Finite Group Databasebeta

← #641800

Subgroups — #641800

Order 512.

4070 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#17057C212yesmaximal, complemented
4#16986C214yesmaximal, complemented
5#16365C212yesmaximal, complemented
6#15092C212yesmaximal, complemented
7#15014C212yesmaximal, complemented
8#14914C212yesmaximal, complemented
9#14786C212yesmaximal, complemented
10#14766C214yesmaximal, complemented
11#14353C212yesmaximal, complemented
12#14281C212yesmaximal, complemented
13#13877C214yesmaximal
14C2×C22⋊#95C211yesmaximal, characteristic
15C25×C4C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23#2848—28no—
24C2×#398C2214yescomplemented
25C2×#385C2212yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#380—28no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2218yes—
39C2×#379C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43#1446C2214yescomplemented
44#1445C2214yescomplemented
45#1445C2214yescomplemented
46#1444C2214yescomplemented
47#1444C2214yescomplemented
48#1444C2214yescomplemented
49#1444C2214yescomplemented
50#1444C2214yescomplemented
51#1442C2218yescomplemented
52#1441C2218yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1439C2218yes—
57#1439C2218yes—
58C22⋊#95C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2218yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C26C2311yescharacteristic
66C23×Q8C2311yescharacteristic
67C24×C4—28no—
68C24×C4—44no—
69C24×C4—44no—
70C24×C4D418yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2311yescharacteristic
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8—44no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2318yes—
101C22×C4⋊C4C2318yes—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—48no—
106C22×C22⋊C4Q8116yescomplemented
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117#398—48no—
118#385—28no—
119#385—28no—
120#385—28no—
121#385—48no—
122#385—48no—
123#380—24no—
124#380—24no—
125#380—24no—
126#380—28no—
127#380—28no—
128#380—28no—
129#380—28no—
130#380—416no—
131#379—28no—
132C2×#95—28no—
133C2×#95—28no—
134C2×#95—44no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2314yes—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C25—24no—
151C25—24no—
152C25—22no—
153C25—24no—
154C25—22no—
155C25—42no—
156C25—42no—
157C25C2×D414yes—
158C25C2×Q818yes—
159C25C2411yescharacteristic
160C25C2411yescharacteristic
161C25C2411yescharacteristic
162C22×Q8—22no—
163C22×Q8—24no—
164C22×Q8—42no—
165C22×Q8—42no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—232no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—416no—
185C23×C4—416no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—464no—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×Q8116yes—
209C23×C4#4118yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2418yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C22⋊Q8—416no—
222C22⋊Q8—416no—
223C22⋊Q8—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4—416no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—48no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—432no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—432no—
275C2×C22⋊C4C2×Q8116yes—
276C2×C22⋊C4C2×Q8116yes—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—44no—
286#95—48no—
287#95—48no—
288#95—48no—
289#95—48no—
290#95—48no—
291#95—48no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—28no—
302C24—28no—
303C24—28no—
304C24—28no—
305C24—216no—
306C24—216no—
307C24—232no—
308C24—22no—
309C24—22no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—21no—
314C24—22no—
315C24—44no—
316C24—44no—
317C24—416no—
318C24—416no—
319C24—48no—
320C24—48no—
321C24—416no—
322C24—48no—
323C24—48no—
324C24C4⋊Q8116yes—
325C24C22×D412yes—
326C24C22×D412yes—
327C24C22×D412yes—
328C24C22×Q814yes—
329C24C22×Q814yes—
330C24C22×Q814yes—
331C24C2×#4114yes—
332C24H(5,𝔽2)14yes—
333C24H(5,𝔽2)14yes—
334C24H(5,𝔽2)12yes—
335C242−1+412yes—
336C24C2511yescharacteristic
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—88no—
341C22×C4—232no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—232no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—4128no—
410C22×C4—416no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—88no—
421C22×C4—816no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4#123116yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×Q8116yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4H(5,𝔽2)14yes—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—816no—
444C4⋊C4—816no—
445C22⋊C4—232no—
446C22⋊C4—416no—
447C22⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—416no—
454C22⋊C4—48no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—232no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—232no—
492C23—416no—
493C23—416no—
494C23—416no—
495C23—416no—
496C23—416no—
497C23—464no—
498C23—464no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23C2×#12314yes—
506C23C2×C4⋊Q814yes—
507C23C2×C4⋊Q814yes—
508C23C2×C4⋊Q814yes—
509C23#53414yes—
510C23#53514yes—
511C23#53714yes—
512C23#54318yes—
513C23#54418yes—
514C23#554116yes—
515C23C23×D411yescharacteristic
516C23C23×Q812yes—
517C23C22×#4112yes—
518C23C2×H(5,𝔽2)12yes—
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23#58512yes—
524Q8—88no—
525C2×C4—232no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—216no—
539C2×C4—216no—
540C2×C4—216no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—416no—
545C2×C4—416no—
546C2×C4—44no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—832no—
580C2×C4—816no—
581C2×C4—816no—
582C2×C4C2×#12314yes—
583C2×C4C2×#12314yes—
584C2×C4#53414yes—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—216no—
598C22—216no—
599C22—216no—
600C22—216no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—464no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—48no—
610C22—48no—
611C22#250218yes—
612C22#265918yes—
613C22C22×#12311yescharacteristic
614C22C22×C4⋊Q811yescharacteristic
615C22C2×#53411yescharacteristic
616C22C2×#53411yescharacteristic
617C22C2×#53511yescharacteristic
618C22C2×#53711yescharacteristic
619C22#344911yescharacteristic
620C22C2×#54312yes—
621C22C2×#54412yes—
622C22C2×#55414yes—
623C22#349014yes—
624C22#349412yes—
625C22#349512yes—
626C22#350114yes—
627C22#353414yes—
628C22#353612yes—
629C22#353712yes—
630C4—24no—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—416no—
635C4—82no—
636C4—84no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4—82no—
641C2—28no—
642C2—28no—
643C2—44no—
644C2C2×#265911yescharacteristic
645C2C2×#250211yescharacteristic
646C2#3499211yescharacteristic
647C2#3501111yescharacteristic
648C2#3526311yescharacteristic
649C2#3743911yescharacteristic
650C2#3744611yescharacteristic
651C2#3787214yes—
652C2#3787314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17057C2semidirect
4. #16986C2semidirect
5. #16365C2semidirect
6. #15092C2semidirect
7. #15014C2semidirect
8. #14914C2semidirect
9. #14786C2semidirect
10. #14766C2semidirect
11. #14353C2semidirect
12. #14281C2semidirect
24. C2×#398C22semidirect
43. #1446C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1444C22semidirect
47. #1444C22semidirect
48. #1444C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
106. C22×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C22×#3802
#170572
#169862
#163652
#150922
#150142
#149142
#147862
#147662
#143532
#142812
#138772
C2×C22⋊#952