Finite Group Databasebeta

← #517248

Subgroups — #517248

Order 512.

7388 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic
3C2×#2322C212yesmaximal, complemented
4C2×C4×#120C211yesmaximal, characteristic
5#14694C211yesmaximal, characteristic
6#14693C211yesmaximal, characteristic, complemented
7#14681C211yesmaximal, characteristic, complemented
8#14263C212yesmaximal, complemented
9#14256C212yesmaximal, complemented
10#14043C214yesmaximal, complemented
11#14039C214yesmaximal, complemented
12#14019C214yesmaximal, complemented
13#14015C214yesmaximal, complemented
14#13776C212yesmaximal, complemented
15C4×#379C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C25×C4C2211yescharacteristic
19C22×#123C2212yes—
20C22×#120C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2212yes—
23C23×C4⋊C4—28no—
24C23×C4⋊C4—216no—
25C23×C4⋊C4C2212yes—
26C23×C4⋊C4C2214yes—
27C23×C4⋊C4C2214yes—
28C23×C4⋊C4C2212yes—
29#2849C4116yescomplemented
30#2322—216no—
31C4×#120—28no—
32C2×#389C2214yescomplemented
33C2×#388C2214yescomplemented
34C2×#387C2214yescomplemented
35C2×#386C2212yes—
36C2×#386C2211yescharacteristic
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#385C2212yes—
41C2×#385C2211yescharacteristic
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#385C2214yescomplemented
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#379C2211yescharacteristic
50C2×#379C2211yescharacteristic
51C2×#379C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2214yes—
56#1444C2218yescomplemented
57#1442C2218yescomplemented
58#1441C2218yescomplemented
59#1439C2218yescomplemented
60#1436C2218yescomplemented
61#1436C2218yescomplemented
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C26C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C23×D4C2312yes—
73C24×C4—24no—
74C24×C4—24no—
75C24×C4—28no—
76C24×C4—28no—
77C24×C4—24no—
78C24×C4—28no—
79C24×C4—28no—
80C24×C4—48no—
81C24×C4—48no—
82C24×C4C2×C418yescomplemented
83C24×C4D418yes—
84C24×C4C2314yes—
85C24×C4C2314yes—
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C2×#123—28no—
94C2×#123—28no—
95C2×#123C2×C4116yescomplemented
96C2×#120—24no—
97C2×#120—24no—
98C2×#120C2×C418yes—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—24no—
102C2×C4×D4—48no—
103C22×C4⋊C4—2128no—
104C22×C4⋊C4—216no—
105C22×C4⋊C4—216no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—28no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—232no—
112C22×C4⋊C4—232no—
113C22×C4⋊C4D4116yescomplemented
114C22×C4⋊C4C2318yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2314yes—
138#389—216no—
139#388—216no—
140#387—216no—
141#386—28no—
142#386—28no—
143#386—24no—
144#386—48no—
145#386C2×C4132yescomplemented
146#385—28no—
147#385—28no—
148#385—24no—
149#385—216no—
150#385—48no—
151#385C2×C4132yescomplemented
152#381—28no—
153#381—28no—
154#381C2×C4116yescomplemented
155#379—24no—
156#379—24no—
157#379C2×C418yescomplemented
158C4×C22⋊C4—28no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—48no—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C25—24no—
176C25—24no—
177C25—28no—
178C25—28no—
179C25C22×C414yes—
180C25C2×D414yes—
181C25#4114yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C25C2412yes—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4C22×C414yes—
194C22×D4C22×C418yes—
195C23×C4—232no—
196C23×C4—264no—
197C23×C4—232no—
198C23×C4—232no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—264no—
206C23×C4—216no—
207C23×C4—216no—
208C23×C4—232no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—232no—
216C23×C4—216no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—432no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—48no—
229C23×C4—48no—
230C23×C4—44no—
231C23×C4—464no—
232C23×C4—464no—
233C23×C4—432no—
234C23×C4C22×C4116yes—
235C23×C4C22×C418yes—
236C23×C4C2×D418yes—
237C23×C4C2×D418yes—
238C23×C4C2×D418yes—
239C23×C4C2×D414yes—
240C23×C4C2×D418yes—
241C23×C4C2×D414yes—
242C23×C4C2×D418yes—
243C23×C4#4118yes—
244C23×C4C2411yescharacteristic
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2414yes—
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2414yes—
253C23×C4C2414yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256#123—264no—
257#120—232no—
258C4×D4—432no—
259C4×D4—416no—
260C2×C4⋊C4—264no—
261C2×C4⋊C4—2128no—
262C2×C4⋊C4—2128no—
263C2×C4⋊C4—264no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4C22×C4116yescomplemented
280C2×C4⋊C4C2×D4116yes—
281C2×C4⋊C4C2×D4116yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4C22×C414yes—
313C2×C22⋊C4C22×C418yes—
314C2×C22⋊C4C22×C418yes—
315C2×C22⋊C4C22×C4116yes—
316C2×C22⋊C4C22×C4116yescomplemented
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—44no—
322C2×C42—48no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—216no—
332#95—48no—
333#95—48no—
334#95—48no—
335#95C22×C4132yescomplemented
336C24—232no—
337C24—232no—
338C24—264no—
339C24—216no—
340C24—232no—
341C24—28no—
342C24—28no—
343C24—216no—
344C24—216no—
345C24—216no—
346C24—232no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—21no—
355C24—22no—
356C24—21no—
357C24—24no—
358C24—24no—
359C24C4×D4116yes—
360C24C23×C412yes—
361C24C23×C412yes—
362C24C23×C414yes—
363C24C22×D412yes—
364C24C22×D414yes—
365C24C22×D414yes—
366C24C2×#4112yes—
367C24C2×#4114yes—
368C24C2×#4114yes—
369C24H(5,𝔽2)18yes—
370C242−1+418yes—
371C24C2511yescharacteristic
372C2×D4—216no—
373C2×D4—28no—
374C2×D4—216no—
375C2×D4—232no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—416no—
382C22×C4—232no—
383C22×C4—264no—
384C22×C4—232no—
385C22×C4—2256no—
386C22×C4—232no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—264no—
400C22×C4—232no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—432no—
448C22×C4—432no—
449C22×C4—432no—
450C22×C4—432no—
451C22×C4—48no—
452C22×C4—4128no—
453C22×C4—4128no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—44no—
460C22×C4—88no—
461C22×C4C4×D4132yescomplemented
462C22×C4#120116yes—
463C22×C4C23×C414yes—
464C22×C4C23×C418yes—
465C22×C4C23×C412yes—
466C22×C4C23×C418yes—
467C22×C4C22×D414yes—
468C22×C4C22×D418yes—
469C22×C4C22×D418yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C2×#4118yes—
475C4⋊C4—264no—
476C4⋊C4—232no—
477C4⋊C4—232no—
478C4⋊C4—232no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—88no—
482C22⋊C4—216no—
483C22⋊C4—28no—
484C22⋊C4—216no—
485C22⋊C4—232no—
486C22⋊C4—232no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C22⋊C4—44no—
495C22⋊C4—88no—
496C42—48no—
497C42—44no—
498C42—88no—
499C23—232no—
500C23—2128no—
501C23—232no—
502C23—2128no—
503C23—264no—
504C23—264no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—216no—
519C23—28no—
520C23—216no—
521C23—216no—
522C23—216no—
523C23—216no—
524C23—232no—
525C23—24no—
526C23—22no—
527C23—28no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—48no—
537C23C2×C4×D418yes—
538C23C2×C4×D414yes—
539C23C2×C4×D418yes—
540C23#520116yes—
541C23C2×#12014yes—
542C23#52518yes—
543C23#52514yes—
544C23#53418yes—
545C23#53618yes—
546C23#539116yes—
547C23C24×C411yescharacteristic
548C23C23×D411yescharacteristic
549C23C23×D412yes—
550C23C22×#4111yescharacteristic
551C23C22×#4112yes—
552C23C2×H(5,𝔽2)14yes—
553C23C2×2−1+414yes—
554D4—432no—
555D4—416no—
556C2×C4—2128no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—264no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—216no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—416no—
581C2×C4—432no—
582C2×C4—432no—
583C2×C4—416no—
584C2×C4—432no—
585C2×C4—464no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—88no—
614C2×C4C2×C4×D4116yes—
615C2×C4C2×#12014yes—
616C2×C4C2×#12014yes—
617C22—2128no—
618C22—28no—
619C22—28no—
620C22—216no—
621C22—232no—
622C22—24no—
623C22—28no—
624C22—24no—
625C22—28no—
626C22—216no—
627C22—232no—
628C22—28no—
629C22—28no—
630C22—28no—
631C22—28no—
632C22—44no—
633C22—48no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—84no—
638C22—82no—
639C22C4×#12018yes—
640C22#2322116yescomplemented
641C22C22×C4×D411yescharacteristic
642C22C22×C4×D412yes—
643C22C2×#52014yes—
644C22C22×#12011yescharacteristic
645C22C2×#52512yes—
646C22C2×#52511yescharacteristic
647C22C2×#53412yes—
648C22C2×#53612yes—
649C22C2×#53914yes—
650C22#348414yes—
651C22#348614yes—
652C22#349218yes—
653C4—24no—
654C4—24no—
655C4—216no—
656C4—44no—
657C4—48no—
658C4—44no—
659C4—48no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—84no—
664C4—82no—
665C2—216no—
666C2—28no—
667C2—82no—
668C2C2×C4×#12011yescharacteristic
669C2C2×#232212yes—
670C2#3420014yes—
671C2#3498011yescharacteristic
672C2#3498914yes—
673C2#3505712yes—
674C2#3524211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#120C22
LCSG ▷ C23 ▷ 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
3. C2×#2322C2semidirect
6. #14693C2semidirect
7. #14681C2semidirect
8. #14263C2semidirect
9. #14256C2semidirect
10. #14043C2semidirect
11. #14039C2semidirect
12. #14019C2semidirect
13. #14015C2semidirect
14. #13776C2semidirect
15. C4×#379C2semidirect
29. #2849C4semidirect
32. C2×#389C22semidirect
33. C2×#388C22semidirect
34. C2×#387C22semidirect
45. C2×#385C22semidirect
56. #1444C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1439C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
82. C24×C4C2×C4semidirect
95. C2×#123C2×C4semidirect
113. C22×C4⋊C4D4semidirect
145. #386C2×C4semidirect
151. #385C2×C4semidirect
154. #381C2×C4semidirect
157. #379C2×C4semidirect
279. C2×C4⋊C4C22×C4semidirect
316. C2×C22⋊C4C22×C4semidirect
335. #95C22×C4semidirect
461. C22×C4C4×D4semidirect
640. C22#2322semidirect

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C2×#28492
C2×#23222
C2×C4×#1202
#146942
#146932
#146812
#142632
#142562
#140432
#140392
#140192
#140152
#137762
C4×#3792