Finite Group Databasebeta

← C2×#24911065 #35572427

Subgroups — C2×#24911065 #35572427

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×#14581768C211yesmaximal, characteristic
2C22×#14581768C211yesmaximal, characteristic
3C22×Q8×Dic83C211yesmaximal, characteristic, complemented
4#24911065C2116yesmaximal, direct product factor
5C2×#19127634C212yesmaximal, complemented
6C2×#19127623C212yesmaximal
7C2×#19127608C211yesmaximal, characteristic, complemented
8C2×#19127599C211yesmaximal, characteristic, complemented
9C2×#19127596C212yesmaximal, complemented
10C2×#19127591C212yesmaximal
11C2×C4×(C2×D83)⋊C4C211yesmaximal, characteristic
12C22×C4⋊Q8×C83C211yesmaximal, characteristic, complemented
13C23×C4×D83C2211yescharacteristic
14C2×#14581768—28no—
15C2×#14581768—28no—
16C2×#14581768C2218yes—
17C2×#14581768C2218yes—
18C2×Q8×Dic83—28no—
19C2×Q8×Dic83C2218yescomplemented
20C22×C4⋊Dic83C2212yes—
21C22×C4⋊Dic83C2211yescharacteristic
22C22×C4⋊Dic83C2211yescharacteristic
23C22×Dic83⋊C4C2212yes—
24C22×Dic83⋊C4C2212yes—
25C22×Dic83⋊C4C2212yes—
26C22×(C2×D83)⋊C4C2212yes—
27C22×(C2×D83)⋊C4C2212yes—
28C22×(C2×D83)⋊C4C2211yescharacteristic
29C22×(C2×D83)⋊C4C2211yescharacteristic
30C22×C4×Dic83C2212yes—
31C22×C4×Dic83C2211yescharacteristic
32#19127634C22116yescomplemented
33#19127623C22116yes—
34#19127608C2218yescomplemented
35#19127599C2218yescomplemented
36#19127596C22116yescomplemented
37#19127591C22116yes—
38C4×(C2×D83)⋊C4C2218yes—
39C2×C83⋊#95C2212yes—
40C2×C83⋊#95C2212yes—
41C2×C83⋊#95C2211yescharacteristic
42C2×C83⋊#95C2212yes—
43C2×C83⋊#95C2211yescharacteristic
44C23×Q8×C83C2211yescharacteristic
45C23×Q8×C83C2211yescharacteristic
46C2×C4⋊Q8×C83—28no—
47C2×C4⋊Q8×C83C2218yescomplemented
48C22×C4⋊C4×C83C2212yes—
49C22×C4⋊C4×C83C2212yes—
50C22×C42×C83C2211yescharacteristic
51C24×D83C2311yescharacteristic
52C23×Dic83C2311yescharacteristic
53C23×Dic83C2312yes—
54C23×Dic83C2311yescharacteristic
55C23×Dic83C2312yes—
56C23×Dic83C2311yescharacteristic
57C22×C4×D83—24no—
58C22×C4×D83—24no—
59C22×C4×D83—24no—
60C22×C4×D83C2314yes—
61#14581768—232no—
62#14581768—232no—
63Q8×Dic83—232no—
64C2×C4⋊Dic83—28no—
65C2×C4⋊Dic83—24no—
66C2×C4⋊Dic83D418yes—
67C2×C4⋊Dic83C2314yes—
68C2×C4⋊Dic83C2318yes—
69C2×C4⋊Dic83C2314yes—
70C2×Dic83⋊C4—28no—
71C2×Dic83⋊C4—28no—
72C2×Dic83⋊C4—28no—
73C2×Dic83⋊C4C2318yes—
74C2×Dic83⋊C4C2318yes—
75C2×Dic83⋊C4C2318yes—
76C2×(C2×D83)⋊C4—28no—
77C2×(C2×D83)⋊C4—28no—
78C2×(C2×D83)⋊C4—24no—
79C2×(C2×D83)⋊C4Q818yescomplemented
80C2×(C2×D83)⋊C4C2318yes—
81C2×(C2×D83)⋊C4C2318yes—
82C2×(C2×D83)⋊C4C2314yes—
83C2×(C2×D83)⋊C4C2314yes—
84C2×C4×Dic83—28no—
85C2×C4×Dic83—24no—
86C2×C4×Dic83C2318yes—
87C2×C4×Dic83C2314yes—
88C83⋊#95C2318yes—
89C83⋊#95C2318yes—
90C83⋊#95C2314yes—
91C83⋊#95C2318yes—
92C83⋊#95C2314yes—
93C22×Q8×C83—24no—
94C22×Q8×C83—24no—
95C22×Q8×C83—24no—
96C22×Q8×C83—24no—
97C22×Q8×C83—24no—
98C22×Q8×C83D418yes—
99C22×Q8×C83C2314yes—
100C22×Q8×C83C2314yes—
101C23×C4×C83C2312yes—
102C23×C4×C83C2312yes—
103C23×C4×C83C2311yescharacteristic
104C23×C4×C83C2311yescharacteristic
105C23×C4×C83C2311yescharacteristic
106C4⋊Q8×C83—232no—
107C2×C4⋊C4×C83—28no—
108C2×C4⋊C4×C83—28no—
109C2×C4⋊C4×C83C2318yes—
110C2×C4⋊C4×C83C2318yes—
111C2×C42×C83—24no—
112C2×C42×C83C2314yes—
113C23×D83—22no—
114C23×D83—22no—
115C23×D83—22no—
116C23×D83—22no—
117C23×D83—22no—
118C23×D83—22no—
119C23×D83C2×Q814yes—
120C23×D83C2412yes—
121C22×Dic83—22no—
122C22×Dic83—24no—
123C22×Dic83—24no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—24no—
129C22×Dic83—22no—
130C22×Dic83—22no—
131C22×Dic83—24no—
132C22×Dic83—22no—
133C22×Dic83—24no—
134C22×Dic83C2×D418yes—
135C22×Dic83C2×Q814yes—
136C22×Dic83C2412yes—
137C22×Dic83C2412yes—
138C22×Dic83C2414yes—
139C22×Dic83C2414yes—
140C22×Dic83C2412yes—
141C2×C4×D83—28no—
142C2×C4×D83—28no—
143C2×C4×D83—28no—
144C2×C4×D83—416no—
145C4⋊Dic83—28no—
146C4⋊Dic83—216no—
147C4⋊Dic83C2×D4116yes—
148Dic83⋊C4—216no—
149Dic83⋊C4—216no—
150Dic83⋊C4—216no—
151(C2×D83)⋊C4—216no—
152(C2×D83)⋊C4—28no—
153(C2×D83)⋊C4—216no—
154(C2×D83)⋊C4C2×Q8116yescomplemented
155C4×Dic83—28no—
156C4×Dic83—216no—
157C24×C83C2411yescharacteristic
158C2×Q8×C83—232no—
159C2×Q8×C83—28no—
160C2×Q8×C83—28no—
161C2×Q8×C83—28no—
162C2×Q8×C83—28no—
163C2×Q8×C83—28no—
164C2×Q8×C83—416no—
165C2×Q8×C83C2×D4116yes—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—24no—
169C22×C4×C83—24no—
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—24no—
173C22×C4×C83—24no—
174C22×C4×C83—24no—
175C22×C4×C83C2×D414yes—
176C22×C4×C83C2×D418yes—
177C22×C4×C83C2×D414yes—
178C22×C4×C83C2×Q814yes—
179C22×C4×C83#4114yes—
180C22×C4×C83#4114yes—
181C22×C4×C83C2414yes—
182C22×C4×C83C2414yes—
183C22×C4×C83C2412yes—
184C22×C4×C83C2412yes—
185C22×C4×C83C2412yes—
186C4⋊C4×C83—216no—
187C4⋊C4×C83—216no—
188C42×C83—28no—
189C23×C83C22×D412yes—
190C23×C83C22×D412yes—
191C23×C83C22×Q812yes—
192C23×C83C2×#4112yes—
193C23×C83C2×#4112yes—
194C23×C832−1+412yes—
195C23×C832−1+412yes—
196C23×C83C2511yescharacteristic
197C22×D83—28no—
198C22×D83—28no—
199C22×D83—28no—
200C22×D83—22no—
201C22×D83—22no—
202C22×D83—22no—
203C22×D83—22no—
204C22×D83—22no—
205C22×D83—22no—
206C22×D83—44no—
207C22×D83—44no—
208C22×D83—44no—
209C22×D83—44no—
210C22×D83C22×Q814yes—
211Q8×C83—232no—
212Q8×C83—416no—
213C2×C4×C83—216no—
214C2×C4×C83—28no—
215C2×C4×C83—28no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—24no—
219C2×C4×C83—24no—
220C2×C4×C83—22no—
221C2×C4×C83—22no—
222C2×C4×C83—24no—
223C2×C4×C83—24no—
224C2×C4×C83—24no—
225C2×C4×C83—48no—
226C2×C4×C83#121116yes—
227C2×C4×C83C22×D414yes—
228C2×C4×C83C22×D418yes—
229C2×C4×C83C22×D414yes—
230C2×C4×C83C22×Q814yes—
231C2×C4×C83C2×#4114yes—
232C2×C4×C83C2×#4114yes—
233C2×Dic83—216no—
234C2×Dic83—28no—
235C2×Dic83—22no—
236C2×Dic83—24no—
237C2×Dic83—24no—
238C2×Dic83—22no—
239C2×Dic83—22no—
240C2×Dic83—24no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—22no—
244C2×Dic83—24no—
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—24no—
248C2×Dic83—44no—
249C2×Dic83—44no—
250C2×Dic83—48no—
251C2×Dic83C22×D418yes—
252C2×Dic83C22×Q814yes—
253C4×D83—416no—
254C22×C83C2×#12114yes—
255C22×C83#53314yes—
256C22×C83#54114yes—
257C22×C83#54714yes—
258C22×C83#54814yes—
259C22×C83D4×Q814yes—
260C22×C83D4×Q814yes—
261C22×C83C23×D411yescharacteristic
262C22×C83C23×D411yescharacteristic
263C22×C83C23×Q811yescharacteristic
264C22×C83C22×#4111yescharacteristic
265C22×C83C22×#4111yescharacteristic
266C22×C83C2×2−1+411yescharacteristic
267C22×C83C2×2−1+411yescharacteristic
268C2×D83—24no—
269C2×D83—24no—
270C2×D83—24no—
271C2×D83—42no—
272C2×D83—416no—
273C2×D83—42no—
274C2×D83—42no—
275C2×D83—42no—
276C4×C83—28no—
277C4×C83—24no—
278C4×C83—24no—
279C4×C83—44no—
280C4×C83C2×#12118yes—
281Dic83—28no—
282Dic83—24no—
283Dic83—42no—
284Dic83—42no—
285Dic83—44no—
286C2×#2429—831nomaximal, Sylow
287C2×C83#242918yescomplemented
288C2×C83C22×#12111yescharacteristic
289C2×C83C2×#53311yescharacteristic
290C2×C83C2×#54111yescharacteristic
291C2×C83C2×#54711yescharacteristic
292C2×C83C2×#54811yescharacteristic
293C2×C83C2×D4×Q811yescharacteristic
294C2×C83C2×D4×Q811yescharacteristic
295D83—44no—
296C22×C4⋊Q8D8311yescharacteristic, complemented
297C22×C22⋊Q8—831no—
298C22×C22⋊Q8—831no—
299C22×C4×Q8—831no—
300#2429—8316no—
301C2×#398—832no—
302C2×#396—832no—
303C2×#391—831no—
304C2×#389—831no—
305C2×#389—832no—
306C2×#388—832no—
307C2×C4×C22⋊C4—831no—
308C83C2×#242911yescharacteristic, complemented, Sylow
309C23×Q8C2×D8311yescharacteristic
310C23×Q8C2×D8311yescharacteristic
311C24×C4—831no—
312C2×C4⋊Q8—28no—
313C2×C4⋊Q8C2×D8318yescomplemented
314C2×C22⋊Q8—838no—
315C2×C22⋊Q8—838no—
316C2×C22⋊Q8—1668no—
317C2×C22⋊Q8—1668no—
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—831no—
326C22×C4⋊C4C2×D8312yes—
327C22×C4⋊C4C2×D8312yes—
328C22×C22⋊C4—831no—
329C22×C22⋊C4—831no—
330C22×C22⋊C4—832no—
331C22×C22⋊C4—832no—
332C22×C42—831no—
333C22×C42—832no—
334C22×C42C2×D8311yescharacteristic
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—
347C25—831no—
348C22×Q8—24no—
349C22×Q8—24no—
350C22×Q8—24no—
351C22×Q8—24no—
352C22×Q8—24no—
353C22×Q8C22⋊D8318yes—
354C22×Q8C22×D8314yes—
355C22×Q8C22×D8314yes—
356C23×C4—832no—
357C23×C4—831no—
358C23×C4—832no—
359C23×C4—831no—
360C23×C4—831no—
361C23×C4—834no—
362C23×C4—1664no—
363C23×C4—1664no—
364C23×C4—1664no—
365C23×C4C22×D8312yes—
366C23×C4C22×D8312yes—
367C23×C4C22×D8311yescharacteristic
368C23×C4C22×D8311yescharacteristic
369C23×C4C22×D8311yescharacteristic
370C4⋊Q8—232no—
371C22⋊Q8—16632no—
372C22⋊Q8—16632no—
373C4×Q8—16632no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—28no—
376C2×C4⋊C4—838no—
377C2×C4⋊C4—834no—
378C2×C4⋊C4—838no—
379C2×C4⋊C4—834no—
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—1664no—
387C2×C4⋊C4—1668no—
388C2×C4⋊C4C22×D8318yes—
389C2×C4⋊C4C22×D8318yes—
390C2×C22⋊C4—838no—
391C2×C22⋊C4—834no—
392C2×C22⋊C4—838no—
393C2×C22⋊C4—838no—
394C2×C22⋊C4—834no—
395C2×C22⋊C4—1668no—
396C2×C22⋊C4—1668no—
397C2×C22⋊C4—1664no—
398C2×C42—24no—
399C2×C42—834no—
400C2×C42—838no—
401C2×C42—1668no—
402C2×C42—1664no—
403C2×C42C22×D8314yes—
404#95—834no—
405#95—834no—
406#95—838no—
407#95—838no—
408#95—838no—
409C24—834no—
410C24—832no—
411C24—1662no—
412C24—1662no—
413C24—1662no—
414C24—1662no—
415C24—1662no—
416C24—1662no—
417C24C23×D8311yescharacteristic
418C2×Q8—232no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—28no—
424C2×Q8—416no—
425C2×Q8C2×C22⋊D83116yes—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—33216no—
436C22×C4—838no—
437C22×C4—834no—
438C22×C4—832no—
439C22×C4—834no—
440C22×C4—832no—
441C22×C4—834no—
442C22×C4—832no—
443C22×C4—1662no—
444C22×C4—1662no—
445C22×C4—1664no—
446C22×C4—1662no—
447C22×C4—1662no—
448C22×C4—1662no—
449C22×C4—1664no—
450C22×C4—1662no—
451C22×C4—1664no—
452C22×C4—1668no—
453C22×C4—1668no—
454C22×C4—1668no—
455C22×C4—1662no—
456C22×C4—1664no—
457C22×C4—1664no—
458C22×C4—1662no—
459C22×C4D4×D8314yes—
460C22×C4C2×C22⋊D8318yes—
461C22×C4C2×C22⋊D8314yes—
462C22×C4Q8×D8314yes—
463C22×C4#1305058914yes—
464C22×C4#1305059114yes—
465C22×C4C23×D8314yes—
466C22×C4C23×D8314yes—
467C22×C4C23×D8312yes—
468C22×C4C23×D8312yes—
469C22×C4C23×D8312yes—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—8316no—
473C4⋊C4—16616no—
474C4⋊C4—16616no—
475C4⋊C4—16616no—
476C4⋊C4—1668no—
477C4⋊C4—16616no—
478C22⋊C4—8316no—
479C22⋊C4—16616no—
480C22⋊C4—16616no—
481C22⋊C4—1668no—
482C42—28no—
483C42—16616no—
484C42—1668no—
485C23—3324no—
486C23—3324no—
487C23—3324no—
488C23—3324no—
489C23—834no—
490C23—1668no—
491C23—1668no—
492C23—1668no—
493C23—1662no—
494C23—1662no—
495C23—1662no—
496C23—1662no—
497C23—1662no—
498C23—1662no—
499C23C2×D4×D8312yes—
500C23C22×C22⋊D8312yes—
501C23C2×Q8×D8312yes—
502C23C2×#1305058912yes—
503C23C2×#1305059112yes—
504C23#1458184012yes—
505C23#1458184012yes—
506C23C24×D8311yescharacteristic
507Q8—232no—
508Q8—416no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—216no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—48no—
522C2×C4—3324no—
523C2×C4—3328no—
524C2×C4—3324no—
525C2×C4—33216no—
526C2×C4—838no—
527C2×C4—834no—
528C2×C4—1662no—
529C2×C4—1662no—
530C2×C4—1664no—
531C2×C4—1662no—
532C2×C4—1662no—
533C2×C4—1662no—
534C2×C4—1664no—
535C2×C4—1662no—
536C2×C4—1664no—
537C2×C4—16616no—
538C2×C4—1668no—
539C2×C4—1662no—
540C2×C4—1664no—
541C2×C4—1664no—
542C2×C4—1662no—
543C2×C4#14581764116yes—
544C2×C4C2×D4×D8314yes—
545C2×C4C22×C22⋊D8318yes—
546C2×C4C22×C22⋊D8314yes—
547C2×C4C2×Q8×D8314yes—
548C2×C4C2×#1305058914yes—
549C2×C4C2×#1305059114yes—
550C22—3322no—
551C22—3322no—
552C22—3322no—
553C22—33216no—
554C22—3322no—
555C22—1664no—
556C22—1664no—
557C22—1664no—
558C22C2×#1458176414yes—
559C22#1912825614yes—
560C22#1912831214yes—
561C22#1912835414yes—
562C22#1912836214yes—
563C22Q8×C22⋊D8314yes—
564C22#1912836614yes—
565C22C22×D4×D8311yescharacteristic
566C22C23×C22⋊D8311yescharacteristic
567C22C22×Q8×D8311yescharacteristic
568C22C22×#1305058911yescharacteristic
569C22C22×#1305059111yescharacteristic
570C22C2×#1458184011yescharacteristic
571C22C2×#1458184011yescharacteristic
572C4—24no—
573C4—24no—
574C4—28no—
575C4—44no—
576C4—3322no—
577C4—3324no—
578C4—3322no—
579C4—1668no—
580C4—1664no—
581C4C2×#1458176418yes—
582C2—3324no—
583C2#2491106518yesdirect product factor
584C2C22×#1458176411yescharacteristic
585C2C2×#1912825611yescharacteristic
586C2C2×#1912831211yescharacteristic
587C2C2×#1912835411yescharacteristic
588C2C2×#1912836211yescharacteristic
589C2C2×Q8×C22⋊D8311yescharacteristic
590C2C2×#1912836611yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC22×C4⋊Q8×C83C2
socleC24×C83C24
prime radicalC24×D83C23
hypercenterC22×C4⋊Q8D83
hypocenterC83C2×#2429
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C22×Q8×Dic83C2semidirect
4. #24911065C2direct (N × H)
5. C2×#19127634C2semidirect
7. C2×#19127608C2semidirect
8. C2×#19127599C2semidirect
9. C2×#19127596C2semidirect
12. C22×C4⋊Q8×C83C2semidirect
19. C2×Q8×Dic83C22semidirect
32. #19127634C22semidirect
34. #19127608C22semidirect
35. #19127599C22semidirect
36. #19127596C22semidirect
47. C2×C4⋊Q8×C83C22semidirect
79. C2×(C2×D83)⋊C4Q8semidirect
154. (C2×D83)⋊C4C2×Q8semidirect
287. C2×C83#2429semidirect
296. C22×C4⋊Q8D83semidirect
308. C83C2×#2429semidirect
313. C2×C4⋊Q8C2×D83semidirect
583. C2#24911065direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817682
C22×Q8×Dic832
#249110652
C2×#191276342
C2×#191276232
C2×#191276082
C2×#191275992
C2×#191275962
C2×#191275912
C2×C4×(C2×D83)⋊C42
C22×C4⋊Q8×C832
C2×#242983