Finite Group Databasebeta

← C22⋊C4×#13050577 #35420238

Subgroups — C22⋊C4×#13050577 #35420238

Order 21248.

1614 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#13050577C211yesmaximal, characteristic
2C2×C4×#13050577C212yesmaximal, complemented
3C22⋊C4×D332C211yesmaximal, characteristic
4#24908320C212yesmaximal, complemented
5#24907818C212yesmaximal, complemented
6#24907800C212yesmaximal, complemented
7#24907486C211yesmaximal, characteristic, complemented
8#24907398C211yesmaximal, characteristic, complemented
9#24907393C211yesmaximal, characteristic
10C22⋊C4×C83⋊C8C211yesmaximal, characteristic, complemented
11Q8×C22⋊C4×C83C211yesmaximal, characteristic, complemented
12C23×D332C2211yescharacteristic
13C22×#13050577—28no—
14C22×#13050577C418yescomplemented
15C22×#13050577C2211yescharacteristic
16C22×#13050577C2211yescharacteristic
17C22×#13050577C2211yescharacteristic
18C22×#13050577C2211yescharacteristic
19C23×C83⋊C8C2211yescharacteristic
20C2×C4×D332C2212yes—
21C2×C22⋊C4×D83—21no—
22C4×#13050577—28no—
23C2×#14581723C2212yescomplemented
24C2×#14581716C2212yescomplemented
25C2×#14581716C2212yescomplemented
26C2×#14581712C2212yes—
27C2×#14581712C2212yes—
28C2×C22⋊(C83⋊C8)C2211yescharacteristic
29C2×C4×C83⋊C8C2212yescomplemented
30#19127603C2212yes—
31#19127579—22no—
32#19127561C2211yescharacteristic
33#19127556—21no—
34#19127463C2212yescomplemented
35#19127456C2212yescomplemented
36C23×Q8×C83C2211yescharacteristic
37C2×C4×Q8×C83C2212yescomplemented
38#391×C83—22no—
39#391×C83C2212yescomplemented
40#380×C83—21no—
41#380×C83C2211yescharacteristic
42C4×C22⋊C4×C83—21no—
43C4×C22⋊C4×C83C2211yescharacteristic
44C24×D83—21no—
45C22×D332—24no—
46C22×D332—21no—
47C22×D332—22no—
48C22×D332—42no—
49C22×D332C2×C414yes—
50C22×D332C2311yescharacteristic
51C22×D332C2311yescharacteristic
52C22×C4×D83—22no—
53C2×#13050577—24no—
54C2×#13050577—232no—
55C2×#13050577—24no—
56C2×#13050577—24no—
57C2×#13050577C2×C414yes—
58C2×#13050577C2×C414yes—
59C2×#13050577C2×C414yescomplemented
60C2×#13050577C2×C414yescomplemented
61C2×#13050577D418yes—
62C22×C83⋊C8—24no—
63C22×C83⋊C8C2×C414yescomplemented
64C22×C83⋊C8C2312yes—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C4×D332—24no—
68C4×D332—42no—
69C2×C4⋊Dic83C2312yes—
70C2×C22⋊Dic83—21no—
71C2×(C2×D83)⋊C4—22no—
72C2×(C2×D83)⋊C4—22no—
73C2×(C2×D83)⋊C4—22no—
74C22⋊C4×D83—44no—
75#14581723—24no—
76#14581716—24no—
77#14581716—42no—
78#14581712—24no—
79#14581712—42no—
80C22⋊(C83⋊C8)—22no—
81C4×C83⋊C8—24no—
82C83⋊#95—21no—
83C22×Q8×C83—21no—
84C22×Q8×C83—22no—
85C22×Q8×C83—24no—
86C22×Q8×C83—42no—
87C22×Q8×C83C2×C414yescomplemented
88C22×Q8×C83C2311yescharacteristic
89C22×Q8×C83C2311yescharacteristic
90C23×C4×C83—21no—
91C23×C4×C83C2311yescharacteristic
92C4×Q8×C83—24no—
93C4×Q8×C83—42no—
94C2×C4⋊C4×C83—22no—
95C2×C4⋊C4×C83C2312yes—
96C2×C22⋊C4×C83—21no—
97C2×C22⋊C4×C83D411yescharacteristic
98C2×C22⋊C4×C83D411yescharacteristic
99C2×C42×C83—22no—
100C2×C42×C83C2312yes—
101#95×C83—22no—
102#95×C83—22no—
103#95×C83—21no—
104#95×C83D412yes—
105C23×D83—21no—
106C23×D83—21no—
107C23×D83—24no—
108C23×D83—44no—
109C23×D83—42no—
110C23×D83—42no—
111C23×D83—82no—
112C2×D332—21no—
113C2×D332—22no—
114C2×D332—22no—
115C2×D332—22no—
116C2×D332—28no—
117C2×D332—24no—
118C2×D332—21no—
119C2×D332—44no—
120C2×D332—48no—
121C2×D332—44no—
122C2×D332C22×C412yes—
123C2×D332C22×C412yes—
124C2×D332C2×D414yes—
125C22×Dic83—22no—
126C2×C4×D83—44no—
127C2×C4×D83—44no—
128C2×C4×D83—82no—
129#13050577—28no—
130#13050577—28no—
131#13050577—28no—
132#13050577C22⋊C4116yesdirect product factor
133C2×C83⋊C8—22no—
134C2×C83⋊C8—28no—
135C2×C83⋊C8—22no—
136C2×C83⋊C8—22no—
137C2×C83⋊C8—22no—
138C2×C83⋊C8C22×C412yes—
139C2×C83⋊C8C22×C412yes—
140C2×C83⋊C8C2×D414yes—
141C4⋊Dic83—22no—
142C4⋊Dic83—22no—
143C22⋊Dic83—42no—
144(C2×D83)⋊C4—44no—
145(C2×D83)⋊C4—44no—
146(C2×D83)⋊C4—82no—
147C24×C83C2×D411yescharacteristic
148C2×Q8×C83—24no—
149C2×Q8×C83—21no—
150C2×Q8×C83—21no—
151C2×Q8×C83—22no—
152C2×Q8×C83—28no—
153C2×Q8×C83—22no—
154C2×Q8×C83—22no—
155C2×Q8×C83—44no—
156C2×Q8×C83—44no—
157C2×Q8×C83—48no—
158C2×Q8×C83C22×C412yes—
159C2×Q8×C83C22×C412yes—
160C2×Q8×C83C2×D414yes—
161C22×C4×C83—21no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—24no—
165C22×C4×C83—21no—
166C22×C4×C83—22no—
167C22×C4×C83—44no—
168C22×C4×C83C22×C412yes—
169C22×C4×C83C2×D412yes—
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D412yes—
172C22×C4×C83#4112yes—
173C22×C4×C83C2411yescharacteristic
174C4⋊C4×C83—22no—
175C4⋊C4×C83—22no—
176C4⋊C4×C83—44no—
177C22⋊C4×C83—22no—
178C22⋊C4×C83—42no—
179C22⋊C4×C83SD1614yescomplemented
180C42×C83—22no—
181C42×C83—22no—
182C42×C83—44no—
183C23×C83—22no—
184C23×C83—22no—
185C23×C83C4×D412yes—
186C23×C83C2×SD1612yes—
187C23×C83#13512yes—
188C23×C83C22×D411yescharacteristic
189C22×D83—22no—
190C22×D83—24no—
191C22×D83—22no—
192C22×D83—48no—
193C22×D83—41no—
194C22×D83—41no—
195C22×D83—42no—
196C22×D83—42no—
197C22×D83—41no—
198C22×D83—41no—
199C22×D83—42no—
200C22×D83—48no—
201C22×D83—84no—
202C22×D83—88no—
203C22×D83—88no—
204Q8×C83—22no—
205Q8×C83—24no—
206Q8×C83—22no—
207Q8×C83—22no—
208Q8×C83—48no—
209Q8×C83—42no—
210Q8×C83C2×C22⋊C414yescomplemented
211C2×C4×C83—22no—
212C2×C4×C83—22no—
213C2×C4×C83—24no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—24no—
219C2×C4×C83—22no—
220C2×C4×C83—22no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—48no—
224C2×C4×C83—42no—
225C2×C4×C83—42no—
226C2×C4×C83—42no—
227C2×C4×C83—42no—
228C2×C4×C83C4×D412yes—
229C2×C4×C83C2×SD1614yescomplemented
230C2×C4×C83C23×C411yescharacteristic
231C2×C4×C83C22×D412yes—
232C2×C4×C83C2×#4111yescharacteristic
233C2×C4×C83H(5,𝔽2)12yes—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83—42no—
237D332—22no—
238D332—22no—
239D332—22no—
240D332—24no—
241D332—48no—
242D332—42no—
243D332C2×C22⋊C414yes—
244C4×D83—88no—
245C83⋊C8—22no—
246C83⋊C8—42no—
247C83⋊C8C2×C22⋊C414yescomplemented
248C22×C83—22no—
249C22×C83—22no—
250C22×C83—24no—
251C22×C83—24no—
252C22×C83C4×SD1614yescomplemented
253C22×C83C2×C4×D411yescharacteristic
254C22×C83D4212yes—
255C22×C83C22×SD1611yescharacteristic
256C22×C83C2×#13511yescharacteristic
257C22×C83#57712yes—
258C2×D83—24no—
259C2×D83—42no—
260C2×D83—42no—
261C2×D83—44no—
262C2×D83—44no—
263C2×D83—42no—
264C2×D83—42no—
265C2×D83—42no—
266C2×D83—816no—
267C2×D83—82no—
268C2×D83—82no—
269C4×C83—22no—
270C4×C83—24no—
271C4×C83—24no—
272C4×C83—42no—
273C4×C83—42no—
274C4×C83—82no—
275C4×C83C22×C22⋊C411yescharacteristic
276C4×C83#51811yescharacteristic
277C4×C83#54612yes—
278Dic83—82no—
279C22⋊C4×SD16—831nomaximal, Sylow
280C2×C83—24no—
281C2×C83D4×C22⋊C411yescharacteristic
282C2×C83C2×C4×SD1611yescharacteristic
283C2×C83#294911yescharacteristic
284C2×C83D4×SD1612yes—
285C2×C83#328212yes—
286D83—44no—
287D83—82no—
288C23×SD16—831no—
289C2×C4×SD16—832no—
290Q8×C22⋊C4D8311yescharacteristic, complemented
291D4×C22⋊C4—831no—
292#1968—832no—
293#1880—832no—
294#1877—832no—
295#1807—831no—
296#1787—831no—
297#1786—831no—
298C8×C22⋊C4—831no—
299C83C22⋊C4×SD1611yescharacteristic, complemented, Sylow
300C23×Q8C2×D8311yescharacteristic
301C23×D4—831no—
302C22×SD16—831no—
303C22×SD16—838no—
304C22×SD16—831no—
305C22×SD16—831no—
306C22×SD16—831no—
307C22×SD16—1668no—
308C23×C8—831no—
309C2×C4×Q8C2×D8312yescomplemented
310C2×C4×D4—832no—
311C22×C22⋊C4—1661no—
312C4×SD16—1668no—
313C2×#106—832no—
314C2×#103—832no—
315C2×#103—832no—
316C2×#102—832no—
317C2×#102—832no—
318C2×C22⋊C8—831no—
319C2×C4×C8—832no—
320#391—22no—
321#391C2×D8312yescomplemented
322#390—832no—
323#386—1662no—
324#380—21no—
325#380—831no—
326#380C2×D8311yescharacteristic
327#379—1661no—
328C4×C22⋊C4—21no—
329C4×C22⋊C4C2×D8311yescharacteristic
330#340—832no—
331#336—832no—
332C25—1661no—
333C22×Q8—21no—
334C22×Q8—22no—
335C22×Q8—24no—
336C22×Q8—42no—
337C22×Q8C4×D8314yescomplemented
338C22×Q8C22×D8311yescharacteristic
339C22×Q8C22×D8311yescharacteristic
340C22×D4—3322no—
341C22×D4—834no—
342C22×D4—831no—
343C22×D4—831no—
344C22×D4—1664no—
345C22×D4—1661no—
346C22×D4—1662no—
347C23×C4—21no—
348C23×C4—1662no—
349C23×C4C22×D8311yescharacteristic
350C2×SD16—834no—
351C2×SD16—838no—
352C2×SD16—834no—
353C2×SD16—834no—
354C2×SD16—834no—
355C2×SD16—1664no—
356C2×SD16—16632no—
357C2×SD16—1664no—
358C2×SD16—1664no—
359C22×C8—831no—
360C22×C8—834no—
361C22×C8—831no—
362C22×C8—832no—
363C22×C8—1664no—
364C4×Q8—24no—
365C4×Q8—42no—
366C4×D4—3322no—
367C4×D4—1664no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—832no—
370C2×C4⋊C4C22×D8312yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—3324no—
373C2×C22⋊C4—1661no—
374C2×C22⋊C4—1662no—
375C2×C22⋊C4—1662no—
376C2×C22⋊C4—1662no—
377C2×C22⋊C4C22⋊D8311yescharacteristic
378C2×C22⋊C4C22⋊D8311yescharacteristic
379C2×C42—22no—
380C2×C42C22×D8312yes—
381#106—1664no—
382#103—3322no—
383#103—1664no—
384#102—3322no—
385#102—1664no—
386C22⋊C8—1662no—
387C4×C8—1664no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—1661no—
392#95C22⋊D8312yes—
393C24—3324no—
394C24—3322no—
395C24—3322no—
396C24—6642no—
397C24—1661no—
398C24—1664no—
399C24—1661no—
400C24C2×C22⋊D8311yescharacteristic
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—28no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8C2×C4×D8312yes—
412C2×Q8C2×C4×D8312yes—
413C2×Q8D4×D8314yes—
414C2×D4—3324no—
415C2×D4—3328no—
416C2×D4—3324no—
417C2×D4—832no—
418C2×D4—832no—
419C2×D4—834no—
420C2×D4—1668no—
421C2×D4—1662no—
422C2×D4—1661no—
423C2×D4—1661no—
424C2×D4—1662no—
425C2×D4—1662no—
426C2×D4—1664no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—3324no—
435C22×C4—3324no—
436C22×C4—6642no—
437C22×C4—1662no—
438C22×C4C2×C4×D8312yes—
439C22×C4C2×C22⋊D8312yes—
440C22×C4C2×C22⋊D8312yes—
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4#1305059012yes—
443C22×C4C23×D8311yescharacteristic
444SD16—8316no—
445SD16—1668no—
446SD16—1668no—
447SD16—1668no—
448C2×C8—832no—
449C2×C8—834no—
450C2×C8—832no—
451C2×C8—1662no—
452C2×C8—1662no—
453C2×C8—1662no—
454C2×C8—1668no—
455C2×C8—1662no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—44no—
459C4⋊C4—1662no—
460C4⋊C4—1662no—
461C22⋊C4—22no—
462C22⋊C4—42no—
463C22⋊C4—3324no—
464C22⋊C4—3322no—
465C22⋊C4—3324no—
466C22⋊C4—6642no—
467C22⋊C4#1305057714yesdirect product factor
468C42—22no—
469C42—22no—
470C42—44no—
471C23—22no—
472C23—22no—
473C23—3328no—
474C23—3321no—
475C23—3321no—
476C23—3322no—
477C23—3322no—
478C23—3321no—
479C23—3321no—
480C23—3322no—
481C23—3328no—
482C23—6644no—
483C23—6648no—
484C23—6648no—
485C23—1662no—
486C23—1664no—
487C23—1662no—
488C23C4×C22⋊D8312yes—
489C23C2×#1305057712yes—
490C23#1458181212yes—
491C23C22×C22⋊D8311yescharacteristic
492Q8—24no—
493Q8—22no—
494Q8—22no—
495Q8—22no—
496Q8—48no—
497Q8—42no—
498Q8C22⋊C4×D8314yescomplemented
499D4—3328no—
500D4—3322no—
501D4—834no—
502D4—1664no—
503D4—1662no—
504D4—1662no—
505D4—1662no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—3322no—
524C2×C4—3322no—
525C2×C4—3322no—
526C2×C4—6648no—
527C2×C4C4×C22⋊D8312yes—
528C2×C4C2×#1305057714yescomplemented
529C2×C4C22×C4×D8311yescharacteristic
530C2×C4C2×D4×D8312yes—
531C2×C4C2×#1305059011yescharacteristic
532C2×C4#1458183712yes—
533C8—3322no—
534C8—834no—
535C8—1662no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—22no—
540C22—3322no—
541C22—3324no—
542C22—3324no—
543C22—3322no—
544C22—3322no—
545C22—3322no—
546C22—3322no—
547C22—66416no—
548C22—6642no—
549C22—6642no—
550C22—1664no—
551C22C4×#1305057714yescomplemented
552C22C2×C4×C22⋊D8311yescharacteristic
553C22D4×C22⋊D8312yes—
554C22C22×#1305057711yescharacteristic
555C22C2×#1458181211yescharacteristic
556C22#1912850912yes—
557C4—24no—
558C4—22no—
559C4—24no—
560C4—42no—
561C4—42no—
562C4—82no—
563C4—6642no—
564C4C2×C22⋊C4×D8311yescharacteristic
565C4#1912817411yescharacteristic
566C4#1912834112yes—
567C2—24no—
568C2—3324no—
569C2—6642no—
570C2C22⋊C4×C22⋊D8311yescharacteristic
571C2C2×C4×#1305057711yescharacteristic
572C2#2491632211yescharacteristic
573C2D4×#1305057712yes—
574C2#2491917112yes—
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingQ8×C22⋊C4×C83C2
socleC23×C83C22×D4
prime radicalC23×D332C22
cosocleC22×C4×C83C24
hypercenterQ8×C22⋊C4D83
hypocenterC83C22⋊C4×SD16
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#13050577C2semidirect
4. #24908320C2semidirect
5. #24907818C2semidirect
6. #24907800C2semidirect
7. #24907486C2semidirect
8. #24907398C2semidirect
10. C22⋊C4×C83⋊C8C2semidirect
11. Q8×C22⋊C4×C83C2semidirect
14. C22×#13050577C4semidirect
23. C2×#14581723C22semidirect
24. C2×#14581716C22semidirect
25. C2×#14581716C22semidirect
29. C2×C4×C83⋊C8C22semidirect
34. #19127463C22semidirect
35. #19127456C22semidirect
37. C2×C4×Q8×C83C22semidirect
39. #391×C83C22semidirect
59. C2×#13050577C2×C4semidirect
60. C2×#13050577C2×C4semidirect
63. C22×C83⋊C8C2×C4semidirect
87. C22×Q8×C83C2×C4semidirect
132. #13050577C22⋊C4direct (N × H)
179. C22⋊C4×C83SD16semidirect
210. Q8×C83C2×C22⋊C4semidirect
229. C2×C4×C83C2×SD16semidirect
247. C83⋊C8C2×C22⋊C4semidirect
252. C22×C83C4×SD16semidirect
290. Q8×C22⋊C4D83semidirect
299. C83C22⋊C4×SD16semidirect
309. C2×C4×Q8C2×D83semidirect
321. #391C2×D83semidirect
337. C22×Q8C4×D83semidirect
467. C22⋊C4#13050577direct (N × H)
498. Q8C22⋊C4×D83semidirect
528. C2×C4C2×#13050577semidirect
551. C22C4×#13050577semidirect

Maximal subgroups

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

subgroupindex
C23×#130505772
C2×C4×#130505772
C22⋊C4×D3322
#249083202
#249078182
#249078002
#249074862
#249073982
#249073932
C22⋊C4×C83⋊C82
Q8×C22⋊C4×C832
C22⋊C4×SD1683