Finite Group Databasebeta

← C22⋊C4×#13050578 #35420233

Subgroups — C22⋊C4×#13050578 #35420233

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×#13050578C211yesmaximal, characteristic
2C2×C4×#13050578C212yesmaximal, complemented
3C22⋊C4×Dic166C211yesmaximal, characteristic, complemented
4#24908316C212yesmaximal, complemented
5#24907821C212yesmaximal, complemented
6#24907797C212yesmaximal, complemented
7#24907486C211yesmaximal, characteristic, complemented
8#24907397C211yesmaximal, characteristic, complemented
9#24907394C211yesmaximal, characteristic
10C22⋊C4×C83⋊C8C211yesmaximal, characteristic, complemented
11D4×C22⋊C4×C83C211yesmaximal, characteristic
12C23×Dic166C2211yescharacteristic
13C22×#13050578—28no—
14C22×#13050578C418yescomplemented
15C22×#13050578C2211yescharacteristic
16C22×#13050578C2211yescharacteristic
17C22×#13050578C2211yescharacteristic
18C22×#13050578C2211yescharacteristic
19C23×C83⋊C8C2211yescharacteristic
20C2×C4×Dic166C2212yescomplemented
21C4×#13050578—28no—
22C2×#14581723C2212yescomplemented
23C2×#14581715C2212yescomplemented
24C2×#14581715C2212yescomplemented
25C2×#14581713C2212yes—
26C2×#14581713C2212yes—
27C2×C22⋊(C83⋊C8)C2211yescharacteristic
28C2×C4×C83⋊C8C2212yescomplemented
29#19127607C2212yescomplemented
30#19127605—22no—
31#19127561C2211yescharacteristic
32#19127559—21no—
33C22⋊C4×Dic83—21no—
34#19127463C2212yescomplemented
35#19127456C2212yescomplemented
36C23×D4×C83C2211yescharacteristic
37C2×C4×D4×C83C2212yes—
38C22×C22⋊C4×C83—21no—
39#390×C83C2212yes—
40#386×C83—22no—
41#380×C83C2211yescharacteristic
42#379×C83—21no—
43C4×C22⋊C4×C83C2211yescharacteristic
44C22×Dic166—24no—
45C22×Dic166—21no—
46C22×Dic166—22no—
47C22×Dic166—42no—
48C22×Dic166C2×C414yescomplemented
49C22×Dic166C2311yescharacteristic
50C22×Dic166C2311yescharacteristic
51C23×Dic83—21no—
52C2×#13050578—24no—
53C2×#13050578—232no—
54C2×#13050578—24no—
55C2×#13050578—24no—
56C2×#13050578C2×C414yes—
57C2×#13050578C2×C414yes—
58C2×#13050578C2×C414yescomplemented
59C2×#13050578C2×C414yescomplemented
60C2×#13050578D418yes—
61C22×C83⋊C8—24no—
62C22×C83⋊C8C2×C414yescomplemented
63C22×C83⋊C8C2312yes—
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C4×Dic166—24no—
67C4×Dic166—42no—
68C2×C4⋊Dic83C2312yes—
69C2×Dic83⋊C4—22no—
70C2×C22⋊Dic83—21no—
71C2×C4×Dic83—22no—
72#14581723—24no—
73#14581715—24no—
74#14581715—42no—
75#14581713—24no—
76#14581713—42no—
77C22⋊(C83⋊C8)—22no—
78C4×C83⋊C8—24no—
79C83⋊#95—22no—
80C83⋊#95—21no—
81C83⋊#95—22no—
82C25×C83—21no—
83C22×D4×C83—24no—
84C22×D4×C83—21no—
85C22×D4×C83—22no—
86C22×D4×C83—42no—
87C22×D4×C83C2×C414yes—
88C22×D4×C83C2311yescharacteristic
89C22×D4×C83C2311yescharacteristic
90C23×C4×C83—22no—
91C23×C4×C83C2311yescharacteristic
92C4×D4×C83—24no—
93C4×D4×C83—42no—
94C2×C4⋊C4×C83C2312yes—
95C2×C22⋊C4×C83—21no—
96C2×C22⋊C4×C83—22no—
97C2×C22⋊C4×C83—22no—
98C2×C22⋊C4×C83—22no—
99C2×C22⋊C4×C83—44no—
100C2×C22⋊C4×C83D411yescharacteristic
101C2×C22⋊C4×C83D411yescharacteristic
102C2×C42×C83C2312yes—
103#95×C83—21no—
104#95×C83D412yes—
105C2×Dic166—21no—
106C2×Dic166—22no—
107C2×Dic166—22no—
108C2×Dic166—22no—
109C2×Dic166—28no—
110C2×Dic166—24no—
111C2×Dic166—21no—
112C2×Dic166—44no—
113C2×Dic166—48no—
114C2×Dic166—44no—
115C2×Dic166C22×C412yes—
116C2×Dic166C22×C412yes—
117C2×Dic166C2×D414yes—
118C22×Dic83—21no—
119C22×Dic83—24no—
120C22×Dic83—21no—
121C22×Dic83—22no—
122C22×Dic83—44no—
123#13050578—28no—
124#13050578—28no—
125#13050578—28no—
126#13050578C22⋊C4116yesdirect product factor
127C2×C83⋊C8—22no—
128C2×C83⋊C8—28no—
129C2×C83⋊C8—22no—
130C2×C83⋊C8—22no—
131C2×C83⋊C8—22no—
132C2×C83⋊C8C22×C412yes—
133C2×C83⋊C8C22×C412yes—
134C2×C83⋊C8C2×D414yes—
135C4⋊Dic83—22no—
136C4⋊Dic83—22no—
137Dic83⋊C4—44no—
138C22⋊Dic83—42no—
139C4×Dic83—44no—
140C24×C83—21no—
141C24×C83—21no—
142C24×C83—24no—
143C24×C83—44no—
144C24×C83—42no—
145C24×C83—42no—
146C24×C83—82no—
147C24×C83C2×D411yescharacteristic
148C2×D4×C83—28no—
149C2×D4×C83—24no—
150C2×D4×C83—22no—
151C2×D4×C83—21no—
152C2×D4×C83—21no—
153C2×D4×C83—22no—
154C2×D4×C83—22no—
155C2×D4×C83—44no—
156C2×D4×C83—48no—
157C2×D4×C83—44no—
158C2×D4×C83C22×C412yes—
159C2×D4×C83C22×C412yes—
160C2×D4×C83C2×D414yes—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—44no—
165C22×C4×C83—44no—
166C22×C4×C83—82no—
167C22×C4×C83C22×C412yes—
168C22×C4×C83C2×D412yes—
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D412yes—
171C22×C4×C83#4112yes—
172C22×C4×C83C2411yescharacteristic
173C4⋊C4×C83—22no—
174C4⋊C4×C83—22no—
175C22⋊C4×C83—22no—
176C22⋊C4×C83—44no—
177C22⋊C4×C83—42no—
178C22⋊C4×C83—44no—
179C22⋊C4×C83—82no—
180C22⋊C4×C83SD1614yescomplemented
181C42×C83—22no—
182C42×C83—22no—
183C23×C83—22no—
184C23×C83—22no—
185C23×C83—22no—
186C23×C83—24no—
187C23×C83—22no—
188C23×C83—48no—
189C23×C83—41no—
190C23×C83—41no—
191C23×C83—42no—
192C23×C83—42no—
193C23×C83—41no—
194C23×C83—41no—
195C23×C83—42no—
196C23×C83—48no—
197C23×C83—84no—
198C23×C83—88no—
199C23×C83—88no—
200C23×C83C4×D412yes—
201C23×C83C2×SD1612yes—
202C23×C83#13512yes—
203C23×C83C22×D411yescharacteristic
204D4×C83—22no—
205D4×C83—22no—
206D4×C83—22no—
207D4×C83—24no—
208D4×C83—48no—
209D4×C83—42no—
210D4×C83C2×C22⋊C414yes—
211C2×C4×C83—22no—
212C2×C4×C83—22no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—24no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—42no—
221C2×C4×C83—42no—
222C2×C4×C83—42no—
223C2×C4×C83—88no—
224C2×C4×C83C4×D412yes—
225C2×C4×C83C2×SD1614yescomplemented
226C2×C4×C83C23×C411yescharacteristic
227C2×C4×C83C22×D412yes—
228C2×C4×C83C2×#4111yescharacteristic
229C2×C4×C83H(5,𝔽2)12yes—
230C2×Dic83—22no—
231C2×Dic83—24no—
232C2×Dic83—22no—
233C2×Dic83—48no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83—42no—
237C2×Dic83—42no—
238Dic166—22no—
239Dic166—24no—
240Dic166—22no—
241Dic166—22no—
242Dic166—48no—
243Dic166—42no—
244Dic166C2×C22⋊C414yescomplemented
245C83⋊C8—22no—
246C83⋊C8—42no—
247C83⋊C8C2×C22⋊C414yescomplemented
248C22×C83—22no—
249C22×C83—22no—
250C22×C83—24no—
251C22×C83—24no—
252C22×C83—24no—
253C22×C83—42no—
254C22×C83—42no—
255C22×C83—44no—
256C22×C83—44no—
257C22×C83—42no—
258C22×C83—42no—
259C22×C83—42no—
260C22×C83—816no—
261C22×C83—82no—
262C22×C83—82no—
263C22×C83C4×SD1614yescomplemented
264C22×C83C2×C4×D411yescharacteristic
265C22×C83D4212yes—
266C22×C83C22×SD1611yescharacteristic
267C22×C83C2×#13511yescharacteristic
268C22×C83#57712yes—
269C4×C83—22no—
270C4×C83—24no—
271C4×C83—42no—
272C4×C83—82no—
273C4×C83C22×C22⋊C411yescharacteristic
274C4×C83#51811yescharacteristic
275C4×C83#54612yes—
276Dic83—24no—
277Dic83—42no—
278Dic83—82no—
279C22⋊C4×SD16—831nomaximal, Sylow
280C2×C83—24no—
281C2×C83—44no—
282C2×C83—82no—
283C2×C83D4×C22⋊C411yescharacteristic
284C2×C83C2×C4×SD1611yescharacteristic
285C2×C83#294911yescharacteristic
286C2×C83D4×SD1612yes—
287C2×C83#328212yes—
288C23×SD16—831no—
289C2×C4×SD16—832no—
290Q8×C22⋊C4—831no—
291D4×C22⋊C4D8311yescharacteristic
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×Q8—831no—
301C23×D4C2×D8311yescharacteristic
302C22×SD16—831no—
303C22×SD16—838no—
304C22×SD16—831no—
305C22×SD16—831no—
306C22×SD16—831no—
307C22×SD16—1668no—
308C23×C8—831no—
309C2×C4×Q8—832no—
310C2×C4×D4C2×D8312yes—
311C22×C22⋊C4—21no—
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—832no—
321#391—1662no—
322#390C2×D8312yes—
323#386—22no—
324#380—831no—
325#380—1661no—
326#380C2×D8311yescharacteristic
327#379—21no—
328C4×C22⋊C4—1661no—
329C4×C22⋊C4C2×D8311yescharacteristic
330#340—832no—
331#336—832no—
332C25—21no—
333C22×Q8—3322no—
334C22×Q8—834no—
335C22×Q8—831no—
336C22×Q8—831no—
337C22×Q8—1664no—
338C22×Q8—1661no—
339C22×Q8—1662no—
340C22×D4—24no—
341C22×D4—21no—
342C22×D4—22no—
343C22×D4—42no—
344C22×D4C4×D8314yes—
345C22×D4C22×D8311yescharacteristic
346C22×D4C22×D8311yescharacteristic
347C23×C4—22no—
348C23×C4—1661no—
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—3322no—
365C4×Q8—1664no—
366C4×D4—24no—
367C4×D4—42no—
368C2×C4⋊C4—832no—
369C2×C4⋊C4—1662no—
370C2×C4⋊C4C22×D8312yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4—1661no—
377C2×C22⋊C4C22⋊D8311yescharacteristic
378C2×C22⋊C4C22⋊D8311yescharacteristic
379C2×C42—1662no—
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—21no—
389#95—1662no—
390#95—1661no—
391#95—1662no—
392#95C22⋊D8312yes—
393C24—21no—
394C24—21no—
395C24—24no—
396C24—44no—
397C24—42no—
398C24—42no—
399C24—82no—
400C24C2×C22⋊D8311yescharacteristic
401C2×Q8—3328no—
402C2×Q8—3324no—
403C2×Q8—3324no—
404C2×Q8—832no—
405C2×Q8—832no—
406C2×Q8—834no—
407C2×Q8—1668no—
408C2×Q8—1662no—
409C2×Q8—1662no—
410C2×Q8—1661no—
411C2×Q8—1661no—
412C2×Q8—1662no—
413C2×Q8—1664no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—44no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4C2×C4×D8312yes—
425C2×D4C2×C4×D8312yes—
426C2×D4D4×D8314yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—82no—
433C22×C4—3324no—
434C22×C4—1662no—
435C22×C4—1661no—
436C22×C4—1664no—
437C22×C4—1661no—
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—3324no—
459C4⋊C4—1662no—
460C4⋊C4—1662no—
461C22⋊C4—22no—
462C22⋊C4—44no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C22⋊C4—3322no—
467C22⋊C4#1305057814yesdirect product factor
468C42—22no—
469C42—22no—
470C42—3324no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—48no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—48no—
485C23—84no—
486C23—88no—
487C23—88no—
488C23C4×C22⋊D8312yes—
489C23C2×#1305057812yes—
490C23#1458181212yes—
491C23C22×C22⋊D8311yescharacteristic
492Q8—3328no—
493Q8—3322no—
494Q8—834no—
495Q8—1662no—
496Q8—1662no—
497Q8—1664no—
498Q8—1662no—
499D4—24no—
500D4—22no—
501D4—22no—
502D4—22no—
503D4—48no—
504D4—42no—
505D4C22⋊C4×D8314yes—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—88no—
519C2×C4—3322no—
520C2×C4—3328no—
521C2×C4—3322no—
522C2×C4—3322no—
523C2×C4—3322no—
524C2×C4—1662no—
525C2×C4—1664no—
526C2×C4—1662no—
527C2×C4C4×C22⋊D8312yes—
528C2×C4C2×#1305057814yescomplemented
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—24no—
541C22—42no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—816no—
549C22—82no—
550C22—82no—
551C22C4×#1305057814yescomplemented
552C22C2×C4×C22⋊D8311yescharacteristic
553C22D4×C22⋊D8312yes—
554C22C22×#1305057811yescharacteristic
555C22C2×#1458181211yescharacteristic
556C22#1912851212yes—
557C4—24no—
558C4—22no—
559C4—42no—
560C4—82no—
561C4—3322no—
562C4—6642no—
563C4—1664no—
564C4C2×C22⋊C4×D8311yescharacteristic
565C4#1912817411yescharacteristic
566C4#1912834112yes—
567C2—24no—
568C2—44no—
569C2—82no—
570C2C22⋊C4×C22⋊D8311yescharacteristic
571C2C2×C4×#1305057811yescharacteristic
572C2#2491632011yescharacteristic
573C2D4×#1305057812yes—
574C2#2491916612yes—
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingD4×C22⋊C4×C83C2
socleC23×C83C22×D4
prime radicalC23×D4×C83C22
cosocleC22×C4×C83C24
hypercenterD4×C22⋊C4D83
hypocenterC83C22⋊C4×SD16
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ D4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#13050578C2semidirect
3. C22⋊C4×Dic166C2semidirect
4. #24908316C2semidirect
5. #24907821C2semidirect
6. #24907797C2semidirect
7. #24907486C2semidirect
8. #24907397C2semidirect
10. C22⋊C4×C83⋊C8C2semidirect
14. C22×#13050578C4semidirect
20. C2×C4×Dic166C22semidirect
22. C2×#14581723C22semidirect
23. C2×#14581715C22semidirect
24. C2×#14581715C22semidirect
28. C2×C4×C83⋊C8C22semidirect
29. #19127607C22semidirect
34. #19127463C22semidirect
35. #19127456C22semidirect
48. C22×Dic166C2×C4semidirect
58. C2×#13050578C2×C4semidirect
59. C2×#13050578C2×C4semidirect
62. C22×C83⋊C8C2×C4semidirect
126. #13050578C22⋊C4direct (N × H)
180. C22⋊C4×C83SD16semidirect
225. C2×C4×C83C2×SD16semidirect
244. Dic166C2×C22⋊C4semidirect
247. C83⋊C8C2×C22⋊C4semidirect
263. C22×C83C4×SD16semidirect
299. C83C22⋊C4×SD16semidirect
467. C22⋊C4#13050578direct (N × H)
528. C2×C4C2×#13050578semidirect
551. C22C4×#13050578semidirect

Maximal subgroups

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

subgroupindex
C23×#130505782
C2×C4×#130505782
C22⋊C4×Dic1662
#249083162
#249078212
#249077972
#249074862
#249073972
#249073942
C22⋊C4×C83⋊C82
D4×C22⋊C4×C832
C22⋊C4×SD1683