Finite Group Databasebeta

← C2×#24907893 #35406083

Subgroups — C2×#24907893 #35406083

Order 21248.

1966 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#13050573C211yesmaximal, characteristic
2C22×#14581768C211yesmaximal, characteristic
3C22×#14581716C211yesmaximal, characteristic, complemented
4C2×#19127558C211yesmaximal, characteristic
5#24907893C218yesmaximal, direct product factor
6C2×#19127464C211yesmaximal, characteristic, complemented
7C2×#19127463C211yesmaximal, characteristic, complemented
8C22×#103×C83C211yesmaximal, characteristic, complemented
9C23×C4×D83C2211yescharacteristic
10C22×#13050573—24no—
11C22×#13050573—24no—
12C22×#13050573—24no—
13C22×#13050573C2214yes—
14C23×C83⋊C8C2211yescharacteristic
15C2×#14581768—22no—
16C2×#14581768—24no—
17C2×#14581768C418yescomplemented
18C2×#14581768C2214yes—
19C22×C4⋊Dic83C2211yescharacteristic
20C22×C4⋊Dic83C2211yescharacteristic
21C22×Dic83⋊C4—21no—
22C22×(C2×D83)⋊C4—21no—
23C22×(C2×D83)⋊C4—21no—
24C2×#14581716—24no—
25C2×#14581716C2214yescomplemented
26#19127558—22no—
27#19127558C2214yes—
28C2×C83⋊#95—21no—
29#19127464C2214yescomplemented
30#19127463C2214yescomplemented
31C23×Q8×C83C2211yescharacteristic
32C23×C8×C83C2211yescharacteristic
33C22×C4⋊C4×C83C2211yescharacteristic
34C2×#103×C83—24no—
35C2×#103×C83C2214yescomplemented
36C24×D83D411yescharacteristic
37C23×Dic83—21no—
38C23×Dic83—21no—
39C23×Dic83D411yescharacteristic
40C22×C4×D83—22no—
41C22×C4×D83—22no—
42C22×C4×D83—22no—
43C22×C4×D83—22no—
44C22×C4×D83C2×C414yes—
45C22×C4×D83D412yes—
46C22×C4×D83D414yes—
47C22×C4×D83C2312yes—
48C2×#13050573—232no—
49C2×#13050573—28no—
50C2×#13050573—28no—
51C2×#13050573—28no—
52C22×C83⋊C8—22no—
53C22×C83⋊C8—22no—
54C22×C83⋊C8D414yes—
55C22×C83⋊C8C2312yes—
56#14581768—216no—
57#14581768—28no—
58#14581768C2×C4116yescomplemented
59C2×C4⋊Dic83—21no—
60C2×C4⋊Dic83—22no—
61C2×C4⋊Dic83—22no—
62C2×C4⋊Dic83—21no—
63C2×C4⋊Dic83—22no—
64C2×C4⋊Dic83C2×C414yes—
65C2×C4⋊Dic83C2312yes—
66C2×C4⋊Dic83C2312yes—
67C2×Dic83⋊C4—28no—
68C2×Dic83⋊C4—24no—
69C2×(C2×D83)⋊C4—28no—
70C2×(C2×D83)⋊C4—24no—
71C2×(C2×D83)⋊C4—24no—
72C2×(C2×D83)⋊C4—44no—
73#14581716—28no—
74C83⋊#95—24no—
75C22×Q8×C83—22no—
76C22×Q8×C83—21no—
77C22×Q8×C83—22no—
78C22×Q8×C83—22no—
79C22×Q8×C83—22no—
80C22×Q8×C83—22no—
81C22×Q8×C83C2×C414yes—
82C22×Q8×C83C2312yes—
83C23×C4×C83—21no—
84C23×C4×C83—21no—
85C23×C4×C83C2311yescharacteristic
86C22×C8×C83—22no—
87C22×C8×C83—22no—
88C22×C8×C83D414yes—
89C22×C8×C83C2312yes—
90C2×C4⋊C4×C83—22no—
91C2×C4⋊C4×C83—22no—
92C2×C4⋊C4×C83—21no—
93C2×C4⋊C4×C83C2312yes—
94#103×C83—28no—
95C23×D83—22no—
96C23×D83—22no—
97C23×D83—22no—
98C23×D83—22no—
99C23×D83—22no—
100C23×D83—22no—
101C23×D83C22⋊C414yes—
102C23×D83C2×D412yes—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—24no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—42no—
109C22×Dic83—42no—
110C22×Dic83—42no—
111C22×Dic83—42no—
112C22×Dic83—42no—
113C22×Dic83C22⋊C414yes—
114C22×Dic83C2×D412yes—
115C2×C4×D83—28no—
116C2×C4×D83—28no—
117C2×C4×D83—28no—
118C2×C4×D83—24no—
119C2×C4×D83—24no—
120C2×C4×D83—28no—
121C2×C4×D83—22no—
122C2×C4×D83—22no—
123C2×C4×D83—22no—
124C2×C4×D83—22no—
125C2×C4×D83C22⋊C418yes—
126C2×C4×D83C22×C414yes—
127C2×C4×D83C2×D414yes—
128#13050573—232no—
129C2×C83⋊C8—28no—
130C2×C83⋊C8—22no—
131C2×C83⋊C8—22no—
132C2×C83⋊C8C2×D414yes—
133C4⋊Dic83—24no—
134C4⋊Dic83—22no—
135C4⋊Dic83—22no—
136C4⋊Dic83—22no—
137C4⋊Dic83—42no—
138C4⋊Dic83C22×C414yes—
139Dic83⋊C4—216no—
140(C2×D83)⋊C4—216no—
141(C2×D83)⋊C4—48no—
142C24×C83C2×D411yescharacteristic
143C2×Q8×C83—24no—
144C2×Q8×C83—22no—
145C2×Q8×C83—28no—
146C2×Q8×C83—28no—
147C2×Q8×C83—22no—
148C2×Q8×C83—22no—
149C2×Q8×C83—22no—
150C2×Q8×C83—22no—
151C2×Q8×C83—44no—
152C2×Q8×C83—42no—
153C2×Q8×C83—42no—
154C2×Q8×C83—44no—
155C2×Q8×C83C22×C414yes—
156C22×C4×C83—24no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—42no—
160C22×C4×C83—42no—
161C22×C4×C83—42no—
162C22×C4×C83—42no—
163C22×C4×C83—42no—
164C22×C4×C83C22×C412yes—
165C22×C4×C83C2×D411yescharacteristic
166C22×C4×C83C2×D412yes—
167C22×C4×C83C2×D412yes—
168C22×C4×C83C2×D412yes—
169C22×C4×C83#4112yes—
170C22×C4×C83#4112yes—
171C22×C4×C83C2411yescharacteristic
172C2×C8×C83—28no—
173C2×C8×C83—22no—
174C2×C8×C83—22no—
175C2×C8×C83C2×D414yes—
176C4⋊C4×C83—22no—
177C4⋊C4×C83—22no—
178C4⋊C4×C83—42no—
179C23×C83C2×C22⋊C412yes—
180C23×C83#12112yes—
181C23×C83#12112yes—
182C23×C83AGL(1,ℤ/8)12yes—
183C23×C83#13712yes—
184C23×C83#13712yes—
185C23×C83#13712yes—
186C23×C83C22×D411yescharacteristic
187C22×D83—22no—
188C22×D83—22no—
189C22×D83—22no—
190C22×D83—22no—
191C22×D83—22no—
192C22×D83—22no—
193C22×D83—28no—
194C22×D83—28no—
195C22×D83—28no—
196C22×D83—44no—
197C22×D83—44no—
198C22×D83—44no—
199C22×D83—44no—
200C22×D83C2×C22⋊C414yes—
201Q8×C83—24no—
202Q8×C83—24no—
203Q8×C83—42no—
204Q8×C83—42no—
205Q8×C83—48no—
206C2×C4×C83—24no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83—48no—
210C2×C4×C83—42no—
211C2×C4×C83—42no—
212C2×C4×C83—42no—
213C2×C4×C83—84no—
214C2×C4×C83C2×C22⋊C412yes—
215C2×C4×C83C4×D414yes—
216C2×C4×C83C4×D414yes—
217C2×C4×C83#12014yes—
218C2×C4×C83#12112yes—
219C2×C4×C83#12112yes—
220C2×C4×C83#12314yes—
221C2×C4×C83C23×C411yescharacteristic
222C2×C4×C83C22×D411yescharacteristic
223C2×C4×C83C22×D411yescharacteristic
224C2×C4×C83C22×D411yescharacteristic
225C2×C4×C83C2×#4111yescharacteristic
226C2×C4×C83C2×#4111yescharacteristic
227C2×Dic83—22no—
228C2×Dic83—22no—
229C2×Dic83—28no—
230C2×Dic83—24no—
231C2×Dic83—42no—
232C2×Dic83—42no—
233C2×Dic83—48no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83—42no—
237C2×Dic83—84no—
238C2×Dic83C2×C22⋊C414yes—
239C4×D83—24no—
240C4×D83—24no—
241C4×D83—24no—
242C4×D83—24no—
243C4×D83—216no—
244C8×C83—24no—
245C83⋊C8—24no—
246C22×C83#38614yes—
247C22×C83#41714yes—
248C22×C83#41914yes—
249C22×C83#46814yes—
250C22×C83#46814yes—
251C22×C83#47014yes—
252C22×C83#47014yes—
253C22×C83C22×C22⋊C411yescharacteristic
254C22×C83C2×#12111yescharacteristic
255C22×C83C2×#12111yescharacteristic
256C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C83C2×#13711yescharacteristic
258C22×C83C2×#13711yescharacteristic
259C22×C83C2×#13711yescharacteristic
260C2×D83—24no—
261C2×D83—24no—
262C2×D83—24no—
263C2×D83—416no—
264C2×D83—42no—
265C2×D83—42no—
266C2×D83—42no—
267C2×D83—42no—
268C4×C83—44no—
269C4×C83—82no—
270C4×C83#38614yes—
271C4×C83C2×C4×D411yescharacteristic
272C4×C83C2×C4×D411yescharacteristic
273C4×C83C2×#12011yescharacteristic
274C4×C83C2×#12311yescharacteristic
275Dic83—24no—
276Dic83—44no—
277Dic83—82no—
278C2×#1894—831nomaximal, Sylow
279C2×C83#189418yescomplemented
280C2×C83C2×#38611yescharacteristic
281C2×C83C2×#41911yescharacteristic
282C2×C83C2×#41711yescharacteristic
283C2×C83C2×#46811yescharacteristic
284C2×C83C2×#46811yescharacteristic
285C2×C83C2×#47011yescharacteristic
286C2×C83C2×#47011yescharacteristic
287D83—44no—
288C23×M16—831no—
289C22×C22⋊Q8—831no—
290C22×#103—831no—
291C22×#103D8311yescharacteristic, complemented
292C2×#380—831no—
293#1894—838no—
294C2×#340—831no—
295C2×#340—831no—
296C83C2×#189411yescharacteristic, complemented, Sylow
297C23×Q8C2×D8311yescharacteristic
298C24×C4—831no—
299C22×M16—834no—
300C22×M16—1664no—
301C22×M16—1664no—
302C22×M16—1664no—
303C23×C8—831no—
304C23×C8C2×D8311yescharacteristic
305C2×C22⋊Q8—838no—
306C2×C22⋊Q8—834no—
307C2×C22⋊Q8—1664no—
308C2×C22⋊Q8—1662no—
309C22×C4⋊C4—831no—
310C22×C4⋊C4—831no—
311C22×C4⋊C4—1661no—
312C22×C4⋊C4C2×D8311yescharacteristic
313C22×C22⋊C4—1661no—
314C22×C22⋊C4—1661no—
315C2×#103—24no—
316C2×#103—834no—
317C2×#103—1664no—
318C2×#103C2×D8314yescomplemented
319#380—834no—
320#380—1662no—
321C2×#95—1661no—
322#340—834no—
323#340—834no—
324C25—831no—
325C22×Q8—21no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8C4×D8314yes—
332C22×Q8C22×D8312yes—
333C23×C4—21no—
334C23×C4—21no—
335C23×C4—831no—
336C23×C4—834no—
337C23×C4—834no—
338C23×C4—832no—
339C23×C4—832no—
340C23×C4—1661no—
341C23×C4—1661no—
342C23×C4—1662no—
343C23×C4—1662no—
344C23×C4—1662no—
345C23×C4—1662no—
346C23×C4C22×D8311yescharacteristic
347C2×M16—1668no—
348C2×M16—1668no—
349C2×M16—16632no—
350C2×M16—1668no—
351C22×C8—22no—
352C22×C8—22no—
353C22×C8—834no—
354C22×C8—832no—
355C22×C8—1662no—
356C22×C8—1662no—
357C22×C8C22⋊D8314yes—
358C22×C8C22×D8312yes—
359C22⋊Q8—8316no—
360C22⋊Q8—1668no—
361C22⋊Q8—16616no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—834no—
366C2×C4⋊C4—832no—
367C2×C4⋊C4—832no—
368C2×C4⋊C4—1661no—
369C2×C4⋊C4—1662no—
370C2×C4⋊C4—1662no—
371C2×C4⋊C4—1664no—
372C2×C4⋊C4—1668no—
373C2×C4⋊C4—1662no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4C22×D8312yes—
376C2×C22⋊C4—3324no—
377C2×C22⋊C4—1664no—
378C2×C22⋊C4—1664no—
379C2×C22⋊C4—1668no—
380#103—28no—
381#103—1668no—
382#95—1664no—
383C24—832no—
384C24—834no—
385C24—1662no—
386C24—1662no—
387C24—1662no—
388C24—1662no—
389C24—1662no—
390C24—1662no—
391C24D4×D8311yescharacteristic
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8C2×C4×D8314yes—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—3322no—
414C22×C4—3322no—
415C22×C4—3322no—
416C22×C4—3322no—
417C22×C4—3322no—
418C22×C4—832no—
419C22×C4—834no—
420C22×C4—834no—
421C22×C4—834no—
422C22×C4—838no—
423C22×C4—1662no—
424C22×C4—1662no—
425C22×C4—1664no—
426C22×C4—1662no—
427C22×C4—1662no—
428C22×C4—1668no—
429C22×C4—1664no—
430C22×C4—1664no—
431C22×C4—1668no—
432C22×C4—1662no—
433C22×C4—1662no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4—1668no—
437C22×C4—1668no—
438C22×C4C2×C4×D8312yes—
439C22×C4D4×D8312yes—
440C22×C4D4×D8312yes—
441C22×C4D4×D8311yescharacteristic
442C22×C4C2×C22⋊D8312yes—
443C22×C4#1305058912yes—
444C22×C4#1305059012yes—
445C22×C4C23×D8311yescharacteristic
446M16—16632no—
447C2×C8—28no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—834no—
451C2×C8—1668no—
452C2×C8—1662no—
453C2×C8—1662no—
454C2×C8C2×C22⋊D8314yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—3322no—
459C4⋊C4—834no—
460C4⋊C4—16616no—
461C4⋊C4—1662no—
462C4⋊C4—1662no—
463C4⋊C4—1662no—
464C4⋊C4—1664no—
465C22⋊C4—3328no—
466C22⋊C4—16616no—
467C23—3324no—
468C23—3324no—
469C23—3324no—
470C23—3324no—
471C23—834no—
472C23—1668no—
473C23—1662no—
474C23—1662no—
475C23—1668no—
476C23—1668no—
477C23—1662no—
478C23—1662no—
479C23—1662no—
480C23—1662no—
481C23C22⋊C4×D8312yes—
482C23#1458176312yes—
483C23#1458176512yes—
484C23#1458181912yes—
485C23#1458182112yes—
486C23#1458182412yes—
487C23#1458182412yes—
488C23C2×D4×D8311yescharacteristic
489Q8—24no—
490Q8—24no—
491Q8—48no—
492Q8—42no—
493Q8—42no—
494C2×C4—24no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—3328no—
503C2×C4—3322no—
504C2×C4—3322no—
505C2×C4—3322no—
506C2×C4—3322no—
507C2×C4—3322no—
508C2×C4—834no—
509C2×C4—6644no—
510C2×C4—1664no—
511C2×C4—1668no—
512C2×C4—1662no—
513C2×C4—1662no—
514C2×C4—16616no—
515C2×C4—1664no—
516C2×C4—1664no—
517C2×C4—1664no—
518C2×C4—1664no—
519C2×C4C22⋊C4×D8312yes—
520C2×C4(C22⋊D83)⋊C414yes—
521C2×C4C4×C22⋊D8314yes—
522C2×C4#1458176014yes—
523C2×C4#1458176312yes—
524C2×C4#1458176512yes—
525C2×C4#1458177414yes—
526C2×C4C22×C4×D8311yescharacteristic
527C2×C4C2×D4×D8311yescharacteristic
528C2×C4C2×D4×D8311yescharacteristic
529C2×C4C22×C22⋊D8311yescharacteristic
530C2×C4C2×#1305058911yescharacteristic
531C2×C4C2×#1305059011yescharacteristic
532C8—24no—
533C8—1664no—
534C22—3322no—
535C22—3322no—
536C22—3322no—
537C22—3322no—
538C22—33216no—
539C22—1664no—
540C22—1664no—
541C22—1664no—
542C22#1912757714yes—
543C22#1912770114yes—
544C22#1912770814yes—
545C22#1912794414yes—
546C22#1912795014yes—
547C22#1912795614yes—
548C22#1912795814yes—
549C22C2×C22⋊C4×D8311yescharacteristic
550C22C2×#1458176311yescharacteristic
551C22C2×#1458176511yescharacteristic
552C22C2×#1458181911yescharacteristic
553C22C2×#1458182111yescharacteristic
554C22C2×#1458182411yescharacteristic
555C22C2×#1458182411yescharacteristic
556C4—44no—
557C4—82no—
558C4—3324no—
559C4—6642no—
560C4—1664no—
561C4#1912757714yes—
562C4C2×(C22⋊D83)⋊C411yescharacteristic
563C4C2×C4×C22⋊D8311yescharacteristic
564C4C2×#1458176011yescharacteristic
565C4C2×#1458177411yescharacteristic
566C2—3324no—
567C2#2490789318yesdirect product factor
568C2C2×#1912757711yescharacteristic
569C2C2×#1912770811yescharacteristic
570C2C2×#1912770111yescharacteristic
571C2C2×#1912794411yescharacteristic
572C2C2×#1912795011yescharacteristic
573C2C2×#1912795611yescharacteristic
574C2C2×#1912795811yescharacteristic
subgroupG/H
centerC24D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC22×#103×C83C2
socleC24×C83C2×D4
prime radicalC24×D83D4
cosocleC22×C4×C83C24
hypercenterC22×#103D83
hypocenterC83C2×#1894
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#14581716C2semidirect
5. #24907893C2direct (N × H)
6. C2×#19127464C2semidirect
7. C2×#19127463C2semidirect
8. C22×#103×C83C2semidirect
17. C2×#14581768C4semidirect
25. C2×#14581716C22semidirect
29. #19127464C22semidirect
30. #19127463C22semidirect
35. C2×#103×C83C22semidirect
58. #14581768C2×C4semidirect
279. C2×C83#1894semidirect
291. C22×#103D83semidirect
296. C83C2×#1894semidirect
318. C2×#103C2×D83semidirect
567. C2#24907893direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#130505732
C22×#145817682
C22×#145817162
C2×#191275582
#249078932
C2×#191274642
C2×#191274632
C22×#103×C832
C2×#189483