Finite Group Databasebeta

← C2×#24911763 #35573924

Subgroups — C2×#24911763 #35573924

Order 21248.

3070 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D332C211yesmaximal, characteristic
2C22×#14581750C211yesmaximal, characteristic, complemented
3#24911763C2116yesmaximal, direct product factor
4C2×#19127627C212yesmaximal, complemented
5C2×#19127619C212yesmaximal
6C2×#19127599C211yesmaximal, characteristic, complemented
7C2×#19127591C212yesmaximal, complemented
8C2×#19127588C212yesmaximal, complemented
9C2×#19127558C211yesmaximal, characteristic, complemented
10C2×#19127558C211yesmaximal, characteristic, complemented
11C2×#19127558C211yesmaximal, characteristic, complemented
12C2×#389×C83C211yesmaximal, characteristic, complemented
13C23×D332C2211yescharacteristic
14C23×C4×D83C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16C2×C4×D332—28no—
17C2×C4×D332C2218yes—
18C2×#14581750—28no—
19C2×#14581750C2218yescomplemented
20C22×C4⋊Dic83C2211yescharacteristic
21C22×C4⋊Dic83C2211yescharacteristic
22C22×C4⋊Dic83C2211yescharacteristic
23C22×Dic83⋊C4C2212yes—
24C22×(C2×D83)⋊C4C2212yes—
25C22×(C2×D83)⋊C4C2212yes—
26C22×(C2×D83)⋊C4C2211yescharacteristic
27C22×(C2×D83)⋊C4C2211yescharacteristic
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2212yes—
30C22×C4×Dic83C2212yes—
31#19127627C22116yescomplemented
32#19127619C22116yes—
33#19127599C2218yescomplemented
34#19127591C22116yescomplemented
35#19127588C22116yescomplemented
36#19127558C2218yescomplemented
37#19127558C2218yescomplemented
38#19127558C2218yescomplemented
39C2×C83⋊#95C2212yes—
40C2×C83⋊#95C2212yes—
41C2×C83⋊#95C2212yes—
42C2×C83⋊#95C2212yes—
43C22×C4⋊C4×C83C2211yescharacteristic
44C22×C4⋊C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2212yes—
46C22×C42×C83C2211yescharacteristic
47#389×C83C2218yescomplemented
48C2×#95×C83C2212yes—
49C24×D83C2311yescharacteristic
50C24×D83C2311yescharacteristic
51C22×D332—24no—
52C22×D332—24no—
53C22×D332—24no—
54C22×D332C2314yes—
55C23×Dic83C2312yes—
56C23×Dic83C2312yes—
57C23×Dic83C2311yescharacteristic
58C23×Dic83C2311yescharacteristic
59C22×C4×D83—24no—
60C22×C4×D83—24no—
61C22×C4×D83—24no—
62C22×C4×D83—24no—
63C22×C4×D83—24no—
64C22×C4×D83D418yes—
65C22×C4×D83C2314yes—
66C22×C4×D83C2314yes—
67C4×D332—232no—
68#14581750—232no—
69C2×C4⋊Dic83—24no—
70C2×C4⋊Dic83—24no—
71C2×C4⋊Dic83—24no—
72C2×C4⋊Dic83C2314yes—
73C2×C4⋊Dic83C2314yes—
74C2×C4⋊Dic83C2314yes—
75C2×Dic83⋊C4—28no—
76C2×Dic83⋊C4C2318yes—
77C2×(C2×D83)⋊C4—28no—
78C2×(C2×D83)⋊C4—24no—
79C2×(C2×D83)⋊C4—28no—
80C2×(C2×D83)⋊C4—24no—
81C2×(C2×D83)⋊C4—44no—
82C2×(C2×D83)⋊C4—44no—
83C2×(C2×D83)⋊C4C2318yes—
84C2×(C2×D83)⋊C4C2314yes—
85C2×(C2×D83)⋊C4C2318yes—
86C2×(C2×D83)⋊C4C2314yes—
87C2×(C2×D83)⋊C4C2318yes—
88C2×(C2×D83)⋊C4C2318yes—
89C2×C4×Dic83—28no—
90C2×C4×Dic83C2318yes—
91C83⋊#95C2318yes—
92C83⋊#95C2318yes—
93C83⋊#95C2318yes—
94C83⋊#95C2318yes—
95C23×C4×C83C2311yescharacteristic
96C23×C4×C83C2312yes—
97C23×C4×C83C2312yes—
98C23×C4×C83C2311yescharacteristic
99C23×C4×C83C2311yescharacteristic
100C2×C4⋊C4×C83—24no—
101C2×C4⋊C4×C83—24no—
102C2×C4⋊C4×C83—44no—
103C2×C4⋊C4×C83C2314yes—
104C2×C4⋊C4×C83C2314yes—
105C2×C4⋊C4×C83C2318yes—
106C2×C42×C83—24no—
107C2×C42×C83C2314yes—
108#95×C83C2318yes—
109C23×D83—22no—
110C23×D83—22no—
111C23×D83—22no—
112C23×D83—22no—
113C23×D83—22no—
114C23×D83—22no—
115C23×D83—22no—
116C23×D83—22no—
117C23×D83—22no—
118C23×D83—22no—
119C23×D83—22no—
120C23×D83—22no—
121C23×D83C2×D414yes—
122C23×D83#4114yes—
123C23×D83C2412yes—
124C23×D83C2412yes—
125C2×D332—232no—
126C2×D332—28no—
127C2×D332—28no—
128C2×D332—28no—
129C22×Dic83—22no—
130C22×Dic83—22no—
131C22×Dic83—22no—
132C22×Dic83—24no—
133C22×Dic83—22no—
134C22×Dic83—22no—
135C22×Dic83—24no—
136C22×Dic83—24no—
137C22×Dic83—24no—
138C22×Dic83—24no—
139C22×Dic83—24no—
140C22×Dic83C2×D414yes—
141C22×Dic83C2412yes—
142C22×Dic83C2414yes—
143C22×Dic83C2414yes—
144C22×Dic83C2412yes—
145C2×C4×D83—232no—
146C2×C4×D83—28no—
147C2×C4×D83—28no—
148C2×C4×D83—28no—
149C2×C4×D83—28no—
150C2×C4×D83—28no—
151C2×C4×D83—416no—
152C2×C4×D83C2×D4116yes—
153C4⋊Dic83—28no—
154C4⋊Dic83—28no—
155C4⋊Dic83—28no—
156Dic83⋊C4—216no—
157(C2×D83)⋊C4—216no—
158(C2×D83)⋊C4—28no—
159(C2×D83)⋊C4—216no—
160(C2×D83)⋊C4—28no—
161(C2×D83)⋊C4—48no—
162(C2×D83)⋊C4—48no—
163C4×Dic83—216no—
164C24×C83C2411yescharacteristic
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—24no—
169C22×C4×C83—24no—
170C22×C4×C83—24no—
171C22×C4×C83—22no—
172C22×C4×C83—24no—
173C22×C4×C83—24no—
174C22×C4×C83—22no—
175C22×C4×C83—24no—
176C22×C4×C83C2×D414yes—
177C22×C4×C83#4114yes—
178C22×C4×C83#4114yes—
179C22×C4×C83#4114yes—
180C22×C4×C83C2414yes—
181C22×C4×C83C2414yes—
182C22×C4×C83C2412yes—
183C22×C4×C83C2412yes—
184C22×C4×C83C2412yes—
185C4⋊C4×C83—28no—
186C4⋊C4×C83—28no—
187C4⋊C4×C83—48no—
188C42×C83—28no—
189C23×C83C22×D412yes—
190C23×C83C2×#4112yes—
191C23×C83C2×#4112yes—
192C23×C83C2×#4112yes—
193C23×C83C2×#4112yes—
194C23×C83H(5,𝔽2)12yes—
195C23×C832−1+412yes—
196C23×C83C2511yescharacteristic
197C22×D83—28no—
198C22×D83—28no—
199C22×D83—28no—
200C22×D83—28no—
201C22×D83—28no—
202C22×D83—28no—
203C22×D83—22no—
204C22×D83—22no—
205C22×D83—22no—
206C22×D83—22no—
207C22×D83—22no—
208C22×D83—22no—
209C22×D83—22no—
210C22×D83—22no—
211C22×D83—22no—
212C22×D83—22no—
213C22×D83—22no—
214C22×D83—22no—
215C22×D83—44no—
216C22×D83—44no—
217C22×D83—44no—
218C22×D83—44no—
219C22×D83—44no—
220C22×D83—44no—
221C22×D83—44no—
222C22×D83—44no—
223C22×D83C22×D414yes—
224C22×D83C2×#4114yes—
225C2×C4×C83—28no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83—22no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—24no—
236C2×C4×C83—22no—
237C2×C4×C83—44no—
238C2×C4×C83—48no—
239C2×C4×C83—48no—
240C2×C4×C83#123116yes—
241C2×C4×C83C22×D414yes—
242C2×C4×C83C2×#4114yes—
243C2×C4×C83C2×#4114yes—
244C2×C4×C83C2×#4114yes—
245C2×Dic83—28no—
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—24no—
249C2×Dic83—22no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—24no—
254C2×Dic83—22no—
255C2×Dic83—22no—
256C2×Dic83—24no—
257C2×Dic83—44no—
258C2×Dic83—48no—
259C2×Dic83—48no—
260C2×Dic83C22×D414yes—
261D332—232no—
262C4×D83—232no—
263C4×D83—416no—
264C22×C83C2×#12314yes—
265C22×C83#52514yes—
266C22×C83#53714yes—
267C22×C83#55214yes—
268C22×C83#55314yes—
269C22×C83#55514yes—
270C22×C83#55614yes—
271C22×C83C23×D411yescharacteristic
272C22×C83C22×#4111yescharacteristic
273C22×C83C22×#4111yescharacteristic
274C22×C83C22×#4111yescharacteristic
275C22×C83C22×#4111yescharacteristic
276C22×C83C2×H(5,𝔽2)11yescharacteristic
277C22×C83C2×2−1+411yescharacteristic
278C2×D83—24no—
279C2×D83—24no—
280C2×D83—24no—
281C2×D83—24no—
282C2×D83—24no—
283C2×D83—24no—
284C2×D83—42no—
285C2×D83—42no—
286C2×D83—42no—
287C2×D83—42no—
288C2×D83—42no—
289C2×D83—42no—
290C2×D83—42no—
291C2×D83—42no—
292C2×D83—416no—
293C2×D83—416no—
294C4×C83—24no—
295C4×C83—42no—
296C4×C83—44no—
297C4×C83—44no—
298C4×C83C2×#12318yes—
299Dic83—24no—
300Dic83—42no—
301Dic83—44no—
302Dic83—44no—
303C2×#2485—831nomaximal, Sylow
304C2×C83#248518yescomplemented
305C2×C83C22×#12311yescharacteristic
306C2×C83C2×#52511yescharacteristic
307C2×C83C2×#53711yescharacteristic
308C2×C83C2×#55211yescharacteristic
309C2×C83C2×#55311yescharacteristic
310C2×C83C2×#55511yescharacteristic
311C2×C83C2×#55611yescharacteristic
312D83—44no—
313D83—44no—
314C22×C4×D4—831no—
315C22×#117—831no—
316#2485—8316no—
317C2×#397—832no—
318C2×#394—832no—
319C2×#389—831no—
320C2×#389D8311yescharacteristic, complemented
321C2×#388—832no—
322C2×#387—832no—
323C2×#380—831no—
324C2×#380—831no—
325C2×#380—831no—
326C83C2×#248511yescharacteristic, complemented, Sylow
327C23×D4—831no—
328C24×C4—831no—
329C24×C4—831no—
330C2×C4×D4—838no—
331C2×C4×D4—1668no—
332C2×#117—838no—
333C2×#117—1668no—
334C22×C4⋊C4—832no—
335C22×C4⋊C4—831no—
336C22×C4⋊C4—831no—
337C22×C4⋊C4—831no—
338C22×C4⋊C4C2×D8311yescharacteristic
339C22×C4⋊C4C2×D8311yescharacteristic
340C22×C4⋊C4C2×D8312yes—
341C22×C22⋊C4—832no—
342C22×C22⋊C4—832no—
343C22×C22⋊C4—832no—
344C22×C22⋊C4—832no—
345C22×C22⋊C4—831no—
346C22×C22⋊C4—831no—
347C22×C42—832no—
348C22×C42C2×D8311yescharacteristic
349#397—8316no—
350#394—8316no—
351#389—838no—
352#389C2×D8318yescomplemented
353#388—8316no—
354#387—8316no—
355#380—838no—
356#380—838no—
357#380—838no—
358C2×#95—832no—
359C2×#95—832no—
360C2×#95—832no—
361C2×#95—832no—
362C2×#95C2×D8312yes—
363C25—831no—
364C25—831no—
365C22×D4—834no—
366C22×D4—1664no—
367C22×D4—1664no—
368C22×D4—1664no—
369C23×C4—831no—
370C23×C4—838no—
371C23×C4—834no—
372C23×C4—832no—
373C23×C4—832no—
374C23×C4—831no—
375C23×C4—834no—
376C23×C4—1664no—
377C23×C4—1664no—
378C23×C4—1664no—
379C23×C4—1664no—
380C23×C4—1664no—
381C23×C4C22×D8312yes—
382C23×C4C22×D8311yescharacteristic
383C23×C4C22×D8312yes—
384C23×C4C22×D8311yescharacteristic
385C23×C4C22×D8311yescharacteristic
386C4×D4—16632no—
387#117—16632no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—44no—
391C2×C4⋊C4—834no—
392C2×C4⋊C4—834no—
393C2×C4⋊C4—834no—
394C2×C4⋊C4—838no—
395C2×C4⋊C4—1664no—
396C2×C4⋊C4—1664no—
397C2×C4⋊C4—1668no—
398C2×C4⋊C4—1664no—
399C2×C4⋊C4C22×D8314yes—
400C2×C4⋊C4C22×D8314yes—
401C2×C4⋊C4C22×D8318yes—
402C2×C22⋊C4—3324no—
403C2×C22⋊C4—3324no—
404C2×C22⋊C4—838no—
405C2×C22⋊C4—838no—
406C2×C22⋊C4—838no—
407C2×C22⋊C4—834no—
408C2×C22⋊C4—834no—
409C2×C22⋊C4—838no—
410C2×C22⋊C4—1664no—
411C2×C22⋊C4—1668no—
412C2×C22⋊C4—1664no—
413C2×C22⋊C4—1668no—
414C2×C42—24no—
415C2×C42—838no—
416C2×C42—1668no—
417C2×C42C22×D8314yes—
418#95—838no—
419#95—838no—
420#95—838no—
421#95—838no—
422#95C22×D8318yes—
423C24—834no—
424C24—834no—
425C24—832no—
426C24—832no—
427C24—1662no—
428C24—1662no—
429C24—1662no—
430C24—1662no—
431C24—1662no—
432C24—1662no—
433C24—1662no—
434C24—1662no—
435C24—1662no—
436C24—1662no—
437C24—1662no—
438C24—1662no—
439C24C23×D8311yescharacteristic
440C2×D4—1668no—
441C2×D4—1668no—
442C2×D4—16632no—
443C2×D4—1668no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—33216no—
456C22×C4—834no—
457C22×C4—832no—
458C22×C4—8316no—
459C22×C4—832no—
460C22×C4—834no—
461C22×C4—834no—
462C22×C4—1664no—
463C22×C4—1662no—
464C22×C4—1664no—
465C22×C4—1662no—
466C22×C4—1664no—
467C22×C4—1664no—
468C22×C4—1664no—
469C22×C4—1662no—
470C22×C4—1664no—
471C22×C4—1668no—
472C22×C4—1668no—
473C22×C4—1668no—
474C22×C4—1662no—
475C22×C4—1662no—
476C22×C4—16632no—
477C22×C4—1668no—
478C22×C4—1668no—
479C22×C4D4×D8314yes—
480C22×C4#1305059014yes—
481C22×C4#1305059114yes—
482C22×C4#1305059114yes—
483C22×C4C23×D8314yes—
484C22×C4C23×D8312yes—
485C22×C4C23×D8314yes—
486C22×C4C23×D8312yes—
487C22×C4C23×D8312yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—48no—
491C4⋊C4—1668no—
492C4⋊C4—1668no—
493C4⋊C4—16616no—
494C4⋊C4—1668no—
495C22⋊C4—3328no—
496C22⋊C4—3328no—
497C22⋊C4—1668no—
498C22⋊C4—16616no—
499C22⋊C4—1668no—
500C22⋊C4—16616no—
501C42—28no—
502C42—16616no—
503C23—3324no—
504C23—3324no—
505C23—3324no—
506C23—3324no—
507C23—3324no—
508C23—3324no—
509C23—3324no—
510C23—3324no—
511C23—834no—
512C23—834no—
513C23—1668no—
514C23—1668no—
515C23—1668no—
516C23—1668no—
517C23—1668no—
518C23—1662no—
519C23—1662no—
520C23—1662no—
521C23—1662no—
522C23—1662no—
523C23—1662no—
524C23—1662no—
525C23—1662no—
526C23—1662no—
527C23—1662no—
528C23—1662no—
529C23—1662no—
530C23—1668no—
531C23C2×D4×D8312yes—
532C23#41×D8312yes—
533C23C2×#1305059012yes—
534C23C2×#1305059112yes—
535C23C2×#1305059112yes—
536C23#1458183712yes—
537C23#1458183912yes—
538C23C24×D8311yescharacteristic
539D4—16632no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—3328no—
556C2×C4—3324no—
557C2×C4—3328no—
558C2×C4—33216no—
559C2×C4—834no—
560C2×C4—1664no—
561C2×C4—1662no—
562C2×C4—1664no—
563C2×C4—1662no—
564C2×C4—1664no—
565C2×C4—1664no—
566C2×C4—1664no—
567C2×C4—1662no—
568C2×C4—1664no—
569C2×C4—1668no—
570C2×C4—1662no—
571C2×C4—1662no—
572C2×C4—16632no—
573C2×C4#14581777116yes—
574C2×C4C2×D4×D8314yes—
575C2×C4C2×#1305059014yes—
576C2×C4C2×#1305059114yes—
577C2×C4C2×#1305059114yes—
578C22—3322no—
579C22—3322no—
580C22—33216no—
581C22—33216no—
582C22—3322no—
583C22—3322no—
584C22—3322no—
585C22—3322no—
586C22—3322no—
587C22—3322no—
588C22—1664no—
589C22—1664no—
590C22—1664no—
591C22—1664no—
592C22—1664no—
593C22—1664no—
594C22C2×#1458177714yes—
595C22#1912821414yes—
596C22#1912828014yes—
597C22#1912838914yes—
598C22#1912840114yes—
599C22#1912841714yes—
600C22#1912842414yes—
601C22C22×D4×D8311yescharacteristic
602C22C2×#41×D8311yescharacteristic
603C22C22×#1305059011yescharacteristic
604C22C22×#1305059111yescharacteristic
605C22C22×#1305059111yescharacteristic
606C22C2×#1458183711yescharacteristic
607C22C2×#1458183911yescharacteristic
608C4—24no—
609C4—44no—
610C4—42no—
611C4—44no—
612C4—3322no—
613C4—3324no—
614C4—3324no—
615C4—1664no—
616C4C2×#1458177718yes—
617C2—3324no—
618C2—3324no—
619C2#2491176318yesdirect product factor
620C2C22×#1458177711yescharacteristic
621C2C2×#1912821411yescharacteristic
622C2C2×#1912828011yescharacteristic
623C2C2×#1912838911yescharacteristic
624C2C2×#1912840111yescharacteristic
625C2C2×#1912841711yescharacteristic
626C2C2×#1912842411yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#389×C83C2
socleC24×C83C24
prime radicalC23×D332C22
hypercenterC2×#389D83
hypocenterC83C2×#2485
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C22×#14581750C2semidirect
3. #24911763C2direct (N × H)
4. C2×#19127627C2semidirect
6. C2×#19127599C2semidirect
7. C2×#19127591C2semidirect
8. C2×#19127588C2semidirect
9. C2×#19127558C2semidirect
10. C2×#19127558C2semidirect
11. C2×#19127558C2semidirect
12. C2×#389×C83C2semidirect
19. C2×#14581750C22semidirect
31. #19127627C22semidirect
33. #19127599C22semidirect
34. #19127591C22semidirect
35. #19127588C22semidirect
36. #19127558C22semidirect
37. #19127558C22semidirect
38. #19127558C22semidirect
47. #389×C83C22semidirect
304. C2×C83#2485semidirect
320. C2×#389D83semidirect
326. C83C2×#2485semidirect
352. #389C2×D83semidirect
619. C2#24911763direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D3322
C22×#145817502
#249117632
C2×#191276272
C2×#191276192
C2×#191275992
C2×#191275912
C2×#191275882
C2×#191275582
C2×#389×C832
C2×#248583