Finite Group Databasebeta

← #35467184

Subgroups — #35467184

Order 21248.

904 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127981C211yesmaximal, characteristic, complemented
2C2×#19127744C211yesmaximal, characteristic, complemented
3#24911165C211yesmaximal, characteristic, complemented
4#24908996C211yesmaximal, characteristic, complemented
5#24908965C211yesmaximal, characteristic, complemented
6#24908837C211yesmaximal, characteristic, complemented
7#24908796C211yesmaximal, characteristic, complemented
8#24908221C211yesmaximal, characteristic, complemented
9#24908055C211yesmaximal, characteristic, complemented
10#24907902C211yesmaximal, characteristic
11#24907781C211yesmaximal, characteristic
12#24907710C211yesmaximal, characteristic, complemented
13#24907662C211yesmaximal, characteristic, complemented
14#24907648C211yesmaximal, characteristic, complemented
15C83×#2363C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581769—21no—
19C22×C4⋊Dic83C2211yescharacteristic
20#19127981—24no—
21#19127744—24no—
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581713C2211yescharacteristic
31C2×#14581713C2211yescharacteristic
32C2×#14581713C2211yescharacteristic
33C2×#14581713C2211yescharacteristic
34C2×C22⋊(C83⋊C8)C2211yescharacteristic
35#19127633—21no—
36#19127622—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127576—21no—
40#19127571C2211yescharacteristic
41#19127561C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×#121×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52C2×#117×C83C2211yescharacteristic
53#390×C83—21no—
54#390×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56#388×C83—21no—
57#386×C83—21no—
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C22⋊C4×C83—21no—
60C4×C22⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C23×Dic83—21no—
63C2×#13050574—22no—
64C2×#13050574—22no—
65C2×#13050574—22no—
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70#14581792—22no—
71#14581792—41no—
72#14581769—44no—
73C2×C4⋊Dic83—22no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83—41no—
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×C22⋊Dic83—21no—
86#14581725—22no—
87#14581725—22no—
88#14581725D414yescomplemented
89#14581723—22no—
90#14581723—22no—
91#14581723—41no—
92C4⋊(C83⋊C8)—22no—
93C4⋊(C83⋊C8)—22no—
94#14581713—22no—
95#14581713—22no—
96#14581713—22no—
97#14581713—41no—
98C22⋊(C83⋊C8)—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C22×D4×C83—21no—
103C22×D4×C83C2311yescharacteristic
104C23×C4×C83—21no—
105C23×C4×C83C2311yescharacteristic
106#121×C83—22no—
107#121×C83Q814yescomplemented
108C4×D4×C83—22no—
109C4×D4×C83—41no—
110#117×C83—22no—
111#117×C83—41no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C22⋊C4×C83—21no—
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83—21no—
119C2×C22⋊C4×C83—21no—
120C2×C22⋊C4×C83—21no—
121C2×C22⋊C4×C83D411yescharacteristic
122C2×C22⋊C4×C83D411yescharacteristic
123C2×C42×C83—21no—
124C2×C42×C83—21no—
125C2×C42×C83C2311yescharacteristic
126C2×C42×C83C2311yescharacteristic
127#95×C83—21no—
128#95×C83—21no—
129#95×C83D411yescharacteristic
130#95×C83D411yescharacteristic
131C2×Dic166—42no—
132C2×Dic166—42no—
133C2×Dic166—81no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—42no—
139C22×Dic83—42no—
140C22×Dic83—81no—
141#13050574—28no—
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⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8C2×D412yes—
153C2×C83⋊C8#4112yes—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—42no—
162C4⋊Dic83—42no—
163C4⋊Dic83—42no—
164C4⋊Dic83—81no—
165C4⋊Dic83C2×D412yes—
166Dic83⋊C4—42no—
167Dic83⋊C4—42no—
168C22⋊Dic83—42no—
169C24×C83—21no—
170C24×C83C2×D411yescharacteristic
171C2×D4×C83—24no—
172C2×D4×C83—21no—
173C2×D4×C83—21no—
174C2×D4×C83—21no—
175C2×D4×C83—21no—
176C2×D4×C83—21no—
177C2×D4×C83—81no—
178C2×D4×C83—81no—
179C2×D4×C83C2×Q812yes—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83—21no—
187C22×C4×C83—42no—
188C22×C4×C83—42no—
189C22×C4×C83—81no—
190C22×C4×C83C2×D411yescharacteristic
191C22×C4×C83C2×D411yescharacteristic
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83C2×D411yescharacteristic
194C22×C4×C83C2×D411yescharacteristic
195C22×C4×C83C2×Q812yes—
196C22×C4×C83#4112yes—
197C22×C4×C83C2411yescharacteristic
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—81no—
203C4⋊C4×C83C2×D412yes—
204C4⋊C4×C83C2×Q812yes—
205C22⋊C4×C83—24no—
206C22⋊C4×C83—22no—
207C22⋊C4×C83—22no—
208C22⋊C4×C83—42no—
209C22⋊C4×C83—42no—
210C22⋊C4×C83—42no—
211C22⋊C4×C83—81no—
212C42×C83—21no—
213C42×C83—21no—
214C42×C83—21no—
215C42×C83—21no—
216C42×C83—42no—
217C42×C83—81no—
218C23×C83—21no—
219C23×C83—21no—
220C23×C83—21no—
221C23×C83—21no—
222C23×C83—21no—
223C23×C83—21no—
224C23×C83—22no—
225C23×C83—41no—
226C23×C83—41no—
227C23×C83—41no—
228C23×C83—41no—
229C23×C83—41no—
230C23×C83—41no—
231C23×C83C22⋊Q812yes—
232C23×C83C22×D411yescharacteristic
233D4×C83—22no—
234D4×C83—22no—
235D4×C83—41no—
236D4×C83—41no—
237D4×C83—84no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—22no—
245C2×C4×C83—22no—
246C2×C4×C83—22no—
247C2×C4×C83—22no—
248C2×C4×C83—22no—
249C2×C4×C83—21no—
250C2×C4×C83—22no—
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×C83C22⋊Q812yes—
265C2×C4×C83AGL(1,ℤ/8)12yes—
266C2×C4×C83#13712yes—
267C2×C4×C83C22×D411yescharacteristic
268C2×C4×C83C22×Q811yescharacteristic
269C2×C4×C83C2×#4111yescharacteristic
270C2×C4×C83C2×#4111yescharacteristic
271C2×C4×C83H(5,𝔽2)11yescharacteristic
272C2×C4×C832−1+411yescharacteristic
273C2×Dic83—22no—
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—41no—
285C2×Dic83—84no—
286Dic166—84no—
287C83⋊C8—22no—
288C83⋊C8—22no—
289C83⋊C8—41no—
290C83⋊C8—41no—
291C22×C83—22no—
292C22×C83—22no—
293C22×C83—22no—
294C22×C83—41no—
295C22×C83—41no—
296C22×C83—41no—
297C22×C83—41no—
298C22×C83—42no—
299C22×C83—42no—
300C22×C83—42no—
301C22×C83—81no—
302C22×C83—81no—
303C22×C83—81no—
304C22×C83—81no—
305C22×C83C2×C22⋊Q811yescharacteristic
306C22×C83D4211yescharacteristic
307C22×C83#54711yescharacteristic
308C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C83C2×#13711yescharacteristic
310C22×C83#57611yescharacteristic
311C22×C83#57711yescharacteristic
312C4×C83—22no—
313C4×C83—22no—
314C4×C83—22no—
315C4×C83—42no—
316C4×C83—41no—
317C4×C83—42no—
318C4×C83—41no—
319C4×C83—81no—
320C4×C83#52511yescharacteristic
321C4×C83#53911yescharacteristic
322C4×C83D4×Q811yescharacteristic
323C4×C83#55411yescharacteristic
324Dic83—42no—
325Dic83—81no—
326Dic83—81no—
327Dic83—81no—
328#26357—831nomaximal, Sylow
329C2×C83—42no—
330C2×C83—82no—
331C2×C83#244911yescharacteristic
332C2×C83#307911yescharacteristic
333C2×C83#308111yescharacteristic
334C2×C83#326411yescharacteristic
335C2×C83#327811yescharacteristic
336C2×C83#334611yescharacteristic
337C2×C83#335411yescharacteristic
338C2×#474—831no—
339C2×#428—831no—
340#2440—831no—
341#2363D8311yescharacteristic
342#2094—831no—
343#2088—831no—
344#2065—831no—
345#2058—831no—
346#1947—831no—
347#1921—831no—
348#1895—831no—
349#1874—831no—
350#1862—831no—
351#1854—831no—
352#1851—831no—
353C83#2635711yescharacteristic, complemented, Sylow
354C22×M16—831no—
355C2×C4⋊Q8—831no—
356C2×C22⋊Q8—1661no—
357C2×#121C2×D8311yescharacteristic
358C2×C4×D4C2×D8311yescharacteristic
359C2×#117C2×D8311yescharacteristic
360C22×C4⋊C4—831no—
361#474—1664no—
362#428—1664no—
363C2×#107—831no—
364C2×#107—831no—
365C2×#107—831no—
366C2×#106—831no—
367C2×#106—831no—
368C2×#106—831no—
369C2×C4⋊C8—831no—
370C2×C4⋊C8—831no—
371C2×#102—831no—
372C2×#102—831no—
373C2×#102—831no—
374C2×#102—831no—
375C2×C22⋊C8—831no—
376#398—1661no—
377#395—1661no—
378#390—21no—
379#390C2×D8311yescharacteristic
380#389—831no—
381#389—831no—
382#389C2×D8311yescharacteristic
383#388—21no—
384#386—21no—
385#385—1661no—
386#384—831no—
387#380—831no—
388C4×C4⋊C4C2×D8311yescharacteristic
389C4×C22⋊C4—21no—
390C4×C22⋊C4C2×D8311yescharacteristic
391#340—831no—
392#340—831no—
393#340—831no—
394#340—831no—
395#340—831no—
396#340—831no—
397#336—831no—
398#336—831no—
399C22×Q8—1661no—
400C22×D4—21no—
401C22×D4C22×D8311yescharacteristic
402C23×C4—21no—
403C23×C4—1661no—
404C23×C4C22×D8311yescharacteristic
405C2×M16—1662no—
406C2×M16—1662no—
407C2×M16—1662no—
408C22×C8—831no—
409C22×C8—831no—
410C22×C8—831no—
411C22×C8—831no—
412C4⋊Q8—3321no—
413C4⋊Q8—1662no—
414C22⋊Q8—3324no—
415#121—22no—
416#121Dic16614yescomplemented
417C4×D4—22no—
418C4×D4—41no—
419#117—22no—
420#117—41no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—3321no—
423C2×C4⋊C4—831no—
424C2×C4⋊C4—831no—
425C2×C4⋊C4—831no—
426C2×C4⋊C4—831no—
427C2×C4⋊C4—1661no—
428C2×C4⋊C4—1661no—
429C2×C4⋊C4—1661no—
430C2×C4⋊C4—1661no—
431C2×C4⋊C4—1661no—
432C2×C4⋊C4—1662no—
433C2×C4⋊C4—1661no—
434C2×C4⋊C4C22×D8311yescharacteristic
435C2×C4⋊C4C22×D8311yescharacteristic
436C2×C4⋊C4C22×D8311yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1661no—
443C2×C22⋊C4C22⋊D8311yescharacteristic
444C2×C22⋊C4C22⋊D8311yescharacteristic
445C2×C42—21no—
446C2×C42—21no—
447C2×C42C22×D8311yescharacteristic
448C2×C42C22×D8311yescharacteristic
449#107—834no—
450#107—1662no—
451#107—1662no—
452#106—3321no—
453#106—1662no—
454#106—1662no—
455C4⋊C8—1662no—
456C4⋊C8—1662no—
457#102—3321no—
458#102—1662no—
459#102—1662no—
460#102—1662no—
461C22⋊C8—1662no—
462#95—21no—
463#95—21no—
464#95—1661no—
465#95—1661no—
466#95—1661no—
467#95C22⋊D8311yescharacteristic
468#95C22⋊D8311yescharacteristic
469C24—21no—
470C24C2×C22⋊D8311yescharacteristic
471C2×Q8—3322no—
472C2×Q8—3322no—
473C2×Q8—6641no—
474C2×D4—24no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—81no—
481C2×D4—81no—
482C2×D4C2×Dic16612yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—81no—
493C22×C4—3322no—
494C22×C4—3322no—
495C22×C4—6641no—
496C22×C4—1661no—
497C22×C4—1661no—
498C22×C4—1661no—
499C22×C4—1661no—
500C22×C4C2×C22⋊D8311yescharacteristic
501C22×C4C2×C22⋊D8311yescharacteristic
502C22×C4C2×C22⋊D8311yescharacteristic
503C22×C4C2×C22⋊D8311yescharacteristic
504C22×C4C2×C22⋊D8311yescharacteristic
505C22×C4C2×Dic16612yes—
506C22×C4#1305059012yes—
507C22×C4C23×D8311yescharacteristic
508M16—1668no—
509C2×C8—832no—
510C2×C8—832no—
511C2×C8—1661no—
512C2×C8—1661no—
513C2×C8—1661no—
514C2×C8—1661no—
515C2×C8—1661no—
516C2×C8—1661no—
517C2×C8—1661no—
518C2×C8—1661no—
519C2×C8—1661no—
520C2×C8—1661no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—81no—
526C4⋊C4—3322no—
527C4⋊C4—3322no—
528C4⋊C4—3322no—
529C4⋊C4—3322no—
530C4⋊C4—3322no—
531C4⋊C4—832no—
532C4⋊C4—6641no—
533C4⋊C4—1661no—
534C4⋊C4—1661no—
535C4⋊C4—1661no—
536C4⋊C4—1661no—
537C4⋊C4—1661no—
538C4⋊C4—1661no—
539C4⋊C4—1661no—
540C4⋊C4D4×D8312yes—
541C4⋊C4C2×Dic16612yes—
542C22⋊C4—24no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—81no—
549C22⋊C4—3322no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—42no—
555C42—81no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23#1458177112yes—
570C23C22×C22⋊D8311yescharacteristic
571Q8—6644no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576D4—84no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—84no—
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—3321no—
613C2×C4—3321no—
614C2×C4—6644no—
615C2×C4—1662no—
616C2×C4#1458177112yes—
617C2×C4#1458181712yes—
618C2×C4#1458182212yes—
619C2×C4C2×D4×D8311yescharacteristic
620C2×C4C22×Dic16611yescharacteristic
621C2×C4#41×D8311yescharacteristic
622C2×C4C2×#1305059011yescharacteristic
623C2×C4#1458183711yescharacteristic
624C2×C4#1458183911yescharacteristic
625C8—3321no—
626C8—3321no—
627C8—1662no—
628C8—1662no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22C2×#1458177111yescharacteristic
644C22D4×C22⋊D8311yescharacteristic
645C22#1912835011yescharacteristic
646C22C2×#1458181711yescharacteristic
647C22C2×#1458182211yescharacteristic
648C22#1912850411yescharacteristic
649C22#1912851211yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—81no—
658C4—3322no—
659C4—6641no—
660C4—6641no—
661C4—6641no—
662C4#1912821011yescharacteristic
663C4#1912829311yescharacteristic
664C4D4×Dic16611yescharacteristic
665C4#1912841011yescharacteristic
666C2—42no—
667C2—82no—
668C2#2491126611yescharacteristic
669C2#2491727811yescharacteristic
670C2#2491729411yescharacteristic
671C2#2491894411yescharacteristic
672C2#2491912911yescharacteristic
673C2#2491994811yescharacteristic
674C2#2492005211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2363C2
socleC23×C83C22×D4
prime radicalC2×#121×C83C22
hypercenter#2363D83
hypocenterC83#26357
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. C2×#19127981C2semidirect
2. C2×#19127744C2semidirect
3. #24911165C2semidirect
4. #24908996C2semidirect
5. #24908965C2semidirect
6. #24908837C2semidirect
7. #24908796C2semidirect
8. #24908221C2semidirect
9. #24908055C2semidirect
12. #24907710C2semidirect
13. #24907662C2semidirect
14. #24907648C2semidirect
88. #14581725D4semidirect
107. #121×C83Q8semidirect
353. C83#26357semidirect
416. #121Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191279812
C2×#191277442
#249111652
#249089962
#249089652
#249088372
#249087962
#249082212
#249080552
#249079022
#249077812
#249077102
#249076622
#249076482
C83×#23632
#2635783