Finite Group Databasebeta

← #613008

Subgroups — #613008

Order 512.

7076 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
1C24×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2579C212yesmaximal, complemented
3C2×C4×#120C211yesmaximal, characteristic
4#15092C212yesmaximal, complemented
5#15091C212yesmaximal, complemented
6#14729C214yesmaximal, complemented
7#14728C214yesmaximal, complemented
8#14694C211yesmaximal, characteristic
9#14694C211yesmaximal, characteristic
10#14693C211yesmaximal, characteristic, complemented
11#14693C211yesmaximal, characteristic, complemented
12#14680C211yesmaximal, characteristic, complemented
13#14039C214yesmaximal, complemented
14#14015C214yesmaximal, complemented
15#13923C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C25×C4C2211yescharacteristic
19C22×#120C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4—28no—
23C23×C4⋊C4—216no—
24C23×C4⋊C4C2214yes—
25C23×C4⋊C4C2214yes—
26C23×C4⋊C4C2212yes—
27C23×C4⋊C4C2212yes—
28#2579—216no—
29C2×#398C2214yescomplemented
30C2×#396C2214yescomplemented
31C4×#120—28no—
32C2×#387C2214yescomplemented
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2214yes—
36C2×#386C2212yes—
37C2×#386C2211yescharacteristic
38C2×#386C2214yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#385C2212yes—
43C2×#385C2211yescharacteristic
44C2×#385C2214yes—
45C2×#385C2214yescomplemented
46C2×#380C2212yes—
47C2×#380C2214yes—
48C2×#379C2211yescharacteristic
49C2×#379C2211yescharacteristic
50C2×#379C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53#1444C2218yescomplemented
54#1442C2218yescomplemented
55#1441C2218yescomplemented
56#1441C2218yescomplemented
57#1439C2218yescomplemented
58#1436C2218yescomplemented
59C22⋊#95C2214yes—
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C26C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C23×D4C2312yes—
72C24×C4—24no—
73C24×C4—28no—
74C24×C4—216no—
75C24×C4—24no—
76C24×C4—28no—
77C24×C4—28no—
78C24×C4—48no—
79C24×C4—48no—
80C24×C4D418yes—
81C24×C4C2312yes—
82C24×C4C2314yes—
83C24×C4C2314yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×#120—24no—
91C2×#120—24no—
92C2×#120Q818yes—
93C2×C4×D4—28no—
94C2×C4×D4—28no—
95C2×C4×D4—24no—
96C2×C4×D4—48no—
97C22×C4⋊C4—2128no—
98C22×C4⋊C4—216no—
99C22×C4⋊C4—216no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—232no—
106C22×C4⋊C4—232no—
107C22×C4⋊C4D4116yescomplemented
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2318yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#398—216no—
133#396—216no—
134#387—216no—
135#386—28no—
136#386—28no—
137#386—28no—
138#386—24no—
139#386—216no—
140#386—48no—
141#385—28no—
142#385—28no—
143#385—28no—
144#385—24no—
145#385—216no—
146#385—216no—
147#385—48no—
148#380—28no—
149#380—48no—
150#379—24no—
151#379—24no—
152#379—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—24no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C25—24no—
169C25—28no—
170C25—28no—
171C25—24no—
172C25C2×D414yes—
173C25C2×Q814yes—
174C25#4114yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C25C2412yes—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—24no—
186C22×D4C2×Q814yes—
187C22×D4C2×Q818yes—
188C23×C4—232no—
189C23×C4—264no—
190C23×C4—2128no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—216no—
196C23×C4—216no—
197C23×C4—216no—
198C23×C4—232no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—232no—
206C23×C4—216no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—432no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—48no—
219C23×C4—48no—
220C23×C4—44no—
221C23×C4—464no—
222C23×C4—464no—
223C23×C4—432no—
224C23×C4C2×D418yes—
225C23×C4C2×D418yes—
226C23×C4C2×D418yes—
227C23×C4C2×D414yes—
228C23×C4C2×D418yes—
229C23×C4C2×D414yes—
230C23×C4C2×D418yes—
231C23×C4#41116yes—
232C23×C4C2411yescharacteristic
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2414yes—
244#120—232no—
245C4×D4—432no—
246C4×D4—416no—
247C2×C4⋊C4—264no—
248C2×C4⋊C4—2128no—
249C2×C4⋊C4—2128no—
250C2×C4⋊C4—264no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4C2×D4116yes—
269C2×C4⋊C4C2×D4116yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4C2×Q814yes—
303C2×C22⋊C4C2×Q818yes—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—44no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—216no—
317#95—216no—
318#95—48no—
319#95—48no—
320#95—48no—
321C24—232no—
322C24—232no—
323C24—264no—
324C24—28no—
325C24—216no—
326C24—216no—
327C24—28no—
328C24—216no—
329C24—216no—
330C24—232no—
331C24—232no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—21no—
340C24—24no—
341C24—21no—
342C24—24no—
343C24—22no—
344C24C22⋊Q8116yes—
345C24C22×D412yes—
346C24C22×D414yes—
347C24C22×D414yes—
348C24C22×Q812yes—
349C24C22×Q812yes—
350C24C22×Q814yes—
351C24C2×#4112yes—
352C24C2×#4114yes—
353C24C2×#4118yes—
354C24H(5,𝔽2)18yes—
355C242−1+414yes—
356C24C2511yescharacteristic
357C2×D4—216no—
358C2×D4—28no—
359C2×D4—216no—
360C2×D4—232no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—416no—
367C22×C4—232no—
368C22×C4—2256no—
369C22×C4—264no—
370C22×C4—232no—
371C22×C4—232no—
372C22×C4—232no—
373C22×C4—216no—
374C22×C4—232no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—232no—
381C22×C4—264no—
382C22×C4—232no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—416no—
430C22×C4—432no—
431C22×C4—432no—
432C22×C4—432no—
433C22×C4—432no—
434C22×C4—48no—
435C22×C4—4128no—
436C22×C4—4128no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—44no—
443C22×C4—88no—
444C22×C4#120116yes—
445C22×C4C22×D414yes—
446C22×C4C22×D418yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D418yes—
452C22×C4C22×Q812yes—
453C22×C4C22×Q814yes—
454C22×C4C2×#41116yes—
455C4⋊C4—264no—
456C4⋊C4—232no—
457C4⋊C4—232no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—416no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C22⋊C4—216no—
464C22⋊C4—28no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—44no—
475C22⋊C4—416no—
476C22⋊C4—88no—
477C42—48no—
478C42—44no—
479C23—232no—
480C23—232no—
481C23—2128no—
482C23—2128no—
483C23—264no—
484C23—264no—
485C23—24no—
486C23—232no—
487C23—24no—
488C23—28no—
489C23—24no—
490C23—216no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—216no—
502C23—216no—
503C23—216no—
504C23—216no—
505C23—22no—
506C23—24no—
507C23—28no—
508C23—41no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—48no—
517C23C2×#12014yes—
518C23C2×C22⋊Q818yes—
519C23C2×C22⋊Q814yes—
520C23#52518yes—
521C23#53418yes—
522C23#53614yes—
523C23#54818yes—
524C23D4×Q818yes—
525C23#553116yes—
526C23#554116yes—
527C23C23×D411yescharacteristic
528C23C23×D412yes—
529C23C23×Q811yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4114yes—
532C23C2×H(5,𝔽2)14yes—
533C23C2×2−1+412yes—
534D4—432no—
535D4—416no—
536C2×C4—264no—
537C2×C4—2128no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—216no—
541C2×C4—216no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—416no—
557C2×C4—432no—
558C2×C4—432no—
559C2×C4—416no—
560C2×C4—432no—
561C2×C4—464no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—48no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—88no—
592C2×C4C2×#12014yes—
593C2×C4C2×#12014yes—
594C2×C4#525116yes—
595C22—2128no—
596C22—24no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—24no—
602C22—28no—
603C22—216no—
604C22—232no—
605C22—216no—
606C22—232no—
607C22—28no—
608C22—28no—
609C22—28no—
610C22—44no—
611C22—42no—
612C22—48no—
613C22—81no—
614C22—81no—
615C22—84no—
616C22—82no—
617C22#240918yes—
618C22#2579116yescomplemented
619C22C22×#12011yescharacteristic
620C22C22×C22⋊Q811yescharacteristic
621C22C2×#52512yes—
622C22C2×#52514yes—
623C22C2×#53412yes—
624C22C2×#53611yescharacteristic
625C22C2×#54812yes—
626C22C2×D4×Q812yes—
627C22C2×#55314yes—
628C22C2×#55414yes—
629C22#348414yes—
630C22#349218yes—
631C4—216no—
632C4—24no—
633C4—24no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—48no—
638C4—81no—
639C4—81no—
640C4—84no—
641C4—84no—
642C4—82no—
643C2—216no—
644C2—28no—
645C2—82no—
646C2C2×#240911yescharacteristic
647C2C2×#257912yes—
648C2#3498011yescharacteristic
649C2#3505712yes—
650C2#3654614yes—
651C2#3658114yes—
652C2#3661411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C24×C4⋊C4C2semidirect
2. C2×#2579C2semidirect
4. #15092C2semidirect
5. #15091C2semidirect
6. #14729C2semidirect
7. #14728C2semidirect
10. #14693C2semidirect
11. #14693C2semidirect
12. #14680C2semidirect
13. #14039C2semidirect
14. #14015C2semidirect
15. #13923C2semidirect
29. C2×#398C22semidirect
30. C2×#396C22semidirect
32. C2×#387C22semidirect
45. C2×#385C22semidirect
53. #1444C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1439C22semidirect
58. #1436C22semidirect
107. C22×C4⋊C4D4semidirect
618. C22#2579semidirect

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C2×#25792
C2×C4×#1202
#150922
#150912
#147292
#147282
#146942
#146932
#146802
#140392
#140152
#139232