Finite Group Databasebeta

← C2×#24911822 #35573502

Subgroups — C2×#24911822 #35573502

Order 21248.

3678 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×#14581791C211yesmaximal, characteristic, complemented
2C22×#14581778C212yesmaximal, complemented
3C22×#14581766C211yesmaximal, characteristic
4#24911822C2116yesmaximal, direct product factor
5C2×#19127616C212yesmaximal, complemented
6C2×#19127614C212yesmaximal, complemented
7C2×#19127602C211yesmaximal, characteristic, complemented
8C2×#19127602C211yesmaximal, characteristic, complemented
9C2×#19127590C212yesmaximal, complemented
10C2×C4⋊C4×Dic83C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
12C2×#380×C83C211yesmaximal, characteristic, complemented
13C23×Dic166C2212yes—
14C23×D332C2211yescharacteristic
15C23×C22⋊D83C2212yes—
16C2×#14581791—28no—
17C2×#14581791C2218yescomplemented
18C2×#14581778—216no—
19C2×#14581778C22116yescomplemented
20C2×#14581766—28no—
21C2×#14581766C2218yes—
22C22×C4⋊Dic83C2211yescharacteristic
23C22×Dic83⋊C4C2212yes—
24C22×C22⋊Dic83C2211yescharacteristic
25C22×C22⋊Dic83C2211yescharacteristic
26C22×(C2×D83)⋊C4C2212yes—
27C22×(C2×D83)⋊C4C2212yes—
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2212yes—
30C22×(C2×D83)⋊C4C2212yes—
31C22×C4×Dic83C2212yes—
32C22×C4×Dic83C2211yescharacteristic
33C22×C4×Dic83C2211yescharacteristic
34#19127616C22116yescomplemented
35#19127614C22116yescomplemented
36#19127602C2218yescomplemented
37#19127602C2218yescomplemented
38#19127590C22116yescomplemented
39C4⋊C4×Dic83C2218yescomplemented
40C4×C22⋊Dic83C2218yescomplemented
41C2×C83⋊#95C2211yescharacteristic
42C2×C83⋊#95C2211yescharacteristic
43C2×C83⋊#95C2212yes—
44C24×C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2211yescharacteristic
46C22×C4⋊C4×C83C2211yescharacteristic
47C22×C22⋊C4×C83C2212yes—
48#380×C83C2218yescomplemented
49C2×#95×C83C2212yes—
50C24×D83C2312yes—
51C22×Dic166—28no—
52C22×Dic166—28no—
53C22×Dic166—44no—
54C22×Dic166C2318yes—
55C22×D332—24no—
56C22×D332—24no—
57C22×D332—24no—
58C22×D332C2314yes—
59C22×C22⋊D83—28no—
60C22×C22⋊D83—28no—
61C22×C22⋊D83—44no—
62C22×C22⋊D83C2318yes—
63C23×Dic83C2311yescharacteristic
64C23×Dic83C2311yescharacteristic
65C23×Dic83C2311yescharacteristic
66C23×Dic83C2312yes—
67C23×Dic83C2311yescharacteristic
68#14581791—232no—
69#14581778—264no—
70#14581766—232no—
71C2×C4⋊Dic83D418yes—
72C2×C4⋊Dic83C2314yes—
73C2×Dic83⋊C4—28no—
74C2×Dic83⋊C4C2318yes—
75C2×C22⋊Dic83—24no—
76C2×C22⋊Dic83D418yescomplemented
77C2×C22⋊Dic83C2314yes—
78C2×C22⋊Dic83C2314yes—
79C2×(C2×D83)⋊C4—28no—
80C2×(C2×D83)⋊C4—28no—
81C2×(C2×D83)⋊C4—28no—
82C2×(C2×D83)⋊C4—28no—
83C2×(C2×D83)⋊C4—44no—
84C2×(C2×D83)⋊C4C2318yes—
85C2×(C2×D83)⋊C4C2318yes—
86C2×(C2×D83)⋊C4C2318yes—
87C2×(C2×D83)⋊C4C2318yes—
88C2×(C2×D83)⋊C4C2318yes—
89C2×C4×Dic83—28no—
90C2×C4×Dic83—24no—
91C2×C4×Dic83—24no—
92C2×C4×Dic83C2318yes—
93C2×C4×Dic83C2314yes—
94C2×C4×Dic83C2314yes—
95C83⋊#95C2318yes—
96C83⋊#95C2314yes—
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×D83—24no—
121C23×D83C2414yes—
122C2×Dic166—216no—
123C2×Dic166—216no—
124C2×Dic166—48no—
125C2×Dic166—432no—
126C2×D332—28no—
127C2×D332—28no—
128C2×D332—28no—
129C2×D332—416no—
130C2×C22⋊D83—216no—
131C2×C22⋊D83—216no—
132C2×C22⋊D83—48no—
133C2×C22⋊D83—432no—
134C22×Dic83—24no—
135C22×Dic83—22no—
136C22×Dic83—22no—
137C22×Dic83—22no—
138C22×Dic83—24no—
139C22×Dic83—22no—
140C22×Dic83—22no—
141C22×Dic83—22no—
142C22×Dic83—22no—
143C22×Dic83—22no—
144C22×Dic83—22no—
145C22×Dic83C2×D418yes—
146C22×Dic83C2×D414yes—
147C22×Dic83C2×D414yes—
148C22×Dic83#4114yes—
149C22×Dic83C2412yes—
150C22×Dic83C2414yes—
151C22×Dic83C2412yes—
152C22×Dic83C2412yes—
153C22×Dic83C2412yes—
154C4⋊Dic83C2×D4116yes—
155Dic83⋊C4—216no—
156C22⋊Dic83—28no—
157C22⋊Dic83C2×D4116yescomplemented
158(C2×D83)⋊C4—216no—
159(C2×D83)⋊C4—216no—
160(C2×D83)⋊C4—216no—
161(C2×D83)⋊C4—216no—
162(C2×D83)⋊C4—48no—
163C4×Dic83—28no—
164C4×Dic83—28no—
165C4×Dic83—216no—
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—24no—
221C22×D83—24no—
222C22×D83—24no—
223C22×D83—24no—
224C22×D83—24no—
225C22×D83—24no—
226C22×D83—24no—
227C22×D83—48no—
228C22×D83—48no—
229C22×D83—48no—
230C22×D83—48no—
231C22×D83—48no—
232C22×D83—48no—
233C22×D83—48no—
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#124116yescomplemented
248C2×C4×C83C22×D414yes—
249C2×C4×C83C2×#4114yes—
250C2×C4×C83C2×#4114yes—
251C22⋊D83—432no—
252C2×Dic83—216no—
253C2×Dic83—28no—
254C2×Dic83—28no—
255C2×Dic83—28no—
256C2×Dic83—22no—
257C2×Dic83—24no—
258C2×Dic83—22no—
259C2×Dic83—22no—
260C2×Dic83—24no—
261C2×Dic83—22no—
262C2×Dic83—22no—
263C2×Dic83—22no—
264C2×Dic83—22no—
265C2×Dic83—22no—
266C2×Dic83—22no—
267C2×Dic83—44no—
268C2×Dic83C22×D414yes—
269C2×Dic83C22×D414yes—
270C2×Dic83C22×D418yes—
271C2×Dic83C2×#4114yes—
272D332—416no—
273Dic166—432no—
274C22×C83—24no—
275C22×C83—24no—
276C22×C83—24no—
277C22×C83—42no—
278C22×C83—42no—
279C22×C83—42no—
280C22×C83—42no—
281C22×C83—416no—
282C22×C83C2×#12414yes—
283C22×C83#53214yes—
284C22×C83#54214yes—
285C22×C83D4214yes—
286C22×C83#54714yes—
287C22×C83#54814yes—
288C22×C83#55514yes—
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—44no—
297C2×D83—44no—
298C2×D83—44no—
299C2×D83—44no—
300C2×D83—44no—
301C2×D83—44no—
302C2×D83—44no—
303C2×D83—816no—
304C4×C83—42no—
305C4×C83—44no—
306C4×C83—44no—
307C4×C83C2×#12418yescomplemented
308Dic83—24no—
309Dic83—28no—
310Dic83—24no—
311Dic83—24no—
312Dic83—42no—
313C2×#2491—831nomaximal, Sylow
314C2×C83—44no—
315C2×C83#249118yescomplemented
316C2×C83C22×#12411yescharacteristic
317C2×C83C2×#53211yescharacteristic
318C2×C83C2×#54211yescharacteristic
319C2×C83C2×D4211yescharacteristic
320C2×C83C2×#54711yescharacteristic
321C2×C83C2×#54811yescharacteristic
322C2×C83C2×#55511yescharacteristic
323D83—84no—
324C22×C4⋊Q8—831no—
325C22×#124—832no—
326C22×#121—831no—
327#2491—8316no—
328C2×#394—832no—
329C2×#393—832no—
330C2×#390—831no—
331C2×#390—831no—
332C2×#388—832no—
333C2×#380D8311yescharacteristic, complemented
334C2×C4×C4⋊C4—831no—
335C2×C4×C22⋊C4—831no—
336C83C2×#249111yescharacteristic, complemented, Sylow
337C23×Q8—832no—
338C23×D4—832no—
339C23×D4—831no—
340C24×C4C2×D8311yescharacteristic
341C2×C4⋊Q8—838no—
342C2×C4⋊Q8—1668no—
343C2×#124—8316no—
344C2×#124—16616no—
345C2×#121—838no—
346C2×#121—1668no—
347C22×C4⋊C4—831no—
348C22×C4⋊C4—832no—
349C22×C4⋊C4C2×D8311yescharacteristic
350C22×C4⋊C4C2×D8311yescharacteristic
351C22×C22⋊C4—832no—
352C22×C22⋊C4—831no—
353C22×C22⋊C4—831no—
354C22×C22⋊C4—832no—
355C22×C22⋊C4—832no—
356C22×C22⋊C4—832no—
357C22×C22⋊C4—832no—
358C22×C22⋊C4C2×D8312yes—
359C22×C42—832no—
360C22×C42—831no—
361C22×C42—831no—
362#394—8316no—
363#393—8316no—
364#390—838no—
365#390—838no—
366#388—8316no—
367#380C2×D8318yescomplemented
368C4×C4⋊C4—838no—
369C4×C22⋊C4—838no—
370C2×#95—831no—
371C2×#95—832no—
372C2×#95—831no—
373C2×#95C2×D8312yes—
374C25—832no—
375C25C22×D8311yescharacteristic
376C22×Q8—3324no—
377C22×Q8—838no—
378C22×Q8—1668no—
379C22×Q8—1668no—
380C22×D4—3324no—
381C22×D4—838no—
382C22×D4—834no—
383C22×D4—1668no—
384C22×D4—1668no—
385C22×D4—1664no—
386C22×D4—1664no—
387C22×D4—1664no—
388C23×C4—24no—
389C23×C4—24no—
390C23×C4—24no—
391C23×C4—831no—
392C23×C4—831no—
393C23×C4—831no—
394C23×C4—832no—
395C23×C4—831no—
396C23×C4C22×D8314yes—
397C23×C4C22×D8312yes—
398C23×C4C22×D8312yes—
399C23×C4C22×D8311yescharacteristic
400C23×C4C22×D8311yescharacteristic
401C4⋊Q8—16632no—
402#124—16664no—
403#121—16632no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—838no—
407C2×C4⋊C4—834no—
408C2×C4⋊C4—838no—
409C2×C4⋊C4—1668no—
410C2×C4⋊C4C22×D8314yes—
411C2×C4⋊C4C22×D8314yes—
412C2×C22⋊C4—28no—
413C2×C22⋊C4—3324no—
414C2×C22⋊C4—838no—
415C2×C22⋊C4—838no—
416C2×C22⋊C4—834no—
417C2×C22⋊C4—834no—
418C2×C22⋊C4—838no—
419C2×C22⋊C4—838no—
420C2×C22⋊C4—838no—
421C2×C22⋊C4—838no—
422C2×C22⋊C4—1668no—
423C2×C22⋊C4—1668no—
424C2×C22⋊C4—1668no—
425C2×C22⋊C4—1668no—
426C2×C22⋊C4—1664no—
427C2×C22⋊C4C22×D8318yes—
428C2×C42—838no—
429C2×C42—834no—
430C2×C42—834no—
431C2×C42—1668no—
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—834no—
445C24—1664no—
446C24—1664no—
447C24—1664no—
448C24—1664no—
449C24—1664no—
450C24—1664no—
451C24—1664no—
452C24D4×D8314yes—
453C24C23×D8312yes—
454C24C23×D8311yescharacteristic
455C2×Q8—3328no—
456C2×Q8—33232no—
457C2×Q8—16616no—
458C2×Q8—16616no—
459C2×D4—33232no—
460C2×D4—3328no—
461C2×D4—33216no—
462C2×D4—16616no—
463C2×D4—16616no—
464C2×D4—1668no—
465C2×D4—1668no—
466C2×D4—1668no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—416no—
480C22×C4—834no—
481C22×C4—834no—
482C22×C4—838no—
483C22×C4—834no—
484C22×C4—832no—
485C22×C4—832no—
486C22×C4—832no—
487C22×C4—834no—
488C22×C4—832no—
489C22×C4—1662no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1662no—
493C22×C4—1664no—
494C22×C4—1662no—
495C22×C4—1662no—
496C22×C4—1664no—
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—8316no—
510C4⋊C4—16616no—
511C22⋊C4—216no—
512C22⋊C4—3328no—
513C22⋊C4—8316no—
514C22⋊C4—16616no—
515C22⋊C4—16616no—
516C22⋊C4—16616no—
517C22⋊C4—16616no—
518C22⋊C4—1668no—
519C42—16616no—
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—3328no—
540C23—3328no—
541C23—3328no—
542C23—1664no—
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
558Q8—33232no—
559D4—33232no—
560D4—33216no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—416no—
574C2×C4—3324no—
575C2×C4—834no—
576C2×C4—834no—
577C2×C4—838no—
578C2×C4—834no—
579C2×C4—1662no—
580C2×C4—1662no—
581C2×C4—1662no—
582C2×C4—16616no—
583C2×C4—1668no—
584C2×C4—1668no—
585C2×C4—1668no—
586C2×C4—1662no—
587C2×C4—1664no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4—1664no—
591C2×C4—1662no—
592C2×C4—1662no—
593C2×C4—1662no—
594C2×C4#14581778116yescomplemented
595C2×C4C2×D4×D8314yes—
596C2×C4C2×#1305058914yes—
597C2×C4C2×#1305059014yes—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—416no—
606C22—3324no—
607C22—3324no—
608C22—3324no—
609C22—3324no—
610C22—3324no—
611C22—3324no—
612C22—3324no—
613C22—66416no—
614C22C2×#1458177814yes—
615C22#1912824614yes—
616C22#1912832414yes—
617C22#1912833514yes—
618C22#1912835214yes—
619C22#1912836114yes—
620C22#1912841414yes—
621C22C22×D4×D8311yescharacteristic
622C22C22×D4×D8311yescharacteristic
623C22C2×#41×D8311yescharacteristic
624C22C22×#1305058911yescharacteristic
625C22C22×#1305059011yescharacteristic
626C22C2×#1458183811yescharacteristic
627C22C2×#1458184011yescharacteristic
628C4—44no—
629C4—42no—
630C4—44no—
631C4—3322no—
632C4—1668no—
633C4—1664no—
634C4—1664no—
635C4—1664no—
636C4C2×#1458177818yescomplemented
637C2—44no—
638C2—6644no—
639C2#2491182218yesdirect product factor
640C2C22×#1458177811yescharacteristic
641C2C2×#1912824611yescharacteristic
642C2C2×#1912832411yescharacteristic
643C2C2×#1912833511yescharacteristic
644C2C2×#1912835211yescharacteristic
645C2C2×#1912836111yescharacteristic
646C2C2×#1912841411yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#380×C83C2
socleC24×C83C24
prime radicalC22×#14581766C2
hypercenterC2×#380D83
hypocenterC83C2×#2491
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#14581791C2semidirect
2. C22×#14581778C2semidirect
4. #24911822C2direct (N × H)
5. C2×#19127616C2semidirect
6. C2×#19127614C2semidirect
7. C2×#19127602C2semidirect
8. C2×#19127602C2semidirect
9. C2×#19127590C2semidirect
10. C2×C4⋊C4×Dic83C2semidirect
11. C2×C4×C22⋊Dic83C2semidirect
12. C2×#380×C83C2semidirect
17. C2×#14581791C22semidirect
19. C2×#14581778C22semidirect
34. #19127616C22semidirect
35. #19127614C22semidirect
36. #19127602C22semidirect
37. #19127602C22semidirect
38. #19127590C22semidirect
39. C4⋊C4×Dic83C22semidirect
40. C4×C22⋊Dic83C22semidirect
48. #380×C83C22semidirect
76. C2×C22⋊Dic83D4semidirect
157. C22⋊Dic83C2×D4semidirect
247. C2×C4×C83#124semidirect
307. C4×C83C2×#124semidirect
315. C2×C83#2491semidirect
333. C2×#380D83semidirect
336. C83C2×#2491semidirect
367. #380C2×D83semidirect
594. C2×C4#14581778semidirect
636. C4C2×#14581778semidirect
639. C2#24911822direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817912
C22×#145817782
C22×#145817662
#249118222
C2×#191276162
C2×#191276142
C2×#191276022
C2×#191275902
C2×C4⋊C4×Dic832
C2×C4×C22⋊Dic832
C2×#380×C832
C2×#249183