Finite Group Databasebeta

← C2×#24911760 #35573921

Subgroups — C2×#24911760 #35573921

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×#14581751C211yesmaximal, characteristic, complemented
2#24911760C2116yesmaximal, direct product factor
3C2×#19127629C212yesmaximal, complemented
4C2×#19127615C212yesmaximal
5C2×#19127599C211yesmaximal, characteristic, complemented
6C2×#19127599C211yesmaximal, characteristic, complemented
7C2×#19127589C212yesmaximal, complemented
8C2×#19127582C212yesmaximal, complemented
9C2×#19127561C211yesmaximal, characteristic, complemented
10C2×#19127561C211yesmaximal, characteristic, complemented
11C2×#19127561C211yesmaximal, characteristic, complemented
12C22×C4×D4×C83C211yesmaximal, characteristic
13C2×#14581751—28no—
14C2×#14581751C2218yescomplemented
15C22×C4⋊Dic83C2211yescharacteristic
16C22×C4⋊Dic83C2211yescharacteristic
17C22×C4⋊Dic83C2211yescharacteristic
18C22×C4⋊Dic83C2211yescharacteristic
19C22×Dic83⋊C4C2212yes—
20C22×Dic83⋊C4C2212yes—
21C22×C22⋊Dic83C2212yes—
22C22×C22⋊Dic83C2212yes—
23C22×C22⋊Dic83C2212yes—
24C22×C22⋊Dic83C2212yes—
25C22×C4×Dic83C2212yes—
26#19127629C22116yescomplemented
27#19127615C22116yes—
28#19127599C2218yescomplemented
29#19127599C2218yescomplemented
30#19127589C22116yescomplemented
31#19127582C22116yescomplemented
32#19127561C2218yescomplemented
33#19127561C2218yescomplemented
34#19127561C2218yescomplemented
35C2×C83⋊#95C2212yes—
36C2×C83⋊#95C2212yes—
37C2×C83⋊#95C2212yes—
38C2×C83⋊#95C2212yes—
39C2×C83⋊#95C2212yes—
40C23×D4×C83C2211yescharacteristic
41C24×C4×C83C2211yescharacteristic
42C24×C4×C83C2211yescharacteristic
43C2×C4×D4×C83—28no—
44C2×C4×D4×C83C2218yes—
45C22×C4⋊C4×C83C2211yescharacteristic
46C22×C22⋊C4×C83C2211yescharacteristic
47C22×C22⋊C4×C83C2211yescharacteristic
48C22×C42×C83C2211yescharacteristic
49C23×Dic83C2312yes—
50C23×Dic83C2312yes—
51C23×Dic83C2312yes—
52C23×Dic83C2312yes—
53#14581751—232no—
54C2×C4⋊Dic83—24no—
55C2×C4⋊Dic83—24no—
56C2×C4⋊Dic83—24no—
57C2×C4⋊Dic83—24no—
58C2×C4⋊Dic83C2314yes—
59C2×C4⋊Dic83C2314yes—
60C2×C4⋊Dic83C2314yes—
61C2×C4⋊Dic83C2314yes—
62C2×Dic83⋊C4—28no—
63C2×Dic83⋊C4—44no—
64C2×Dic83⋊C4C2318yes—
65C2×Dic83⋊C4C2318yes—
66C2×C22⋊Dic83—28no—
67C2×C22⋊Dic83—28no—
68C2×C22⋊Dic83—44no—
69C2×C22⋊Dic83—44no—
70C2×C22⋊Dic83C2318yes—
71C2×C22⋊Dic83C2318yes—
72C2×C22⋊Dic83C2318yes—
73C2×C22⋊Dic83C2318yes—
74C2×C4×Dic83—28no—
75C2×C4×Dic83C2318yes—
76C83⋊#95C2318yes—
77C83⋊#95C2318yes—
78C83⋊#95C2318yes—
79C83⋊#95C2318yes—
80C83⋊#95C2318yes—
81C25×C83C2311yescharacteristic
82C25×C83C2311yescharacteristic
83C22×D4×C83—24no—
84C22×D4×C83—24no—
85C22×D4×C83—24no—
86C22×D4×C83C2314yes—
87C23×C4×C83—24no—
88C23×C4×C83—24no—
89C23×C4×C83—24no—
90C23×C4×C83—24no—
91C23×C4×C83—24no—
92C23×C4×C83D418yes—
93C23×C4×C83C2314yes—
94C23×C4×C83C2311yescharacteristic
95C23×C4×C83C2311yescharacteristic
96C23×C4×C83C2314yes—
97C23×C4×C83C2311yescharacteristic
98C23×C4×C83C2311yescharacteristic
99C23×C4×C83C2311yescharacteristic
100C4×D4×C83—232no—
101C2×C4⋊C4×C83—24no—
102C2×C4⋊C4×C83C2314yes—
103C2×C22⋊C4×C83—24no—
104C2×C22⋊C4×C83—24no—
105C2×C22⋊C4×C83C2314yes—
106C2×C22⋊C4×C83C2314yes—
107C2×C42×C83—24no—
108C2×C42×C83C2314yes—
109C22×Dic83—24no—
110C22×Dic83—24no—
111C22×Dic83—24no—
112C22×Dic83—24no—
113C22×Dic83—24no—
114C22×Dic83—24no—
115C22×Dic83—24no—
116C22×Dic83—24no—
117C22×Dic83—24no—
118C22×Dic83—24no—
119C22×Dic83—24no—
120C22×Dic83—24no—
121C22×Dic83C2414yes—
122C22×Dic83C2414yes—
123C22×Dic83C2414yes—
124C22×Dic83C2414yes—
125C4⋊Dic83—28no—
126C4⋊Dic83—28no—
127C4⋊Dic83—28no—
128C4⋊Dic83—28no—
129Dic83⋊C4—216no—
130Dic83⋊C4—48no—
131C22⋊Dic83—216no—
132C22⋊Dic83—216no—
133C22⋊Dic83—48no—
134C22⋊Dic83—48no—
135C4×Dic83—216no—
136C24×C83—22no—
137C24×C83—22no—
138C24×C83—22no—
139C24×C83—22no—
140C24×C83—22no—
141C24×C83—22no—
142C24×C83—22no—
143C24×C83—22no—
144C24×C83—22no—
145C24×C83—22no—
146C24×C83—22no—
147C24×C83—22no—
148C24×C83C2×D414yes—
149C24×C83#4114yes—
150C24×C83C2412yes—
151C24×C83C2412yes—
152C24×C83C2411yescharacteristic
153C2×D4×C83—232no—
154C2×D4×C83—28no—
155C2×D4×C83—28no—
156C2×D4×C83—28no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—232no—
162C22×C4×C83—28no—
163C22×C4×C83—28no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83—28no—
171C22×C4×C83—28no—
172C22×C4×C83—28no—
173C22×C4×C83—416no—
174C22×C4×C83C2×D414yes—
175C22×C4×C83C2×D4116yes—
176C22×C4×C83C2×D414yes—
177C22×C4×C83#4114yes—
178C22×C4×C83#4114yes—
179C22×C4×C83#4114yes—
180C22×C4×C83C2412yes—
181C22×C4×C83C2412yes—
182C22×C4×C83C2412yes—
183C22×C4×C83C2412yes—
184C22×C4×C83C2412yes—
185C4⋊C4×C83—28no—
186C22⋊C4×C83—28no—
187C22⋊C4×C83—28no—
188C42×C83—28no—
189C23×C83—28no—
190C23×C83—28no—
191C23×C83—28no—
192C23×C83—28no—
193C23×C83—28no—
194C23×C83—28no—
195C23×C83—22no—
196C23×C83—22no—
197C23×C83—22no—
198C23×C83—22no—
199C23×C83—22no—
200C23×C83—22no—
201C23×C83—22no—
202C23×C83—22no—
203C23×C83—22no—
204C23×C83—22no—
205C23×C83—22no—
206C23×C83—22no—
207C23×C83—44no—
208C23×C83—44no—
209C23×C83—44no—
210C23×C83—44no—
211C23×C83—44no—
212C23×C83—44no—
213C23×C83—44no—
214C23×C83—44no—
215C23×C83C22×D414yes—
216C23×C83C22×D412yes—
217C23×C83C2×#4112yes—
218C23×C83C2×#4112yes—
219C23×C83C2×#4114yes—
220C23×C83C2×#4112yes—
221C23×C83C2×#4112yes—
222C23×C83H(5,𝔽2)12yes—
223C23×C832−1+412yes—
224C23×C83C2511yescharacteristic
225D4×C83—232no—
226C2×C4×C83—28no—
227C2×C4×C83—28no—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—232no—
233C2×C4×C83—22no—
234C2×C4×C83—22no—
235C2×C4×C83—22no—
236C2×C4×C83—22no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—44no—
240C2×C4×C83—44no—
241C2×C4×C83—416no—
242C2×C4×C83#123116yes—
243C2×C4×C83C22×D414yes—
244C2×C4×C83C22×D414yes—
245C2×C4×C83C2×#4114yes—
246C2×C4×C83C2×#4114yes—
247C2×C4×C83C2×#4114yes—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—24no—
254C2×Dic83—24no—
255C2×Dic83—24no—
256C2×Dic83—24no—
257C2×Dic83—24no—
258C2×Dic83—24no—
259C2×Dic83—24no—
260C2×Dic83—48no—
261C2×Dic83—48no—
262C2×Dic83—48no—
263C2×Dic83—48no—
264C22×C83—24no—
265C22×C83—24no—
266C22×C83—24no—
267C22×C83—24no—
268C22×C83—24no—
269C22×C83—24no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—42no—
273C22×C83—42no—
274C22×C83—42no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—42no—
278C22×C83—416no—
279C22×C83—416no—
280C22×C83C2×#12314yes—
281C22×C83#52514yes—
282C22×C83#53714yes—
283C22×C83#55214yes—
284C22×C83#55314yes—
285C22×C83#55514yes—
286C22×C83#55614yes—
287C22×C83C23×D411yescharacteristic
288C22×C83C22×#4111yescharacteristic
289C22×C83C22×#4111yescharacteristic
290C22×C83C22×#4111yescharacteristic
291C22×C83C22×#4111yescharacteristic
292C22×C83C2×H(5,𝔽2)11yescharacteristic
293C22×C83C2×2−1+411yescharacteristic
294C4×C83—24no—
295C4×C83—24no—
296C4×C83—42no—
297C4×C83—42no—
298C4×C83C2×#12318yes—
299Dic83—44no—
300Dic83—44no—
301Dic83—44no—
302Dic83—44no—
303C2×#2485—831nomaximal, Sylow
304C2×C83—44no—
305C2×C83—44no—
306C2×C83#248518yescomplemented
307C2×C83C22×#12311yescharacteristic
308C2×C83C2×#52511yescharacteristic
309C2×C83C2×#53711yescharacteristic
310C2×C83C2×#55211yescharacteristic
311C2×C83C2×#55311yescharacteristic
312C2×C83C2×#55511yescharacteristic
313C2×C83C2×#55611yescharacteristic
314C22×C4×D4D8311yescharacteristic
315C22×#117—831no—
316#2485—8316no—
317C2×#397—832no—
318C2×#394—832no—
319C2×#389—831no—
320C2×#389—831no—
321C2×#388—832no—
322C2×#387—832no—
323C2×#380—831no—
324C2×#380—831no—
325C2×#380—831no—
326C83C2×#248511yescharacteristic, complemented, Sylow
327C23×D4C2×D8311yescharacteristic
328C24×C4C2×D8311yescharacteristic
329C24×C4C2×D8311yescharacteristic
330C2×C4×D4—28no—
331C2×C4×D4C2×D8318yes—
332C2×#117—838no—
333C2×#117—1668no—
334C22×C4⋊C4—832no—
335C22×C4⋊C4—832no—
336C22×C4⋊C4—831no—
337C22×C4⋊C4—831no—
338C22×C4⋊C4—831no—
339C22×C4⋊C4—831no—
340C22×C4⋊C4C2×D8311yescharacteristic
341C22×C22⋊C4—832no—
342C22×C22⋊C4—832no—
343C22×C22⋊C4—832no—
344C22×C22⋊C4—832no—
345C22×C22⋊C4C2×D8311yescharacteristic
346C22×C22⋊C4C2×D8311yescharacteristic
347C22×C42—832no—
348C22×C42C2×D8311yescharacteristic
349#397—8316no—
350#394—8316no—
351#389—838no—
352#389—838no—
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×#95—832no—
363C25C22×D8311yescharacteristic
364C25C22×D8311yescharacteristic
365C22×D4—24no—
366C22×D4—24no—
367C22×D4—24no—
368C22×D4C22×D8314yes—
369C23×C4—24no—
370C23×C4—24no—
371C23×C4—24no—
372C23×C4—24no—
373C23×C4—24no—
374C23×C4—832no—
375C23×C4—832no—
376C23×C4—832no—
377C23×C4—832no—
378C23×C4C22⋊D8318yes—
379C23×C4C22×D8314yes—
380C23×C4C22×D8311yescharacteristic
381C23×C4C22×D8311yescharacteristic
382C23×C4C22×D8314yes—
383C23×C4C22×D8311yescharacteristic
384C23×C4C22×D8311yescharacteristic
385C23×C4C22×D8311yescharacteristic
386C4×D4—232no—
387#117—16632no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—3324no—
390C2×C4⋊C4—838no—
391C2×C4⋊C4—834no—
392C2×C4⋊C4—834no—
393C2×C4⋊C4—834no—
394C2×C4⋊C4—834no—
395C2×C4⋊C4—838no—
396C2×C4⋊C4—1664no—
397C2×C4⋊C4—1664no—
398C2×C4⋊C4—1664no—
399C2×C4⋊C4—1664no—
400C2×C4⋊C4—1668no—
401C2×C4⋊C4C22×D8314yes—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—3324no—
405C2×C22⋊C4—3324no—
406C2×C22⋊C4—838no—
407C2×C22⋊C4—838no—
408C2×C22⋊C4—838no—
409C2×C22⋊C4—838no—
410C2×C22⋊C4—1668no—
411C2×C22⋊C4—1668no—
412C2×C22⋊C4C22×D8314yes—
413C2×C22⋊C4C22×D8314yes—
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#95—838no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24C2×C22⋊D8314yes—
436C24#1305059014yes—
437C24C23×D8312yes—
438C24C23×D8312yes—
439C24C23×D8311yescharacteristic
440C2×D4—232no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—232no—
449C22×C4—28no—
450C22×C4—28no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—416no—
461C22×C4—834no—
462C22×C4—834no—
463C22×C4—834no—
464C22×C4—834no—
465C22×C4—1664no—
466C22×C4—1664no—
467C22×C4—1664no—
468C22×C4—1664no—
469C22×C4—1664no—
470C22×C4—1664no—
471C22×C4—1664no—
472C22×C4—1664no—
473C22×C4—1664no—
474C22×C4—1664no—
475C22×C4—1664no—
476C22×C4—1664no—
477C22×C4C2×C22⋊D8314yes—
478C22×C4C2×C22⋊D83116yes—
479C22×C4C2×C22⋊D8314yes—
480C22×C4#1305058914yes—
481C22×C4#1305058914yes—
482C22×C4#1305059014yes—
483C22×C4C23×D8312yes—
484C22×C4C23×D8312yes—
485C22×C4C23×D8312yes—
486C22×C4C23×D8312yes—
487C22×C4C23×D8312yes—
488C4⋊C4—28no—
489C4⋊C4—3328no—
490C4⋊C4—1668no—
491C4⋊C4—1668no—
492C4⋊C4—1668no—
493C4⋊C4—1668no—
494C4⋊C4—16616no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—3328no—
498C22⋊C4—3328no—
499C22⋊C4—16616no—
500C22⋊C4—16616no—
501C42—28no—
502C42—16616no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23C22×C22⋊D8314yes—
530C23C22×C22⋊D8312yes—
531C23C2×#1305058912yes—
532C23C2×#1305058912yes—
533C23C2×#1305059014yes—
534C23C2×#1305059012yes—
535C23C2×#1305059012yes—
536C23#1458183812yes—
537C23#1458183912yes—
538C23C24×D8311yescharacteristic
539D4—232no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—232no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—416no—
556C2×C4—3328no—
557C2×C4—3328no—
558C2×C4—3328no—
559C2×C4—3328no—
560C2×C4—1664no—
561C2×C4—1664no—
562C2×C4—1664no—
563C2×C4—1664no—
564C2×C4—1664no—
565C2×C4—1664no—
566C2×C4—1664no—
567C2×C4—1664no—
568C2×C4—1664no—
569C2×C4—1664no—
570C2×C4—1664no—
571C2×C4—1664no—
572C2×C4#14581773116yes—
573C2×C4C22×C22⋊D8314yes—
574C2×C4C22×C22⋊D8314yes—
575C2×C4C2×#1305058914yes—
576C2×C4C2×#1305058914yes—
577C2×C4C2×#1305059014yes—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—416no—
587C22—416no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22C2×#1458177314yes—
595C22#1912821214yes—
596C22#1912827914yes—
597C22#1912839114yes—
598C22#1912839714yes—
599C22#1912841614yes—
600C22#1912842114yes—
601C22C23×C22⋊D8311yescharacteristic
602C22C22×#1305058911yescharacteristic
603C22C22×#1305058911yescharacteristic
604C22C22×#1305059011yescharacteristic
605C22C22×#1305059011yescharacteristic
606C22C2×#1458183811yescharacteristic
607C22C2×#1458183911yescharacteristic
608C4—24no—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—3324no—
613C4—3324no—
614C4—3324no—
615C4—3324no—
616C4C2×#1458177318yes—
617C2—44no—
618C2—44no—
619C2#2491176018yesdirect product factor
620C2C22×#1458177311yescharacteristic
621C2C2×#1912821211yescharacteristic
622C2C2×#1912827911yescharacteristic
623C2C2×#1912839111yescharacteristic
624C2C2×#1912839711yescharacteristic
625C2C2×#1912841611yescharacteristic
626C2C2×#1912842111yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC22×C4×D4×C83C2
socleC24×C83C24
prime radicalC23×D4×C83C22
hypercenterC22×C4×D4D83
hypocenterC83C2×#2485
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14581751C2semidirect
2. #24911760C2direct (N × H)
3. C2×#19127629C2semidirect
5. C2×#19127599C2semidirect
6. C2×#19127599C2semidirect
7. C2×#19127589C2semidirect
8. C2×#19127582C2semidirect
9. C2×#19127561C2semidirect
10. C2×#19127561C2semidirect
11. C2×#19127561C2semidirect
14. C2×#14581751C22semidirect
26. #19127629C22semidirect
28. #19127599C22semidirect
29. #19127599C22semidirect
30. #19127589C22semidirect
31. #19127582C22semidirect
32. #19127561C22semidirect
33. #19127561C22semidirect
34. #19127561C22semidirect
306. C2×C83#2485semidirect
326. C83C2×#2485semidirect
619. C2#24911760direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817512
#249117602
C2×#191276292
C2×#191276152
C2×#191275992
C2×#191275892
C2×#191275822
C2×#191275612
C22×C4×D4×C832
C2×#248583