Finite Group Databasebeta

← C2×#24918631 #36276434

Subgroups — C2×#24918631 #36276434

Order 21248.

1822 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128235C211yesmaximal, characteristic
2#24918631C2116yesmaximal, direct product factor
3C2×#19128100C212yesmaximal, complemented
4C2×#19128058C212yesmaximal, complemented
5C2×#19128036C211yesmaximal, characteristic, complemented
6C2×#19128019C211yesmaximal, characteristic
7C2×#19128031C211yesmaximal, characteristic, complemented
8C2×#19128014C211yesmaximal, characteristic
9C2×#19128005C212yesmaximal, complemented
10C2×#19127998C212yesmaximal, complemented
11C2×#19127754C211yesmaximal, characteristic, complemented
12C22×#125×C83C211yesmaximal, characteristic, complemented
13#19128235—24no—
14#19128235C2218yes—
15C2×#14581786—21no—
16C2×#14581784C2211yescharacteristic
17C2×#14581780C2211yescharacteristic
18C2×#14581775—21no—
19C2×#14581771C2211yescharacteristic
20C2×#14581766C2211yescharacteristic
21C2×C4×Dic166C2211yescharacteristic
22C2×C4×D332C2211yescharacteristic
23C2×C4×C22⋊D83—21no—
24C2×#14581749—21no—
25#19128100C22116yescomplemented
26#19128058C22116yescomplemented
27#19128036C2218yescomplemented
28#19128031C2218yescomplemented
29#19128019C2218yes—
30#19128014C2218yes—
31#19128005C22116yescomplemented
32#19127998C22116yescomplemented
33#19127754C2218yescomplemented
34C2×#14581725C2212yes—
35C2×#14581725C2212yes—
36C2×#14581723C2212yes—
37C2×#14581723C2212yes—
38C2×C4⋊(C83⋊C8)C2212yes—
39C2×#14581715C2212yes—
40C2×#14581715C2212yes—
41C2×#14581712C2212yes—
42C2×#14581712C2212yes—
43C2×C22⋊(C83⋊C8)C2211yescharacteristic
44C2×C22⋊(C83⋊C8)C2211yescharacteristic
45C2×C4×C83⋊C8C2212yes—
46C2×#125×C83—21no—
47C2×#125×C83—21no—
48C2×#125×C83—24no—
49C2×#125×C83C2212yes—
50C2×#125×C83C2212yes—
51C2×#125×C83C2218yescomplemented
52C22×C4⋊C4×C83—21no—
53C22×C4⋊C4×C83—21no—
54C22×C4⋊C4×C83C2211yescharacteristic
55C22×C4⋊C4×C83C2211yescharacteristic
56C22×C42×C83C2211yescharacteristic
57C22×Dic166C2311yescharacteristic
58C22×D332C2311yescharacteristic
59C22×C22⋊D83—21no—
60C22×C4×D83—21no—
61C22×C83⋊C8C2312yes—
62C22×C83⋊C8C2312yes—
63#14581786—28no—
64#14581784—22no—
65#14581784C2314yes—
66#14581780—22no—
67#14581780C2314yes—
68#14581775—28no—
69#14581771—22no—
70#14581771C2314yes—
71#14581766—22no—
72#14581766C2314yes—
73C4×Dic166—22no—
74C4×Dic166C2314yes—
75C4×D332—22no—
76C4×D332C2314yes—
77C4×C22⋊D83—28no—
78#14581749—28no—
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×C22⋊Dic83—21no—
86C2×(C2×D83)⋊C4—21no—
87C2×(C2×D83)⋊C4—21no—
88C2×(C2×D83)⋊C4—21no—
89C2×(C2×D83)⋊C4—21no—
90C2×C4×Dic83—21no—
91#14581725C2318yes—
92#14581725C2318yes—
93#14581723C2318yes—
94#14581723C2318yes—
95C4⋊(C83⋊C8)C2318yes—
96#14581715C2318yes—
97#14581715C2318yes—
98#14581712C2318yes—
99#14581712C2318yes—
100C22⋊(C83⋊C8)C2314yes—
101C22⋊(C83⋊C8)C2314yes—
102C4×C83⋊C8C2318yes—
103C23×C4×C83—21no—
104C23×C4×C83—21no—
105C23×C4×C83D411yescharacteristic
106C23×C4×C83D411yescharacteristic
107C23×C4×C83C2311yescharacteristic
108#125×C83—24no—
109#125×C83—24no—
110#125×C83—28no—
111#125×C83—28no—
112#125×C83C2318yes—
113#125×C83C2318yes—
114C2×C4⋊C4×C83—22no—
115C2×C4⋊C4×C83—22no—
116C2×C4⋊C4×C83—22no—
117C2×C4⋊C4×C83—22no—
118C2×C4⋊C4×C83—28no—
119C2×C4⋊C4×C83—28no—
120C2×C4⋊C4×C83—22no—
121C2×C4⋊C4×C83—22no—
122C2×C4⋊C4×C83—21no—
123C2×C4⋊C4×C83—21no—
124C2×C4⋊C4×C83C2312yes—
125C2×C4⋊C4×C83C2312yes—
126C2×C4⋊C4×C83C2314yes—
127C2×C4⋊C4×C83C2314yes—
128C2×C42×C83D411yescharacteristic
129C2×C42×C83D411yescharacteristic
130C2×C42×C83D414yes—
131C2×C42×C83C2311yescharacteristic
132C2×C42×C83C2311yescharacteristic
133C2×C42×C83C2314yes—
134C23×D83—21no—
135C2×Dic166—24no—
136C2×Dic166—21no—
137C2×Dic166C2412yes—
138C2×D332—24no—
139C2×D332—21no—
140C2×D332C2412yes—
141C2×C22⋊D83—24no—
142C2×C22⋊D83—44no—
143C22×Dic83—21no—
144C22×Dic83—21no—
145C22×Dic83—21no—
146C2×C4×D83—24no—
147C2×C4×D83—44no—
148C2×C83⋊C8—24no—
149C2×C83⋊C8—24no—
150C2×C83⋊C8C2414yes—
151C2×C83⋊C8C2414yes—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83C2412yes—
155C4⋊Dic83C2412yes—
156Dic83⋊C4—24no—
157Dic83⋊C4—24no—
158Dic83⋊C4—24no—
159Dic83⋊C4—24no—
160C22⋊Dic83—24no—
161(C2×D83)⋊C4—24no—
162(C2×D83)⋊C4—24no—
163(C2×D83)⋊C4—24no—
164(C2×D83)⋊C4—24no—
165C4×Dic83—24no—
166C24×C83C2×D411yescharacteristic
167C22×C4×C83—24no—
168C22×C4×C83—24no—
169C22×C4×C83—24no—
170C22×C4×C83—24no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83—42no—
174C22×C4×C83—44no—
175C22×C4×C83—44no—
176C22×C4×C83C2×D412yes—
177C22×C4×C83C2×D414yes—
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D414yes—
184C22×C4×C83C2412yes—
185C22×C4×C83C2411yescharacteristic
186C4⋊C4×C83—28no—
187C4⋊C4×C83—28no—
188C4⋊C4×C83—28no—
189C4⋊C4×C83—28no—
190C4⋊C4×C83—24no—
191C4⋊C4×C83—24no—
192C4⋊C4×C83—22no—
193C4⋊C4×C83—22no—
194C4⋊C4×C83C2414yes—
195C4⋊C4×C83C2414yes—
196C42×C83C2×D412yes—
197C42×C83C2×D414yes—
198C42×C83C2×D414yes—
199C42×C83C2×D412yes—
200C42×C83C2412yes—
201C42×C83C2412yes—
202C23×C83—22no—
203C23×C83—22no—
204C23×C83#13514yes—
205C23×C83C22×D412yes—
206C23×C83C22×D411yescharacteristic
207C22×D83—22no—
208C22×D83—42no—
209C22×D83—42no—
210C22×D83—42no—
211C2×C4×C83—22no—
212C2×C4×C83—28no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—24no—
218C2×C4×C83—24no—
219C2×C4×C83—24no—
220C2×C4×C83—24no—
221C2×C4×C83—44no—
222C2×C4×C83—48no—
223C2×C4×C83—48no—
224C2×C4×C83—44no—
225C2×C4×C83—44no—
226C2×C4×C83C22×D412yes—
227C2×C4×C83C22×D412yes—
228C2×C4×C83C22×D412yes—
229C2×C4×C83C22×D412yes—
230C2×C4×C83C22×D412yes—
231C2×C4×C83C22×D411yescharacteristic
232C2×C4×C832−1+414yes—
233C2×C4×C83C2511yescharacteristic
234C22⋊D83—48no—
235C2×Dic83—22no—
236C2×Dic83—22no—
237C2×Dic83—22no—
238C2×Dic83—42no—
239C2×Dic83—42no—
240C2×Dic83—42no—
241D332—24no—
242D332—42no—
243C4×D83—48no—
244Dic166—24no—
245Dic166—42no—
246C83⋊C8—24no—
247C83⋊C8—24no—
248C22×C83—22no—
249C22×C83—28no—
250C22×C83—22no—
251C22×C83#53612yes—
252C22×C83C2×#13514yes—
253C22×C83C2×#13512yes—
254C22×C83#57512yes—
255C22×C83C23×D411yescharacteristic
256C2×D83—42no—
257C2×D83—42no—
258C2×D83—42no—
259C2×D83—84no—
260C4×C83—22no—
261C4×C83—22no—
262C4×C83—22no—
263C4×C83—22no—
264C4×C83—22no—
265C4×C83—44no—
266C4×C83—44no—
267C4×C83#53612yes—
268C4×C83C2×2−1+412yes—
269Dic83—42no—
270Dic83—42no—
271Dic83—42no—
272C2×#3232—831nomaximal, Sylow
273C2×C83—24no—
274C2×C83#323214yescomplemented
275C2×C83C2×#53611yescharacteristic
276C2×C83C22×#13511yescharacteristic
277C2×C83C2×#57511yescharacteristic
278D83—82no—
279C2×#529—831no—
280C22×#125D8311yescharacteristic, complemented
281#3232—8316no—
282C2×#499—832no—
283C2×#488—832no—
284C2×#484—831no—
285C2×#481—831no—
286C2×#483—831no—
287C2×#480—831no—
288C2×#479—832no—
289C2×#478—832no—
290C2×#431—831no—
291C83C2×#323211yescharacteristic, complemented, Sylow
292#529—838no—
293#529—1664no—
294C2×#126—1661no—
295C2×#125—21no—
296C2×#125—21no—
297C2×#125—24no—
298C2×#125—831no—
299C2×#125C2×D8312yes—
300C2×#125C2×D8312yes—
301C2×#125C2×D8318yescomplemented
302C2×#124—831no—
303C2×#123—1661no—
304C2×C22⋊Q8—831no—
305C2×#121—831no—
306C2×C4×Q8—831no—
307C2×C4×D4—831no—
308C2×C4×D4—1661no—
309C2×#117—1661no—
310C22×C4⋊C4—21no—
311C22×C4⋊C4—21no—
312C22×C4⋊C4C2×D8311yescharacteristic
313C22×C4⋊C4C2×D8311yescharacteristic
314C22×C42C2×D8311yescharacteristic
315#499—8316no—
316#488—8316no—
317#484—838no—
318#483—838no—
319#481—838no—
320#480—838no—
321#479—8316no—
322#478—8316no—
323#431—838no—
324C2×#107—832no—
325C2×#107—832no—
326C2×#106—832no—
327C2×#106—832no—
328C2×C4⋊C8—832no—
329C2×#103—832no—
330C2×#103—832no—
331C2×#102—832no—
332C2×#102—832no—
333C2×C22⋊C8—831no—
334C2×C22⋊C8—831no—
335C2×C4×C8—832no—
336C22×Q8—831no—
337C22×D4—831no—
338C22×D4—1661no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—1661no—
342C23×C4C22⋊D8311yescharacteristic
343C23×C4C22⋊D8311yescharacteristic
344C23×C4C22×D8311yescharacteristic
345C22×C8—832no—
346C22×C8—832no—
347#126—1668no—
348#125—24no—
349#125—24no—
350#125—28no—
351#125—28no—
352#125—834no—
353#125—1662no—
354#125C22×D8318yes—
355#125C22×D8318yes—
356#124—834no—
357#124—1662no—
358#123—1668no—
359C22⋊Q8—834no—
360C22⋊Q8—1662no—
361#121—834no—
362#121—1662no—
363C4×Q8—834no—
364C4×Q8—1662no—
365C4×D4—834no—
366C4×D4—1668no—
367C4×D4—1662no—
368#117—1668no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—831no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4—1661no—
385C2×C4⋊C4C22×D8312yes—
386C2×C4⋊C4C22×D8312yes—
387C2×C4⋊C4C22×D8314yes—
388C2×C4⋊C4C22×D8314yes—
389C2×C22⋊C4—1661no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C22⋊C4—1661no—
393C2×C22⋊C4—1661no—
394C2×C42—1661no—
395C2×C42C22⋊D8311yescharacteristic
396C2×C42C22⋊D8311yescharacteristic
397C2×C42C22⋊D8314yes—
398C2×C42C22×D8311yescharacteristic
399C2×C42C22×D8311yescharacteristic
400C2×C42C22×D8314yes—
401#107—838no—
402#107—838no—
403#106—838no—
404#106—838no—
405C4⋊C8—838no—
406#103—838no—
407#103—838no—
408#102—838no—
409#102—838no—
410C22⋊C8—834no—
411C22⋊C8—834no—
412C4×C8—838no—
413C24—1661no—
414C24C2×C22⋊D8311yescharacteristic
415C2×Q8—832no—
416C2×Q8—1664no—
417C2×Q8—1661no—
418C2×D4—3324no—
419C2×D4—832no—
420C2×D4—1664no—
421C2×D4—1661no—
422C2×D4—1664no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—3324no—
433C22×C4—1661no—
434C22×C4—1661no—
435C22×C4—1661no—
436C22×C4—1664no—
437C22×C4C2×C22⋊D8312yes—
438C22×C4C2×C22⋊D8314yes—
439C22×C4C2×C22⋊D8311yescharacteristic
440C22×C4C2×C22⋊D8311yescharacteristic
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4C2×C22⋊D8311yescharacteristic
443C22×C4C2×C22⋊D8311yescharacteristic
444C22×C4C2×C22⋊D8314yes—
445C22×C4C23×D8312yes—
446C22×C4C23×D8311yescharacteristic
447C2×C8—834no—
448C2×C8—834no—
449C2×C8—1664no—
450C2×C8—1664no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—832no—
460C4⋊C4—832no—
461C4⋊C4—1661no—
462C4⋊C4—1661no—
463C4⋊C4—1664no—
464C4⋊C4—1664no—
465C4⋊C4—1664no—
466C4⋊C4—1664no—
467C4⋊C4C23×D8314yes—
468C4⋊C4C23×D8314yes—
469C22⋊C4—1664no—
470C22⋊C4—1664no—
471C22⋊C4—1664no—
472C22⋊C4—1664no—
473C22⋊C4—1664no—
474C42—1664no—
475C42C2×C22⋊D8314yes—
476C42C2×C22⋊D8312yes—
477C42C2×C22⋊D8312yes—
478C42C2×C22⋊D8314yes—
479C42C23×D8312yes—
480C42C23×D8312yes—
481C23—22no—
482C23—22no—
483C23—3322no—
484C23—3322no—
485C23—3322no—
486C23—1662no—
487C23#1458181214yes—
488C23C22×C22⋊D8312yes—
489C23C22×C22⋊D8311yescharacteristic
490Q8—3322no—
491Q8—1664no—
492D4—3328no—
493D4—3322no—
494D4—1664no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—3328no—
511C2×C4—3322no—
512C2×C4—3322no—
513C2×C4—3322no—
514C2×C4—1662no—
515C2×C4—1662no—
516C2×C4—1662no—
517C2×C4C22×C22⋊D8312yes—
518C2×C4C22×C22⋊D8312yes—
519C2×C4C22×C22⋊D8312yes—
520C2×C4C22×C22⋊D8312yes—
521C2×C4C22×C22⋊D8312yes—
522C2×C4C22×C22⋊D8311yescharacteristic
523C2×C4#1458184014yes—
524C2×C4C24×D8311yescharacteristic
525C8—1664no—
526C8—1664no—
527C22—22no—
528C22—28no—
529C22—22no—
530C22—3322no—
531C22—3322no—
532C22—3322no—
533C22—6644no—
534C22#1912827512yes—
535C22C2×#1458181214yes—
536C22C2×#1458181212yes—
537C22#1912849912yes—
538C22C23×C22⋊D8311yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—44no—
545C4—44no—
546C4—3322no—
547C4—3322no—
548C4—3322no—
549C4#1912827512yes—
550C4C2×#1458184012yes—
551C2—24no—
552C2—6642no—
553C2#2491863114yesdirect product factor
554C2C2×#1912827511yescharacteristic
555C2C22×#1458181211yescharacteristic
556C2C2×#1912849911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC22×#125×C83C2
socleC23×C83C22×D4
prime radicalC2×#14581766C22
hypercenterC22×#125D83
hypocenterC83C2×#3232
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #24918631C2direct (N × H)
3. C2×#19128100C2semidirect
4. C2×#19128058C2semidirect
5. C2×#19128036C2semidirect
7. C2×#19128031C2semidirect
9. C2×#19128005C2semidirect
10. C2×#19127998C2semidirect
11. C2×#19127754C2semidirect
12. C22×#125×C83C2semidirect
25. #19128100C22semidirect
26. #19128058C22semidirect
27. #19128036C22semidirect
28. #19128031C22semidirect
31. #19128005C22semidirect
32. #19127998C22semidirect
33. #19127754C22semidirect
51. C2×#125×C83C22semidirect
274. C2×C83#3232semidirect
280. C22×#125D83semidirect
291. C83C2×#3232semidirect
301. C2×#125C2×D83semidirect
553. C2#24918631direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282352
#249186312
C2×#191281002
C2×#191280582
C2×#191280362
C2×#191280192
C2×#191280312
C2×#191280142
C2×#191280052
C2×#191279982
C2×#191277542
C22×#125×C832
C2×#323283