Finite Group Databasebeta

← C2×#24910979 #35572280

Subgroups — C2×#24910979 #35572280

Order 21248.

3830 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581774C212yesmaximal, complemented
2C22×#14581766C211yesmaximal, characteristic
3C22×#14581754C211yesmaximal, characteristic, complemented
4#24910979C2116yesmaximal, direct product factor
5C2×#19127630C212yesmaximal, complemented
6C2×#19127616C212yesmaximal, complemented
7C2×#19127602C211yesmaximal, characteristic, complemented
8C2×#19127600C211yesmaximal, characteristic, complemented
9C2×#19127575C212yesmaximal, complemented
10C2×#19127570C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
12C2×#380×C83C211yesmaximal, characteristic, complemented
13C23×D332C2211yescharacteristic
14C23×C22⋊D83C2212yes—
15C23×C4×D83C2212yes—
16C2×#14581774—216no—
17C2×#14581774C22116yescomplemented
18C2×#14581766—28no—
19C2×#14581766C2218yes—
20C2×#14581754—28no—
21C2×#14581754C2218yescomplemented
22C22×C4⋊Dic83C2211yescharacteristic
23C22×C4⋊Dic83C2212yes—
24C22×Dic83⋊C4C2212yes—
25C22×Dic83⋊C4C2212yes—
26C22×C22⋊Dic83C2211yescharacteristic
27C22×C22⋊Dic83C2211yescharacteristic
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2212yes—
30C22×(C2×D83)⋊C4C2212yes—
31C22×(C2×D83)⋊C4C2212yes—
32C22×C4×Dic83C2211yescharacteristic
33C22×C4×Dic83C2211yescharacteristic
34#19127630C22116yescomplemented
35#19127616C22116yescomplemented
36#19127602C2218yescomplemented
37#19127600C2218yescomplemented
38#19127575C22116yescomplemented
39#19127570C2218yescomplemented
40C4×C22⋊Dic83C2218yescomplemented
41C2×C83⋊#95C2211yescharacteristic
42C2×C83⋊#95C2212yes—
43C2×C83⋊#95C2211yescharacteristic
44C24×C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2211yescharacteristic
46C22×C4⋊C4×C83C2211yescharacteristic
47C22×C22⋊C4×C83C2212yes—
48#380×C83C2218yescomplemented
49C2×#95×C83C2212yes—
50C24×D83C2312yes—
51C22×D332—24no—
52C22×D332—24no—
53C22×D332D418yes—
54C22×D332C2314yes—
55C22×C22⋊D83—28no—
56C22×C22⋊D83—28no—
57C22×C22⋊D83—44no—
58C22×C22⋊D83C2318yes—
59C23×Dic83C2311yescharacteristic
60C23×Dic83C2312yes—
61C23×Dic83C2311yescharacteristic
62C23×Dic83C2311yescharacteristic
63C23×Dic83C2311yescharacteristic
64C22×C4×D83—28no—
65C22×C4×D83—28no—
66C22×C4×D83—44no—
67C22×C4×D83C2318yes—
68#14581774—264no—
69#14581766—232no—
70#14581754—232no—
71C2×C4⋊Dic83—28no—
72C2×C4⋊Dic83—24no—
73C2×C4⋊Dic83C2314yes—
74C2×C4⋊Dic83C2318yes—
75C2×Dic83⋊C4—28no—
76C2×Dic83⋊C4—44no—
77C2×Dic83⋊C4C2318yes—
78C2×Dic83⋊C4C2318yes—
79C2×C22⋊Dic83—24no—
80C2×C22⋊Dic83D418yescomplemented
81C2×C22⋊Dic83C2314yes—
82C2×C22⋊Dic83C2314yes—
83C2×(C2×D83)⋊C4—28no—
84C2×(C2×D83)⋊C4—28no—
85C2×(C2×D83)⋊C4—28no—
86C2×(C2×D83)⋊C4—28no—
87C2×(C2×D83)⋊C4C2318yes—
88C2×(C2×D83)⋊C4C2318yes—
89C2×(C2×D83)⋊C4C2318yes—
90C2×(C2×D83)⋊C4C2318yes—
91C2×C4×Dic83—24no—
92C2×C4×Dic83—24no—
93C2×C4×Dic83C2314yes—
94C2×C4×Dic83C2314yes—
95C83⋊#95C2314yes—
96C83⋊#95C2318yes—
97C83⋊#95C2314yes—
98C25×C83C2311yescharacteristic
99C23×C4×C83—24no—
100C23×C4×C83—24no—
101C23×C4×C83—24no—
102C23×C4×C83C2312yes—
103C23×C4×C83C2312yes—
104C23×C4×C83C2311yescharacteristic
105C23×C4×C83C2314yes—
106C23×C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83—24no—
108C2×C4⋊C4×C83—24no—
109C2×C4⋊C4×C83C2314yes—
110C2×C4⋊C4×C83C2314yes—
111C2×C22⋊C4×C83—28no—
112C2×C22⋊C4×C83C2318yes—
113#95×C83C2318yes—
114C23×D83—24no—
115C23×D83—24no—
116C23×D83—24no—
117C23×D83—24no—
118C23×D83—24no—
119C23×D83—24no—
120C23×D83C2×D418yes—
121C23×D83C2414yes—
122C2×D332—232no—
123C2×D332—28no—
124C2×D332—28no—
125C2×D332C2×D4116yes—
126C2×C22⋊D83—216no—
127C2×C22⋊D83—216no—
128C2×C22⋊D83—48no—
129C2×C22⋊D83—432no—
130C22×Dic83—22no—
131C22×Dic83—22no—
132C22×Dic83—22no—
133C22×Dic83—22no—
134C22×Dic83—22no—
135C22×Dic83—22no—
136C22×Dic83—24no—
137C22×Dic83—24no—
138C22×Dic83—22no—
139C22×Dic83—22no—
140C22×Dic83—24no—
141C22×Dic83—22no—
142C22×Dic83C2×D414yes—
143C22×Dic83C2×D414yes—
144C22×Dic83#4114yes—
145C22×Dic83C2414yes—
146C22×Dic83C2412yes—
147C22×Dic83C2412yes—
148C22×Dic83C2412yes—
149C22×Dic83C2412yes—
150C2×C4×D83—216no—
151C2×C4×D83—216no—
152C2×C4×D83—48no—
153C2×C4×D83—432no—
154C4⋊Dic83—28no—
155C4⋊Dic83—216no—
156Dic83⋊C4—216no—
157Dic83⋊C4—48no—
158C22⋊Dic83—28no—
159C22⋊Dic83C2×D4116yescomplemented
160(C2×D83)⋊C4—216no—
161(C2×D83)⋊C4—216no—
162(C2×D83)⋊C4—216no—
163(C2×D83)⋊C4—216no—
164C4×Dic83—28no—
165C4×Dic83—28no—
166C24×C83—22no—
167C24×C83—22no—
168C24×C83—22no—
169C24×C83—22no—
170C24×C83—22no—
171C24×C83—22no—
172C24×C83C2×D414yes—
173C24×C83C2412yes—
174C24×C83C2411yescharacteristic
175C22×C4×C83—24no—
176C22×C4×C83—24no—
177C22×C4×C83—24no—
178C22×C4×C83—22no—
179C22×C4×C83—24no—
180C22×C4×C83—22no—
181C22×C4×C83—22no—
182C22×C4×C83—24no—
183C22×C4×C83—24no—
184C22×C4×C83—28no—
185C22×C4×C83—28no—
186C22×C4×C83—28no—
187C22×C4×C83—416no—
188C22×C4×C83C2×D414yes—
189C22×C4×C83#4114yes—
190C22×C4×C83#4114yes—
191C22×C4×C83C2414yes—
192C22×C4×C83C2412yes—
193C22×C4×C83C2414yes—
194C22×C4×C83C2412yes—
195C4⋊C4×C83—28no—
196C4⋊C4×C83—28no—
197C22⋊C4×C83—216no—
198C23×C83—28no—
199C23×C83—28no—
200C23×C83—28no—
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×C83C22×D414yes—
212C23×C83C22×D412yes—
213C23×C83C22×D412yes—
214C23×C83C2×#4112yes—
215C23×C83C2×#4112yes—
216C23×C83C2×#4112yes—
217C23×C83H(5,𝔽2)12yes—
218C23×C832−1+412yes—
219C23×C83C2511yescharacteristic
220C22×D83—216no—
221C22×D83—216no—
222C22×D83—216no—
223C22×D83—24no—
224C22×D83—24no—
225C22×D83—24no—
226C22×D83—24no—
227C22×D83—24no—
228C22×D83—24no—
229C22×D83—48no—
230C22×D83—48no—
231C22×D83—48no—
232C22×D83—48no—
233C22×D83C22×D418yes—
234C2×C4×C83—24no—
235C2×C4×C83—24no—
236C2×C4×C83—24no—
237C2×C4×C83—22no—
238C2×C4×C83—24no—
239C2×C4×C83—22no—
240C2×C4×C83—22no—
241C2×C4×C83—24no—
242C2×C4×C83—24no—
243C2×C4×C83—44no—
244C2×C4×C83—48no—
245C2×C4×C83—48no—
246C2×C4×C83—416no—
247C2×C4×C83#123116yescomplemented
248C2×C4×C83C22×D414yes—
249C2×C4×C83C2×#4114yes—
250C2×C4×C83C2×#4114yes—
251C22⋊D83—432no—
252C2×Dic83—28no—
253C2×Dic83—28no—
254C2×Dic83—28no—
255C2×Dic83—22no—
256C2×Dic83—22no—
257C2×Dic83—22no—
258C2×Dic83—22no—
259C2×Dic83—22no—
260C2×Dic83—22no—
261C2×Dic83—22no—
262C2×Dic83—24no—
263C2×Dic83—24no—
264C2×Dic83—22no—
265C2×Dic83—24no—
266C2×Dic83—22no—
267C2×Dic83—48no—
268C2×Dic83—44no—
269C2×Dic83C22×D414yes—
270C2×Dic83C22×D414yes—
271C2×Dic83C2×#4114yes—
272D332—232no—
273C4×D83—432no—
274C22×C83—24no—
275C22×C83—24no—
276C22×C83—24no—
277C22×C83—42no—
278C22×C83—416no—
279C22×C83—42no—
280C22×C83—42no—
281C22×C83—42no—
282C22×C83C2×#12314yes—
283C22×C83#53214yes—
284C22×C83#54214yes—
285C22×C83D4214yes—
286C22×C83#54714yes—
287C22×C83#54714yes—
288C22×C83#55314yes—
289C22×C83C23×D411yescharacteristic
290C22×C83C23×D411yescharacteristic
291C22×C83C22×#4111yescharacteristic
292C22×C83C22×#4111yescharacteristic
293C22×C83C22×#4111yescharacteristic
294C22×C83C2×H(5,𝔽2)11yescharacteristic
295C22×C83C2×2−1+411yescharacteristic
296C2×D83—28no—
297C2×D83—28no—
298C2×D83—28no—
299C2×D83—44no—
300C2×D83—432no—
301C2×D83—44no—
302C2×D83—44no—
303C2×D83—44no—
304C4×C83—42no—
305C4×C83—44no—
306C4×C83—44no—
307C4×C83C2×#12318yescomplemented
308Dic83—24no—
309Dic83—24no—
310Dic83—24no—
311Dic83—44no—
312Dic83—42no—
313C2×#2422—831nomaximal, Sylow
314C2×C83—44no—
315C2×C83#242218yescomplemented
316C2×C83C22×#12311yescharacteristic
317C2×C83C2×#53211yescharacteristic
318C2×C83C2×#54211yescharacteristic
319C2×C83C2×D4211yescharacteristic
320C2×C83C2×#54711yescharacteristic
321C2×C83C2×#54711yescharacteristic
322C2×C83C2×#55311yescharacteristic
323D83—48no—
324C22×#123—832no—
325C22×#121—831no—
326C22×C4×D4—831no—
327#2422—8316no—
328C2×#397—832no—
329C2×#394—832no—
330C2×#390—831no—
331C2×#389—831no—
332C2×#385—832no—
333C2×#384—831no—
334C2×#380D8311yescharacteristic, complemented
335C2×C4×C22⋊C4—831no—
336C83C2×#242211yescharacteristic, complemented, Sylow
337C23×D4—832no—
338C23×D4—831no—
339C24×C4—832no—
340C24×C4C2×D8311yescharacteristic
341C2×#123—8316no—
342C2×#123—16616no—
343C2×#121—838no—
344C2×#121—1668no—
345C2×C4×D4—838no—
346C2×C4×D4—1668no—
347C22×C4⋊C4—832no—
348C22×C4⋊C4—832no—
349C22×C4⋊C4—831no—
350C22×C4⋊C4—832no—
351C22×C4⋊C4C2×D8311yescharacteristic
352C22×C4⋊C4C2×D8311yescharacteristic
353C22×C22⋊C4—831no—
354C22×C22⋊C4—831no—
355C22×C22⋊C4—832no—
356C22×C22⋊C4—832no—
357C22×C22⋊C4—832no—
358C22×C22⋊C4—832no—
359C22×C22⋊C4C2×D8312yes—
360C22×C42—831no—
361C22×C42—831no—
362#397—8316no—
363#394—8316no—
364#390—838no—
365#389—838no—
366#385—8316no—
367#384—838no—
368#380C2×D8318yescomplemented
369C4×C22⋊C4—838no—
370C2×#95—832no—
371C2×#95—831no—
372C2×#95—831no—
373C2×#95C2×D8312yes—
374C25—832no—
375C25C22×D8311yescharacteristic
376C22×D4—3324no—
377C22×D4—838no—
378C22×D4—838no—
379C22×D4—834no—
380C22×D4—1668no—
381C22×D4—1668no—
382C22×D4—1664no—
383C22×D4—1664no—
384C23×C4—24no—
385C23×C4—24no—
386C23×C4—24no—
387C23×C4—3324no—
388C23×C4—838no—
389C23×C4—831no—
390C23×C4—831no—
391C23×C4—831no—
392C23×C4—831no—
393C23×C4—832no—
394C23×C4—1668no—
395C23×C4—1668no—
396C23×C4C22×D8314yes—
397C23×C4C22×D8312yes—
398C23×C4C22×D8311yescharacteristic
399C23×C4C22×D8312yes—
400C23×C4C22×D8311yescharacteristic
401#123—16664no—
402#121—16632no—
403C4×D4—16632no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—3324no—
407C2×C4⋊C4—838no—
408C2×C4⋊C4—834no—
409C2×C4⋊C4—838no—
410C2×C4⋊C4—838no—
411C2×C4⋊C4—1668no—
412C2×C4⋊C4—1668no—
413C2×C4⋊C4—1664no—
414C2×C4⋊C4C22×D8314yes—
415C2×C4⋊C4C22×D8314yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—838no—
418C2×C22⋊C4—834no—
419C2×C22⋊C4—834no—
420C2×C22⋊C4—838no—
421C2×C22⋊C4—838no—
422C2×C22⋊C4—838no—
423C2×C22⋊C4—838no—
424C2×C22⋊C4—1668no—
425C2×C22⋊C4—1668no—
426C2×C22⋊C4—1668no—
427C2×C22⋊C4—1668no—
428C2×C22⋊C4—1664no—
429C2×C22⋊C4C22×D8318yes—
430C2×C42—834no—
431C2×C42—834no—
432C2×C42—1664no—
433C2×C42—1664no—
434#95—834no—
435#95—834no—
436#95—838no—
437#95C22×D8318yes—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—838no—
445C24—834no—
446C24—1664no—
447C24—1664no—
448C24—1664no—
449C24—1664no—
450C24—1664no—
451C24—1664no—
452C24D4×D8314yes—
453C24C23×D8312yes—
454C24C23×D8311yescharacteristic
455C2×D4—33232no—
456C2×D4—3328no—
457C2×D4—8316no—
458C2×D4—16616no—
459C2×D4—16616no—
460C2×D4—16632no—
461C2×D4—1668no—
462C2×D4—1668no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—416no—
476C22×C4—3328no—
477C22×C4—33232no—
478C22×C4—834no—
479C22×C4—834no—
480C22×C4—834no—
481C22×C4—832no—
482C22×C4—832no—
483C22×C4—832no—
484C22×C4—832no—
485C22×C4—834no—
486C22×C4—16616no—
487C22×C4—16616no—
488C22×C4—1662no—
489C22×C4—1664no—
490C22×C4—1664no—
491C22×C4—1662no—
492C22×C4—1664no—
493C22×C4—1662no—
494C22×C4—1662no—
495C22×C4—1662no—
496C22×C4—1662no—
497C22×C4—1662no—
498C22×C4—1662no—
499C22×C4—1662no—
500C22×C4D4×D8314yes—
501C22×C4#1305058914yes—
502C22×C4#1305059014yes—
503C22×C4C23×D8312yes—
504C22×C4C23×D8314yes—
505C22×C4C23×D8314yes—
506C22×C4C23×D8312yes—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—3328no—
510C4⋊C4—16616no—
511C4⋊C4—16616no—
512C4⋊C4—1668no—
513C22⋊C4—216no—
514C22⋊C4—8316no—
515C22⋊C4—16616no—
516C22⋊C4—16616no—
517C22⋊C4—16616no—
518C22⋊C4—16616no—
519C22⋊C4—1668no—
520C42—1668no—
521C42—1668no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—3328no—
536C23—3328no—
537C23—3328no—
538C23—3328no—
539C23—838no—
540C23—16616no—
541C23—16616no—
542C23—16616no—
543C23—1664no—
544C23—1664no—
545C23—1664no—
546C23—1664no—
547C23—1664no—
548C23—1664no—
549C23C2×D4×D8314yes—
550C23C2×D4×D8312yes—
551C23C2×D4×D8312yes—
552C23#41×D8312yes—
553C23C2×#1305058912yes—
554C23C2×#1305059012yes—
555C23#1458183812yes—
556C23#1458184012yes—
557C23C24×D8311yescharacteristic
558D4—33232no—
559D4—16632no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—416no—
573C2×C4—33232no—
574C2×C4—3328no—
575C2×C4—3324no—
576C2×C4—834no—
577C2×C4—834no—
578C2×C4—834no—
579C2×C4—1662no—
580C2×C4—1664no—
581C2×C4—1664no—
582C2×C4—1662no—
583C2×C4—1664no—
584C2×C4—1662no—
585C2×C4—1668no—
586C2×C4—1668no—
587C2×C4—1668no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4—1662no—
591C2×C4—1662no—
592C2×C4—1662no—
593C2×C4—1662no—
594C2×C4#14581774116yescomplemented
595C2×C4C2×D4×D8314yes—
596C2×C4C2×#1305058914yes—
597C2×C4C2×#1305059014yes—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—416no—
605C22—42no—
606C22—3324no—
607C22—3324no—
608C22—33232no—
609C22—3324no—
610C22—3324no—
611C22—1668no—
612C22—1668no—
613C22—1668no—
614C22C2×#1458177414yes—
615C22#1912824614yes—
616C22#1912832414yes—
617C22#1912833514yes—
618C22#1912835214yes—
619C22#1912835314yes—
620C22#1912839514yes—
621C22C22×D4×D8311yescharacteristic
622C22C22×D4×D8311yescharacteristic
623C22C2×#41×D8311yescharacteristic
624C22C22×#1305058911yescharacteristic
625C22C22×#1305059011yescharacteristic
626C22C2×#1458183811yescharacteristic
627C22C2×#1458184011yescharacteristic
628C4—44no—
629C4—42no—
630C4—44no—
631C4—3324no—
632C4—3322no—
633C4—1664no—
634C4—1664no—
635C4—1664no—
636C4C2×#1458177418yescomplemented
637C2—44no—
638C2—3328no—
639C2#2491097918yesdirect product factor
640C2C22×#1458177411yescharacteristic
641C2C2×#1912824611yescharacteristic
642C2C2×#1912832411yescharacteristic
643C2C2×#1912833511yescharacteristic
644C2C2×#1912835211yescharacteristic
645C2C2×#1912835311yescharacteristic
646C2C2×#1912839511yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#380×C83C2
socleC24×C83C24
prime radicalC22×#14581766C2
hypercenterC2×#380D83
hypocenterC83C2×#2422
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#14581774C2semidirect
3. C22×#14581754C2semidirect
4. #24910979C2direct (N × H)
5. C2×#19127630C2semidirect
6. C2×#19127616C2semidirect
7. C2×#19127602C2semidirect
8. C2×#19127600C2semidirect
9. C2×#19127575C2semidirect
10. C2×#19127570C2semidirect
11. C2×C4×C22⋊Dic83C2semidirect
12. C2×#380×C83C2semidirect
17. C2×#14581774C22semidirect
21. C2×#14581754C22semidirect
34. #19127630C22semidirect
35. #19127616C22semidirect
36. #19127602C22semidirect
37. #19127600C22semidirect
38. #19127575C22semidirect
39. #19127570C22semidirect
40. C4×C22⋊Dic83C22semidirect
48. #380×C83C22semidirect
80. C2×C22⋊Dic83D4semidirect
159. C22⋊Dic83C2×D4semidirect
247. C2×C4×C83#123semidirect
307. C4×C83C2×#123semidirect
315. C2×C83#2422semidirect
334. C2×#380D83semidirect
336. C83C2×#2422semidirect
368. #380C2×D83semidirect
594. C2×C4#14581774semidirect
636. C4C2×#14581774semidirect
639. C2#24910979direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817742
C22×#145817662
C22×#145817542
#249109792
C2×#191276302
C2×#191276162
C2×#191276022
C2×#191276002
C2×#191275752
C2×#191275702
C2×C4×C22⋊Dic832
C2×#380×C832
C2×#242283