912844911yescharacteristic
586C22#1912845511yescharacteristic
587C4—24no—
588C4—24no—
589C4—24no—
590C4—44no—
591C4—82no—
592C4—3324no—
593C4—3324no—
594C4—6642no—
595C4—1668no—
596C2#2491007611yescharacteristic
597C2#2491019311yescharacteristic
598C2#2491019611yescharacteristic
599C2#2491037511yescharacteristic
600C2C4⋊C4×Dic16611yescharacteristic
601C2#2491055211yescharacteristic
602C2#2491063411yescharacteristic
603C2#2491447911yescharacteristic
604C2#2491448011yescharacteristic
605C2#2491448811yescharacteristic
606C2#2491449411yescharacteristic
607C2#2491542211yescharacteristic
608C2#2491542511yescharacteristic
609C2#2491544811yescharacteristic
610C2#2491545611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×C4⋊#95C2
socle, prime radical, cosocleC24×C83C24
hypercenterC4⋊#95D83
hypocenterC83#14467
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
17. #19127607C4semidirect
52. C2×C4⋊Dic83Q8semidirect
81. C2×C4⋊C4×C83Q8semidirect
93. C2×Dic166C4⋊C4semidirect
315. C83#14467semidirect
375. C2×C4⋊C4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191276072
C2×#191276052
#249057272
#249057262
#249057152
#249056972
#249056952
#249056932
C83×C4⋊#952
#1446783