Finite Group Databasebeta

← C2×#25584 #6449038

Subgroups — C2×#25584 #6449038

Order 512.

3636 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C22×#491C211yesmaximal, characteristic, complemented
3C22×#462C211yesmaximal, characteristic, complemented
4C2×#2542C211yesmaximal, characteristic
5#25584C2116yesmaximal, direct product factor
6C2×#2001C212yesmaximal, complemented
7C2×#2001C212yesmaximal, complemented
8C2×#1989C211yesmaximal, characteristic, complemented
9C2×#1955C211yesmaximal, characteristic
10C2×#1873C212yesmaximal, complemented
11C2×#1872C212yesmaximal, complemented
12C2×#1763C211yesmaximal, characteristic, complemented
13C23×Q16C2212yes—
14C2×#552—28no—
15C2×#552—28no—
16C2×#552—24no—
17C2×#552C2218yes—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126—21no—
20C22×#125—21no—
21C22×#125C2211yescharacteristic
22C22×#123—21no—
23C22×C22⋊Q8C2211yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2212yes—
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4C2211yescharacteristic
29C22×#117—21no—
30C22×#117C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C2×#491—28no—
33C2×#491C2218yescomplemented
34C2×#462—28no—
35C2×#462C2218yescomplemented
36C22×C4⋊C8C2211yescharacteristic
37C22×#103C2212yes—
38C22×#103C2212yes—
39C22×#103C2212yes—
40C22×#103C2212yes—
41C22×C22⋊C8C2212yes—
42C22×C22⋊C8C2212yes—
43C22×#97C2211yescharacteristic
44#2542—24no—
45#2542C2218yes—
46C2×#399—21no—
47C2×#399—21no—
48C2×#390—21no—
49C2×#384—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2001C22116yescomplemented
52#2001C22116yescomplemented
53#1989C2218yescomplemented
54#1955C2218yes—
55#1873C22116yescomplemented
56#1872C22116yescomplemented
57#1763C2218yescomplemented
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#340C2212yes—
61C23×Q8C2312yes—
62C23×Q8C2311yescharacteristic
63C23×D4D411yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C22×Q16—28no—
68C22×Q16—44no—
69C22×Q16—44no—
70C22×Q16C2318yes—
71C23×C8C2312yes—
72C23×C8C2312yes—
73#552—264no—
74#552—232no—
75#552—232no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8C2314yes—
80C2×#126—216no—
81C2×#126—28no—
82C2×#125—216no—
83C2×#125—28no—
84C2×#125—22no—
85C2×#125—24no—
86C2×#125D418yes—
87C2×#125C2314yes—
88C2×#123—216no—
89C2×#123—28no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—44no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2314yes—
104C2×C22⋊Q8C2318yes—
105C2×C22⋊Q8C2318yes—
106C2×C4×Q8—24no—
107C2×C4×Q8—22no—
108C2×C4×Q8D418yes—
109C2×C4×Q8C2314yes—
110C2×C4×D4—24no—
111C2×C4×D4—24no—
112C2×C4×D4D414yes—
113C2×C4×D4C2314yes—
114C2×#117—216no—
115C2×#117—22no—
116C2×#117—24no—
117C2×#117—24no—
118C2×#117—28no—
119C2×#117C2314yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—24no—
128C22×C4⋊C4—24no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C22⋊C4—22no—
143C22×C22⋊C4—22no—
144C22×C22⋊C4D411yescharacteristic
145C22×C22⋊C4D411yescharacteristic
146C22×C42—21no—
147C22×C42—21no—
148C22×C42—21no—
149C22×C42C2311yescharacteristic
150#491—232no—
151#462—232no—
152C2×C4⋊C8—24no—
153C2×C4⋊C8C2314yes—
154C2×#103—28no—
155C2×#103—28no—
156C2×#103—28no—
157C2×#103—28no—
158C2×#103C2318yes—
159C2×#103C2318yes—
160C2×#103C2318yes—
161C2×#103C2318yes—
162C2×C22⋊C8—44no—
163C2×C22⋊C8—44no—
164C2×C22⋊C8C2318yes—
165C2×C22⋊C8C2318yes—
166C2×#97—24no—
167C2×#97C2314yes—
168#399—28no—
169#399—28no—
170#390—28no—
171#384—28no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2314yes—
174C2×#95—21no—
175#340C2318yes—
176#340C2318yes—
177#340C2318yes—
178C25C2×D411yescharacteristic
179C25C2×D411yescharacteristic
180C22×Q8—22no—
181C22×Q8—21no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8C2×D414yes—
192C22×Q8C2414yes—
193C22×Q8C2412yes—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—42no—
197C22×D4C2×D414yes—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C2×Q16—216no—
226C2×Q16—48no—
227C2×Q16—48no—
228C2×Q16—432no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8—24no—
232C22×C8—24no—
233C22×C8—24no—
234C22×C8—24no—
235C22×C8C2414yes—
236C22×C8C2414yes—
237C4⋊Q8—28no—
238C4⋊Q8—28no—
239C4⋊Q8—48no—
240#126—264no—
241#125—264no—
242#125—216no—
243#125—28no—
244#125C2×D4116yes—
245#123—264no—
246C22⋊Q8—216no—
247C22⋊Q8—216no—
248C22⋊Q8—28no—
249C22⋊Q8—28no—
250C22⋊Q8—28no—
251C22⋊Q8—28no—
252C22⋊Q8—216no—
253C22⋊Q8—216no—
254C22⋊Q8—48no—
255C22⋊Q8—48no—
256C22⋊Q8—416no—
257C22⋊Q8—416no—
258C4×Q8—216no—
259C4×Q8—28no—
260C4×Q8C2×D4116yes—
261C4×D4—216no—
262C4×D4—28no—
263C4×D4—28no—
264#117—216no—
265#117—264no—
266#117—28no—
267#117—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—216no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4#4118yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4C2×D414yes—
322C2×C22⋊C4C2×D414yes—
323C2×C42—28no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42C2×D412yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2412yes—
333C4⋊C8—28no—
334#103—216no—
335#103—216no—
336#103—216no—
337#103—216no—
338C22⋊C8—48no—
339C22⋊C8—48no—
340#97—28no—
341#95—24no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24#13714yes—
352C24C22×D412yes—
353C24C22×D412yes—
354C24C22×D411yescharacteristic
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8C22×D414yes—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—44no—
380C2×D4—416no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—216no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—432no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4#12014yes—
433C22×C4#12014yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C4C2511yescharacteristic
444Q16—432no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—48no—
452C2×C8—48no—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4C22×D414yes—
473C4⋊C4C22×D414yes—
474C4⋊C4C22×D414yes—
475C4⋊C4C2×#4118yes—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C42—216no—
484C42—216no—
485C42—48no—
486C42#12018yes—
487C42C22×D414yes—
488C42C22×D414yes—
489C23—22no—
490C23—24no—
491C23—28no—
492C23—216no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—44no—
501C23—44no—
502C23—48no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—48no—
507C23—48no—
508C23C2×#12012yes—
509C23#54614yes—
510C23C2×#13714yes—
511C23C2×#13712yes—
512C23#57512yes—
513C23#57814yes—
514C23C23×D411yescharacteristic
515Q8—24no—
516Q8—24no—
517Q8—48no—
518Q8—44no—
519Q8—44no—
520Q8—44no—
521Q8—44no—
522Q8—44no—
523Q8—88no—
524D4—416no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—432no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—416no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—88no—
567C2×C4C2×#12014yes—
568C2×C4C2×#12014yes—
569C2×C4C2×#12012yes—
570C2×C4#53214yes—
571C2×C4#53514yes—
572C2×C4#54614yes—
573C2×C4#55018yes—
574C2×C4C23×D411yescharacteristic
575C2×C4C23×D411yescharacteristic
576C2×C4C22×#4112yes—
577C2×C4C2×H(5,𝔽2)12yes—
578C8—44no—
579C8—44no—
580C22—24no—
581C22—28no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—416no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—88no—
592C22#243114yes—
593C22#300714yes—
594C22#301114yes—
595C22#331218yes—
596C22#331518yes—
597C22C22×#12011yescharacteristic
598C22C2×#54612yes—
599C22C22×#13711yescharacteristic
600C22C2×#57511yescharacteristic
601C22C2×#57812yes—
602C4—24no—
603C4—24no—
604C4—42no—
605C4—42no—
606C4—44no—
607C4—44no—
608C4—44no—
609C4—44no—
610C4—48no—
611C4—84no—
612C4#243114yes—
613C4C2×#53211yescharacteristic
614C4C2×#53511yescharacteristic
615C4C2×#55012yes—
616C2—44no—
617C2—82no—
618C2#2558418yesdirect product factor
619C2C2×#243111yescharacteristic
620C2C2×#300711yescharacteristic
621C2C2×#301111yescharacteristic
622C2C2×#331212yes—
623C2C2×#331512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#491C2semidirect
3. C22×#462C2semidirect
5. #25584C2direct (N × H)
6. C2×#2001C2semidirect
7. C2×#2001C2semidirect
8. C2×#1989C2semidirect
10. C2×#1873C2semidirect
11. C2×#1872C2semidirect
12. C2×#1763C2semidirect
33. C2×#491C22semidirect
35. C2×#462C22semidirect
51. #2001C22semidirect
52. #2001C22semidirect
53. #1989C22semidirect
55. #1873C22semidirect
56. #1872C22semidirect
57. #1763C22semidirect
618. C2#25584direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5522
C22×#4912
C22×#4622
C2×#25422
#255842
C2×#20012
C2×#19892
C2×#19552
C2×#18732
C2×#18722
C2×#17632