Finite Group Databasebeta

← C2×#24911064 #35572425

Subgroups — C2×#24911064 #35572425

Order 21248.

3246 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581790C211yesmaximal, characteristic, complemented
2C22×#14581771C211yesmaximal, characteristic
3C22×Q8×Dic83C211yesmaximal, characteristic, complemented
4#24911064C2116yesmaximal, direct product factor
5C2×#19127634C212yesmaximal, complemented
6C2×#19127624C212yesmaximal
7C2×#19127606C211yesmaximal, characteristic, complemented
8C2×#19127597C211yesmaximal, characteristic, complemented
9C2×#19127596C212yesmaximal, complemented
10C2×#19127595C212yesmaximal
11C2×C4×C22⋊Dic83C211yesmaximal, characteristic
12C22×C22⋊Q8×C83C211yesmaximal, characteristic
13C23×Dic166C2211yescharacteristic
14C2×#14581790—28no—
15C2×#14581790C2218yescomplemented
16C2×#14581771—28no—
17C2×#14581771C2218yes—
18C2×Q8×Dic83—28no—
19C2×Q8×Dic83C2218yescomplemented
20C22×C4⋊Dic83C2212yes—
21C22×C4⋊Dic83C2211yescharacteristic
22C22×Dic83⋊C4C2212yes—
23C22×Dic83⋊C4C2212yes—
24C22×Dic83⋊C4C2212yes—
25C22×Dic83⋊C4C2212yes—
26C22×C22⋊Dic83C2212yes—
27C22×C22⋊Dic83C2211yescharacteristic
28C22×C22⋊Dic83C2211yescharacteristic
29C22×C4×Dic83C2212yes—
30C22×C4×Dic83C2211yescharacteristic
31C22×C4×Dic83C2211yescharacteristic
32#19127634C22116yescomplemented
33#19127624C22116yes—
34#19127606C2218yescomplemented
35#19127597C2218yescomplemented
36#19127596C22116yescomplemented
37#19127595C22116yes—
38C4×C22⋊Dic83C2218yes—
39C2×C83⋊#95C2212yes—
40C2×C83⋊#95C2212yes—
41C2×C83⋊#95C2211yescharacteristic
42C2×C83⋊#95C2212yes—
43C2×C83⋊#95C2211yescharacteristic
44C23×Q8×C83C2211yescharacteristic
45C24×C4×C83C2211yescharacteristic
46C2×C22⋊Q8×C83—28no—
47C2×C22⋊Q8×C83C2218yes—
48C22×C4⋊C4×C83C2212yes—
49C22×C4⋊C4×C83C2211yescharacteristic
50C22×C22⋊C4×C83C2212yes—
51C22×Dic166—24no—
52C22×Dic166—24no—
53C22×Dic166—24no—
54C22×Dic166C2314yes—
55C23×Dic83C2311yescharacteristic
56C23×Dic83C2312yes—
57C23×Dic83C2312yes—
58C23×Dic83C2311yescharacteristic
59C23×Dic83C2311yescharacteristic
60C23×Dic83C2311yescharacteristic
61#14581790—232no—
62#14581771—232no—
63Q8×Dic83—232no—
64C2×C4⋊Dic83—28no—
65C2×C4⋊Dic83D418yes—
66C2×C4⋊Dic83C2318yes—
67C2×C4⋊Dic83C2314yes—
68C2×Dic83⋊C4—28no—
69C2×Dic83⋊C4—28no—
70C2×Dic83⋊C4—28no—
71C2×Dic83⋊C4—28no—
72C2×Dic83⋊C4C2318yes—
73C2×Dic83⋊C4C2318yes—
74C2×Dic83⋊C4C2318yes—
75C2×Dic83⋊C4C2318yes—
76C2×C22⋊Dic83—28no—
77C2×C22⋊Dic83—24no—
78C2×C22⋊Dic83Q818yescomplemented
79C2×C22⋊Dic83C2318yes—
80C2×C22⋊Dic83C2314yes—
81C2×C22⋊Dic83C2314yes—
82C2×C4×Dic83—28no—
83C2×C4×Dic83—24no—
84C2×C4×Dic83—24no—
85C2×C4×Dic83C2318yes—
86C2×C4×Dic83C2314yes—
87C2×C4×Dic83C2314yes—
88C83⋊#95C2318yes—
89C83⋊#95C2318yes—
90C83⋊#95C2314yes—
91C83⋊#95C2318yes—
92C83⋊#95C2314yes—
93C25×C83C2311yescharacteristic
94C22×Q8×C83—24no—
95C22×Q8×C83—24no—
96C22×Q8×C83D418yes—
97C22×Q8×C83C2314yes—
98C23×C4×C83—24no—
99C23×C4×C83—24no—
100C23×C4×C83—24no—
101C23×C4×C83C2312yes—
102C23×C4×C83C2311yescharacteristic
103C23×C4×C83C2314yes—
104C23×C4×C83C2312yes—
105C23×C4×C83C2311yescharacteristic
106C22⋊Q8×C83—232no—
107C2×C4⋊C4×C83—24no—
108C2×C4⋊C4×C83—28no—
109C2×C4⋊C4×C83C2318yes—
110C2×C4⋊C4×C83C2314yes—
111C2×C22⋊C4×C83—28no—
112C2×C22⋊C4×C83C2318yes—
113C2×Dic166—28no—
114C2×Dic166—28no—
115C2×Dic166—28no—
116C2×Dic166—416no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—24no—
121C22×Dic83—22no—
122C22×Dic83—22no—
123C22×Dic83—24no—
124C22×Dic83—22no—
125C22×Dic83—24no—
126C22×Dic83—24no—
127C22×Dic83—22no—
128C22×Dic83—24no—
129C22×Dic83—22no—
130C22×Dic83—22no—
131C22×Dic83C2×D418yes—
132C22×Dic83C2×Q814yes—
133C22×Dic83C2×Q814yes—
134C22×Dic83#4114yes—
135C22×Dic83C2414yes—
136C22×Dic83C2412yes—
137C22×Dic83C2414yes—
138C22×Dic83C2412yes—
139C22×Dic83C2412yes—
140C22×Dic83C2412yes—
141C4⋊Dic83—216no—
142C4⋊Dic83C2×D4116yes—
143Dic83⋊C4—216no—
144Dic83⋊C4—216no—
145Dic83⋊C4—216no—
146Dic83⋊C4—216no—
147C22⋊Dic83—216no—
148C22⋊Dic83—28no—
149C22⋊Dic83C2×Q8116yescomplemented
150C4×Dic83—28no—
151C4×Dic83—28no—
152C4×Dic83—216no—
153C24×C83—22no—
154C24×C83—22no—
155C24×C83—22no—
156C24×C83—22no—
157C24×C83—22no—
158C24×C83—22no—
159C24×C83C2×Q814yes—
160C24×C83C2412yes—
161C24×C83C2411yescharacteristic
162C2×Q8×C83—232no—
163C2×Q8×C83—28no—
164C2×Q8×C83—28no—
165C2×Q8×C83C2×D4116yes—
166C22×C4×C83—24no—
167C22×C4×C83—24no—
168C22×C4×C83—24no—
169C22×C4×C83—22no—
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—24no—
173C22×C4×C83—24no—
174C22×C4×C83—28no—
175C22×C4×C83—28no—
176C22×C4×C83—28no—
177C22×C4×C83—416no—
178C22×C4×C83C2×D414yes—
179C22×C4×C83C2×D418yes—
180C22×C4×C83C2×D414yes—
181C22×C4×C83#4114yes—
182C22×C4×C83C2414yes—
183C22×C4×C83C2412yes—
184C22×C4×C83C2414yes—
185C22×C4×C83C2412yes—
186C4⋊C4×C83—28no—
187C4⋊C4×C83—216no—
188C22⋊C4×C83—216no—
189C23×C83—28no—
190C23×C83—28no—
191C23×C83—28no—
192C23×C83—22no—
193C23×C83—22no—
194C23×C83—22no—
195C23×C83—22no—
196C23×C83—22no—
197C23×C83—22no—
198C23×C83—44no—
199C23×C83—44no—
200C23×C83—44no—
201C23×C83—44no—
202C23×C83C22×D412yes—
203C23×C83C22×D412yes—
204C23×C83C22×Q814yes—
205C23×C83C22×Q812yes—
206C23×C83C2×#4112yes—
207C23×C83C2×#4112yes—
208C23×C832−1+412yes—
209C23×C832−1+412yes—
210C23×C83C2511yescharacteristic
211Q8×C83—232no—
212C2×C4×C83—216no—
213C2×C4×C83—24no—
214C2×C4×C83—24no—
215C2×C4×C83—24no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—24no—
220C2×C4×C83—24no—
221C2×C4×C83—44no—
222C2×C4×C83—48no—
223C2×C4×C83—416no—
224C2×C4×C83#121116yes—
225C2×C4×C83C22×D414yes—
226C2×C4×C83C22×D418yes—
227C2×C4×C83C22×D414yes—
228C2×C4×C83C2×#4114yes—
229C2×Dic83—216no—
230C2×Dic83—28no—
231C2×Dic83—28no—
232C2×Dic83—28no—
233C2×Dic83—22no—
234C2×Dic83—22no—
235C2×Dic83—22no—
236C2×Dic83—24no—
237C2×Dic83—22no—
238C2×Dic83—22no—
239C2×Dic83—24no—
240C2×Dic83—22no—
241C2×Dic83—22no—
242C2×Dic83—24no—
243C2×Dic83—24no—
244C2×Dic83—22no—
245C2×Dic83—24no—
246C2×Dic83—22no—
247C2×Dic83—48no—
248C2×Dic83—44no—
249C2×Dic83C22×D418yes—
250C2×Dic83C22×Q814yes—
251C2×Dic83C22×Q814yes—
252C2×Dic83C2×#4114yes—
253Dic166—416no—
254C22×C83—24no—
255C22×C83—24no—
256C22×C83—24no—
257C22×C83—42no—
258C22×C83—416no—
259C22×C83—42no—
260C22×C83—42no—
261C22×C83—42no—
262C22×C83C2×#12114yes—
263C22×C83#53314yes—
264C22×C83#54114yes—
265C22×C83#54714yes—
266C22×C83#54814yes—
267C22×C83D4×Q814yes—
268C22×C83D4×Q814yes—
269C22×C83C23×D411yescharacteristic
270C22×C83C23×D411yescharacteristic
271C22×C83C23×Q811yescharacteristic
272C22×C83C22×#4111yescharacteristic
273C22×C83C22×#4111yescharacteristic
274C22×C83C2×2−1+411yescharacteristic
275C22×C83C2×2−1+411yescharacteristic
276C4×C83—28no—
277C4×C83—42no—
278C4×C83—44no—
279C4×C83C2×#12118yes—
280Dic83—28no—
281Dic83—24no—
282Dic83—24no—
283Dic83—24no—
284Dic83—44no—
285Dic83—42no—
286C2×#2429—831nomaximal, Sylow
287C2×C83—44no—
288C2×C83#242918yescomplemented
289C2×C83C22×#12111yescharacteristic
290C2×C83C2×#53311yescharacteristic
291C2×C83C2×#54111yescharacteristic
292C2×C83C2×#54711yescharacteristic
293C2×C83C2×#54811yescharacteristic
294C2×C83C2×D4×Q811yescharacteristic
295C2×C83C2×D4×Q811yescharacteristic
296C22×C4⋊Q8—831no—
297C22×C22⋊Q8—831no—
298C22×C22⋊Q8D8311yescharacteristic
299C22×C4×Q8—831no—
300#2429—8316no—
301C2×#398—832no—
302C2×#396—832no—
303C2×#391—831no—
304C2×#389—832no—
305C2×#389—831no—
306C2×#388—832no—
307C2×C4×C22⋊C4—831no—
308C83C2×#242911yescharacteristic, complemented, Sylow
309C23×Q8—831no—
310C23×Q8C2×D8311yescharacteristic
311C24×C4C2×D8311yescharacteristic
312C2×C4⋊Q8—838no—
313C2×C4⋊Q8—1668no—
314C2×C22⋊Q8—28no—
315C2×C22⋊Q8—838no—
316C2×C22⋊Q8—1668no—
317C2×C22⋊Q8C2×D8318yes—
318C2×C4×Q8—838no—
319C2×C4×Q8—1668no—
320C22×C4⋊C4—831no—
321C22×C4⋊C4—832no—
322C22×C4⋊C4—832no—
323C22×C4⋊C4—832no—
324C22×C4⋊C4—832no—
325C22×C4⋊C4—832no—
326C22×C4⋊C4C2×D8312yes—
327C22×C4⋊C4C2×D8311yescharacteristic
328C22×C22⋊C4—831no—
329C22×C22⋊C4—831no—
330C22×C22⋊C4—832no—
331C22×C22⋊C4C2×D8312yes—
332C22×C42—831no—
333C22×C42—832no—
334C22×C42—831no—
335#398—8316no—
336#396—8316no—
337#391—838no—
338#389—8316no—
339#389—838no—
340#388—8316no—
341C4×C22⋊C4—838no—
342C2×#95—832no—
343C2×#95—831no—
344C2×#95—832no—
345C2×#95—831no—
346C2×#95—832no—
347C25C22×D8311yescharacteristic
348C22×Q8—24no—
349C22×Q8—24no—
350C22×Q8—834no—
351C22×Q8—1664no—
352C22×Q8—1664no—
353C22×Q8—1664no—
354C22×Q8C22⋊D8318yes—
355C22×Q8C22×D8314yes—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—832no—
360C23×C4—831no—
361C23×C4—831no—
362C23×C4—831no—
363C23×C4—832no—
364C23×C4—831no—
365C23×C4C22×D8314yes—
366C23×C4C22×D8312yes—
367C23×C4C22×D8311yescharacteristic
368C23×C4C22×D8312yes—
369C23×C4C22×D8311yescharacteristic
370C4⋊Q8—16632no—
371C22⋊Q8—232no—
372C22⋊Q8—16632no—
373C4×Q8—16632no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—28no—
376C2×C4⋊C4—838no—
377C2×C4⋊C4—834no—
378C2×C4⋊C4—838no—
379C2×C4⋊C4—838no—
380C2×C4⋊C4—838no—
381C2×C4⋊C4—838no—
382C2×C4⋊C4—838no—
383C2×C4⋊C4—1668no—
384C2×C4⋊C4—1668no—
385C2×C4⋊C4—1668no—
386C2×C4⋊C4—1668no—
387C2×C4⋊C4—1668no—
388C2×C4⋊C4C22×D8314yes—
389C2×C4⋊C4C22×D8318yes—
390C2×C22⋊C4—28no—
391C2×C22⋊C4—838no—
392C2×C22⋊C4—834no—
393C2×C22⋊C4—838no—
394C2×C22⋊C4—834no—
395C2×C22⋊C4—1668no—
396C2×C22⋊C4—1664no—
397C2×C22⋊C4C22×D8318yes—
398C2×C42—834no—
399C2×C42—838no—
400C2×C42—834no—
401C2×C42—1668no—
402C2×C42—1664no—
403C2×C42—1664no—
404#95—834no—
405#95—834no—
406#95—838no—
407#95—838no—
408#95—838no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24Q8×D8314yes—
416C24C23×D8312yes—
417C24C23×D8311yescharacteristic
418C2×Q8—232no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—33216no—
422C2×Q8—1668no—
423C2×Q8—1668no—
424C2×Q8—1668no—
425C2×Q8C2×C22⋊D83116yes—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—416no—
438C22×C4—838no—
439C22×C4—834no—
440C22×C4—834no—
441C22×C4—834no—
442C22×C4—832no—
443C22×C4—832no—
444C22×C4—834no—
445C22×C4—832no—
446C22×C4—832no—
447C22×C4—834no—
448C22×C4—1662no—
449C22×C4—1664no—
450C22×C4—1664no—
451C22×C4—1662no—
452C22×C4—1664no—
453C22×C4—1662no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4—1662no—
457C22×C4—1664no—
458C22×C4—1662no—
459C22×C4—1662no—
460C22×C4—1664no—
461C22×C4—1662no—
462C22×C4D4×D8314yes—
463C22×C4C2×C22⋊D8318yes—
464C22×C4C2×C22⋊D8314yes—
465C22×C4#1305058914yes—
466C22×C4C23×D8314yes—
467C22×C4C23×D8312yes—
468C22×C4C23×D8314yes—
469C22×C4C23×D8312yes—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—8316no—
473C4⋊C4—16616no—
474C4⋊C4—16616no—
475C4⋊C4—16616no—
476C4⋊C4—16616no—
477C4⋊C4—16616no—
478C22⋊C4—216no—
479C22⋊C4—8316no—
480C22⋊C4—16616no—
481C22⋊C4—1668no—
482C42—16616no—
483C42—1668no—
484C42—1668no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23C2×D4×D8312yes—
499C23C22×C22⋊D8312yes—
500C23C2×Q8×D8314yes—
501C23C2×Q8×D8312yes—
502C23#41×D8312yes—
503C23C2×#1305058912yes—
504C23#1458183912yes—
505C23#1458184012yes—
506C23C24×D8311yescharacteristic
507Q8—232no—
508Q8—33216no—
509C2×C4—216no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—416no—
521C2×C4—3324no—
522C2×C4—3328no—
523C2×C4—838no—
524C2×C4—834no—
525C2×C4—834no—
526C2×C4—834no—
527C2×C4—1662no—
528C2×C4—1664no—
529C2×C4—1664no—
530C2×C4—1662no—
531C2×C4—1664no—
532C2×C4—1662no—
533C2×C4—16616no—
534C2×C4—1668no—
535C2×C4—1668no—
536C2×C4—1668no—
537C2×C4—1662no—
538C2×C4—1662no—
539C2×C4—1662no—
540C2×C4—1664no—
541C2×C4—1662no—
542C2×C4—1662no—
543C2×C4—1664no—
544C2×C4—1662no—
545C2×C4#14581764116yes—
546C2×C4C2×D4×D8314yes—
547C2×C4C22×C22⋊D8318yes—
548C2×C4C22×C22⋊D8314yes—
549C2×C4C2×#1305058914yes—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—416no—
557C22—42no—
558C22C2×#1458176414yes—
559C22#1912825314yes—
560C22#1912831114yes—
561C22#1912835314yes—
562C22#1912835814yes—
563C22#1912836414yes—
564C22Q8×C22⋊D8314yes—
565C22C22×D4×D8311yescharacteristic
566C22C23×C22⋊D8311yescharacteristic
567C22C22×Q8×D8311yescharacteristic
568C22C2×#41×D8311yescharacteristic
569C22C22×#1305058911yescharacteristic
570C22C2×#1458183911yescharacteristic
571C22C2×#1458184011yescharacteristic
572C4—28no—
573C4—44no—
574C4—42no—
575C4—3322no—
576C4—3324no—
577C4—1668no—
578C4—1664no—
579C4—1664no—
580C4—1664no—
581C4C2×#1458176418yes—
582C2—44no—
583C2#2491106418yesdirect product factor
584C2C22×#1458176411yescharacteristic
585C2C2×#1912825311yescharacteristic
586C2C2×#1912831111yescharacteristic
587C2C2×#1912835311yescharacteristic
588C2C2×#1912835811yescharacteristic
589C2C2×#1912836411yescharacteristic
590C2C2×Q8×C22⋊D8311yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC22×C22⋊Q8×C83C2
socleC24×C83C24
prime radicalC25×C83C23
hypercenterC22×C22⋊Q8D83
hypocenterC83C2×#2429
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14581790C2semidirect
3. C22×Q8×Dic83C2semidirect
4. #24911064C2direct (N × H)
5. C2×#19127634C2semidirect
7. C2×#19127606C2semidirect
8. C2×#19127597C2semidirect
9. C2×#19127596C2semidirect
15. C2×#14581790C22semidirect
19. C2×Q8×Dic83C22semidirect
32. #19127634C22semidirect
34. #19127606C22semidirect
35. #19127597C22semidirect
36. #19127596C22semidirect
78. C2×C22⋊Dic83Q8semidirect
149. C22⋊Dic83C2×Q8semidirect
288. C2×C83#2429semidirect
308. C83C2×#2429semidirect
583. C2#24911064direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817902
C22×#145817712
C22×Q8×Dic832
#249110642
C2×#191276342
C2×#191276242
C2×#191276062
C2×#191275972
C2×#191275962
C2×#191275952
C2×C4×C22⋊Dic832
C22×C22⋊Q8×C832
C2×#242983