Finite Group Databasebeta

← C2×#24915814 #35610339

Subgroups — C2×#24915814 #35610339

Order 21248.

3790 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581791C211yesmaximal, characteristic, complemented
2C22×#14581770C212yesmaximal
3C22×C42×D83C211yesmaximal, characteristic
4#24915814C2116yesmaximal, direct product factor
5C2×#19127605C212yesmaximal, complemented
6C2×#19127599C211yesmaximal, characteristic, complemented
7C2×#19127596C212yesmaximal, complemented
8C2×#19127570C211yesmaximal, characteristic, complemented
9C2×#19127560C212yesmaximal
10C2×#19127558C211yesmaximal, characteristic
11C2×#19127558C211yesmaximal, characteristic
12C2×#389×C83C211yesmaximal, characteristic, complemented
13C23×Dic166C2212yes—
14C23×C4×D83C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16C23×C4×D83C2211yescharacteristic
17C2×#14581791—28no—
18C2×#14581791C2218yescomplemented
19C2×#14581770—216no—
20C2×#14581770C22116yes—
21C22×C4⋊Dic83C2211yescharacteristic
22C22×C4⋊Dic83C2211yescharacteristic
23C22×Dic83⋊C4C2212yes—
24C22×Dic83⋊C4C2212yes—
25C22×Dic83⋊C4C2212yes—
26C22×Dic83⋊C4C2212yes—
27C22×(C2×D83)⋊C4C2212yes—
28C22×(C2×D83)⋊C4C2212yes—
29C22×C4×Dic83C2211yescharacteristic
30C22×C4×Dic83C2211yescharacteristic
31C22×C4×Dic83C2211yescharacteristic
32C2×C42×D83—28no—
33C2×C42×D83C2218yes—
34#19127605C22116yescomplemented
35#19127599C2218yescomplemented
36#19127596C22116yescomplemented
37#19127570C2218yescomplemented
38#19127560C22116yes—
39#19127558C2218yes—
40#19127558C2218yes—
41C2×C83⋊#95C2212yes—
42C2×C83⋊#95C2212yes—
43C2×C83⋊#95C2212yes—
44C22×C4⋊C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2211yescharacteristic
46C22×C4⋊C4×C83C2212yes—
47C22×C42×C83C2211yescharacteristic
48#389×C83C2218yescomplemented
49C2×#95×C83C2212yes—
50C24×D83C2311yescharacteristic
51C22×Dic166—28no—
52C22×Dic166—28no—
53C22×Dic166—28no—
54C22×Dic166C2318yes—
55C23×Dic83C2312yes—
56C23×Dic83C2312yes—
57C23×Dic83C2311yescharacteristic
58C23×Dic83C2311yescharacteristic
59C23×Dic83C2311yescharacteristic
60C22×C4×D83—24no—
61C22×C4×D83—24no—
62C22×C4×D83—24no—
63C22×C4×D83—24no—
64C22×C4×D83—24no—
65C22×C4×D83—24no—
66C22×C4×D83D418yes—
67C22×C4×D83Q818yescomplemented
68C22×C4×D83Q818yescomplemented
69C22×C4×D83C2314yes—
70C22×C4×D83C2314yes—
71C22×C4×D83C2314yes—
72#14581791—232no—
73#14581770—264no—
74C2×C4⋊Dic83—24no—
75C2×C4⋊Dic83—24no—
76C2×C4⋊Dic83C2314yes—
77C2×C4⋊Dic83C2314yes—
78C2×Dic83⋊C4—28no—
79C2×Dic83⋊C4—28no—
80C2×Dic83⋊C4—28no—
81C2×Dic83⋊C4—28no—
82C2×Dic83⋊C4C2318yes—
83C2×Dic83⋊C4C2318yes—
84C2×Dic83⋊C4C2318yes—
85C2×Dic83⋊C4C2318yes—
86C2×(C2×D83)⋊C4—28no—
87C2×(C2×D83)⋊C4—28no—
88C2×(C2×D83)⋊C4C2318yes—
89C2×(C2×D83)⋊C4C2318yes—
90C2×C4×Dic83—24no—
91C2×C4×Dic83—24no—
92C2×C4×Dic83D418yes—
93C2×C4×Dic83C2314yes—
94C2×C4×Dic83C2314yes—
95C2×C4×Dic83C2314yes—
96C42×D83—232no—
97C83⋊#95C2318yes—
98C83⋊#95C2318yes—
99C83⋊#95C2318yes—
100C23×C4×C83C2312yes—
101C23×C4×C83C2312yes—
102C23×C4×C83C2311yescharacteristic
103C23×C4×C83C2311yescharacteristic
104C23×C4×C83C2311yescharacteristic
105C2×C4⋊C4×C83—28no—
106C2×C4⋊C4×C83—24no—
107C2×C4⋊C4×C83—24no—
108C2×C4⋊C4×C83C2314yes—
109C2×C4⋊C4×C83C2314yes—
110C2×C4⋊C4×C83C2318yes—
111C2×C42×C83—24no—
112C2×C42×C83C2314yes—
113#95×C83C2318yes—
114C23×D83—22no—
115C23×D83—22no—
116C23×D83—22no—
117C23×D83—22no—
118C23×D83C2×D414yes—
119C23×D83C2×Q814yes—
120C23×D83C2×Q814yes—
121C23×D83C2412yes—
122C2×Dic166—216no—
123C2×Dic166—216no—
124C2×Dic166—216no—
125C2×Dic166—432no—
126C22×Dic83—22no—
127C22×Dic83—22no—
128C22×Dic83—24no—
129C22×Dic83—24no—
130C22×Dic83—24no—
131C22×Dic83—24no—
132C22×Dic83—24no—
133C22×Dic83—24no—
134C22×Dic83C2×D414yes—
135C22×Dic83C2×D414yes—
136C22×Dic83C2×D414yes—
137C22×Dic83C2×Q814yes—
138C22×Dic83C2×Q814yes—
139C22×Dic83#4114yes—
140C22×Dic83#4114yes—
141C22×Dic83C2414yes—
142C22×Dic83C2414yes—
143C22×Dic83C2412yes—
144C22×Dic83C2412yes—
145C22×Dic83C2412yes—
146C2×C4×D83—232no—
147C2×C4×D83—28no—
148C2×C4×D83—28no—
149C2×C4×D83—232no—
150C2×C4×D83—232no—
151C2×C4×D83—28no—
152C2×C4×D83—28no—
153C2×C4×D83—28no—
154C2×C4×D83—28no—
155C2×C4×D83C2×D4116yes—
156C2×C4×D83C2×Q8116yescomplemented
157C2×C4×D83C2×Q8116yescomplemented
158C4⋊Dic83—28no—
159C4⋊Dic83—28no—
160Dic83⋊C4—216no—
161Dic83⋊C4—216no—
162Dic83⋊C4—216no—
163Dic83⋊C4—216no—
164(C2×D83)⋊C4—216no—
165(C2×D83)⋊C4—216no—
166C4×Dic83—28no—
167C4×Dic83—28no—
168C4×Dic83C2×D4116yes—
169C24×C83C2411yescharacteristic
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—24no—
175C22×C4×C83—24no—
176C22×C4×C83—24no—
177C22×C4×C83—24no—
178C22×C4×C83—24no—
179C22×C4×C83—24no—
180C22×C4×C83C2×D414yes—
181C22×C4×C83C2×D414yes—
182C22×C4×C83C2×Q814yes—
183C22×C4×C83C2×Q814yes—
184C22×C4×C83#4114yes—
185C22×C4×C83C2414yes—
186C22×C4×C83C2414yes—
187C22×C4×C83C2412yes—
188C22×C4×C83C2412yes—
189C22×C4×C83C2412yes—
190C4⋊C4×C83—216no—
191C4⋊C4×C83—28no—
192C4⋊C4×C83—28no—
193C42×C83—28no—
194C23×C83C22×D412yes—
195C23×C83C22×D412yes—
196C23×C83C22×Q812yes—
197C23×C83C22×Q812yes—
198C23×C83C2×#4112yes—
199C23×C83C2×#4112yes—
200C23×C83C2×#4112yes—
201C23×C83C2511yescharacteristic
202C22×D83—28no—
203C22×D83—28no—
204C22×D83—28no—
205C22×D83—28no—
206C22×D83—28no—
207C22×D83—28no—
208C22×D83—22no—
209C22×D83—22no—
210C22×D83—22no—
211C22×D83—22no—
212C22×D83C4⋊Q8116yescomplemented
213C22×D83C22×D414yes—
214C22×D83C22×Q814yes—
215C22×D83C22×Q814yes—
216C2×C4×C83—28no—
217C2×C4×C83—28no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—22no—
221C2×C4×C83—22no—
222C2×C4×C83—24no—
223C2×C4×C83—24no—
224C2×C4×C83—24no—
225C2×C4×C83—24no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—48no—
229C2×C4×C83—48no—
230C2×C4×C83C22⋊Q8116yescomplemented
231C2×C4×C83C22×D414yes—
232C2×C4×C83C22×D414yes—
233C2×C4×C83C22×Q814yes—
234C2×C4×C83C22×Q814yes—
235C2×C4×C83C2×#4114yes—
236C2×Dic83—28no—
237C2×Dic83—22no—
238C2×Dic83—22no—
239C2×Dic83—24no—
240C2×Dic83—24no—
241C2×Dic83—24no—
242C2×Dic83—24no—
243C2×Dic83—24no—
244C2×Dic83—24no—
245C2×Dic83—48no—
246C2×Dic83—48no—
247C2×Dic83#121116yes—
248C2×Dic83C22⋊Q8116yescomplemented
249C2×Dic83C22×D414yes—
250C2×Dic83C22×D414yes—
251C2×Dic83C22×D414yes—
252C2×Dic83C22×Q814yes—
253C2×Dic83C22×Q814yes—
254C2×Dic83C2×#4114yes—
255C2×Dic83C2×#4114yes—
256C4×D83—232no—
257C4×D83—232no—
258C4×D83—232no—
259Dic166—432no—
260C22×C83C2×#12114yes—
261C22×C83C2×C22⋊Q814yes—
262C22×C83C2×C22⋊Q814yes—
263C22×C83C2×C4⋊Q814yes—
264C22×C83#53214yes—
265C22×C83#53314yes—
266C22×C83#53314yes—
267C22×C83C23×D411yescharacteristic
268C22×C83C23×D411yescharacteristic
269C22×C83C23×Q811yescharacteristic
270C22×C83C23×Q811yescharacteristic
271C22×C83C22×#4111yescharacteristic
272C22×C83C22×#4111yescharacteristic
273C22×C83C22×#4111yescharacteristic
274C2×D83—232no—
275C2×D83—24no—
276C2×D83—24no—
277C2×D83—24no—
278C2×D83—24no—
279C2×D83—24no—
280C2×D83—24no—
281C2×D83C2×C4⋊Q818yescomplemented
282C4×C83—24no—
283C4×C83—24no—
284C4×C83—44no—
285C4×C83—44no—
286C4×C83C2×C22⋊Q818yescomplemented
287Dic83—24no—
288Dic83—44no—
289Dic83—44no—
290Dic83C2×#12118yes—
291Dic83C2×C22⋊Q818yescomplemented
292C2×#2857—831nomaximal, Sylow
293C2×C83#285718yescomplemented
294C2×C83C22×#12111yescharacteristic
295C2×C83C22×C22⋊Q811yescharacteristic
296C2×C83C22×C22⋊Q811yescharacteristic
297C2×C83C22×C4⋊Q811yescharacteristic
298C2×C83C2×#53211yescharacteristic
299C2×C83C2×#53311yescharacteristic
300C2×C83C2×#53311yescharacteristic
301D83—28no—
302C22×C4⋊Q8—831no—
303C22×C22⋊Q8—832no—
304C23×C42—831no—
305#2857—8316no—
306C2×#391—832no—
307C2×#389—831no—
308C2×#389—832no—
309C2×#389D8311yescharacteristic, complemented
310C2×#384—831no—
311C2×#380—832no—
312C2×#380—831no—
313C2×#380—831no—
314C83C2×#285711yescharacteristic, complemented, Sylow
315C23×Q8—832no—
316C24×C4—831no—
317C24×C4—831no—
318C24×C4—831no—
319C2×C4⋊Q8—838no—
320C2×C4⋊Q8—1668no—
321C2×C22⋊Q8—8316no—
322C2×C22⋊Q8—16616no—
323C22×C4⋊C4—832no—
324C22×C4⋊C4—832no—
325C22×C4⋊C4—832no—
326C22×C4⋊C4—832no—
327C22×C4⋊C4—831no—
328C22×C4⋊C4—831no—
329C22×C4⋊C4C2×D8312yes—
330C22×C4⋊C4C2×D8311yescharacteristic
331C22×C4⋊C4C2×D8311yescharacteristic
332C22×C22⋊C4—832no—
333C22×C22⋊C4—832no—
334C22×C42—831no—
335C22×C42—831no—
336C22×C42—831no—
337C22×C42—838no—
338C22×C42—1668no—
339C22×C42C2×D8311yescharacteristic
340#391—8316no—
341#389—8316no—
342#389—838no—
343#389C2×D8318yescomplemented
344#384—838no—
345#380—838no—
346#380—838no—
347#380—8316no—
348C2×#95—832no—
349C2×#95—832no—
350C2×#95—832no—
351C2×#95C2×D8312yes—
352C25—831no—
353C22×Q8—838no—
354C22×Q8—1668no—
355C22×Q8—1668no—
356C22×Q8—1668no—
357C23×C4—831no—
358C23×C4—831no—
359C23×C4—831no—
360C23×C4—838no—
361C23×C4—838no—
362C23×C4—838no—
363C23×C4—834no—
364C23×C4—834no—
365C23×C4—834no—
366C23×C4—832no—
367C23×C4—832no—
368C23×C4—1664no—
369C23×C4—1664no—
370C23×C4—1664no—
371C23×C4—1664no—
372C23×C4—1664no—
373C23×C4—1664no—
374C23×C4C22×D8312yes—
375C23×C4C22×D8312yes—
376C23×C4C22×D8311yescharacteristic
377C23×C4C22×D8311yescharacteristic
378C23×C4C22×D8311yescharacteristic
379C4⋊Q8—16632no—
380C22⋊Q8—16664no—
381C2×C4⋊C4—28no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—838no—
385C2×C4⋊C4—838no—
386C2×C4⋊C4—838no—
387C2×C4⋊C4—834no—
388C2×C4⋊C4—838no—
389C2×C4⋊C4—834no—
390C2×C4⋊C4—1668no—
391C2×C4⋊C4—1668no—
392C2×C4⋊C4—1668no—
393C2×C4⋊C4—1668no—
394C2×C4⋊C4—1664no—
395C2×C4⋊C4—1664no—
396C2×C4⋊C4C22×D8318yes—
397C2×C4⋊C4C22×D8314yes—
398C2×C4⋊C4C22×D8314yes—
399C2×C22⋊C4—838no—
400C2×C22⋊C4—838no—
401C2×C22⋊C4—1668no—
402C2×C22⋊C4—1668no—
403C2×C42—24no—
404C2×C42—838no—
405C2×C42—834no—
406C2×C42—834no—
407C2×C42—834no—
408C2×C42—16632no—
409C2×C42—1664no—
410C2×C42—1664no—
411C2×C42C22×D8314yes—
412#95—838no—
413#95—838no—
414#95—838no—
415#95C22×D8318yes—
416C24—834no—
417C24—834no—
418C24—834no—
419C24—832no—
420C24—1662no—
421C24—1662no—
422C24—1662no—
423C24—1662no—
424C24C23×D8311yescharacteristic
425C2×Q8—33232no—
426C2×Q8—16616no—
427C2×Q8—16616no—
428C2×Q8—16616no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—834no—
440C22×C4—834no—
441C22×C4—834no—
442C22×C4—834no—
443C22×C4—834no—
444C22×C4—834no—
445C22×C4—832no—
446C22×C4—832no—
447C22×C4—834no—
448C22×C4—832no—
449C22×C4—8316no—
450C22×C4—8316no—
451C22×C4—8316no—
452C22×C4—834no—
453C22×C4—834no—
454C22×C4—1668no—
455C22×C4—1668no—
456C22×C4—16632no—
457C22×C4—16632no—
458C22×C4—1668no—
459C22×C4—1668no—
460C22×C4—1664no—
461C22×C4—1664no—
462C22×C4—1664no—
463C22×C4—1664no—
464C22×C4—1664no—
465C22×C4—1664no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—16632no—
469C22×C4—1668no—
470C22×C4—1668no—
471C22×C4D4×D8314yes—
472C22×C4D4×D8314yes—
473C22×C4Q8×D8314yes—
474C22×C4Q8×D8314yes—
475C22×C4#1305059014yes—
476C22×C4C23×D8314yes—
477C22×C4C23×D8314yes—
478C22×C4C23×D8312yes—
479C22×C4C23×D8312yes—
480C22×C4C23×D8312yes—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—16616no—
485C4⋊C4—16616no—
486C4⋊C4—16616no—
487C4⋊C4—16616no—
488C4⋊C4—1668no—
489C4⋊C4—1668no—
490C22⋊C4—16616no—
491C22⋊C4—16616no—
492C42—28no—
493C42—8316no—
494C42—1668no—
495C42—1668no—
496C23—8316no—
497C23—834no—
498C23—834no—
499C23—834no—
500C23—1668no—
501C23—1668no—
502C23—1668no—
503C23—1668no—
504C23—1668no—
505C23—1668no—
506C23—1662no—
507C23—1662no—
508C23—1662no—
509C23—1662no—
510C23C2×D4×D8312yes—
511C23C2×D4×D8312yes—
512C23C2×Q8×D8312yes—
513C23C2×Q8×D8312yes—
514C23#41×D8312yes—
515C23#41×D8312yes—
516C23C2×#1305059012yes—
517C23C24×D8311yescharacteristic
518Q8—33232no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—3328no—
534C2×C4—3328no—
535C2×C4—8316no—
536C2×C4—8316no—
537C2×C4—834no—
538C2×C4—834no—
539C2×C4—834no—
540C2×C4—834no—
541C2×C4—834no—
542C2×C4—834no—
543C2×C4—834no—
544C2×C4—16632no—
545C2×C4—16632no—
546C2×C4—1664no—
547C2×C4—1664no—
548C2×C4—1664no—
549C2×C4—1664no—
550C2×C4—1664no—
551C2×C4—1664no—
552C2×C4—1668no—
553C2×C4—1662no—
554C2×C4—1662no—
555C2×C4—16632no—
556C2×C4#14581770116yescomplemented
557C2×C4C2×D4×D8314yes—
558C2×C4C2×D4×D8314yes—
559C2×C4C2×Q8×D8314yes—
560C2×C4C2×Q8×D8314yes—
561C2×C4C2×#1305059014yes—
562C22—838no—
563C22—16632no—
564C22—1664no—
565C22—1664no—
566C22—1664no—
567C22—1664no—
568C22—1664no—
569C22—1664no—
570C22#121×D8314yes—
571C22C22⋊Q8×D8314yes—
572C22C2×#1458177014yes—
573C22C4⋊Q8×D8314yes—
574C22#1912824814yes—
575C22#1912825514yes—
576C22#1912825714yes—
577C22C22×D4×D8311yescharacteristic
578C22C22×D4×D8311yescharacteristic
579C22C22×Q8×D8311yescharacteristic
580C22C22×Q8×D8311yescharacteristic
581C22C2×#41×D8311yescharacteristic
582C22C2×#41×D8311yescharacteristic
583C22C22×#1305059011yescharacteristic
584C4—24no—
585C4—24no—
586C4—44no—
587C4—44no—
588C4—3324no—
589C4—3324no—
590C4—838no—
591C4—838no—
592C4—1664no—
593C4C2×#1458177018yescomplemented
594C2—1668no—
595C2#2491581418yesdirect product factor
596C2C2×#121×D8311yescharacteristic
597C2C2×C22⋊Q8×D8311yescharacteristic
598C2C22×#1458177011yescharacteristic
599C2C2×C4⋊Q8×D8311yescharacteristic
600C2C2×#1912824811yescharacteristic
601C2C2×#1912825511yescharacteristic
602C2C2×#1912825711yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#389×C83C2
socleC24×C83C24
prime radicalC24×D83C23
hypercenterC2×#389D83
hypocenterC83C2×#2857
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#14581791C2semidirect
4. #24915814C2direct (N × H)
5. C2×#19127605C2semidirect
6. C2×#19127599C2semidirect
7. C2×#19127596C2semidirect
8. C2×#19127570C2semidirect
12. C2×#389×C83C2semidirect
18. C2×#14581791C22semidirect
34. #19127605C22semidirect
35. #19127599C22semidirect
36. #19127596C22semidirect
37. #19127570C22semidirect
48. #389×C83C22semidirect
67. C22×C4×D83Q8semidirect
68. C22×C4×D83Q8semidirect
156. C2×C4×D83C2×Q8semidirect
157. C2×C4×D83C2×Q8semidirect
212. C22×D83C4⋊Q8semidirect
230. C2×C4×C83C22⋊Q8semidirect
248. C2×Dic83C22⋊Q8semidirect
281. C2×D83C2×C4⋊Q8semidirect
286. C4×C83C2×C22⋊Q8semidirect
291. Dic83C2×C22⋊Q8semidirect
293. C2×C83#2857semidirect
309. C2×#389D83semidirect
314. C83C2×#2857semidirect
343. #389C2×D83semidirect
556. C2×C4#14581770semidirect
593. C4C2×#14581770semidirect
595. C2#24915814direct (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×#145817702
C22×C42×D832
#249158142
C2×#191276052
C2×#191275992
C2×#191275962
C2×#191275702
C2×#191275602
C2×#191275582
C2×#389×C832
C2×#285783