Finite Group Databasebeta

← C2×#24912383 #35576172

Subgroups — C2×#24912383 #35576172

Order 21248.

3614 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×#14581769C212yesmaximal, complemented
2C22×C4×Dic166C211yesmaximal, characteristic, complemented
3C22×D4×Dic83C211yesmaximal, characteristic
4#24912383C2116yesmaximal, direct product factor
5C2×#19127624C212yesmaximal, complemented
6C2×#19127624C212yesmaximal, complemented
7C2×#19127604C211yesmaximal, characteristic
8C2×#19127600C211yesmaximal, characteristic, complemented
9C2×#19127599C211yesmaximal, characteristic, complemented
10C2×#19127570C211yesmaximal, characteristic, complemented
11C2×#19127559C212yesmaximal, complemented
12C2×#390×C83C211yesmaximal, characteristic
13C23×Dic166C2211yescharacteristic
14C24×Dic83C2212yes—
15C2×#14581769—216no—
16C2×#14581769C22116yescomplemented
17C2×C4×Dic166—28no—
18C2×C4×Dic166C2218yescomplemented
19C2×D4×Dic83—28no—
20C2×D4×Dic83C2218yes—
21C22×C4⋊Dic83C2211yescharacteristic
22C22×C4⋊Dic83C2211yescharacteristic
23C22×C4⋊Dic83C2211yescharacteristic
24C22×C4⋊Dic83C2212yes—
25C22×Dic83⋊C4C2211yescharacteristic
26C22×Dic83⋊C4C2212yes—
27C22×Dic83⋊C4C2211yescharacteristic
28C22×Dic83⋊C4C2212yes—
29C22×Dic83⋊C4C2212yes—
30C22×C22⋊Dic83C2212yes—
31C22×C22⋊Dic83C2212yes—
32C22×C22⋊Dic83C2212yes—
33C22×C4×Dic83C2211yescharacteristic
34C22×C4×Dic83C2211yescharacteristic
35#19127624C22116yescomplemented
36#19127624C22116yescomplemented
37#19127604C2218yes—
38#19127600C2218yescomplemented
39#19127599C2218yescomplemented
40#19127570C2218yescomplemented
41#19127559C22116yescomplemented
42C2×C83⋊#95C2212yes—
43C2×C83⋊#95C2212yes—
44C23×D4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2211yescharacteristic
46C22×C22⋊C4×C83C2212yes—
47C22×C22⋊C4×C83C2212yes—
48C22×C42×C83C2211yescharacteristic
49#390×C83C2218yes—
50C22×Dic166—24no—
51C22×Dic166—24no—
52C22×Dic166D418yescomplemented
53C22×Dic166C2314yes—
54C23×Dic83—28no—
55C23×Dic83—28no—
56C23×Dic83—28no—
57C23×Dic83C2318yes—
58C23×Dic83C2311yescharacteristic
59C23×Dic83C2312yes—
60C23×Dic83C2312yes—
61C23×Dic83C2311yescharacteristic
62C23×Dic83C2311yescharacteristic
63C23×Dic83C2311yescharacteristic
64#14581769—264no—
65C4×Dic166—232no—
66D4×Dic83—232no—
67C2×C4⋊Dic83—24no—
68C2×C4⋊Dic83—28no—
69C2×C4⋊Dic83—24no—
70C2×C4⋊Dic83—24no—
71C2×C4⋊Dic83C2314yes—
72C2×C4⋊Dic83C2314yes—
73C2×C4⋊Dic83C2318yes—
74C2×C4⋊Dic83C2314yes—
75C2×Dic83⋊C4—24no—
76C2×Dic83⋊C4—24no—
77C2×Dic83⋊C4—28no—
78C2×Dic83⋊C4—28no—
79C2×Dic83⋊C4—44no—
80C2×Dic83⋊C4C2314yes—
81C2×Dic83⋊C4C2318yes—
82C2×Dic83⋊C4C2318yes—
83C2×Dic83⋊C4C2314yes—
84C2×Dic83⋊C4C2318yes—
85C2×C22⋊Dic83—28no—
86C2×C22⋊Dic83—28no—
87C2×C22⋊Dic83—44no—
88C2×C22⋊Dic83C2318yes—
89C2×C22⋊Dic83C2318yes—
90C2×C22⋊Dic83C2318yes—
91C2×C4×Dic83—24no—
92C2×C4×Dic83D418yes—
93C2×C4×Dic83C2314yes—
94C2×C4×Dic83C2314yes—
95C83⋊#95C2318yes—
96C83⋊#95C2318yes—
97C25×C83C2312yes—
98C22×D4×C83—24no—
99C22×D4×C83—24no—
100C22×D4×C83Q818yescomplemented
101C22×D4×C83C2314yes—
102C23×C4×C83C2311yescharacteristic
103C23×C4×C83C2312yes—
104C23×C4×C83C2311yescharacteristic
105C23×C4×C83C2311yescharacteristic
106C2×C4⋊C4×C83—24no—
107C2×C4⋊C4×C83C2314yes—
108C2×C22⋊C4×C83—28no—
109C2×C22⋊C4×C83—44no—
110C2×C22⋊C4×C83C2318yes—
111C2×C22⋊C4×C83C2318yes—
112C2×C42×C83—24no—
113C2×C42×C83C2314yes—
114C2×Dic166—232no—
115C2×Dic166—28no—
116C2×Dic166—28no—
117C2×Dic166C2×D4116yescomplemented
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—22no—
122C22×Dic83—24no—
123C22×Dic83—24no—
124C22×Dic83—22no—
125C22×Dic83—24no—
126C22×Dic83—24no—
127C22×Dic83—24no—
128C22×Dic83—22no—
129C22×Dic83—24no—
130C22×Dic83—22no—
131C22×Dic83—216no—
132C22×Dic83—216no—
133C22×Dic83—216no—
134C22×Dic83—432no—
135C22×Dic83C2×D414yes—
136C22×Dic83C2×D414yes—
137C22×Dic83C2×D414yes—
138C22×Dic83C2×D414yes—
139C22×Dic83#4114yes—
140C22×Dic83C2414yes—
141C22×Dic83C2412yes—
142C22×Dic83C2414yes—
143C22×Dic83C2412yes—
144C22×Dic83C2412yes—
145C22×Dic83C2412yes—
146C4⋊Dic83—28no—
147C4⋊Dic83—28no—
148C4⋊Dic83—216no—
149C4⋊Dic83—28no—
150Dic83⋊C4—216no—
151Dic83⋊C4—216no—
152Dic83⋊C4—28no—
153Dic83⋊C4—28no—
154Dic83⋊C4—48no—
155C22⋊Dic83—216no—
156C22⋊Dic83—216no—
157C22⋊Dic83—48no—
158C4×Dic83—28no—
159C4×Dic83C2×D4116yes—
160C24×C83—24no—
161C24×C83—24no—
162C24×C83—24no—
163C24×C83—24no—
164C24×C83—24no—
165C24×C83—24no—
166C24×C83C2×Q818yes—
167C24×C83C2414yes—
168C24×C83C2411yescharacteristic
169C2×D4×C83—232no—
170C2×D4×C83—28no—
171C2×D4×C83—28no—
172C2×D4×C83C2×Q8116yescomplemented
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—24no—
177C22×C4×C83—24no—
178C22×C4×C83—22no—
179C22×C4×C83—24no—
180C22×C4×C83—22no—
181C22×C4×C83C2×D414yes—
182C22×C4×C83C2×D414yes—
183C22×C4×C83C2×Q814yes—
184C22×C4×C83#4114yes—
185C22×C4×C83C2412yes—
186C22×C4×C83C2414yes—
187C22×C4×C83C2412yes—
188C22×C4×C83C2412yes—
189C4⋊C4×C83—28no—
190C22⋊C4×C83—216no—
191C22⋊C4×C83—48no—
192C42×C83—28no—
193C23×C83—216no—
194C23×C83—216no—
195C23×C83—216no—
196C23×C83—24no—
197C23×C83—24no—
198C23×C83—24no—
199C23×C83—24no—
200C23×C83—24no—
201C23×C83—24no—
202C23×C83—48no—
203C23×C83—48no—
204C23×C83—48no—
205C23×C83—48no—
206C23×C83C22×D412yes—
207C23×C83C22×D412yes—
208C23×C83C22×Q818yes—
209C23×C83C22×Q812yes—
210C23×C83C2×#4112yes—
211C23×C83C2×#4112yes—
212C23×C83H(5,𝔽2)12yes—
213C23×C832−1+412yes—
214C23×C83C2511yescharacteristic
215D4×C83—232no—
216C2×C4×C83—28no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—24no—
221C2×C4×C83—24no—
222C2×C4×C83—22no—
223C2×C4×C83—24no—
224C2×C4×C83—22no—
225C2×C4×C83—44no—
226C2×C4×C83—48no—
227C2×C4×C83C22⋊Q8116yescomplemented
228C2×C4×C83C22×D414yes—
229C2×C4×C83C22×D414yes—
230C2×C4×C83C22×Q814yes—
231C2×C4×C83C2×#4114yes—
232C2×Dic83—28no—
233C2×Dic83—28no—
234C2×Dic83—22no—
235C2×Dic83—22no—
236C2×Dic83—22no—
237C2×Dic83—22no—
238C2×Dic83—22no—
239C2×Dic83—24no—
240C2×Dic83—24no—
241C2×Dic83—24no—
242C2×Dic83—24no—
243C2×Dic83—24no—
244C2×Dic83—22no—
245C2×Dic83—24no—
246C2×Dic83—22no—
247C2×Dic83—48no—
248C2×Dic83—44no—
249C2×Dic83—48no—
250C2×Dic83—432no—
251C2×Dic83#121116yes—
252C2×Dic83C22×D414yes—
253C2×Dic83C22×D414yes—
254C2×Dic83C22×D414yes—
255C2×Dic83C22×D414yes—
256C2×Dic83C2×#4114yes—
257Dic166—232no—
258C22×C83—28no—
259C22×C83—28no—
260C22×C83—28no—
261C22×C83—44no—
262C22×C83—44no—
263C22×C83—44no—
264C22×C83—44no—
265C22×C83—432no—
266C22×C83C2×#12114yes—
267C22×C83C2×C22⋊Q814yes—
268C22×C83#53714yes—
269C22×C83D4×Q814yes—
270C22×C83#55014yes—
271C22×C83#55414yes—
272C22×C83#55614yes—
273C22×C83C23×D411yescharacteristic
274C22×C83C23×D411yescharacteristic
275C22×C83C23×Q811yescharacteristic
276C22×C83C22×#4111yescharacteristic
277C22×C83C22×#4111yescharacteristic
278C22×C83C2×H(5,𝔽2)11yescharacteristic
279C22×C83C2×2−1+411yescharacteristic
280C4×C83—24no—
281C4×C83—42no—
282C4×C83—44no—
283C4×C83C2×C22⋊Q818yescomplemented
284Dic83—24no—
285Dic83—24no—
286Dic83—44no—
287Dic83—42no—
288Dic83—44no—
289Dic83C2×#12118yes—
290C2×#2538—831nomaximal, Sylow
291C2×C83—48no—
292C2×C83#253818yescomplemented
293C2×C83C22×#12111yescharacteristic
294C2×C83C22×C22⋊Q811yescharacteristic
295C2×C83C2×#53711yescharacteristic
296C2×C83C2×D4×Q811yescharacteristic
297C2×C83C2×#55011yescharacteristic
298C2×C83C2×#55411yescharacteristic
299C2×C83C2×#55611yescharacteristic
300C22×C22⋊Q8—832no—
301C22×C4×Q8—831no—
302C22×C4×D4—831no—
303#2538—8316no—
304C2×#396—832no—
305C2×#396—832no—
306C2×#390—831no—
307C2×#390D8311yescharacteristic
308C2×#389—831no—
309C2×#389—831no—
310C2×#384—831no—
311C2×#380—832no—
312C83C2×#253811yescharacteristic, complemented, Sylow
313C23×Q8—831no—
314C23×D4C2×D8311yescharacteristic
315C24×C4—832no—
316C2×C22⋊Q8—8316no—
317C2×C22⋊Q8—16616no—
318C2×C4×Q8—838no—
319C2×C4×Q8—1668no—
320C2×C4×D4—838no—
321C2×C4×D4—1668no—
322C22×C4⋊C4—832no—
323C22×C4⋊C4—831no—
324C22×C4⋊C4—831no—
325C22×C4⋊C4—832no—
326C22×C4⋊C4—832no—
327C22×C4⋊C4—831no—
328C22×C4⋊C4—832no—
329C22×C4⋊C4—831no—
330C22×C4⋊C4—831no—
331C22×C4⋊C4C2×D8311yescharacteristic
332C22×C22⋊C4—832no—
333C22×C22⋊C4—832no—
334C22×C22⋊C4—832no—
335C22×C22⋊C4C2×D8312yes—
336C22×C22⋊C4C2×D8312yes—
337C22×C42—831no—
338C22×C42—831no—
339C22×C42C2×D8311yescharacteristic
340#396—8316no—
341#396—8316no—
342#390—838no—
343#390C2×D8318yes—
344#389—838no—
345#389—838no—
346#384—838no—
347#380—8316no—
348C2×#95—832no—
349C2×#95—832no—
350C25C22×D8312yes—
351C22×Q8—838no—
352C22×Q8—834no—
353C22×Q8—1664no—
354C22×Q8—1664no—
355C22×D4—24no—
356C22×D4—24no—
357C22×D4Dic16618yescomplemented
358C22×D4C22×D8314yes—
359C23×C4—831no—
360C23×C4—831no—
361C23×C4—831no—
362C23×C4—838no—
363C23×C4—832no—
364C23×C4—831no—
365C23×C4—832no—
366C23×C4—1668no—
367C23×C4—1668no—
368C23×C4—1668no—
369C23×C4C22×D8312yes—
370C23×C4C22×D8311yescharacteristic
371C23×C4C22×D8311yescharacteristic
372C23×C4C22×D8311yescharacteristic
373C22⋊Q8—16664no—
374C4×Q8—16632no—
375C4×D4—16632no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—3324no—
378C2×C4⋊C4—838no—
379C2×C4⋊C4—834no—
380C2×C4⋊C4—834no—
381C2×C4⋊C4—834no—
382C2×C4⋊C4—834no—
383C2×C4⋊C4—838no—
384C2×C4⋊C4—838no—
385C2×C4⋊C4—838no—
386C2×C4⋊C4—834no—
387C2×C4⋊C4—1664no—
388C2×C4⋊C4—1664no—
389C2×C4⋊C4—1664no—
390C2×C4⋊C4—1668no—
391C2×C4⋊C4—1668no—
392C2×C4⋊C4—1664no—
393C2×C4⋊C4—1668no—
394C2×C4⋊C4—1664no—
395C2×C4⋊C4C22×D8314yes—
396C2×C22⋊C4—28no—
397C2×C22⋊C4—44no—
398C2×C22⋊C4—3324no—
399C2×C22⋊C4—838no—
400C2×C22⋊C4—838no—
401C2×C22⋊C4—838no—
402C2×C22⋊C4—1668no—
403C2×C22⋊C4—1668no—
404C2×C22⋊C4C22×D8318yes—
405C2×C22⋊C4C22×D8318yes—
406C2×C42—24no—
407C2×C42—838no—
408C2×C42—834no—
409C2×C42—834no—
410C2×C42—1664no—
411C2×C42C22×D8314yes—
412#95—838no—
413#95—838no—
414C24—24no—
415C24—24no—
416C24—24no—
417C24—24no—
418C24—24no—
419C24—24no—
420C24C2×Dic16618yes—
421C24C23×D8314yes—
422C24C23×D8311yescharacteristic
423C2×Q8—8316no—
424C2×Q8—1668no—
425C2×Q8—16632no—
426C2×Q8—1668no—
427C2×D4—232no—
428C2×D4—28no—
429C2×D4—28no—
430C2×D4C2×Dic166116yescomplemented
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—33232no—
440C22×C4—834no—
441C22×C4—834no—
442C22×C4—834no—
443C22×C4—832no—
444C22×C4—834no—
445C22×C4—834no—
446C22×C4—832no—
447C22×C4—832no—
448C22×C4—832no—
449C22×C4—834no—
450C22×C4—834no—
451C22×C4—16616no—
452C22×C4—16616no—
453C22×C4—16616no—
454C22×C4—1662no—
455C22×C4—1664no—
456C22×C4—1664no—
457C22×C4—1664no—
458C22×C4—1664no—
459C22×C4—1664no—
460C22×C4—1662no—
461C22×C4—1664no—
462C22×C4—1662no—
463C22×C4—1662no—
464C22×C4—1662no—
465C22×C4—1662no—
466C22×C4—1662no—
467C22×C4D4×D8314yes—
468C22×C4D4×D8314yes—
469C22×C4C2×Dic16614yes—
470C22×C4#1305058914yes—
471C22×C4C23×D8312yes—
472C22×C4C23×D8314yes—
473C22×C4C23×D8312yes—
474C22×C4C23×D8312yes—
475C4⋊C4—28no—
476C4⋊C4—3328no—
477C4⋊C4—1668no—
478C4⋊C4—1668no—
479C4⋊C4—1668no—
480C4⋊C4—16616no—
481C4⋊C4—16616no—
482C4⋊C4—1668no—
483C4⋊C4—16616no—
484C4⋊C4—1668no—
485C22⋊C4—216no—
486C22⋊C4—48no—
487C22⋊C4—3328no—
488C22⋊C4—16616no—
489C22⋊C4—16616no—
490C42—28no—
491C42—8316no—
492C42—1668no—
493C23—216no—
494C23—216no—
495C23—216no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23C2×D4×D8312yes—
507C23C2×D4×D8312yes—
508C23C22×Dic16618yes—
509C23C22×Dic16612yes—
510C23#41×D8312yes—
511C23C2×#1305058912yes—
512C23#1458183712yes—
513C23#1458183912yes—
514C23C24×D8311yescharacteristic
515Q8—16632no—
516D4—232no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—3328no—
529C2×C4—3328no—
530C2×C4—3324no—
531C2×C4—33232no—
532C2×C4—8316no—
533C2×C4—834no—
534C2×C4—834no—
535C2×C4—834no—
536C2×C4—834no—
537C2×C4—834no—
538C2×C4—1662no—
539C2×C4—1664no—
540C2×C4—1664no—
541C2×C4—1664no—
542C2×C4—1664no—
543C2×C4—1664no—
544C2×C4—1662no—
545C2×C4—1664no—
546C2×C4—1662no—
547C2×C4—1668no—
548C2×C4—1662no—
549C2×C4—1662no—
550C2×C4—1662no—
551C2×C4—1662no—
552C2×C4—1668no—
553C2×C4#14581769116yescomplemented
554C2×C4C2×D4×D8314yes—
555C2×C4C2×D4×D8314yes—
556C2×C4C22×Dic16614yes—
557C2×C4C2×#1305058914yes—
558C22—28no—
559C22—28no—
560C22—28no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—432no—
566C22#121×D8314yes—
567C22C2×#1458176914yes—
568C22#1912828014yes—
569C22D4×Dic16614yes—
570C22#1912837114yes—
571C22#1912841014yes—
572C22#1912842014yes—
573C22C22×D4×D8311yescharacteristic
574C22C22×D4×D8311yescharacteristic
575C22C23×Dic16611yescharacteristic
576C22C2×#41×D8311yescharacteristic
577C22C22×#1305058911yescharacteristic
578C22C2×#1458183711yescharacteristic
579C22C2×#1458183911yescharacteristic
580C4—24no—
581C4—44no—
582C4—42no—
583C4—3324no—
584C4—3324no—
585C4—3322no—
586C4—838no—
587C4—1664no—
588C4—1664no—
589C4C2×#1458176918yescomplemented
590C2—48no—
591C2#2491238318yesdirect product factor
592C2C2×#121×D8311yescharacteristic
593C2C22×#1458176911yescharacteristic
594C2C2×#1912828011yescharacteristic
595C2C2×D4×Dic16611yescharacteristic
596C2C2×#1912837111yescharacteristic
597C2C2×#1912841011yescharacteristic
598C2C2×#1912842011yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#390×C83C2
socleC24×C83C24
prime radicalC23×D4×C83C22
hypercenterC2×#390D83
hypocenterC83C2×#2538
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C22×#14581769C2semidirect
2. C22×C4×Dic166C2semidirect
4. #24912383C2direct (N × H)
5. C2×#19127624C2semidirect
6. C2×#19127624C2semidirect
8. C2×#19127600C2semidirect
9. C2×#19127599C2semidirect
10. C2×#19127570C2semidirect
11. C2×#19127559C2semidirect
16. C2×#14581769C22semidirect
18. C2×C4×Dic166C22semidirect
35. #19127624C22semidirect
36. #19127624C22semidirect
38. #19127600C22semidirect
39. #19127599C22semidirect
40. #19127570C22semidirect
41. #19127559C22semidirect
52. C22×Dic166D4semidirect
100. C22×D4×C83Q8semidirect
117. C2×Dic166C2×D4semidirect
172. C2×D4×C83C2×Q8semidirect
227. C2×C4×C83C22⋊Q8semidirect
283. C4×C83C2×C22⋊Q8semidirect
292. C2×C83#2538semidirect
312. C83C2×#2538semidirect
357. C22×D4Dic166semidirect
430. C2×D4C2×Dic166semidirect
553. C2×C4#14581769semidirect
589. C4C2×#14581769semidirect
591. C2#24912383direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817692
C22×C4×Dic1662
C22×D4×Dic832
#249123832
C2×#191276242
C2×#191276042
C2×#191276002
C2×#191275992
C2×#191275702
C2×#191275592
C2×#390×C832
C2×#253883