Finite Group Databasebeta

← #35467079

Subgroups — #35467079

Order 21248.

922 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128001C211yesmaximal, characteristic, complemented
2C22×#14581725C211yesmaximal, characteristic, complemented
3#24911187C211yesmaximal, characteristic, complemented
4#24908984C211yesmaximal, characteristic, complemented
5#24908959C211yesmaximal, characteristic, complemented
6#24908851C211yesmaximal, characteristic, complemented
7#24908801C211yesmaximal, characteristic, complemented
8#24908231C211yesmaximal, characteristic, complemented
9#24908055C211yesmaximal, characteristic, complemented
10#24907892C211yesmaximal, characteristic
11#24907781C211yesmaximal, characteristic
12#24907710C211yesmaximal, characteristic, complemented
13#24907656C211yesmaximal, characteristic, complemented
14#24907643C211yesmaximal, characteristic, complemented
15C83×#2361C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C22×C4⋊Dic83C2211yescharacteristic
19#19128001—24no—
20C2×#14581725—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581713C2211yescharacteristic
30C2×#14581713C2211yescharacteristic
31C2×#14581713C2211yescharacteristic
32C2×#14581713C2211yescharacteristic
33C2×C22⋊(C83⋊C8)C2211yescharacteristic
34#19127643—21no—
35#19127635—21no—
36#19127599C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127573—21no—
39#19127567C2211yescharacteristic
40#19127561C2211yescharacteristic
41#19127559—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×#121×C83C2211yescharacteristic
51C2×C4×D4×C83—21no—
52C2×C4×D4×C83C2211yescharacteristic
53C22×C4⋊C4×C83C2211yescharacteristic
54#390×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56#388×C83—21no—
57#386×C83—21no—
58#385×C83—21no—
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C22⋊C4×C83C2211yescharacteristic
61C23×Dic83—21no—
62C22×C83⋊C8—22no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—22no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581784—22no—
70#14581784—41no—
71C2×C4⋊Dic83—22no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—41no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×C22⋊Dic83—21no—
83#14581725—22no—
84#14581725—22no—
85#14581725—22no—
86#14581725D414yescomplemented
87#14581723—22no—
88#14581723—41no—
89C4⋊(C83⋊C8)—22no—
90C4⋊(C83⋊C8)—22no—
91#14581713—22no—
92#14581713—22no—
93#14581713—22no—
94#14581713—41no—
95C22⋊(C83⋊C8)—22no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×D4×C83—21no—
102C22×D4×C83C2311yescharacteristic
103C23×C4×C83—21no—
104C23×C4×C83—21no—
105C23×C4×C83C2311yescharacteristic
106#121×C83—22no—
107#121×C83Q814yes—
108C4×D4×C83—22no—
109C4×D4×C83—41no—
110C4×D4×C83—44no—
111C2×C4⋊C4×C83—22no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83—41no—
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C22⋊C4×C83—21no—
119C2×C22⋊C4×C83—21no—
120C2×C22⋊C4×C83—21no—
121C2×C22⋊C4×C83—21no—
122C2×C22⋊C4×C83D411yescharacteristic
123C2×C22⋊C4×C83D411yescharacteristic
124C2×C42×C83—21no—
125C2×C42×C83C2311yescharacteristic
126C2×C42×C83C2311yescharacteristic
127#95×C83—21no—
128#95×C83—21no—
129#95×C83D411yescharacteristic
130#95×C83D411yescharacteristic
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—42no—
136C22×Dic83—42no—
137C22×Dic83—81no—
138C2×C83⋊C8—28no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8C2×D412yes—
150C2×C83⋊C8#4112yes—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—42no—
159C4⋊Dic83C2×D412yes—
160Dic83⋊C4—42no—
161Dic83⋊C4—42no—
162Dic83⋊C4—81no—
163Dic83⋊C4—81no—
164C22⋊Dic83—42no—
165C24×C83—21no—
166C24×C83C2×D411yescharacteristic
167C2×D4×C83—24no—
168C2×D4×C83—21no—
169C2×D4×C83—21no—
170C2×D4×C83—21no—
171C2×D4×C83—21no—
172C2×D4×C83—21no—
173C2×D4×C83—42no—
174C2×D4×C83—42no—
175C2×D4×C83C2×Q812yes—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—42no—
184C22×C4×C83—42no—
185C22×C4×C83—42no—
186C22×C4×C83—42no—
187C22×C4×C83—81no—
188C22×C4×C83—81no—
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2×D411yescharacteristic
191C22×C4×C83C2×D411yescharacteristic
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83C2×D411yescharacteristic
194C22×C4×C83C2×Q812yes—
195C22×C4×C83#4112yes—
196C22×C4×C83C2411yescharacteristic
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—42no—
202C4⋊C4×C83—42no—
203C4⋊C4×C83C2×D412yes—
204C4⋊C4×C83C2×Q812yes—
205C22⋊C4×C83—22no—
206C22⋊C4×C83—22no—
207C22⋊C4×C83—24no—
208C22⋊C4×C83—42no—
209C22⋊C4×C83—42no—
210C22⋊C4×C83—81no—
211C42×C83—21no—
212C42×C83—21no—
213C42×C83—21no—
214C42×C83—21no—
215C42×C83—42no—
216C23×C83—21no—
217C23×C83—21no—
218C23×C83—21no—
219C23×C83—21no—
220C23×C83—22no—
221C23×C83—41no—
222C23×C83—41no—
223C23×C83—41no—
224C23×C83—41no—
225C23×C83—41no—
226C23×C83—41no—
227C23×C83C22⋊Q812yes—
228C23×C83#13512yes—
229C23×C83AGL(1,ℤ/8)12yes—
230C23×C83C22×D411yescharacteristic
231D4×C83—22no—
232D4×C83—22no—
233D4×C83—48no—
234D4×C83—41no—
235D4×C83—41no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—22no—
239C2×C4×C83—22no—
240C2×C4×C83—22no—
241C2×C4×C83—22no—
242C2×C4×C83—22no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—22no—
246C2×C4×C83—21no—
247C2×C4×C83—21no—
248C2×C4×C83—21no—
249C2×C4×C83—21no—
250C2×C4×C83—21no—
251C2×C4×C83—21no—
252C2×C4×C83—21no—
253C2×C4×C83—21no—
254C2×C4×C83—21no—
255C2×C4×C83—21no—
256C2×C4×C83—41no—
257C2×C4×C83—41no—
258C2×C4×C83—41no—
259C2×C4×C83—41no—
260C2×C4×C83—41no—
261C2×C4×C83—41no—
262C2×C4×C83—41no—
263C2×C4×C83—84no—
264C2×C4×C83—84no—
265C2×C4×C83C22⋊Q812yes—
266C2×C4×C83C22×D411yescharacteristic
267C2×C4×C83C22×Q811yescharacteristic
268C2×C4×C83C2×#4111yescharacteristic
269C2×C4×C83C2×#4111yescharacteristic
270C2×C4×C83H(5,𝔽2)11yescharacteristic
271C2×C4×C832−1+411yescharacteristic
272C2×Dic83—22no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275C2×Dic83—41no—
276C2×Dic83—41no—
277C2×Dic83—41no—
278C2×Dic83—41no—
279C2×Dic83—41no—
280C2×Dic83—41no—
281C2×Dic83—41no—
282C2×Dic83—41no—
283C2×Dic83—41no—
284C2×Dic83—84no—
285C83⋊C8—22no—
286C83⋊C8—22no—
287C83⋊C8—41no—
288C83⋊C8—41no—
289C22×C83—22no—
290C22×C83—22no—
291C22×C83—22no—
292C22×C83—22no—
293C22×C83—22no—
294C22×C83—22no—
295C22×C83—42no—
296C22×C83—42no—
297C22×C83—42no—
298C22×C83—81no—
299C22×C83—81no—
300C22×C83—81no—
301C22×C83—81no—
302C22×C83#47814yescomplemented
303C22×C83C2×C22⋊Q811yescharacteristic
304C22×C83D4211yescharacteristic
305C22×C83#54711yescharacteristic
306C22×C83C2×#13511yescharacteristic
307C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C83#57611yescharacteristic
309C22×C83#57811yescharacteristic
310C4×C83—22no—
311C4×C83—42no—
312C4×C83—41no—
313C4×C83—41no—
314C4×C83—41no—
315C4×C83—42no—
316C4×C83—41no—
317C4×C83—81no—
318C4×C83#52511yescharacteristic
319C4×C83#53911yescharacteristic
320C4×C83D4×Q811yescharacteristic
321C4×C83#55411yescharacteristic
322Dic83—42no—
323Dic83—81no—
324Dic83—81no—
325Dic83—81no—
326#26350—831nomaximal, Sylow
327C2×C83—24no—
328C2×C83—82no—
329C2×C83#244911yescharacteristic
330C2×C83C2×#47811yescharacteristic
331C2×C83#308311yescharacteristic
332C2×C83#326411yescharacteristic
333C2×C83#328511yescharacteristic
334C2×C83#333911yescharacteristic
335C2×C83#334311yescharacteristic
336C2×#478—831no—
337C22×#107—831no—
338#2442—831no—
339#2361D8311yescharacteristic
340#2092—831no—
341#2087—831no—
342#2067—831no—
343#2059—831no—
344#1949—831no—
345#1921—831no—
346#1893—831no—
347#1874—831no—
348#1862—831no—
349#1853—831no—
350#1850—831no—
351C83#2635011yescharacteristic, complemented, Sylow
352C23×C8—831no—
353C2×#125—831no—
354C2×#121C2×D8311yescharacteristic
355C2×C4×D4—21no—
356C2×C4×D4C2×D8311yescharacteristic
357C22×C4⋊C4—831no—
358C22×C4⋊C4C2×D8311yescharacteristic
359#478—1664no—
360C2×#107—831no—
361C2×#107—831no—
362C2×#107—831no—
363C2×#107—831no—
364C2×#107—1664no—
365C2×#106—831no—
366C2×#106—831no—
367C2×C4⋊C8—831no—
368C2×C4⋊C8—831no—
369C2×#102—831no—
370C2×#102—831no—
371C2×#102—831no—
372C2×#102—831no—
373C2×C22⋊C8—831no—
374#400—1661no—
375#398—1661no—
376#390C2×D8311yescharacteristic
377#389—831no—
378#389—831no—
379#389C2×D8311yescharacteristic
380#388—21no—
381#386—21no—
382#385—21no—
383#385—1661no—
384#382—831no—
385#380—831no—
386#380—1661no—
387C4×C4⋊C4C2×D8311yescharacteristic
388C4×C22⋊C4C2×D8311yescharacteristic
389#340—831no—
390#340—831no—
391#340—831no—
392#340—831no—
393#340—831no—
394#340—831no—
395#336—831no—
396#336—831no—
397C22×D4—21no—
398C22×D4C22×D8311yescharacteristic
399C23×C4—21no—
400C23×C4—21no—
401C23×C4—1661no—
402C23×C4C22×D8311yescharacteristic
403C22×C8—831no—
404C22×C8—831no—
405C22×C8—831no—
406C22×C8—831no—
407C22×C8—1662no—
408C22×C8—1662no—
409C22×C8—1662no—
410#125—3321no—
411#125—1662no—
412#121—22no—
413#121Dic16614yes—
414C4×D4—22no—
415C4×D4—41no—
416C4×D4—44no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—41no—
421C2×C4⋊C4—3321no—
422C2×C4⋊C4—831no—
423C2×C4⋊C4—831no—
424C2×C4⋊C4—831no—
425C2×C4⋊C4—831no—
426C2×C4⋊C4—1661no—
427C2×C4⋊C4—1661no—
428C2×C4⋊C4—1661no—
429C2×C4⋊C4—1662no—
430C2×C4⋊C4—1661no—
431C2×C4⋊C4—1661no—
432C2×C4⋊C4C22×D8311yescharacteristic
433C2×C4⋊C4C22×D8311yescharacteristic
434C2×C4⋊C4C22×D8311yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1661no—
440C2×C22⋊C4C22⋊D8311yescharacteristic
441C2×C22⋊C4C22⋊D8311yescharacteristic
442C2×C42—21no—
443C2×C42C22×D8311yescharacteristic
444C2×C42C22×D8311yescharacteristic
445#107—834no—
446#107—1662no—
447#107—1662no—
448#107—1662no—
449#106—3321no—
450#106—1662no—
451C4⋊C8—1662no—
452C4⋊C8—1662no—
453#102—3321no—
454#102—1662no—
455#102—1662no—
456#102—1662no—
457C22⋊C8—1662no—
458#95—21no—
459#95—21no—
460#95—1661no—
461#95—1661no—
462#95—1661no—
463#95—1661no—
464#95—1661no—
465#95C22⋊D8311yescharacteristic
466#95C22⋊D8311yescharacteristic
467C24—21no—
468C24C2×C22⋊D8311yescharacteristic
469C2×D4—24no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4C2×Dic16612yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—3322no—
492C22×C4—3322no—
493C22×C4—6641no—
494C22×C4—1661no—
495C22×C4—1661no—
496C22×C4—1661no—
497C22×C4—1661no—
498C22×C4C2×C22⋊D8311yescharacteristic
499C22×C4C2×C22⋊D8311yescharacteristic
500C22×C4C2×C22⋊D8311yescharacteristic
501C22×C4C2×C22⋊D8311yescharacteristic
502C22×C4C2×C22⋊D8311yescharacteristic
503C22×C4C2×Dic16612yes—
504C22×C4#1305059012yes—
505C22×C4C23×D8311yescharacteristic
506C2×C8—832no—
507C2×C8—832no—
508C2×C8—1661no—
509C2×C8—1661no—
510C2×C8—1661no—
511C2×C8—1661no—
512C2×C8—1661no—
513C2×C8—1661no—
514C2×C8—1668no—
515C2×C8—1661no—
516C2×C8—1661no—
517C2×C8—1661no—
518C2×C8—1661no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—3322no—
526C4⋊C4—3322no—
527C4⋊C4—3322no—
528C4⋊C4—832no—
529C4⋊C4—6641no—
530C4⋊C4—6641no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4—1661no—
534C4⋊C4—1661no—
535C4⋊C4—1661no—
536C4⋊C4—1661no—
537C4⋊C4—1661no—
538C4⋊C4D4×D8312yes—
539C4⋊C4C2×Dic16612yes—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—24no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—81no—
546C22⋊C4—3322no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—42no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23#1458177112yes—
564C23#1458181212yes—
565C23#1458181812yes—
566C23C22×C22⋊D8311yescharacteristic
567D4—22no—
568D4—22no—
569D4—48no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—3321no—
602C2×C4—3321no—
603C2×C4—3321no—
604C2×C4—3321no—
605C2×C4—3321no—
606C2×C4—3321no—
607C2×C4—3321no—
608C2×C4—3321no—
609C2×C4—3321no—
610C2×C4—3321no—
611C2×C4—3321no—
612C2×C4—6644no—
613C2×C4—1662no—
614C2×C4#1458177112yes—
615C2×C4C2×D4×D8311yescharacteristic
616C2×C4C22×Dic16611yescharacteristic
617C2×C4#41×D8311yescharacteristic
618C2×C4C2×#1305059011yescharacteristic
619C2×C4#1458183711yescharacteristic
620C2×C4#1458183911yescharacteristic
621C8—3321no—
622C8—3321no—
623C8—1662no—
624C8—1662no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22#1912800114yescomplemented
639C22C2×#1458177111yescharacteristic
640C22D4×C22⋊D8311yescharacteristic
641C22#1912835011yescharacteristic
642C22C2×#1458181211yescharacteristic
643C22C2×#1458181811yescharacteristic
644C22#1912850411yescharacteristic
645C22#1912851611yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—81no—
654C4—3322no—
655C4—6641no—
656C4—6641no—
657C4—6641no—
658C4#1912821011yescharacteristic
659C4#1912829311yescharacteristic
660C4D4×Dic16611yescharacteristic
661C4#1912841011yescharacteristic
662C2—24no—
663C2—82no—
664C2#2491126611yescharacteristic
665C2C2×#1912800111yescharacteristic
666C2#2491731411yescharacteristic
667C2#2491893911yescharacteristic
668C2#2491920011yescharacteristic
669C2#2491985911yescharacteristic
670C2#2491989711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2361C2
socleC23×C83C22×D4
prime radicalC2×#121×C83C22
hypercenter#2361D83
hypocenterC83#26350
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. C2×#19128001C2semidirect
2. C22×#14581725C2semidirect
3. #24911187C2semidirect
4. #24908984C2semidirect
5. #24908959C2semidirect
6. #24908851C2semidirect
7. #24908801C2semidirect
8. #24908231C2semidirect
9. #24908055C2semidirect
12. #24907710C2semidirect
13. #24907656C2semidirect
14. #24907643C2semidirect
86. #14581725D4semidirect
302. C22×C83#478semidirect
351. C83#26350semidirect
638. C22#19128001semidirect

Maximal subgroups

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

subgroupindex
C2×#191280012
C22×#145817252
#249111872
#249089842
#249089592
#249088512
#249088012
#249082312
#249080552
#249078922
#249077812
#249077102
#249076562
#249076432
C83×#23612
#2635083