Finite Group Databasebeta

← C2×#24917861 #36273699

Subgroups — C2×#24917861 #36273699

Order 21248.

2182 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128247C211yesmaximal, characteristic
2C22×#14581783C211yesmaximal, characteristic, complemented
3#24917861C2116yesmaximal, direct product factor
4C2×#19128068C212yesmaximal, complemented
5C2×#19128064C212yesmaximal, complemented
6C2×#19128057C212yesmaximal, complemented
7C2×#19128054C212yesmaximal, complemented
8C2×#19127703C211yesmaximal, characteristic, complemented
9C2×#19127702C211yesmaximal, characteristic, complemented
10C2×#19127700C211yesmaximal, characteristic, complemented
11C2×M16×Dic83C211yesmaximal, characteristic, complemented
12C2×#417×C83C211yesmaximal, characteristic, complemented
13C22×#13050590C2211yescharacteristic
14C22×#13050574C2211yescharacteristic
15#19128247—24no—
16#19128247C2218yes—
17C2×#14581790C2211yescharacteristic
18C2×#14581788C2211yescharacteristic
19C2×#14581783—24no—
20C2×#14581783—21no—
21C2×#14581783—21no—
22C2×#14581783C2212yes—
23C2×#14581783C2212yes—
24C2×#14581783C2218yescomplemented
25C2×#14581781C2211yescharacteristic
26C2×#14581779C2211yescharacteristic
27C2×#14581764—21no—
28C2×#14581764—21no—
29C2×C4×C22⋊D83—21no—
30C2×C4×C22⋊D83—21no—
31C22×C4⋊Dic83—21no—
32C22×C4⋊Dic83C2211yescharacteristic
33C22×Dic83⋊C4—21no—
34C22×C4×Dic83C2211yescharacteristic
35#19128068C22116yescomplemented
36#19128064C22116yescomplemented
37#19128057C22116yescomplemented
38#19128054C22116yescomplemented
39#19127703C2218yescomplemented
40#19127702C2218yescomplemented
41#19127700C2218yescomplemented
42C2×#14581717C2212yes—
43C2×#14581716C2212yes—
44C2×#14581715C2212yes—
45C2×#14581714C2212yes—
46C2×#14581713C2212yes—
47C2×#14581712C2212yes—
48M16×Dic83C2218yescomplemented
49C2×#14581702C2212yes—
50C2×C8×Dic83C2212yes—
51C22×#41×C83C2211yescharacteristic
52C22×M16×C83C2211yescharacteristic
53C22×C4⋊C4×C83C2211yescharacteristic
54#417×C83C2218yescomplemented
55C2×#103×C83C2212yes—
56C2×#102×C83C2212yes—
57C2×#13050590—22no—
58C2×#13050590—24no—
59C2×#13050590—24no—
60C2×#13050590C2314yes—
61C22×Dic166C2311yescharacteristic
62C22×D332C2311yescharacteristic
63C22×C22⋊D83—21no—
64C22×C22⋊D83—21no—
65C22×C22⋊D83—21no—
66C23×Dic83—21no—
67C23×Dic83D411yescharacteristic
68C23×Dic83D411yescharacteristic
69C22×C4×D83—21no—
70C2×#13050574—24no—
71C2×#13050574C2314yes—
72C22×C83⋊C8C2312yes—
73#14581790—22no—
74#14581790C2314yes—
75#14581788—22no—
76#14581788C2314yes—
77#14581783—24no—
78#14581783—24no—
79#14581783—28no—
80#14581783—28no—
81#14581783C2318yes—
82#14581783C2318yes—
83#14581781—22no—
84#14581781C2314yes—
85#14581779—22no—
86#14581779C2314yes—
87#14581764—28no—
88#14581764—28no—
89C4×C22⋊D83—28no—
90C4×C22⋊D83—28no—
91C2×C4⋊Dic83—22no—
92C2×C4⋊Dic83—22no—
93C2×C4⋊Dic83—28no—
94C2×C4⋊Dic83—21no—
95C2×C4⋊Dic83—22no—
96C2×C4⋊Dic83C2312yes—
97C2×C4⋊Dic83C2314yes—
98C2×Dic83⋊C4—22no—
99C2×Dic83⋊C4—22no—
100C2×Dic83⋊C4—28no—
101C2×Dic83⋊C4—21no—
102C2×Dic83⋊C4—21no—
103C2×C22⋊Dic83—21no—
104C2×C22⋊Dic83—21no—
105C2×(C2×D83)⋊C4—21no—
106C2×(C2×D83)⋊C4—21no—
107C2×C4×Dic83D411yescharacteristic
108C2×C4×Dic83D411yescharacteristic
109C2×C4×Dic83D414yes—
110C2×C4×Dic83C2311yescharacteristic
111C2×C4×Dic83C2311yescharacteristic
112C2×C4×Dic83C2314yes—
113#14581717C2318yes—
114#14581716C2318yes—
115#14581715C2318yes—
116#14581714C2318yes—
117#14581713C2318yes—
118#14581712C2318yes—
119#14581702C2318yes—
120C8×Dic83C2318yes—
121C2×#41×C83—22no—
122C2×#41×C83—24no—
123C2×#41×C83—24no—
124C2×#41×C83C2314yes—
125C22×Q8×C83C2311yescharacteristic
126C22×D4×C83—21no—
127C22×D4×C83C2311yescharacteristic
128C23×C4×C83—21no—
129C23×C4×C83—21no—
130C23×C4×C83C2311yescharacteristic
131C2×M16×C83—24no—
132C2×M16×C83C2314yes—
133C22×C8×C83C2312yes—
134C2×C4⋊C4×C83—21no—
135C2×C4⋊C4×C83—22no—
136C2×C4⋊C4×C83C2312yes—
137C2×C4⋊C4×C83C2314yes—
138#103×C83C2318yes—
139#102×C83C2318yes—
140C23×D83—21no—
141#13050590—28no—
142#13050590—28no—
143#13050590—48no—
144C2×Dic166—22no—
145C2×Dic166—22no—
146C2×Dic166—21no—
147C2×Dic166C2412yes—
148C2×D332—22no—
149C2×D332—22no—
150C2×D332—21no—
151C2×D332C2412yes—
152C2×C22⋊D83—24no—
153C2×C22⋊D83—24no—
154C2×C22⋊D83—24no—
155C2×C22⋊D83—44no—
156C2×C22⋊D83—44no—
157C2×C22⋊D83—44no—
158C22×Dic83—21no—
159C22×Dic83—24no—
160C22×Dic83—22no—
161C22×Dic83—42no—
162C22×Dic83—42no—
163C22×Dic83—44no—
164C22×Dic83C2×D411yescharacteristic
165C22×Dic83C2×D411yescharacteristic
166C22×Dic83C2×D411yescharacteristic
167C22×Dic83C2×D411yescharacteristic
168C22×Dic83C2×D414yes—
169C22×Dic83C2×D414yes—
170C2×C4×D83—24no—
171C2×C4×D83—44no—
172#13050574—28no—
173C2×C83⋊C8—24no—
174C2×C83⋊C8C2414yes—
175C4⋊Dic83—24no—
176C4⋊Dic83—22no—
177C4⋊Dic83—28no—
178C4⋊Dic83—28no—
179C4⋊Dic83C2414yes—
180Dic83⋊C4—24no—
181Dic83⋊C4—24no—
182Dic83⋊C4—28no—
183Dic83⋊C4—28no—
184C22⋊Dic83—24no—
185C22⋊Dic83—24no—
186(C2×D83)⋊C4—24no—
187(C2×D83)⋊C4—24no—
188C4×Dic83C2×D412yes—
189C4×Dic83C2×D412yes—
190C4×Dic83C2×D414yes—
191C4×Dic83C2×D414yes—
192C4×Dic83C2412yes—
193C4×Dic83C2412yes—
194C24×C83—21no—
195C24×C83C2×D411yescharacteristic
196#41×C83—28no—
197#41×C83—28no—
198#41×C83—48no—
199C2×Q8×C83—21no—
200C2×Q8×C83—22no—
201C2×Q8×C83—22no—
202C2×Q8×C83C2412yes—
203C2×D4×C83—24no—
204C2×D4×C83—21no—
205C2×D4×C83—22no—
206C2×D4×C83—22no—
207C2×D4×C83—44no—
208C2×D4×C83C2412yes—
209C22×C4×C83—24no—
210C22×C4×C83—22no—
211C22×C4×C83—21no—
212C22×C4×C83—24no—
213C22×C4×C83—44no—
214C22×C4×C83—44no—
215C22×C4×C83C2×D411yescharacteristic
216C22×C4×C83C2×D412yes—
217C22×C4×C83#4114yes—
218C22×C4×C83#4114yes—
219C22×C4×C83C2411yescharacteristic
220C22×C4×C83C2412yes—
221M16×C83—28no—
222C2×C8×C83—24no—
223C2×C8×C83C2414yes—
224C4⋊C4×C83—24no—
225C4⋊C4×C83—22no—
226C4⋊C4×C83C2414yes—
227C23×C83—22no—
228C23×C83—22no—
229C23×C83—22no—
230C23×C83—42no—
231C23×C83—42no—
232C23×C83—42no—
233C23×C83#12414yes—
234C23×C83C22×D412yes—
235C23×C83C22×D411yescharacteristic
236C22×D83—22no—
237C22×D83—42no—
238C22×D83—42no—
239C22×D83—42no—
240Q8×C83—22no—
241Q8×C83—22no—
242Q8×C83—42no—
243D4×C83—22no—
244D4×C83—22no—
245D4×C83—48no—
246D4×C83—42no—
247C2×C4×C83—24no—
248C2×C4×C83—22no—
249C2×C4×C83—48no—
250C2×C4×C83—42no—
251C2×C4×C83—44no—
252C2×C4×C83—48no—
253C2×C4×C83#12414yes—
254C2×C4×C83#12418yes—
255C2×C4×C83C22×D412yes—
256C2×C4×C83C22×D411yescharacteristic
257C2×C4×C83C2×#4112yes—
258C2×C4×C83C2×#4114yes—
259C2×C4×C83C2×#4112yes—
260C2×C4×C83C2×#4114yes—
261C2×C4×C83C2511yescharacteristic
262C22⋊D83—48no—
263C22⋊D83—48no—
264C22⋊D83—48no—
265C2×Dic83—22no—
266C2×Dic83—22no—
267C2×Dic83—22no—
268C2×Dic83—22no—
269C2×Dic83—22no—
270C2×Dic83—24no—
271C2×Dic83—44no—
272C2×Dic83—44no—
273C2×Dic83—48no—
274C2×Dic83—42no—
275C2×Dic83—44no—
276C2×Dic83C22×D412yes—
277C2×Dic83C22×D412yes—
278C2×Dic83C22×D412yes—
279C2×Dic83C22×D412yes—
280D332—22no—
281D332—22no—
282D332—42no—
283C4×D83—48no—
284Dic166—22no—
285Dic166—22no—
286Dic166—42no—
287C8×C83—24no—
288C83⋊C8—24no—
289C22×C83—28no—
290C22×C83—22no—
291C22×C83—22no—
292C22×C83—42no—
293C22×C83—42no—
294C22×C83—42no—
295C22×C83—84no—
296C22×C83C2×#12414yes—
297C22×C83C2×#12412yes—
298C22×C83#57512yes—
299C22×C83#57512yes—
300C22×C83C23×D411yescharacteristic
301C2×D83—42no—
302C2×D83—42no—
303C2×D83—42no—
304C2×D83—84no—
305C4×C83—42no—
306C4×C83—44no—
307C4×C83C2×#12412yes—
308C4×C83C2×#12414yes—
309C4×C83C22×#4111yescharacteristic
310C4×C83C22×#4111yescharacteristic
311Dic83—22no—
312Dic83—22no—
313Dic83—22no—
314Dic83—22no—
315Dic83—42no—
316Dic83—44no—
317C2×#3142—831nomaximal, Sylow
318C2×C83—24no—
319C2×C83—82no—
320C2×C83#314214yescomplemented
321C2×C83C22×#12411yescharacteristic
322C2×C83C2×#57511yescharacteristic
323C2×C83C2×#57511yescharacteristic
324D83—82no—
325C2×#532—831no—
326C22×#125—831no—
327#3142—8316no—
328C2×#491—832no—
329C2×#490—832no—
330C2×#488—832no—
331C2×#488—832no—
332C2×#417—831no—
333C2×#417—831no—
334C2×#417—831no—
335C2×#417D8311yescharacteristic, complemented
336C2×C4×M16—831no—
337C83C2×#314211yescharacteristic, complemented, Sylow
338C22×#41—831no—
339C22×#41C2×D8311yescharacteristic
340C22×M16—831no—
341C22×M16C2×D8311yescharacteristic
342#532—838no—
343#532—1664no—
344C2×C4⋊Q8—831no—
345C2×#127—831no—
346C2×#125—832no—
347C2×#125—832no—
348C2×#125—838no—
349C2×#125—1661no—
350C2×#125—1661no—
351C2×#125—1664no—
352C2×#124—831no—
353C2×#124—831no—
354C2×#121—1661no—
355C2×#121—1661no—
356C2×C4×D4—1661no—
357C2×C4×D4—1661no—
358C22×C4⋊C4—831no—
359C22×C4⋊C4—1661no—
360C22×C4⋊C4—1661no—
361C22×C4⋊C4C2×D8311yescharacteristic
362C22×C42—831no—
363#491—8316no—
364#490—8316no—
365#488—8316no—
366#488—8316no—
367#417—838no—
368#417—838no—
369#417—838no—
370#417C2×D8318yescomplemented
371C2×#103—832no—
372C2×#103—832no—
373C2×#103—832no—
374C2×#103C2×D8312yes—
375C2×#102—832no—
376C2×#102—832no—
377C2×#102—832no—
378C2×#102C2×D8312yes—
379C4×M16—838no—
380C2×#97—832no—
381C2×C4×C8—832no—
382C2×#41—22no—
383C2×#41—24no—
384C2×#41—24no—
385C2×#41—834no—
386C2×#41—1664no—
387C2×#41—1664no—
388C2×#41—1662no—
389C2×#41C22×D8314yes—
390C22×Q8—831no—
391C22×Q8C22×D8311yescharacteristic
392C22×D4—21no—
393C22×D4—831no—
394C22×D4—1661no—
395C22×D4—1661no—
396C22×D4—1661no—
397C22×D4C22×D8311yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4—831no—
401C23×C4—831no—
402C23×C4—1661no—
403C23×C4—1661no—
404C23×C4C22×D8311yescharacteristic
405C2×M16—24no—
406C2×M16—834no—
407C2×M16—1664no—
408C2×M16C22×D8314yes—
409C22×C8—832no—
410C22×C8C22×D8312yes—
411C4⋊Q8—834no—
412C4⋊Q8—1662no—
413#127—834no—
414#127—1662no—
415#125—838no—
416#125—838no—
417#125—1668no—
418#125—1668no—
419#125—1664no—
420#125—1664no—
421#124—834no—
422#124—834no—
423#124—1662no—
424#124—1662no—
425#121—1668no—
426#121—1668no—
427C4×D4—1668no—
428C4×D4—1668no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—832no—
432C2×C4⋊C4—834no—
433C2×C4⋊C4—1668no—
434C2×C4⋊C4—1662no—
435C2×C4⋊C4—1662no—
436C2×C4⋊C4—1668no—
437C2×C4⋊C4—1662no—
438C2×C4⋊C4—1662no—
439C2×C4⋊C4—1661no—
440C2×C4⋊C4—1661no—
441C2×C4⋊C4—1662no—
442C2×C4⋊C4—1661no—
443C2×C4⋊C4C22×D8312yes—
444C2×C4⋊C4C22×D8314yes—
445C2×C22⋊C4—1661no—
446C2×C22⋊C4—1661no—
447C2×C22⋊C4—1661no—
448C2×C22⋊C4—1661no—
449C2×C42—831no—
450C2×C42—831no—
451C2×C42—834no—
452C2×C42—831no—
453C2×C42—831no—
454C2×C42—834no—
455#103—838no—
456#103—838no—
457#103—838no—
458#103C22×D8318yes—
459#102—838no—
460#102—838no—
461#102—838no—
462#102C22×D8318yes—
463#97—838no—
464C4×C8—838no—
465C24—21no—
466C24—1661no—
467C24D4×D8311yescharacteristic
468#41—28no—
469#41—28no—
470#41—48no—
471#41—3328no—
472#41—1668no—
473#41—1668no—
474C2×Q8—21no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—832no—
478C2×Q8—1662no—
479C2×Q8—1662no—
480C2×Q8—1661no—
481C2×Q8C23×D8312yes—
482C2×D4—24no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—44no—
487C2×D4—3324no—
488C2×D4—3324no—
489C2×D4—3324no—
490C2×D4—832no—
491C2×D4—1664no—
492C2×D4—1664no—
493C2×D4—1664no—
494C2×D4—1661no—
495C2×D4—1662no—
496C2×D4—1662no—
497C2×D4C23×D8312yes—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—3324no—
505C22×C4—3324no—
506C22×C4—3322no—
507C22×C4—3322no—
508C22×C4—834no—
509C22×C4—834no—
510C22×C4—831no—
511C22×C4—831no—
512C22×C4—831no—
513C22×C4—831no—
514C22×C4—1664no—
515C22×C4—1661no—
516C22×C4—1662no—
517C22×C4—1664no—
518C22×C4D4×D8312yes—
519C22×C4D4×D8311yescharacteristic
520C22×C4#1305058914yes—
521C22×C4#1305059014yes—
522C22×C4C23×D8312yes—
523C22×C4C23×D8311yescharacteristic
524M16—28no—
525M16—1668no—
526C2×C8—24no—
527C2×C8—834no—
528C2×C8—1664no—
529C2×C8C23×D8314yes—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—834no—
533C4⋊C4—1668no—
534C4⋊C4—1668no—
535C4⋊C4—1668no—
536C4⋊C4—1668no—
537C4⋊C4—1664no—
538C4⋊C4—1664no—
539C4⋊C4—1664no—
540C4⋊C4—1662no—
541C4⋊C4C23×D8314yes—
542C22⋊C4—1664no—
543C22⋊C4—1664no—
544C22⋊C4—1664no—
545C22⋊C4—1664no—
546C42—834no—
547C42—834no—
548C42—832no—
549C42—832no—
550C42—832no—
551C42—832no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—3322no—
559C23—3322no—
560C23—3322no—
561C23—1662no—
562C23#1458177814yes—
563C23C2×D4×D8312yes—
564C23C2×D4×D8311yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—42no—
568Q8—3322no—
569Q8—1662no—
570Q8—1662no—
571D4—22no—
572D4—22no—
573D4—48no—
574D4—42no—
575D4—3328no—
576D4—3328no—
577D4—3328no—
578D4—3322no—
579D4—1662no—
580D4—1662no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—3328no—
588C2×C4—3322no—
589C2×C4—3324no—
590C2×C4—3328no—
591C2×C4—3324no—
592C2×C4—3324no—
593C2×C4—832no—
594C2×C4—832no—
595C2×C4—832no—
596C2×C4—832no—
597C2×C4—1662no—
598C2×C4—1664no—
599C2×C4—1662no—
600C2×C4—1662no—
601C2×C4—1662no—
602C2×C4—1662no—
603C2×C4#1458177814yes—
604C2×C4#1458177818yes—
605C2×C4C2×D4×D8312yes—
606C2×C4C2×D4×D8311yescharacteristic
607C2×C4C2×#1305058912yes—
608C2×C4C2×#1305058914yes—
609C2×C4C2×#1305059012yes—
610C2×C4C2×#1305059014yes—
611C2×C4C24×D8311yescharacteristic
612C8—24no—
613C8—1664no—
614C22—22no—
615C22—22no—
616C22—28no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—84no—
621C22—3322no—
622C22—3322no—
623C22—3322no—
624C22—6644no—
625C22C2×#1458177814yes—
626C22C2×#1458177812yes—
627C22#1912849712yes—
628C22#1912850012yes—
629C22C22×D4×D8311yescharacteristic
630C4—44no—
631C4—42no—
632C4—3324no—
633C4—3322no—
634C4—1662no—
635C4—1662no—
636C4—1662no—
637C4—1662no—
638C4C2×#1458177814yes—
639C4C2×#1458177812yes—
640C4C22×#1305058911yescharacteristic
641C4C22×#1305059011yescharacteristic
642C2—24no—
643C2—82no—
644C2—6642no—
645C2#2491786114yesdirect product factor
646C2C22×#1458177811yescharacteristic
647C2C2×#1912849711yescharacteristic
648C2C2×#1912850011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#417×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128247C2
hypercenterC2×#417D83
hypocenterC83C2×#3142
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#14581783C2semidirect
3. #24917861C2direct (N × H)
4. C2×#19128068C2semidirect
5. C2×#19128064C2semidirect
6. C2×#19128057C2semidirect
7. C2×#19128054C2semidirect
8. C2×#19127703C2semidirect
9. C2×#19127702C2semidirect
10. C2×#19127700C2semidirect
11. C2×M16×Dic83C2semidirect
12. C2×#417×C83C2semidirect
24. C2×#14581783C22semidirect
35. #19128068C22semidirect
36. #19128064C22semidirect
37. #19128057C22semidirect
38. #19128054C22semidirect
39. #19127703C22semidirect
40. #19127702C22semidirect
41. #19127700C22semidirect
48. M16×Dic83C22semidirect
54. #417×C83C22semidirect
320. C2×C83#3142semidirect
335. C2×#417D83semidirect
337. C83C2×#3142semidirect
370. #417C2×D83semidirect
645. C2#24917861direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282472
C22×#145817832
#249178612
C2×#191280682
C2×#191280642
C2×#191280572
C2×#191280542
C2×#191277032
C2×#191277022
C2×#191277002
C2×M16×Dic832
C2×#417×C832
C2×#314283