Finite Group Databasebeta

← C2×#24913206 #35570768

Subgroups — C2×#24913206 #35570768

Order 21248.

3718 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581777C212yesmaximal, complemented
2C22×#14581767C211yesmaximal, characteristic
3C22×C4⋊C4×D83C211yesmaximal, characteristic, complemented
4#24913206C2116yesmaximal, direct product factor
5C2×#19127632C211yesmaximal, characteristic, complemented
6C2×#19127625C212yesmaximal, complemented
7C2×#19127623C212yesmaximal, complemented
8C2×#19127619C211yesmaximal, characteristic, complemented
9C2×#19127618C211yesmaximal, characteristic, complemented
10C2×#19127579C212yesmaximal, complemented
11C2×#19127560C211yesmaximal, characteristic, complemented
12C2×#398×C83C211yesmaximal, characteristic, complemented
13C23×D332C2212yes—
14C23×D332C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16C23×C4×D83C2211yescharacteristic
17C23×C4×D83C2211yescharacteristic
18C2×#14581777—216no—
19C2×#14581777C22116yescomplemented
20C2×#14581767—28no—
21C2×#14581767C2218yes—
22C22×C4⋊Dic83C2211yescharacteristic
23C22×Dic83⋊C4C2211yescharacteristic
24C22×Dic83⋊C4C2211yescharacteristic
25C22×Dic83⋊C4C2212yes—
26C22×Dic83⋊C4C2212yes—
27C2×C4⋊C4×D83—28no—
28C2×C4⋊C4×D83C2218yescomplemented
29C22×(C2×D83)⋊C4C2212yes—
30C22×(C2×D83)⋊C4C2212yes—
31C22×(C2×D83)⋊C4C2212yes—
32C22×(C2×D83)⋊C4C2212yes—
33C22×(C2×D83)⋊C4C2212yes—
34C22×(C2×D83)⋊C4C2212yes—
35#19127632C2218yescomplemented
36#19127625C22116yescomplemented
37#19127623C22116yescomplemented
38#19127619C2218yescomplemented
39#19127618C2218yescomplemented
40#19127579C22116yescomplemented
41#19127560C2218yescomplemented
42C2×C83⋊#95C2211yescharacteristic
43C2×C83⋊#95C2212yes—
44C22×C4⋊C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2212yes—
46C22×C4⋊C4×C83C2211yescharacteristic
47#398×C83C2218yescomplemented
48C2×#95×C83C2211yescharacteristic
49C2×#95×C83C2212yes—
50C24×D83C2312yes—
51C24×D83C2311yescharacteristic
52C22×D332—28no—
53C22×D332—28no—
54C22×D332—28no—
55C22×D332—24no—
56C22×D332—44no—
57C22×D332C2318yes—
58C22×D332C2314yes—
59C23×Dic83C2312yes—
60C23×Dic83C2311yescharacteristic
61C23×Dic83C2311yescharacteristic
62C23×Dic83C2311yescharacteristic
63C22×C4×D83—24no—
64C22×C4×D83—24no—
65C22×C4×D83—24no—
66C22×C4×D83—44no—
67C22×C4×D83—44no—
68C22×C4×D83D418yes—
69C22×C4×D83D418yescomplemented
70C22×C4×D83C2314yes—
71C22×C4×D83C2314yes—
72C22×C4×D83C2314yes—
73#14581777—264no—
74#14581767—232no—
75C2×C4⋊Dic83—24no—
76C2×C4⋊Dic83C2314yes—
77C2×Dic83⋊C4—24no—
78C2×Dic83⋊C4—28no—
79C2×Dic83⋊C4—24no—
80C2×Dic83⋊C4—44no—
81C2×Dic83⋊C4C2314yes—
82C2×Dic83⋊C4C2314yes—
83C2×Dic83⋊C4C2318yes—
84C2×Dic83⋊C4C2318yes—
85C4⋊C4×D83—232no—
86C2×(C2×D83)⋊C4—28no—
87C2×(C2×D83)⋊C4—28no—
88C2×(C2×D83)⋊C4—28no—
89C2×(C2×D83)⋊C4—28no—
90C2×(C2×D83)⋊C4—44no—
91C2×(C2×D83)⋊C4—44no—
92C2×(C2×D83)⋊C4C2318yes—
93C2×(C2×D83)⋊C4C2318yes—
94C2×(C2×D83)⋊C4C2318yes—
95C2×(C2×D83)⋊C4C2318yes—
96C2×(C2×D83)⋊C4C2318yes—
97C2×(C2×D83)⋊C4C2318yes—
98C83⋊#95C2314yes—
99C83⋊#95C2318yes—
100C23×C4×C83C2312yes—
101C23×C4×C83C2312yes—
102C23×C4×C83C2311yescharacteristic
103C23×C4×C83C2311yescharacteristic
104C23×C4×C83C2311yescharacteristic
105C2×C4⋊C4×C83—24no—
106C2×C4⋊C4×C83—24no—
107C2×C4⋊C4×C83—28no—
108C2×C4⋊C4×C83C2314yes—
109C2×C4⋊C4×C83C2318yes—
110C2×C4⋊C4×C83C2314yes—
111#95×C83C2314yes—
112#95×C83C2318yes—
113C23×D83—24no—
114C23×D83—24no—
115C23×D83—24no—
116C23×D83—24no—
117C23×D83—24no—
118C23×D83—24no—
119C23×D83—24no—
120C23×D83—24no—
121C23×D83—24no—
122C23×D83C2×D414yes—
123C23×D83C2×D414yes—
124C23×D83#4114yes—
125C23×D83C2414yes—
126C23×D83C2412yes—
127C2×D332—216no—
128C2×D332—216no—
129C2×D332—216no—
130C2×D332—28no—
131C2×D332—48no—
132C2×D332—416no—
133C2×D332—432no—
134C22×Dic83—22no—
135C22×Dic83—22no—
136C22×Dic83—24no—
137C22×Dic83—22no—
138C22×Dic83—24no—
139C22×Dic83—24no—
140C22×Dic83—24no—
141C22×Dic83—24no—
142C22×Dic83C2×D414yes—
143C22×Dic83C2×D414yes—
144C22×Dic83C2412yes—
145C22×Dic83C2414yes—
146C22×Dic83C2412yes—
147C22×Dic83C2412yes—
148C2×C4×D83—232no—
149C2×C4×D83—28no—
150C2×C4×D83—28no—
151C2×C4×D83—28no—
152C2×C4×D83—416no—
153C2×C4×D83—48no—
154C2×C4×D83—48no—
155C2×C4×D83—416no—
156C2×C4×D83C2×D4116yes—
157C2×C4×D83C2×D4116yescomplemented
158C4⋊Dic83—28no—
159Dic83⋊C4—28no—
160Dic83⋊C4—216no—
161Dic83⋊C4—28no—
162Dic83⋊C4—48no—
163(C2×D83)⋊C4—216no—
164(C2×D83)⋊C4—216no—
165(C2×D83)⋊C4—216no—
166(C2×D83)⋊C4—216no—
167(C2×D83)⋊C4—48no—
168(C2×D83)⋊C4—48no—
169C24×C83C2411yescharacteristic
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—24no—
173C22×C4×C83—22no—
174C22×C4×C83—24no—
175C22×C4×C83—24no—
176C22×C4×C83—24no—
177C22×C4×C83—24no—
178C22×C4×C83—24no—
179C22×C4×C83—24no—
180C22×C4×C83—24no—
181C22×C4×C83C2×D414yes—
182C22×C4×C83C2×D414yes—
183C22×C4×C83C2412yes—
184C22×C4×C83C2414yes—
185C22×C4×C83C2414yes—
186C22×C4×C83C2412yes—
187C22×C4×C83C2412yes—
188C4⋊C4×C83—28no—
189C4⋊C4×C83—216no—
190C4⋊C4×C83—28no—
191C23×C83C22×D412yes—
192C23×C83C22×D412yes—
193C23×C83C2×#4112yes—
194C23×C83H(5,𝔽2)14yes—
195C23×C832−1+414yes—
196C23×C83C2511yescharacteristic
197C22×D83—28no—
198C22×D83—28no—
199C22×D83—216no—
200C22×D83—216no—
201C22×D83—24no—
202C22×D83—24no—
203C22×D83—24no—
204C22×D83—24no—
205C22×D83—24no—
206C22×D83—24no—
207C22×D83—24no—
208C22×D83—24no—
209C22×D83—24no—
210C22×D83—48no—
211C22×D83—48no—
212C22×D83—48no—
213C22×D83—48no—
214C22×D83—48no—
215C22×D83—48no—
216C22×D83—48no—
217C22×D83#121116yes—
218C22×D83C22×D414yes—
219C22×D83C22×D414yes—
220C22×D83C2×#4114yes—
221C2×C4×C83—28no—
222C2×C4×C83—28no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—24no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—22no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83—24no—
233C2×C4×C83—24no—
234C2×C4×C83—48no—
235C2×C4×C83—44no—
236C2×C4×C83—48no—
237C2×C4×C83C22×D414yes—
238C2×C4×C83C22×D414yes—
239C2×Dic83—28no—
240C2×Dic83—28no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—24no—
244C2×Dic83—24no—
245C2×Dic83—22no—
246C2×Dic83—24no—
247C2×Dic83—24no—
248C2×Dic83—24no—
249C2×Dic83—44no—
250C2×Dic83—48no—
251C2×Dic83C22×D414yes—
252C2×Dic83C22×D414yes—
253D332—416no—
254D332—432no—
255C4×D83—232no—
256C4×D83—416no—
257C4×D83—416no—
258C22×C83C2×#12114yes—
259C22×C83#53414yes—
260C22×C83#53614yes—
261C22×C83#53718yes—
262C22×C83#53918yes—
263C22×C83C23×D411yescharacteristic
264C22×C83C23×D411yescharacteristic
265C22×C83C22×#4111yescharacteristic
266C22×C83C2×H(5,𝔽2)12yes—
267C22×C83C2×2−1+412yes—
268C2×D83—232no—
269C2×D83—24no—
270C2×D83—24no—
271C2×D83—28no—
272C2×D83—28no—
273C2×D83—44no—
274C2×D83—44no—
275C2×D83—44no—
276C2×D83—44no—
277C2×D83—44no—
278C2×D83—44no—
279C2×D83—44no—
280C2×D83—816no—
281C2×D83C2×#12118yes—
282C4×C83—24no—
283C4×C83—24no—
284C4×C83—44no—
285C4×C83—42no—
286C4×C83—44no—
287Dic83—24no—
288Dic83—24no—
289Dic83—42no—
290Dic83—44no—
291C2×#2618—831nomaximal, Sylow
292C2×C83#261818yescomplemented
293C2×C83C22×#12111yescharacteristic
294C2×C83C2×#53411yescharacteristic
295C2×C83C2×#53611yescharacteristic
296C2×C83C2×#53712yes—
297C2×C83C2×#53912yes—
298D83—28no—
299D83—84no—
300C22×#123—832no—
301C22×#121—831no—
302C23×C4⋊C4—831no—
303#2618—8316no—
304C2×#398—831no—
305C2×#398D8311yescharacteristic, complemented
306C2×#396—832no—
307C2×#396—832no—
308C2×#394—831no—
309C2×#394—831no—
310C2×#386—832no—
311C2×#380—831no—
312C83C2×#261811yescharacteristic, complemented, Sylow
313C23×D4—831no—
314C23×D4—832no—
315C24×C4—831no—
316C24×C4—831no—
317C24×C4—831no—
318C2×#123—8316no—
319C2×#123—16616no—
320C2×#121—838no—
321C2×#121—1668no—
322C22×C4⋊C4—832no—
323C22×C4⋊C4—831no—
324C22×C4⋊C4—831no—
325C22×C4⋊C4—831no—
326C22×C4⋊C4—832no—
327C22×C4⋊C4—838no—
328C22×C4⋊C4—1668no—
329C22×C4⋊C4C2×D8311yescharacteristic
330C22×C4⋊C4C2×D8312yes—
331C22×C4⋊C4C2×D8311yescharacteristic
332C22×C22⋊C4—832no—
333C22×C22⋊C4—832no—
334C22×C22⋊C4—832no—
335C22×C22⋊C4—832no—
336C22×C22⋊C4—832no—
337C22×C22⋊C4—832no—
338#398—838no—
339#398C2×D8318yescomplemented
340#396—8316no—
341#396—8316no—
342#394—838no—
343#394—838no—
344#386—8316no—
345#380—838no—
346C2×#95—831no—
347C2×#95—832no—
348C2×#95C2×D8311yescharacteristic
349C2×#95C2×D8312yes—
350C25—831no—
351C25—832no—
352C22×D4—3324no—
353C22×D4—834no—
354C22×D4—838no—
355C22×D4—1668no—
356C22×D4—1668no—
357C22×D4—1668no—
358C22×D4—1664no—
359C23×C4—3324no—
360C23×C4—3324no—
361C23×C4—838no—
362C23×C4—834no—
363C23×C4—834no—
364C23×C4—831no—
365C23×C4—831no—
366C23×C4—838no—
367C23×C4—834no—
368C23×C4—832no—
369C23×C4—831no—
370C23×C4—1664no—
371C23×C4—1664no—
372C23×C4—1664no—
373C23×C4C22×D8312yes—
374C23×C4C22×D8311yescharacteristic
375C23×C4C22×D8312yes—
376C23×C4C22×D8311yescharacteristic
377C23×C4C22×D8311yescharacteristic
378#123—16664no—
379#121—16632no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—28no—
383C2×C4⋊C4—3324no—
384C2×C4⋊C4—838no—
385C2×C4⋊C4—834no—
386C2×C4⋊C4—834no—
387C2×C4⋊C4—834no—
388C2×C4⋊C4—838no—
389C2×C4⋊C4—16632no—
390C2×C4⋊C4—1664no—
391C2×C4⋊C4—1664no—
392C2×C4⋊C4—1668no—
393C2×C4⋊C4—1664no—
394C2×C4⋊C4C22×D8314yes—
395C2×C4⋊C4C22×D8314yes—
396C2×C4⋊C4C22×D8318yes—
397C2×C22⋊C4—3324no—
398C2×C22⋊C4—3324no—
399C2×C22⋊C4—838no—
400C2×C22⋊C4—838no—
401C2×C22⋊C4—838no—
402C2×C22⋊C4—838no—
403C2×C22⋊C4—838no—
404C2×C22⋊C4—838no—
405C2×C22⋊C4—1668no—
406C2×C22⋊C4—1668no—
407C2×C22⋊C4—1668no—
408C2×C22⋊C4—1668no—
409#95—834no—
410#95—838no—
411#95C22×D8314yes—
412#95C22×D8318yes—
413C24—834no—
414C24—834no—
415C24—834no—
416C24—832no—
417C24—834no—
418C24—1664no—
419C24—1664no—
420C24—1664no—
421C24—1664no—
422C24—1664no—
423C24—1664no—
424C24—1664no—
425C24—1664no—
426C24—1664no—
427C24C23×D8311yescharacteristic
428C2×D4—33216no—
429C2×D4—3328no—
430C2×D4—33232no—
431C2×D4—16616no—
432C2×D4—16616no—
433C2×D4—16616no—
434C2×D4—1668no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—33216no—
447C22×C4—3328no—
448C22×C4—3328no—
449C22×C4—33216no—
450C22×C4—8316no—
451C22×C4—834no—
452C22×C4—834no—
453C22×C4—832no—
454C22×C4—832no—
455C22×C4—8316no—
456C22×C4—834no—
457C22×C4—832no—
458C22×C4—1668no—
459C22×C4—1668no—
460C22×C4—16632no—
461C22×C4—1668no—
462C22×C4—1664no—
463C22×C4—1662no—
464C22×C4—1664no—
465C22×C4—1664no—
466C22×C4—1664no—
467C22×C4—1662no—
468C22×C4—1662no—
469C22×C4—1664no—
470C22×C4D4×D8314yes—
471C22×C4D4×D8314yes—
472C22×C4C23×D8314yes—
473C22×C4C23×D8314yes—
474C22×C4C23×D8312yes—
475C22×C4C23×D8312yes—
476C22×C4C23×D8312yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—216no—
480C4⋊C4—3328no—
481C4⋊C4—1668no—
482C4⋊C4—1668no—
483C4⋊C4—16616no—
484C4⋊C4—1668no—
485C22⋊C4—3328no—
486C22⋊C4—3328no—
487C22⋊C4—16616no—
488C22⋊C4—16616no—
489C22⋊C4—16616no—
490C22⋊C4—16616no—
491C23—3328no—
492C23—3328no—
493C23—3328no—
494C23—3328no—
495C23—3328no—
496C23—3328no—
497C23—3328no—
498C23—8316no—
499C23—834no—
500C23—834no—
501C23—834no—
502C23—1664no—
503C23—1664no—
504C23—1664no—
505C23—1664no—
506C23—1664no—
507C23—1664no—
508C23—1664no—
509C23—1668no—
510C23—1668no—
511C23—1664no—
512C23—1664no—
513C23—16616no—
514C23—16616no—
515C23C2×D4×D8312yes—
516C23C2×D4×D8312yes—
517C23#41×D8312yes—
518C23#1458183814yes—
519C23#1458184014yes—
520C23C24×D8311yescharacteristic
521D4—33216no—
522D4—33232no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—48no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—33216no—
540C2×C4—33216no—
541C2×C4—3324no—
542C2×C4—3328no—
543C2×C4—834no—
544C2×C4—834no—
545C2×C4—16632no—
546C2×C4—1664no—
547C2×C4—1662no—
548C2×C4—1664no—
549C2×C4—1664no—
550C2×C4—1664no—
551C2×C4—1668no—
552C2×C4—1668no—
553C2×C4—1662no—
554C2×C4—1662no—
555C2×C4—1664no—
556C2×C4C2×D4×D8314yes—
557C2×C4C2×D4×D8314yes—
558C22—3324no—
559C22—3324no—
560C22—3324no—
561C22—3324no—
562C22—3324no—
563C22—3324no—
564C22—3324no—
565C22—838no—
566C22—66416no—
567C22—16632no—
568C22—1664no—
569C22—1664no—
570C22—1668no—
571C22—1668no—
572C22#121×D8314yes—
573C22#1912826214yes—
574C22#1912827614yes—
575C22#1912828118yes—
576C22#1912829918yes—
577C22C22×D4×D8311yescharacteristic
578C22C22×D4×D8311yescharacteristic
579C22C2×#41×D8311yescharacteristic
580C22C2×#1458183812yes—
581C22C2×#1458184012yes—
582C4—24no—
583C4—24no—
584C4—44no—
585C4—42no—
586C4—44no—
587C4—3322no—
588C4—3324no—
589C4—1664no—
590C4—1664no—
591C2—6644no—
592C2—1668no—
593C2#2491320618yesdirect product factor
594C2C2×#121×D8311yescharacteristic
595C2C2×#1912826211yescharacteristic
596C2C2×#1912827611yescharacteristic
597C2C2×#1912828112yes—
598C2C2×#1912829912yes—
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#398×C83C2
socleC24×C83C24
prime radicalC22×#14581767C2
hypercenterC2×#398D83
hypocenterC83C2×#2618
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
1. C22×#14581777C2semidirect
3. C22×C4⋊C4×D83C2semidirect
4. #24913206C2direct (N × H)
5. C2×#19127632C2semidirect
6. C2×#19127625C2semidirect
7. C2×#19127623C2semidirect
8. C2×#19127619C2semidirect
9. C2×#19127618C2semidirect
10. C2×#19127579C2semidirect
11. C2×#19127560C2semidirect
12. C2×#398×C83C2semidirect
19. C2×#14581777C22semidirect
28. C2×C4⋊C4×D83C22semidirect
35. #19127632C22semidirect
36. #19127625C22semidirect
37. #19127623C22semidirect
38. #19127619C22semidirect
39. #19127618C22semidirect
40. #19127579C22semidirect
41. #19127560C22semidirect
47. #398×C83C22semidirect
69. C22×C4×D83D4semidirect
157. C2×C4×D83C2×D4semidirect
292. C2×C83#2618semidirect
305. C2×#398D83semidirect
312. C83C2×#2618semidirect
339. #398C2×D83semidirect
593. C2#24913206direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817772
C22×#145817672
C22×C4⋊C4×D832
#249132062
C2×#191276322
C2×#191276252
C2×#191276232
C2×#191276192
C2×#191276182
C2×#191275792
C2×#191275602
C2×#398×C832
C2×#261883