Finite Group Databasebeta

← C2×#24918802 #36277211

Subgroups — C2×#24918802 #36277211

Order 21248.

1930 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
1C2×#19128256C211yesmaximal, characteristic, complemented
2C2×#19128233C211yesmaximal, characteristic
3#24918802C2116yesmaximal, direct product factor
4C2×#19128007C212yesmaximal, complemented
5C2×#19127983C211yesmaximal, characteristic, complemented
6C2×#19127943C212yesmaximal, complemented
7C2×#19127929C212yesmaximal
8C2×#19127922C212yesmaximal, complemented
9C2×#19127912C211yesmaximal, characteristic, complemented
10C2×#19127906C211yesmaximal, characteristic, complemented
11C2×C4⋊C8×D83C211yesmaximal, characteristic, complemented
12C2×#474×C83C211yesmaximal, characteristic, complemented
13C22×Dic332C2212yes—
14C22×#13050578C2212yes—
15C22×C8×D83C2212yes—
16#19128256—24no—
17#19128256C2218yescomplemented
18C2×#14581792C2211yescharacteristic
19#19128233—24no—
20#19128233C2218yes—
21C2×#14581785—21no—
22C2×#14581783—21no—
23C2×#14581783C2211yescharacteristic
24C2×#14581779C2211yescharacteristic
25C2×#14581774—21no—
26C2×#14581772C2212yes—
27C2×#14581772C2211yescharacteristic
28C2×#14581768—21no—
29C2×#14581763C2211yescharacteristic
30C2×Q8×Dic83—21no—
31C2×C4×Dic166C2211yescharacteristic
32C2×#14581757C2212yes—
33C2×C4×C22⋊D83—21no—
34C2×#14581751—21no—
35C2×#14581750—21no—
36C2×C42×D83C2211yescharacteristic
37#19128007C22116yescomplemented
38#19127983C2218yescomplemented
39#19127943C22116yescomplemented
40#19127929C22116yes—
41#19127922C22116yescomplemented
42#19127912C2218yescomplemented
43#19127906C2218yescomplemented
44C2×#14581723C2212yes—
45C2×C4⋊(C83⋊C8)C2211yescharacteristic
46C2×Dic83⋊C8C2212yes—
47C4⋊C8×D83C2218yescomplemented
48C2×#14581717C2212yes—
49C2×#14581715C2212yes—
50C2×#14581715C2212yes—
51C2×C4⋊Q8×C83C2211yescharacteristic
52C2×C4×D4×C83C2211yescharacteristic
53#474×C83C2218yescomplemented
54C2×#107×C83C2212yes—
55C2×C4⋊C8×C83C2211yescharacteristic
56C2×#102×C83C2212yes—
57C22×Dic166C2312yes—
58C22×Dic166C2311yescharacteristic
59C22×C22⋊D83—21no—
60C22×C4×D83D411yescharacteristic
61C22×C4×D83D411yescharacteristic
62C22×C4×D83C2311yescharacteristic
63C2×Dic332—44no—
64C2×Dic332C2318yes—
65C2×#13050578—44no—
66C2×#13050578C2318yes—
67C22×C83⋊C8C2312yes—
68C2×C8×D83—44no—
69C2×C8×D83C2318yes—
70#14581792—22no—
71#14581792C2314yes—
72#14581785—28no—
73#14581783—28no—
74#14581783—22no—
75#14581783C2314yes—
76#14581779—22no—
77#14581779C2314yes—
78#14581774—28no—
79#14581772—22no—
80#14581772—24no—
81#14581772C2318yes—
82#14581772C2314yes—
83#14581768—28no—
84#14581763—22no—
85#14581763C2314yes—
86Q8×Dic83—28no—
87C4×Dic166—22no—
88C4×Dic166C2314yes—
89#14581757—24no—
90#14581757C2318yes—
91C4×C22⋊D83—28no—
92#14581751—28no—
93#14581750—28no—
94C2×C4⋊Dic83—21no—
95C2×C4⋊Dic83—21no—
96C2×C4⋊Dic83—22no—
97C2×C4⋊Dic83—21no—
98C2×C4⋊Dic83—21no—
99C2×C4⋊Dic83C2311yescharacteristic
100C2×Dic83⋊C4—21no—
101C2×Dic83⋊C4—22no—
102C2×Dic83⋊C4—21no—
103C2×C22⋊Dic83—21no—
104C2×C22⋊Dic83—21no—
105C2×C22⋊Dic83—21no—
106C2×(C2×D83)⋊C4—22no—
107C2×(C2×D83)⋊C4—21no—
108C2×C4×Dic83—22no—
109C2×C4×Dic83—21no—
110C2×C4×Dic83D411yescharacteristic
111C2×C4×Dic83D411yescharacteristic
112C2×C4×Dic83C2311yescharacteristic
113C42×D83D414yes—
114C42×D83C2314yes—
115#14581723C2318yes—
116C4⋊(C83⋊C8)C2314yes—
117Dic83⋊C8C2318yes—
118#14581717C2318yes—
119#14581715C2318yes—
120#14581715C2318yes—
121C22×Q8×C83—21no—
122C22×D4×C83C2311yescharacteristic
123C23×C4×C83—21no—
124C22×C8×C83C2312yes—
125C4⋊Q8×C83—22no—
126C4⋊Q8×C83C2314yes—
127C4×D4×C83—22no—
128C4×D4×C83C2314yes—
129C2×C4⋊C4×C83—21no—
130C2×C4⋊C4×C83C2311yescharacteristic
131C2×C4⋊C4×C83C2312yes—
132C2×C22⋊C4×C83—21no—
133C2×C42×C83C2311yescharacteristic
134#107×C83C2318yes—
135C4⋊C8×C83C2314yes—
136#102×C83C2318yes—
137C23×D83C2×D411yescharacteristic
138C2×Dic166—24no—
139C2×Dic166—22no—
140C2×Dic166—24no—
141C2×Dic166—22no—
142C2×Dic166—21no—
143C2×Dic166—22no—
144C2×Dic166C2412yes—
145C2×Dic166C2414yes—
146C2×C22⋊D83—24no—
147C2×C22⋊D83—44no—
148C22×Dic83—21no—
149C22×Dic83—21no—
150C22×Dic83—22no—
151C22×Dic83C2×D411yescharacteristic
152C22×Dic83C2×D411yescharacteristic
153C22×Dic83C2×D411yescharacteristic
154C2×C4×D83—24no—
155C2×C4×D83—24no—
156C2×C4×D83—22no—
157C2×C4×D83—22no—
158C2×C4×D83C2×D412yes—
159C2×C4×D83C2×D414yes—
160C2×C4×D83C2×D414yes—
161C2×C4×D83C2412yes—
162Dic332—48no—
163#13050578—48no—
164C2×C83⋊C8—24no—
165C2×C83⋊C8C2414yes—
166C8×D83—48no—
167C4⋊Dic83—21no—
168C4⋊Dic83—24no—
169C4⋊Dic83—24no—
170C4⋊Dic83—28no—
171C4⋊Dic83—24no—
172C4⋊Dic83—24no—
173C4⋊Dic83C2412yes—
174Dic83⋊C4—24no—
175Dic83⋊C4—28no—
176Dic83⋊C4—24no—
177C22⋊Dic83—24no—
178C22⋊Dic83—24no—
179C22⋊Dic83—24no—
180(C2×D83)⋊C4—28no—
181(C2×D83)⋊C4—24no—
182C4×Dic83—24no—
183C4×Dic83—28no—
184C4×Dic83C2×D412yes—
185C4×Dic83C2×D414yes—
186C4×Dic83C2×D414yes—
187C4×Dic83C2412yes—
188C24×C83—21no—
189C2×Q8×C83—24no—
190C2×Q8×C83—44no—
191C2×D4×C83—21no—
192C2×D4×C83—22no—
193C2×D4×C83—22no—
194C2×D4×C83C2412yes—
195C22×C4×C83—21no—
196C22×C4×C83—22no—
197C22×C4×C83—24no—
198C22×C4×C83—44no—
199C22×C4×C83C2×D411yescharacteristic
200C22×C4×C83C2×D411yescharacteristic
201C22×C4×C83C2411yescharacteristic
202C2×C8×C83—24no—
203C2×C8×C83C2414yes—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—22no—
206C4⋊C4×C83—24no—
207C4⋊C4×C83C2412yes—
208C4⋊C4×C83C2414yes—
209C22⋊C4×C83—24no—
210C42×C83C2×D412yes—
211C42×C83C2412yes—
212C23×C83—22no—
213C23×C83—42no—
214C23×C83—42no—
215C23×C83—42no—
216C23×C83C22×D411yescharacteristic
217C22×D83—22no—
218C22×D83—22no—
219C22×D83#13714yes—
220C22×D83C22×D412yes—
221Q8×C83—48no—
222D4×C83—22no—
223D4×C83—22no—
224D4×C83—42no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—24no—
228C2×C4×C83—42no—
229C2×C4×C83—44no—
230C2×C4×C83—48no—
231C2×C4×C83#13514yes—
232C2×C4×C83C22×D412yes—
233C2×C4×C83C22×D411yescharacteristic
234C2×C4×C83C22×D412yes—
235C2×C4×C83H(5,𝔽2)12yes—
236C2×C4×C832−1+412yes—
237C2×C4×C83C2511yescharacteristic
238C22⋊D83—48no—
239C2×Dic83—22no—
240C2×Dic83—22no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—24no—
244C2×Dic83—22no—
245C2×Dic83—42no—
246C2×Dic83—44no—
247C2×Dic83—42no—
248C2×Dic83C22×D412yes—
249C2×Dic83C22×D412yes—
250C2×Dic83C22×D412yes—
251C4×D83—28no—
252C4×D83—24no—
253C4×D83—28no—
254C4×D83—22no—
255C4×D83—22no—
256Dic166—24no—
257Dic166—22no—
258Dic166—22no—
259Dic166—24no—
260Dic166—44no—
261Dic166—42no—
262C8×C83—24no—
263C83⋊C8—24no—
264C22×C83—42no—
265C22×C83—42no—
266C22×C83—42no—
267C22×C83—84no—
268C22×C83#53712yes—
269C22×C83C2×#13512yes—
270C22×C83C2×#13712yes—
271C22×C83C23×D411yescharacteristic
272C2×D83—22no—
273C2×D83—28no—
274C2×D83—22no—
275C2×D83C2×#13714yes—
276C4×C83—22no—
277C4×C83—42no—
278C4×C83—44no—
279C4×C83#53712yes—
280C4×C83C2×#13514yes—
281C4×C83C2×H(5,𝔽2)11yescharacteristic
282C4×C83C2×2−1+411yescharacteristic
283Dic83—22no—
284Dic83—22no—
285Dic83—22no—
286Dic83—44no—
287Dic83—42no—
288Dic83—42no—
289C2×#3248—831nomaximal, Sylow
290C2×C83—82no—
291C2×C83#324814yescomplemented
292C2×C83C2×#53711yescharacteristic
293C2×C83C22×#13511yescharacteristic
294C2×C83C22×#13711yescharacteristic
295D83—24no—
296C2×#533—831no—
297C2×#529—831no—
298#3248—8316no—
299C2×#479—832no—
300C2×#475—831no—
301C2×#474D8311yescharacteristic, complemented
302C2×#467—832no—
303C2×#465—832no—
304C2×#464—832no—
305C2×#462—831no—
306C2×#461—831no—
307C22×C4⋊C8—831no—
308C83C2×#324811yescharacteristic, complemented, Sylow
309C22×Q16—832no—
310C22×SD16—832no—
311C23×C8—832no—
312#533—838no—
313#533—1664no—
314C2×C4⋊Q8—831no—
315C2×C4⋊Q8C2×D8311yescharacteristic
316#529—838no—
317#529—1664no—
318C2×#126—1661no—
319C2×#125—831no—
320C2×#125—1661no—
321C2×#124—831no—
322C2×#123—1661no—
323C2×C22⋊Q8—831no—
324C2×C22⋊Q8—832no—
325C2×C22⋊Q8—1661no—
326C2×#121—831no—
327C2×C4×Q8—832no—
328C2×C4×Q8—831no—
329C2×C4×Q8—1661no—
330C2×C4×D4—1661no—
331C2×C4×D4C2×D8311yescharacteristic
332C2×#117—1661no—
333C2×#117—1661no—
334C22×C42—831no—
335#479—8316no—
336#475—838no—
337#474C2×D8318yescomplemented
338#467—8316no—
339#465—8316no—
340#464—8316no—
341#462—838no—
342#461—838no—
343C2×#107C2×D8312yes—
344C2×#106—832no—
345C2×C4⋊C8—832no—
346C2×C4⋊C8—831no—
347C2×C4⋊C8—838no—
348C2×C4⋊C8C2×D8311yescharacteristic
349C2×#103—832no—
350C2×#103—832no—
351C2×#103—832no—
352C2×#102C2×D8312yes—
353C22×Q8—21no—
354C22×Q8—831no—
355C22×Q8—832no—
356C22×D4—1661no—
357C22×D4C22×D8311yescharacteristic
358C23×C4—21no—
359C23×C4—831no—
360C23×C4—831no—
361C23×C4—831no—
362C2×Q16—3324no—
363C2×Q16—838no—
364C2×SD16—3324no—
365C2×SD16—838no—
366C22×C8—3324no—
367C22×C8—838no—
368C22×C8—832no—
369C22×C8C22×D8312yes—
370C4⋊Q8—22no—
371C4⋊Q8—834no—
372C4⋊Q8—1662no—
373C4⋊Q8C22×D8314yes—
374#126—1668no—
375#125—834no—
376#125—1662no—
377#125—1668no—
378#124—834no—
379#124—1662no—
380#123—1668no—
381C22⋊Q8—834no—
382C22⋊Q8—838no—
383C22⋊Q8—1662no—
384C22⋊Q8—1664no—
385C22⋊Q8—1668no—
386#121—834no—
387#121—1662no—
388C4×Q8—838no—
389C4×Q8—834no—
390C4×Q8—1664no—
391C4×Q8—1668no—
392C4×Q8—1662no—
393C4×D4—22no—
394C4×D4—1668no—
395C4×D4C22×D8314yes—
396#117—1668no—
397#117—1668no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—831no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1661no—
403C2×C4⋊C4—1662no—
404C2×C4⋊C4—1662no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4C22×D8311yescharacteristic
409C2×C4⋊C4C22×D8312yes—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1662no—
412C2×C22⋊C4—1661no—
413C2×C22⋊C4—1661no—
414C2×C22⋊C4—1661no—
415C2×C22⋊C4—1661no—
416C2×C42—831no—
417C2×C42—831no—
418C2×C42—831no—
419C2×C42—834no—
420C2×C42—834no—
421C2×C42—1662no—
422C2×C42—1661no—
423C2×C42C22×D8311yescharacteristic
424#107C22×D8318yes—
425#106—838no—
426C4⋊C8—838no—
427C4⋊C8—834no—
428C4⋊C8C22×D8314yes—
429#103—838no—
430#103—838no—
431#103—838no—
432#102C22×D8318yes—
433C24—21no—
434C24—831no—
435C2×Q8—24no—
436C2×Q8—44no—
437C2×Q8—832no—
438C2×Q8—834no—
439C2×Q8—1664no—
440C2×Q8—1662no—
441C2×Q8—1662no—
442C2×Q8—1664no—
443C2×Q8—1661no—
444C2×Q8—1662no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—3324no—
449C2×D4—1664no—
450C2×D4C23×D8312yes—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—831no—
456C22×C4—831no—
457C22×C4—831no—
458C22×C4—834no—
459C22×C4—834no—
460C22×C4—832no—
461C22×C4—832no—
462C22×C4—1664no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1662no—
466C22×C4—1664no—
467C22×C4—1662no—
468C22×C4—1662no—
469C22×C4D4×D8311yescharacteristic
470C22×C4D4×D8311yescharacteristic
471C22×C4C23×D8311yescharacteristic
472Q16—3328no—
473SD16—3328no—
474C2×C8—24no—
475C2×C8—3328no—
476C2×C8—834no—
477C2×C8—1664no—
478C2×C8C23×D8314yes—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—832no—
483C4⋊C4—1664no—
484C4⋊C4—1664no—
485C4⋊C4—1664no—
486C4⋊C4—1668no—
487C4⋊C4—1668no—
488C4⋊C4—1661no—
489C4⋊C4—1664no—
490C4⋊C4—1664no—
491C4⋊C4—1664no—
492C4⋊C4C23×D8314yes—
493C4⋊C4C23×D8312yes—
494C22⋊C4—24no—
495C22⋊C4—1668no—
496C22⋊C4—1664no—
497C22⋊C4—1664no—
498C22⋊C4—1664no—
499C22⋊C4—1664no—
500C42—834no—
501C42—832no—
502C42—834no—
503C42—832no—
504C42—1668no—
505C42—1664no—
506C42D4×D8312yes—
507C42C23×D8312yes—
508C23—22no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—834no—
513C23—832no—
514C23—1662no—
515C23—1662no—
516C23C2×D4×D8311yescharacteristic
517Q8—48no—
518Q8—3324no—
519Q8—3322no—
520Q8—1664no—
521Q8—1662no—
522Q8—1662no—
523Q8—1664no—
524D4—22no—
525D4—22no—
526D4—42no—
527D4—3328no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—3322no—
535C2×C4—3322no—
536C2×C4—3324no—
537C2×C4—832no—
538C2×C4—832no—
539C2×C4—832no—
540C2×C4—1668no—
541C2×C4—1662no—
542C2×C4—1662no—
543C2×C4—1664no—
544C2×C4—1662no—
545C2×C4—1662no—
546C2×C4—1662no—
547C2×C4—1668no—
548C2×C4—1662no—
549C2×C4—1662no—
550C2×C4—1664no—
551C2×C4#1458181114yes—
552C2×C4C2×D4×D8312yes—
553C2×C4C2×D4×D8312yes—
554C2×C4C2×D4×D8311yescharacteristic
555C2×C4#1458183812yes—
556C2×C4#1458184012yes—
557C2×C4C24×D8311yescharacteristic
558C8—24no—
559C8—1664no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—84no—
564C22—834no—
565C22—1668no—
566C22—1662no—
567C22—1662no—
568C22#1912828112yes—
569C22C2×#1458181112yes—
570C22#137×D8312yes—
571C22C22×D4×D8311yescharacteristic
572C4—22no—
573C4—42no—
574C4—44no—
575C4—3322no—
576C4—3322no—
577C4—3324no—
578C4—1662no—
579C4—1662no—
580C4—1662no—
581C4#1912828112yes—
582C4C2×#1458181114yes—
583C4C2×#1458183811yescharacteristic
584C4C2×#1458184011yescharacteristic
585C2—82no—
586C2—1664no—
587C2#2491880214yesdirect product factor
588C2C2×#1912828111yescharacteristic
589C2C22×#1458181111yescharacteristic
590C2C2×#137×D8311yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#474×C83C2
socleC23×C83C22×D4
prime radicalC2×#14581763C22
hypercenterC2×#474D83
hypocenterC83C2×#3248
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#19128256C2semidirect
3. #24918802C2direct (N × H)
4. C2×#19128007C2semidirect
5. C2×#19127983C2semidirect
6. C2×#19127943C2semidirect
8. C2×#19127922C2semidirect
9. C2×#19127912C2semidirect
10. C2×#19127906C2semidirect
11. C2×C4⋊C8×D83C2semidirect
12. C2×#474×C83C2semidirect
17. #19128256C22semidirect
37. #19128007C22semidirect
38. #19127983C22semidirect
39. #19127943C22semidirect
41. #19127922C22semidirect
42. #19127912C22semidirect
43. #19127906C22semidirect
47. C4⋊C8×D83C22semidirect
53. #474×C83C22semidirect
291. C2×C83#3248semidirect
301. C2×#474D83semidirect
308. C83C2×#3248semidirect
337. #474C2×D83semidirect
587. C2#24918802direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282562
C2×#191282332
#249188022
C2×#191280072
C2×#191279832
C2×#191279432
C2×#191279292
C2×#191279222
C2×#191279122
C2×#191279062
C2×C4⋊C8×D832
C2×#474×C832
C2×#324883