Finite Group Databasebeta

← #35405082

Subgroups — #35405082

Order 21248.

1646 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D83C211yesmaximal, characteristic
2C2×#19127724C211yesmaximal, characteristic, complemented
3C2×#19127664C211yesmaximal, characteristic
4C2×C4×(C2×D83)⋊C4C211yesmaximal, characteristic
5#24907644C214yesmaximal, complemented
6#24907639C214yesmaximal, complemented
7#24907157C211yesmaximal, characteristic, complemented
8#24907156C211yesmaximal, characteristic, complemented
9C2×#423×C83C211yesmaximal, characteristic, complemented
10C23×C4×D83C2211yescharacteristic
11C22×#13050574C2211yescharacteristic
12C22×#13050574C2211yescharacteristic
13C22×#13050573C2212yes—
14C2×M16×D83—24no—
15C2×M16×D83—24no—
16C2×M16×D83—24no—
17C22×C8×D83C2212yes—
18C22×(C2×D83)⋊C4C412yes—
19C22×C4×Dic83C2211yescharacteristic
20#19127724—24no—
21C2×Dic83⋊C8C2214yescomplemented
22#19127664—24no—
23C2×(C2×D83)⋊C8C2212yes—
24C2×(C2×D83)⋊C8C2212yes—
25C4×(C2×D83)⋊C4C411yescharacteristic
26C4×(C2×D83)⋊C4C411yescharacteristic
27C4×(C2×D83)⋊C4C411yescharacteristic
28C4×(C2×D83)⋊C4C411yescharacteristic
29C4×(C2×D83)⋊C4C2212yes—
30C4×(C2×D83)⋊C4C2212yes—
31C2×C83⋊#95C412yes—
32#19127456C2214yescomplemented
33#19127455C2214yescomplemented
34C22×M16×C83C2211yescharacteristic
35C22×M16×C83C2211yescharacteristic
36C22×C42×C83C2211yescharacteristic
37#423×C83—24no—
38C2×C4⋊C8×C83C2214yescomplemented
39C24×D83C2×C411yescharacteristic
40C23×Dic83C2×C412yes—
41C23×Dic83C2×C411yescharacteristic
42C22×C4×D83—22no—
43C22×C4×D83—22no—
44C22×C4×D83—22no—
45C22×C4×D83—22no—
46C22×C4×D83C2×C411yescharacteristic
47C22×C4×D83C2×C411yescharacteristic
48C22×C4×D83D414yes—
49C22×C4×D83Q814yes—
50C22×C4×D83C2311yescharacteristic
51C22×C4×D83C2311yescharacteristic
52C2×#13050574—22no—
53C2×#13050574—22no—
54C2×#13050574—22no—
55C2×#13050574—22no—
56C2×#13050574—22no—
57C2×#13050574D414yes—
58C2×#13050573—24no—
59C2×#13050573—24no—
60C2×#13050573—24no—
61M16×D83—232no—
62C22×C83⋊C8C2312yes—
63C22×C83⋊C8C2312yes—
64C2×C8×D83—24no—
65C2×C8×D83—24no—
66C2×C8×D83—24no—
67C2×(C2×D83)⋊C4—28no—
68C2×(C2×D83)⋊C4C2×C412yes—
69C2×(C2×D83)⋊C4C2×C412yes—
70C2×(C2×D83)⋊C4C2×C412yes—
71C2×(C2×D83)⋊C4C2×C412yes—
72C2×C4×Dic83—22no—
73C2×C4×Dic83—22no—
74C2×C4×Dic83C2×C411yescharacteristic
75C2×C4×Dic83C2×C411yescharacteristic
76C2×C4×Dic83C2312yes—
77Dic83⋊C8—28no—
78(C2×D83)⋊C8—24no—
79(C2×D83)⋊C8—24no—
80C83⋊#95C2×C412yes—
81C83⋊#95C2×C412yes—
82C83⋊#95C2×C412yes—
83C83⋊#95C2×C412yes—
84C23×C4×C83C2×C412yes—
85C23×C4×C83C2311yescharacteristic
86C2×M16×C83—22no—
87C2×M16×C83—22no—
88C2×M16×C83—22no—
89C2×M16×C83—22no—
90C2×M16×C83—22no—
91C2×M16×C83D414yes—
92C22×C8×C83C2312yes—
93C22×C8×C83C2312yes—
94C2×C42×C83—22no—
95C2×C42×C83—22no—
96C2×C42×C83C2×C411yescharacteristic
97C2×C42×C83C2×C411yescharacteristic
98C2×C42×C83C2312yes—
99C4⋊C8×C83—28no—
100C23×D83—22no—
101C23×D83—22no—
102C23×D83—24no—
103C23×D83—42no—
104C23×D83C4⋊C414yes—
105C23×D83C22×C411yescharacteristic
106C23×D83C22×C411yescharacteristic
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—24no—
110C22×Dic83—24no—
111C22×Dic83—42no—
112C22×Dic83C4⋊C414yes—
113C22×Dic83C22×C412yes—
114C22×Dic83C22×C411yescharacteristic
115C22×Dic83C22×C411yescharacteristic
116C22×Dic83C22×C412yes—
117C2×C4×D83—22no—
118C2×C4×D83—22no—
119C2×C4×D83—22no—
120C2×C4×D83—22no—
121C2×C4×D83—28no—
122C2×C4×D83—28no—
123C2×C4×D83—28no—
124C2×C4×D83—28no—
125C2×C4×D83—21no—
126C2×C4×D83—21no—
127C2×C4×D83—21no—
128C2×C4×D83—21no—
129C2×C4×D83—21no—
130C2×C4×D83—21no—
131C2×C4×D83—21no—
132C2×C4×D83—21no—
133C2×C4×D83C4⋊C414yes—
134C2×C4×D83C4⋊C414yes—
135C2×C4×D83C2×D412yes—
136C2×C4×D83C2×D412yes—
137C2×C4×D83C2×Q812yes—
138C2×C4×D83C2×Q812yes—
139#13050574—28no—
140#13050574—44no—
141#13050573—216no—
142C2×C83⋊C8—22no—
143C2×C83⋊C8—22no—
144C2×C83⋊C8—22no—
145C2×C83⋊C8—22no—
146C2×C83⋊C8—22no—
147C2×C83⋊C8C2×D414yes—
148C8×D83—216no—
149(C2×D83)⋊C4—24no—
150(C2×D83)⋊C4—24no—
151(C2×D83)⋊C4—24no—
152(C2×D83)⋊C4M1618yescomplemented
153C4×Dic83—22no—
154C4×Dic83—22no—
155C4×Dic83—22no—
156C4×Dic83—22no—
157C24×C83C22×C411yescharacteristic
158C22×C4×C83—24no—
159C22×C4×C83—24no—
160C22×C4×C83—42no—
161C22×C4×C83C22×C412yes—
162C22×C4×C83C22×C412yes—
163C22×C4×C83C22×C411yescharacteristic
164C22×C4×C83C2×D412yes—
165C22×C4×C83C2×D412yes—
166C22×C4×C83C2×D412yes—
167C22×C4×C83C2×Q812yes—
168C22×C4×C83#4112yes—
169C22×C4×C83#4112yes—
170C22×C4×C83C2411yescharacteristic
171M16×C83—28no—
172M16×C83—44no—
173C2×C8×C83—22no—
174C2×C8×C83—22no—
175C2×C8×C83—22no—
176C2×C8×C83—22no—
177C2×C8×C83—22no—
178C2×C8×C83C2×D414yes—
179C42×C83—22no—
180C42×C83—22no—
181C42×C83—22no—
182C42×C83—22no—
183C23×C83—22no—
184C23×C83—22no—
185C23×C83C2×C4⋊C412yes—
186C23×C83C4×D412yes—
187C23×C83C4×D412yes—
188C23×C83C23×C411yescharacteristic
189C22×D83—21no—
190C22×D83—21no—
191C22×D83—22no—
192C22×D83—21no—
193C22×D83—21no—
194C22×D83—22no—
195C22×D83—22no—
196C22×D83—28no—
197C22×D83—28no—
198C22×D83—44no—
199C22×D83—48no—
200C22×D83—48no—
201C22×D83C2×C4⋊C412yes—
202C22×D83C2×C4⋊C412yes—
203C22×D83C2×M1614yes—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—22no—
207C2×C4×C83—22no—
208C2×C4×C83—22no—
209C2×C4×C83—48no—
210C2×C4×C83C2×C4⋊C412yes—
211C2×C4×C83C4×D412yes—
212C2×C4×C83C4×D412yes—
213C2×C4×C83#12014yes—
214C2×C4×C83C22⋊Q814yes—
215C2×C4×C83C22⋊Q814yes—
216C2×C4×C83#12314yes—
217C2×C4×C83C2×M1614yescomplemented
218C2×C4×C83C22×D411yescharacteristic
219C2×C4×C83C22×D411yescharacteristic
220C2×C4×C83C22×D411yescharacteristic
221C2×C4×C83C22×Q811yescharacteristic
222C2×C4×C83C2×#4111yescharacteristic
223C2×C4×C83C2×#4111yescharacteristic
224C2×Dic83—22no—
225C2×Dic83—21no—
226C2×Dic83—21no—
227C2×Dic83—21no—
228C2×Dic83—21no—
229C2×Dic83—22no—
230C2×Dic83—28no—
231C2×Dic83—22no—
232C2×Dic83—22no—
233C2×Dic83—22no—
234C2×Dic83—48no—
235C2×Dic83C2×C4⋊C412yes—
236C2×Dic83C2×C4⋊C412yes—
237C2×Dic83C2×M1614yescomplemented
238C4×D83—22no—
239C4×D83—22no—
240C4×D83—22no—
241C4×D83—22no—
242C4×D83—22no—
243C4×D83—22no—
244C4×D83—22no—
245C4×D83—22no—
246C4×D83—28no—
247C4×D83—28no—
248C8×C83—24no—
249C8×C83—42no—
250C83⋊C8—24no—
251C83⋊C8—42no—
252C22×C83—22no—
253C22×C83—22no—
254C22×C83—24no—
255C22×C83—24no—
256C22×C83#38514yes—
257C22×C83C22×C4⋊C411yescharacteristic
258C22×C83C2×C4×D411yescharacteristic
259C22×C83C2×C4×D411yescharacteristic
260C22×C83C22×M1612yes—
261C22×C83#56812yes—
262C2×D83—22no—
263C2×D83—24no—
264C2×D83—24no—
265C2×D83—22no—
266C2×D83—22no—
267C2×D83—22no—
268C2×D83—22no—
269C2×D83—416no—
270C2×D83—42no—
271C2×D83—42no—
272C2×D83#42314yescomplemented
273C4×C83—24no—
274C4×C83—42no—
275C4×C83#38514yes—
276C4×C83C2×#12011yescharacteristic
277C4×C83C2×C22⋊Q811yescharacteristic
278C4×C83C2×C22⋊Q811yescharacteristic
279C4×C83C2×#12311yescharacteristic
280Dic83—24no—
281Dic83—22no—
282Dic83—22no—
283Dic83—42no—
284#20837—831nomaximal, Sylow
285C2×C83—24no—
286C2×C83C2×#38511yescharacteristic
287C2×C83C2×#42311yescharacteristic
288C2×C83#290511yescharacteristic
289C2×C83D4×M1612yes—
290C2×C83D4×M1612yes—
291D83—24no—
292D83—42no—
293C23×M16—831no—
294C2×#423—831no—
295C2×#423D8311yescharacteristic, complemented
296C2×#407—831no—
297C2×C4×C22⋊C4—831no—
298#1851—834no—
299#1850—834no—
300#1728—831no—
301#1728—831no—
302C83#2083711yescharacteristic, complemented, Sylow
303C24×C4—831no—
304C22×M16—831no—
305C22×M16—831no—
306C22×M16—832no—
307C22×M16—1664no—
308C22×M16—1664no—
309C22×M16—1664no—
310C22×M16C2×D8311yescharacteristic
311C22×M16C2×D8311yescharacteristic
312C23×C8—832no—
313C22×C22⋊C4—832no—
314C22×C42—831no—
315C22×C42C2×D8311yescharacteristic
316#423—24no—
317#423—1664no—
318C2×C4⋊C8—834no—
319C2×C4⋊C8C2×D8314yescomplemented
320#407—1664no—
321C2×C22⋊C8—832no—
322C2×C22⋊C8—832no—
323C4×C22⋊C4—831no—
324C4×C22⋊C4—831no—
325C4×C22⋊C4—831no—
326C4×C22⋊C4—831no—
327C4×C22⋊C4—832no—
328C4×C22⋊C4—832no—
329C2×#95—832no—
330#336—834no—
331#336—834no—
332C25—831no—
333C23×C4—832no—
334C23×C4—831no—
335C23×C4—831no—
336C23×C4—831no—
337C23×C4—834no—
338C23×C4—834no—
339C23×C4—831no—
340C23×C4—831no—
341C23×C4—1662no—
342C23×C4—1662no—
343C23×C4—1662no—
344C23×C4—1662no—
345C23×C4C4×D8312yes—
346C23×C4C22×D8311yescharacteristic
347C2×M16—22no—
348C2×M16—22no—
349C2×M16—22no—
350C2×M16—22no—
351C2×M16—22no—
352C2×M16—834no—
353C2×M16—1664no—
354C2×M16—1664no—
355C2×M16—16632no—
356C2×M16—1662no—
357C2×M16—1662no—
358C2×M16—1662no—
359C2×M16—1662no—
360C2×M16—1662no—
361C2×M16—1664no—
362C2×M16C22⋊D8314yes—
363C22×C8—832no—
364C22×C8—832no—
365C22×C8—1664no—
366C22×C8—1664no—
367C22×C8—1664no—
368C22×C8C22×D8312yes—
369C22×C8C22×D8312yes—
370C2×C22⋊C4—832no—
371C2×C22⋊C4—832no—
372C2×C22⋊C4—832no—
373C2×C22⋊C4—832no—
374C2×C22⋊C4—1668no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—831no—
378C2×C42—831no—
379C2×C42—832no—
380C2×C42—1662no—
381C2×C42—1662no—
382C2×C42C4×D8311yescharacteristic
383C2×C42C4×D8311yescharacteristic
384C2×C42C22×D8312yes—
385C4⋊C8—28no—
386C4⋊C8—1668no—
387C22⋊C8—1664no—
388C22⋊C8—1664no—
389#95—832no—
390#95—832no—
391#95—832no—
392#95—832no—
393C24—3322no—
394C24—834no—
395C24—831no—
396C24—831no—
397C24—1664no—
398C24—1662no—
399C24—1662no—
400C24C2×C4×D8311yescharacteristic
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—42no—
404C22×C4—3322no—
405C22×C4—831no—
406C22×C4—831no—
407C22×C4—832no—
408C22×C4—834no—
409C22×C4—832no—
410C22×C4—832no—
411C22×C4—832no—
412C22×C4—832no—
413C22×C4—834no—
414C22×C4—834no—
415C22×C4—832no—
416C22×C4—1664no—
417C22×C4—1664no—
418C22×C4—1662no—
419C22×C4—1662no—
420C22×C4—1668no—
421C22×C4—1668no—
422C22×C4—1668no—
423C22×C4—1668no—
424C22×C4—1661no—
425C22×C4—1661no—
426C22×C4—1661no—
427C22×C4—1661no—
428C22×C4—1661no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1662no—
433C22×C4—1662no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4C2×C4×D8312yes—
437C22×C4C2×C4×D8312yes—
438C22×C4C2×C4×D8311yescharacteristic
439C22×C4D4×D8312yes—
440C22×C4D4×D8312yes—
441C22×C4C2×C22⋊D8312yes—
442C22×C4Q8×D8312yes—
443C22×C4#1305059012yes—
444C22×C4#1305059112yes—
445C22×C4C23×D8311yescharacteristic
446M16—28no—
447M16—44no—
448M16—3324no—
449M16—16616no—
450M16—1668no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—834no—
457C2×C8—16616no—
458C2×C8—1662no—
459C2×C8—1662no—
460C2×C8—1662no—
461C2×C8—1662no—
462C2×C8—1662no—
463C2×C8C2×C22⋊D8314yes—
464C22⋊C4—838no—
465C22⋊C4—1664no—
466C22⋊C4—1664no—
467C22⋊C4—1664no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—1662no—
473C42—1662no—
474C42—1662no—
475C42—1662no—
476C23—22no—
477C23—22no—
478C23—3328no—
479C23—3328no—
480C23—3324no—
481C23—832no—
482C23—834no—
483C23—832no—
484C23—1668no—
485C23—1668no—
486C23—1661no—
487C23—1661no—
488C23—1662no—
489C23—1662no—
490C23—1661no—
491C23—1661no—
492C23—1662no—
493C23C4⋊C4×D8312yes—
494C23C4×C22⋊D8312yes—
495C23#1458175412yes—
496C23C22×C4×D8311yescharacteristic
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—48no—
503C2×C4—3328no—
504C2×C4—832no—
505C2×C4—832no—
506C2×C4—834no—
507C2×C4—1662no—
508C2×C4—1662no—
509C2×C4—1662no—
510C2×C4—1668no—
511C2×C4—1661no—
512C2×C4—1661no—
513C2×C4—1662no—
514C2×C4—1661no—
515C2×C4—1661no—
516C2×C4—1662no—
517C2×C4—1668no—
518C2×C4—1668no—
519C2×C4—1662no—
520C2×C4—1662no—
521C2×C4—1662no—
522C2×C4—1662no—
523C2×C4—1662no—
524C2×C4—1662no—
525C2×C4—1662no—
526C2×C4—1662no—
527C2×C4C4⋊C4×D8312yes—
528C2×C4C4×C22⋊D8312yes—
529C2×C4#1458175412yes—
530C2×C4#1458176014yes—
531C2×C4#1458176814yes—
532C2×C4#1458177014yes—
533C2×C4#1458177714yes—
534C2×C4M16×D8314yescomplemented
535C2×C4C2×D4×D8311yescharacteristic
536C2×C4C2×D4×D8311yescharacteristic
537C2×C4C22×C22⋊D8311yescharacteristic
538C2×C4C2×Q8×D8311yescharacteristic
539C2×C4C2×#1305059011yescharacteristic
540C2×C4C2×#1305059111yescharacteristic
541C8—24no—
542C8—42no—
543C8—3322no—
544C8—1664no—
545C22—22no—
546C22—22no—
547C22—24no—
548C22—24no—
549C22—33216no—
550C22—3322no—
551C22—3322no—
552C22—834no—
553C22—1662no—
554C22—1662no—
555C22—1662no—
556C22—1662no—
557C22—1662no—
558C22—1664no—
559C22—1664no—
560C22#1912757214yes—
561C22C2×C4⋊C4×D8311yescharacteristic
562C22C2×C4×C22⋊D8311yescharacteristic
563C22C2×#1458175411yescharacteristic
564C22C2×M16×D8312yes—
565C22#1912846712yes—
566C4—24no—
567C4—42no—
568C4—3322no—
569C4—1662no—
570C4—1662no—
571C4—1664no—
572C4#1912757214yes—
573C4C2×#1458176011yescharacteristic
574C4C2×#1458176811yescharacteristic
575C4C2×#1458177011yescharacteristic
576C4C2×#1458177711yescharacteristic
577C2—24no—
578C2—3322no—
579C2—1664no—
580C2C2×#1912757211yescharacteristic
581C2#423×D8311yescharacteristic
582C2#2491602511yescharacteristic
583C2M16×C22⋊D8312yes—
584C2#2491619812yes—
subgroupG/H
center, FrattiniC22×C4C23×D83
commutator, socleC23×C83C23×C4
FittingC2×#423×C83C2
prime radicalC24×D83C2×C4
cosocleC22×C4×C83C24
hypercenterC2×#423D83
hypocenterC83#20837
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
2. C2×#19127724C2semidirect
5. #24907644C2semidirect
6. #24907639C2semidirect
7. #24907157C2semidirect
8. #24907156C2semidirect
9. C2×#423×C83C2semidirect
21. C2×Dic83⋊C8C22semidirect
32. #19127456C22semidirect
33. #19127455C22semidirect
38. C2×C4⋊C8×C83C22semidirect
152. (C2×D83)⋊C4M16semidirect
217. C2×C4×C83C2×M16semidirect
237. C2×Dic83C2×M16semidirect
272. C2×D83#423semidirect
295. C2×#423D83semidirect
302. C83#20837semidirect
319. C2×C4⋊C8C2×D83semidirect
534. C2×C4M16×D83semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D832
C2×#191277242
C2×#191276642
C2×C4×(C2×D83)⋊C42
#249076442
#249076392
#249071572
#249071562
C2×#423×C832
#2083783