Finite Group Databasebeta

← C4×#1859 #354446

Subgroups — C4×#1859 #354446

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#520C211yesmaximal, characteristic
2C2×#1859C212yesmaximal
3C4×#428C211yesmaximal, characteristic, complemented
4C4×#417C211yesmaximal, characteristic
5C2×C4×#104C211yesmaximal, characteristic
6C4×#384C211yesmaximal, characteristic, complemented
7#10899C212yesmaximal, complemented
8#10891C212yesmaximal, complemented
9#10781C212yesmaximal
10C4×#343C211yesmaximal, characteristic, complemented
11C4×#339C211yesmaximal, characteristic, complemented
12C2×#520C2212yes—
13C2×C4×#41C2211yescharacteristic
14C2×#428C2212yes—
15C22×#104C2211yescharacteristic
16C22×#104C2211yescharacteristic
17C2×#417C2212yes—
18C2×C4×M16C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20C2×#384C2212yes—
21#2276—21no—
22#2275—22no—
23#2275—21no—
24#2274—22no—
25#2274C2212yes—
26C42×Q8—21no—
27C42×D4—22no—
28C42×D4—21no—
29C4×#117—21no—
30C4×#117C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4—22no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1859C4116yesdirect product factor
37#1776—22no—
38C4×#107—21no—
39C4×#106—21no—
40#1763—21no—
41#1762—21no—
42C4×#103—21no—
43C4×#102—21no—
44C4×#104C418yescomplemented
45C2×#343C2212yes—
46C2×#343C2212yes—
47#1736C2212yes—
48#1736C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C4⋊#95—22no—
54C4⋊#95C2212yes—
55C22×#41C2311yescharacteristic
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59#520—24no—
60#520—24no—
61#520—24no—
62#520C2×C418yes—
63C4×#41—22no—
64C4×#41—22no—
65C4×#41C2×C414yes—
66C4×#41Q814yes—
67C2×C4×Q8—22no—
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—24no—
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×#117—22no—
77C2×#117—21no—
78C2×#117C2311yescharacteristic
79C2×#117C2312yes—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42D412yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#428—24no—
94#428C2×C418yescomplemented
95C2×#107—22no—
96C2×#106—22no—
97C2×#104—24no—
98C2×#104C2×C414yes—
99C2×#104C2×C418yes—
100C2×#104C2×C414yes—
101#417—24no—
102#417C2×C418yes—
103C2×#103—22no—
104C2×#102—22no—
105C4×M16—22no—
106C4×M16—22no—
107C4×M16—22no—
108C4×M16C2×C414yescomplemented
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113#384—24no—
114#384C2×C418yescomplemented
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—48no—
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D412yes—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—24no—
142#376—22no—
143#376—21no—
144C2×#95—22no—
145C2×#95—21no—
146C2×#95—24no—
147C2×#95D412yes—
148C43—22no—
149C43—21no—
150C43—22no—
151C43C2×C414yescomplemented
152#343—24no—
153#343C2×C418yescomplemented
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#339—24no—
159#339C2×C418yescomplemented
160#339C2×C414yescomplemented
161#339C2×C414yescomplemented
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41C22×C412yes—
167C2×#41C22×C414yes—
168C2×#41C2×Q812yes—
169C2×#41#4114yes—
170C22×Q8C2×D411yescharacteristic
171C22×D4—21no—
172C22×D4C2×D411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—24no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D414yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C2×M16—21no—
185C2×M16—22no—
186C2×M16—21no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16C22×C414yes—
195C2×M16C22×C412yes—
196C2×M16C22×C412yes—
197C2×M16C22×C414yes—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C4×Q8—28no—
202C4×Q8—28no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8C4⋊C414yescomplemented
206C4×Q8C4⋊C414yescomplemented
207C4×D4—216no—
208C4×D4—28no—
209C4×D4—28no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—216no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4C4⋊C414yescomplemented
218C4×D4C4⋊C414yescomplemented
219#117—28no—
220#117—21no—
221#117—21no—
222#117—22no—
223#117—22no—
224#117—28no—
225#117—24no—
226#117—24no—
227#117C22×C412yes—
228#117C22×C414yes—
229#117C2×Q812yes—
230#117#4114yes—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4#4114yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—41no—
287C2×C42—41no—
288C2×C42—84no—
289C2×C42C22×C412yes—
290C2×C42C22×C412yes—
291C2×C42C22×C414yes—
292C2×C42C22×C412yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×Q812yes—
298C2×C42#4112yes—
299C2×C42#4112yes—
300#107—28no—
301#106—28no—
302#104—28no—
303#104C42116yescomplemented
304#103—28no—
305#102—28no—
306#97—24no—
307#97—42no—
308C4×C8—24no—
309C4×C8—42no—
310#95—24no—
311#95—24no—
312#95—21no—
313#95—22no—
314#95—22no—
315#95—21no—
316#95—22no—
317#95—48no—
318#95—41no—
319#95—41no—
320C24—21no—
321C24C22×D411yescharacteristic
322#41—22no—
323#41—24no—
324#41—22no—
325#41—24no—
326#41—24no—
327#41—24no—
328#41—24no—
329#41C2×C4214yes—
330#41#11714yes—
331#41C4×Q818yes—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8C2×C4⋊C412yes—
335C2×Q8C4×D414yes—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4C2×C4⋊C412yes—
343C2×D4C4×D414yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—216no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—82no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4C4×D412yes—
384C22×C4C4×D418yes—
385C22×C4C4×D412yes—
386C22×C4#12011yescharacteristic
387C22×C4#12011yescharacteristic
388C22×C4#12011yescharacteristic
389C22×C4#12011yescharacteristic
390C22×C4C22⋊Q812yes—
391C22×C4C22⋊Q812yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C23×C412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×Q811yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402M16—22no—
403M16—24no—
404M16—22no—
405M16—22no—
406M16—24no—
407M16—24no—
408M16—42no—
409M16—42no—
410M16—42no—
411M16C2×C4214yescomplemented
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C4⋊C4—216no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—432no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4C2×C4⋊C412yes—
436C4⋊C4C2×C4⋊C412yes—
437C4⋊C4C4×D414yes—
438C4⋊C4C4×D414yes—
439C22⋊C4—216no—
440C22⋊C4—22no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C42—216no—
447C42—216no—
448C42—22no—
449C42—24no—
450C42—22no—
451C42—24no—
452C42—22no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—22no—
457C42—22no—
458C42—48no—
459C42—42no—
460C42—44no—
461C42—42no—
462C42—42no—
463C42—82no—
464C42—82no—
465C42—82no—
466C42—84no—
467C42—84no—
468C42—82no—
469C42C2×C4214yescomplemented
470C42C2×C4⋊C412yes—
471C42C2×C4⋊C412yes—
472C42C4×D412yes—
473C42C4×D412yes—
474C42C22⋊Q812yes—
475C42C22⋊Q812yes—
476C42#12312yes—
477C42#12312yes—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23C2×C4×D411yescharacteristic
487C23C2×C4×D412yes—
488C23C2×#12011yescharacteristic
489C23C2×C22⋊Q811yescharacteristic
490C23#52512yes—
491Q8—22no—
492Q8—24no—
493Q8C4×C4⋊C414yescomplemented
494D4—24no—
495D4—24no—
496D4—28no—
497D4—22no—
498D4—24no—
499D4C4×C4⋊C414yescomplemented
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—816no—
544C2×C4—84no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#38512yes—
548C2×C4#38512yes—
549C2×C4#38512yes—
550C2×C4#38512yes—
551C2×C4#45312yes—
552C2×C4#45612yes—
553C2×C4C22×C4211yescharacteristic
554C2×C4C22×C4⋊C411yescharacteristic
555C2×C4C2×#11711yescharacteristic
556C2×C4C2×C4×D412yes—
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C2×C4×D412yes—
559C2×C4C2×C4×Q812yes—
560C2×C4C4×#4111yescharacteristic
561C2×C4C4×#4112yes—
562C2×C4C4×#4111yescharacteristic
563C2×C4C4×#4112yes—
564C2×C4C2×C22⋊Q811yescharacteristic
565C2×C4C2×#12311yescharacteristic
566C2×C4#52512yes—
567C2×C4#52512yes—
568C2×C4#52912yes—
569C2×C4#52912yes—
570C2×C4#53312yes—
571C8—24no—
572C8—42no—
573C8—44no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—44no—
587C22C42×D411yescharacteristic
588C22C2×#38511yescharacteristic
589C22#229011yescharacteristic
590C22C4×#12012yes—
591C22C4×C22⋊Q812yes—
592C22C2×#45311yescharacteristic
593C22C2×#45611yescharacteristic
594C22#301812yes—
595C4—28no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—24no—
600C4—44no—
601C4—42no—
602C4—42no—
603C4—48no—
604C4—44no—
605C4—44no—
606C4—84no—
607C4#185914yesdirect product factor
608C4C2×C4×C4⋊C411yescharacteristic
609C4C42×D411yescharacteristic
610C4#229011yescharacteristic
611C4#229611yescharacteristic
612C4C4×#12312yes—
613C4C4×C22⋊Q812yes—
614C2—21no—
615C2—21no—
616C2—22no—
617C2—44no—
618C2C4×#38511yescharacteristic
619C2C2×#185911yescharacteristic
620C2#2086311yescharacteristic
621C2C4×#45312yes—
622C2C4×#45612yes—
subgroupG/H
centerC22×C4#120
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#428C2semidirect
6. C4×#384C2semidirect
7. #10899C2semidirect
8. #10891C2semidirect
10. C4×#343C2semidirect
11. C4×#339C2semidirect
36. #1859C4direct (N × H)
44. C4×#104C4semidirect
94. #428C2×C4semidirect
108. C4×M16C2×C4semidirect
114. #384C2×C4semidirect
151. C43C2×C4semidirect
153. #343C2×C4semidirect
159. #339C2×C4semidirect
160. #339C2×C4semidirect
161. #339C2×C4semidirect
205. C4×Q8C4⋊C4semidirect
206. C4×Q8C4⋊C4semidirect
217. C4×D4C4⋊C4semidirect
218. C4×D4C4⋊C4semidirect
303. #104C42semidirect
411. M16C2×C42semidirect
469. C42C2×C42semidirect
493. Q8C4×C4⋊C4semidirect
499. D4C4×C4⋊C4semidirect
607. C4#1859direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#5202
C2×#18592
C4×#4282
C4×#4172
C2×C4×#1042
C4×#3842
#108992
#108912
#107812
C4×#3432
C4×#3392