Finite Group Databasebeta

← C2×#24912823 #35575619

Subgroups — C2×#24912823 #35575619

Order 21248.

3222 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic166C211yesmaximal, characteristic, complemented
2C22×#14581754C211yesmaximal, characteristic
3#24912823C2116yesmaximal, direct product factor
4C2×#19127636C212yesmaximal, complemented
5C2×#19127614C212yesmaximal, complemented
6C2×#19127603C211yesmaximal, characteristic
7C2×#19127602C211yesmaximal, characteristic
8C2×#19127597C211yesmaximal, characteristic, complemented
9C2×#19127575C212yesmaximal, complemented
10C2×#19127572C212yesmaximal, complemented
11C2×#19127566C211yesmaximal, characteristic, complemented
12C2×#389×C83C211yesmaximal, characteristic, complemented
13C23×Dic166C2211yescharacteristic
14C23×D332C2211yescharacteristic
15C23×C4×D83C2212yes—
16C2×C4×Dic166—28no—
17C2×C4×Dic166C2218yescomplemented
18C2×#14581754—28no—
19C2×#14581754C2218yes—
20C22×C4⋊Dic83C2211yescharacteristic
21C22×C4⋊Dic83C2211yescharacteristic
22C22×Dic83⋊C4C2211yescharacteristic
23C22×Dic83⋊C4C2211yescharacteristic
24C22×Dic83⋊C4C2212yes—
25C22×(C2×D83)⋊C4C2212yes—
26C22×(C2×D83)⋊C4C2212yes—
27C22×(C2×D83)⋊C4C2212yes—
28C22×(C2×D83)⋊C4C2212yes—
29C22×(C2×D83)⋊C4C2212yes—
30C22×C4×Dic83C2211yescharacteristic
31C22×C4×Dic83C2211yescharacteristic
32#19127636C22116yescomplemented
33#19127614C22116yescomplemented
34#19127603C2218yes—
35#19127602C2218yes—
36#19127597C2218yescomplemented
37#19127575C22116yescomplemented
38#19127572C22116yescomplemented
39#19127566C2218yescomplemented
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×D83C2312yes—
50C22×Dic166—24no—
51C22×Dic166—24no—
52C22×Dic166—24no—
53C22×Dic166C2314yes—
54C22×D332—24no—
55C22×D332—24no—
56C22×D332Q818yescomplemented
57C22×D332C2314yes—
58C23×Dic83C2311yescharacteristic
59C23×Dic83C2312yes—
60C23×Dic83C2311yescharacteristic
61C23×Dic83C2311yescharacteristic
62C23×Dic83C2311yescharacteristic
63C22×C4×D83—28no—
64C22×C4×D83—44no—
65C22×C4×D83—44no—
66C22×C4×D83C2318yes—
67C4×Dic166—232no—
68#14581754—232no—
69C2×C4⋊Dic83—24no—
70C2×C4⋊Dic83—24no—
71C2×C4⋊Dic83C2314yes—
72C2×C4⋊Dic83C2314yes—
73C2×Dic83⋊C4—24no—
74C2×Dic83⋊C4—24no—
75C2×Dic83⋊C4—44no—
76C2×Dic83⋊C4C2314yes—
77C2×Dic83⋊C4C2314yes—
78C2×Dic83⋊C4C2318yes—
79C2×(C2×D83)⋊C4—28no—
80C2×(C2×D83)⋊C4—28no—
81C2×(C2×D83)⋊C4—28no—
82C2×(C2×D83)⋊C4—44no—
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—24no—
90C2×C4×Dic83D418yes—
91C2×C4×Dic83C2314yes—
92C2×C4×Dic83C2314yes—
93C83⋊#95C2318yes—
94C83⋊#95C2318yes—
95C83⋊#95C2318yes—
96C23×C4×C83C2311yescharacteristic
97C23×C4×C83C2312yes—
98C23×C4×C83C2312yes—
99C23×C4×C83C2311yescharacteristic
100C23×C4×C83C2311yescharacteristic
101C2×C4⋊C4×C83—24no—
102C2×C4⋊C4×C83—24no—
103C2×C4⋊C4×C83—44no—
104C2×C4⋊C4×C83C2314yes—
105C2×C4⋊C4×C83C2314yes—
106C2×C4⋊C4×C83C2318yes—
107C2×C42×C83—24no—
108C2×C42×C83C2314yes—
109#95×C83C2318yes—
110C23×D83—24no—
111C23×D83—24no—
112C23×D83—24no—
113C23×D83—24no—
114C23×D83—24no—
115C23×D83—24no—
116C23×D83C2×Q818yes—
117C23×D83C2414yes—
118C2×Dic166—232no—
119C2×Dic166—28no—
120C2×Dic166—28no—
121C2×Dic166—28no—
122C2×D332—232no—
123C2×D332—28no—
124C2×D332—28no—
125C2×D332C2×Q8116yescomplemented
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—22no—
129C22×Dic83—22no—
130C22×Dic83—22no—
131C22×Dic83—22no—
132C22×Dic83—22no—
133C22×Dic83—24no—
134C22×Dic83—24no—
135C22×Dic83—22no—
136C22×Dic83—24no—
137C22×Dic83—22no—
138C22×Dic83C2×D414yes—
139C22×Dic83C2×D414yes—
140C22×Dic83#4114yes—
141C22×Dic83C2414yes—
142C22×Dic83C2412yes—
143C22×Dic83C2412yes—
144C22×Dic83C2412yes—
145C22×Dic83C2412yes—
146C2×C4×D83—216no—
147C2×C4×D83—48no—
148C2×C4×D83—48no—
149C2×C4×D83—432no—
150C4⋊Dic83—28no—
151C4⋊Dic83—28no—
152Dic83⋊C4—28no—
153Dic83⋊C4—28no—
154Dic83⋊C4—48no—
155(C2×D83)⋊C4—216no—
156(C2×D83)⋊C4—216no—
157(C2×D83)⋊C4—216no—
158(C2×D83)⋊C4—48no—
159(C2×D83)⋊C4—48no—
160C4×Dic83—28no—
161C4×Dic83C2×D4116yes—
162C24×C83C2411yescharacteristic
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—24no—
166C22×C4×C83—22no—
167C22×C4×C83—24no—
168C22×C4×C83—24no—
169C22×C4×C83—24no—
170C22×C4×C83—24no—
171C22×C4×C83—22no—
172C22×C4×C83—24no—
173C22×C4×C83—22no—
174C22×C4×C83C2×D414yes—
175C22×C4×C83C2×Q814yes—
176C22×C4×C83#4114yes—
177C22×C4×C83#4114yes—
178C22×C4×C83C2414yes—
179C22×C4×C83C2414yes—
180C22×C4×C83C2412yes—
181C22×C4×C83C2412yes—
182C22×C4×C83C2412yes—
183C4⋊C4×C83—28no—
184C4⋊C4×C83—28no—
185C4⋊C4×C83—48no—
186C42×C83—28no—
187C23×C83C22×D412yes—
188C23×C83C22×Q812yes—
189C23×C83C2×#4112yes—
190C23×C83C2×#4112yes—
191C23×C83C2×#4112yes—
192C23×C83H(5,𝔽2)12yes—
193C23×C83H(5,𝔽2)12yes—
194C23×C83C2511yescharacteristic
195C22×D83—216no—
196C22×D83—216no—
197C22×D83—216no—
198C22×D83—24no—
199C22×D83—24no—
200C22×D83—24no—
201C22×D83—24no—
202C22×D83—24no—
203C22×D83—24no—
204C22×D83—48no—
205C22×D83—48no—
206C22×D83—48no—
207C22×D83—48no—
208C22×D83C22×Q818yes—
209C2×C4×C83—28no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—22no—
213C2×C4×C83—24no—
214C2×C4×C83—24no—
215C2×C4×C83—24no—
216C2×C4×C83—22no—
217C2×C4×C83—24no—
218C2×C4×C83—24no—
219C2×C4×C83—22no—
220C2×C4×C83—24no—
221C2×C4×C83—44no—
222C2×C4×C83—48no—
223C2×C4×C83—48no—
224C2×C4×C83C22⋊Q8116yes—
225C2×C4×C83C22×D414yes—
226C2×C4×C83C22×Q814yes—
227C2×C4×C83C2×#4114yes—
228C2×C4×C83C2×#4114yes—
229C2×Dic83—28no—
230C2×Dic83—28no—
231C2×Dic83—28no—
232C2×Dic83—22no—
233C2×Dic83—22no—
234C2×Dic83—22no—
235C2×Dic83—22no—
236C2×Dic83—22no—
237C2×Dic83—22no—
238C2×Dic83—22no—
239C2×Dic83—24no—
240C2×Dic83—24no—
241C2×Dic83—22no—
242C2×Dic83—24no—
243C2×Dic83—22no—
244C2×Dic83—44no—
245C2×Dic83—48no—
246C2×Dic83C22×D414yes—
247C2×Dic83C22×D414yes—
248C2×Dic83C2×#4114yes—
249D332—232no—
250C4×D83—432no—
251Dic166—232no—
252C22×C83C2×C22⋊Q814yes—
253C22×C83#52514yes—
254C22×C83#53514yes—
255C22×C83#55414yes—
256C22×C83#55414yes—
257C22×C83#55914yes—
258C22×C83#55914yes—
259C22×C83C23×D411yescharacteristic
260C22×C83C23×Q811yescharacteristic
261C22×C83C22×#4111yescharacteristic
262C22×C83C22×#4111yescharacteristic
263C22×C83C22×#4111yescharacteristic
264C22×C83C2×H(5,𝔽2)11yescharacteristic
265C22×C83C2×H(5,𝔽2)11yescharacteristic
266C2×D83—28no—
267C2×D83—28no—
268C2×D83—28no—
269C2×D83—432no—
270C2×D83—44no—
271C2×D83—44no—
272C2×D83—44no—
273C2×D83—44no—
274C4×C83—24no—
275C4×C83—44no—
276C4×C83—42no—
277C4×C83—44no—
278C4×C83C2×C22⋊Q818yes—
279Dic83—24no—
280Dic83—24no—
281Dic83—24no—
282Dic83—42no—
283Dic83—44no—
284C2×#2580—831nomaximal, Sylow
285C2×C83#258018yescomplemented
286C2×C83C22×C22⋊Q811yescharacteristic
287C2×C83C2×#52511yescharacteristic
288C2×C83C2×#53511yescharacteristic
289C2×C83C2×#55411yescharacteristic
290C2×C83C2×#55411yescharacteristic
291C2×C83C2×#55911yescharacteristic
292C2×C83C2×#55911yescharacteristic
293D83—48no—
294C22×C4×Q8—831no—
295C22×C4×D4—831no—
296#2580—8316no—
297C2×#399—832no—
298C2×#393—832no—
299C2×#390—831no—
300C2×#390—831no—
301C2×#389—831no—
302C2×#389D8311yescharacteristic, complemented
303C2×#385—832no—
304C2×#385—832no—
305C2×#382—831no—
306C83C2×#258011yescharacteristic, complemented, Sylow
307C23×Q8—831no—
308C23×D4—831no—
309C24×C4—832no—
310C2×C4×Q8—838no—
311C2×C4×Q8—1668no—
312C2×C4×D4—838no—
313C2×C4×D4—1668no—
314C22×C4⋊C4—832no—
315C22×C4⋊C4—831no—
316C22×C4⋊C4—831no—
317C22×C4⋊C4—831no—
318C22×C4⋊C4—831no—
319C22×C4⋊C4C2×D8311yescharacteristic
320C22×C4⋊C4C2×D8311yescharacteristic
321C22×C4⋊C4C2×D8312yes—
322C22×C22⋊C4—832no—
323C22×C22⋊C4—832no—
324C22×C22⋊C4—832no—
325C22×C22⋊C4—832no—
326C22×C22⋊C4—832no—
327C22×C42—831no—
328C22×C42—831no—
329C22×C42C2×D8311yescharacteristic
330#399—8316no—
331#393—8316no—
332#390—838no—
333#390—838no—
334#389—838no—
335#389C2×D8318yescomplemented
336#385—8316no—
337#385—8316no—
338#382—838no—
339C2×#95—832no—
340C2×#95—832no—
341C2×#95—832no—
342C2×#95C2×D8312yes—
343C25—832no—
344C22×Q8—834no—
345C22×Q8—1664no—
346C22×Q8—1664no—
347C22×Q8—1664no—
348C22×D4—838no—
349C22×D4—834no—
350C22×D4—1664no—
351C22×D4—1664no—
352C23×C4—3324no—
353C23×C4—3324no—
354C23×C4—838no—
355C23×C4—831no—
356C23×C4—831no—
357C23×C4—831no—
358C23×C4—831no—
359C23×C4—832no—
360C23×C4—1668no—
361C23×C4C22×D8311yescharacteristic
362C23×C4C22×D8312yes—
363C23×C4C22×D8312yes—
364C23×C4C22×D8311yescharacteristic
365C23×C4C22×D8311yescharacteristic
366C4×Q8—16632no—
367C4×D4—16632no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—44no—
371C2×C4⋊C4—3324no—
372C2×C4⋊C4—838no—
373C2×C4⋊C4—834no—
374C2×C4⋊C4—834no—
375C2×C4⋊C4—834no—
376C2×C4⋊C4—834no—
377C2×C4⋊C4—1664no—
378C2×C4⋊C4—1664no—
379C2×C4⋊C4—1664no—
380C2×C4⋊C4—1664no—
381C2×C4⋊C4C22×D8314yes—
382C2×C4⋊C4C22×D8314yes—
383C2×C4⋊C4C22×D8318yes—
384C2×C22⋊C4—3324no—
385C2×C22⋊C4—3324no—
386C2×C22⋊C4—838no—
387C2×C22⋊C4—838no—
388C2×C22⋊C4—838no—
389C2×C22⋊C4—838no—
390C2×C22⋊C4—838no—
391C2×C22⋊C4—1668no—
392C2×C22⋊C4—1668no—
393C2×C22⋊C4—1668no—
394C2×C42—24no—
395C2×C42—838no—
396C2×C42—834no—
397C2×C42—834no—
398C2×C42—1664no—
399C2×C42C22×D8314yes—
400#95—838no—
401#95—838no—
402#95—838no—
403#95C22×D8318yes—
404C24—838no—
405C24—834no—
406C24—1664no—
407C24—1664no—
408C24—1664no—
409C24—1664no—
410C24—1664no—
411C24—1664no—
412C24C23×D8311yescharacteristic
413C2×Q8—1668no—
414C2×Q8—1668no—
415C2×Q8—16632no—
416C2×Q8—1668no—
417C2×D4—8316no—
418C2×D4—16632no—
419C2×D4—1668no—
420C2×D4—1668no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—3328no—
433C22×C4—3328no—
434C22×C4—33232no—
435C22×C4—834no—
436C22×C4—834no—
437C22×C4—834no—
438C22×C4—832no—
439C22×C4—832no—
440C22×C4—832no—
441C22×C4—834no—
442C22×C4—832no—
443C22×C4—16616no—
444C22×C4—1662no—
445C22×C4—1664no—
446C22×C4—1664no—
447C22×C4—1662no—
448C22×C4—1664no—
449C22×C4—1662no—
450C22×C4—1662no—
451C22×C4—1662no—
452C22×C4—1662no—
453C22×C4—1662no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4D4×D8314yes—
457C22×C4Q8×D8314yes—
458C22×C4#1305059014yes—
459C22×C4#1305059114yes—
460C22×C4C23×D8314yes—
461C22×C4C23×D8314yes—
462C22×C4C23×D8312yes—
463C22×C4C23×D8312yes—
464C22×C4C23×D8312yes—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—48no—
468C4⋊C4—3328no—
469C4⋊C4—1668no—
470C4⋊C4—1668no—
471C4⋊C4—1668no—
472C4⋊C4—1668no—
473C22⋊C4—3328no—
474C22⋊C4—3328no—
475C22⋊C4—16616no—
476C22⋊C4—16616no—
477C22⋊C4—16616no—
478C42—28no—
479C42—8316no—
480C42—1668no—
481C23—3328no—
482C23—3328no—
483C23—3328no—
484C23—3328no—
485C23—838no—
486C23—16616no—
487C23—16616no—
488C23—16616no—
489C23—1664no—
490C23—1664no—
491C23—1664no—
492C23—1664no—
493C23—1664no—
494C23—1664no—
495C23C2×D4×D8312yes—
496C23C2×Q8×D8312yes—
497C23#41×D8312yes—
498C23C2×#1305059012yes—
499C23C2×#1305059112yes—
500C23#1458183712yes—
501C23#1458183812yes—
502C23C24×D8311yescharacteristic
503Q8—16632no—
504D4—16632no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—33232no—
521C2×C4—3328no—
522C2×C4—3324no—
523C2×C4—834no—
524C2×C4—834no—
525C2×C4—834no—
526C2×C4—1662no—
527C2×C4—1664no—
528C2×C4—1664no—
529C2×C4—1662no—
530C2×C4—1664no—
531C2×C4—1662no—
532C2×C4—1668no—
533C2×C4—1668no—
534C2×C4—1662no—
535C2×C4—1668no—
536C2×C4—1662no—
537C2×C4—1662no—
538C2×C4—1662no—
539C2×C4—1662no—
540C2×C4—1662no—
541C2×C4#14581770116yes—
542C2×C4C2×D4×D8314yes—
543C2×C4C2×Q8×D8314yes—
544C2×C4C2×#1305059014yes—
545C2×C4C2×#1305059114yes—
546C22—3324no—
547C22—33232no—
548C22—3324no—
549C22—3324no—
550C22—3324no—
551C22—1668no—
552C22—1668no—
553C22—1668no—
554C22C2×#1458177014yes—
555C22#1912821414yes—
556C22#1912826714yes—
557C22#1912840714yes—
558C22#1912841214yes—
559C22#1912843414yes—
560C22#1912843614yes—
561C22C22×D4×D8311yescharacteristic
562C22C22×Q8×D8311yescharacteristic
563C22C2×#41×D8311yescharacteristic
564C22C22×#1305059011yescharacteristic
565C22C22×#1305059111yescharacteristic
566C22C2×#1458183711yescharacteristic
567C22C2×#1458183811yescharacteristic
568C4—24no—
569C4—44no—
570C4—42no—
571C4—44no—
572C4—3322no—
573C4—3324no—
574C4—1664no—
575C4—1664no—
576C4—1664no—
577C4C2×#1458177018yes—
578C2—3328no—
579C2#2491282318yesdirect product factor
580C2C22×#1458177011yescharacteristic
581C2C2×#1912821411yescharacteristic
582C2C2×#1912826711yescharacteristic
583C2C2×#1912840711yescharacteristic
584C2C2×#1912841211yescharacteristic
585C2C2×#1912843411yescharacteristic
586C2C2×#1912843611yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#389×C83C2
socleC24×C83C24
prime radicalC23×D332C22
hypercenterC2×#389D83
hypocenterC83C2×#2580
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic166C2semidirect
3. #24912823C2direct (N × H)
4. C2×#19127636C2semidirect
5. C2×#19127614C2semidirect
8. C2×#19127597C2semidirect
9. C2×#19127575C2semidirect
10. C2×#19127572C2semidirect
11. C2×#19127566C2semidirect
12. C2×#389×C83C2semidirect
17. C2×C4×Dic166C22semidirect
32. #19127636C22semidirect
33. #19127614C22semidirect
36. #19127597C22semidirect
37. #19127575C22semidirect
38. #19127572C22semidirect
39. #19127566C22semidirect
47. #389×C83C22semidirect
56. C22×D332Q8semidirect
125. C2×D332C2×Q8semidirect
285. C2×C83#2580semidirect
302. C2×#389D83semidirect
306. C83C2×#2580semidirect
335. #389C2×D83semidirect
579. C2#24912823direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Dic1662
C22×#145817542
#249128232
C2×#191276362
C2×#191276142
C2×#191276032
C2×#191276022
C2×#191275972
C2×#191275752
C2×#191275722
C2×#191275662
C2×#389×C832
C2×#258083