Finite Group Databasebeta

← #525323

Subgroups — #525323

Order 512.

2938 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3C2×#2295C211yesmaximal, characteristic, complemented
4#14707C211yesmaximal, characteristic, complemented
5#14553C214yesmaximal, complemented
6#14551C214yesmaximal, complemented
7#14266C212yesmaximal, complemented
8#14131C214yesmaximal, complemented
9#14046C212yesmaximal, complemented
10#14018C212yesmaximal
11#14011C214yesmaximal, complemented
12#13925C211yesmaximal, characteristic, complemented
13#13804C211yesmaximal, characteristic, complemented
14#13785C212yesmaximal
15C4×#390C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19#2580C4116yescomplemented
20C2×#399C2212yes—
21C2×#393C2212yes—
22#2349—28no—
23#2295—28no—
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×#390C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#382C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2214yes—
46C2×#376C2212yes—
47C22×#95C2212yes—
48C2×C43C2211yescharacteristic
49#1446C2212yes—
50#1445C2214yes—
51#1445C2212yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73#1434C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C4×Q8—24no—
83C2×C4×Q8—24no—
84C2×C4×Q8C2×C418yescomplemented
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—28no—
88C2×C4×D4—24no—
89C2×C4×D4—44no—
90C2×C4×D4C2×C418yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119#399C2×C4116yescomplemented
120#393C2×C4116yescomplemented
121#391—28no—
122#390—24no—
123#390C2×C4116yes—
124#390Q818yescomplemented
125#389—28no—
126#389—28no—
127#389—48no—
128#389—44no—
129#389C2×C4116yescomplemented
130#387—28no—
131#387—28no—
132#385—216no—
133#385—48no—
134#385C2×C4132yescomplemented
135#382—24no—
136#382C2×C418yescomplemented
137#380—28no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—44no—
141C4×C4⋊C4D418yes—
142C4×C22⋊C4—48no—
143#376—28no—
144C2×#95—28no—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C43—24no—
173C25C2412yes—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8C22×C414yes—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4C22×C414yes—
184C22×D4C2×Q814yes—
185C22×D4#4114yes—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—28no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C22×C418yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C4×Q8—232no—
213C4×D4—232no—
214C4×D4—432no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C22×C418yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C4116yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×Q814yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C4116yes—
270C2×C22⋊C4C22×C4116yes—
271C2×C22⋊C4C2×Q8116yes—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C22×C418yes—
293C2×C42C22×C414yes—
294C2×C42C2×D418yes—
295C2×C42C2×D414yes—
296C2×C42C2×Q814yes—
297C2×C42#4114yes—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C22×C418yes—
321#95C22×C418yes—
322#95C22×C4116yes—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24C23×C414yes—
334C24C22×Q814yes—
335C24C2×#4114yes—
336C24C2511yescharacteristic
337C2×Q8—28no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4C4×Q8116yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—88no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4C23×C418yes—
428C22×C4C23×C414yes—
429C22×C4C23×C414yes—
430C22×C4C23×C414yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C23×C412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×Q814yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q812yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+412yes—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—416no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C22⋊C4—216no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C42—216no—
469C42—216no—
470C42—28no—
471C42—48no—
472C42—48no—
473C42—44no—
474C42—84no—
475C42—88no—
476C42C4×D4116yes—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23C2×C4×Q818yes—
500C23C24×C411yescharacteristic
501C23C23×D411yescharacteristic
502C23C23×Q811yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4111yescharacteristic
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×H(5,𝔽2)12yes—
507C23C2×2−1+411yescharacteristic
508C23#58512yes—
509C23#58512yes—
510C23#58512yes—
511Q8—416no—
512D4—232no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—816no—
593C2×C4—832no—
594C2×C4C2×C4×D414yes—
595C2×C4C2×C4×D414yes—
596C2×C4C2×C4×D414yes—
597C2×C4C2×C4×Q814yes—
598C2×C4#52014yes—
599C2×C4#54614yes—
600C2×C4D4×Q814yes—
601C2×C4#55214yes—
602C2×C4#55414yes—
603C22—216no—
604C22—28no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—48no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—48no—
614C22C22×C4×D411yescharacteristic
615C22C22×C4×Q811yescharacteristic
616C22C2×#52011yescharacteristic
617C22#342812yes—
618C22C4×H(5,𝔽2)12yes—
619C22#344912yes—
620C22#345511yescharacteristic
621C22#345511yescharacteristic
622C22C2×#54611yescharacteristic
623C22C2×D4×Q811yescharacteristic
624C22C2×#55211yescharacteristic
625C22C2×#55411yescharacteristic
626C22#348812yes—
627C22#350112yes—
628C22#350412yes—
629C22#350811yescharacteristic
630C22#351011yescharacteristic
631C22#351312yes—
632C22#352112yes—
633C22#355412yes—
634C22#355712yes—
635C22#355812yes—
636C22#356212yes—
637C4—24no—
638C4—24no—
639C4—44no—
640C4—44no—
641C4—48no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—82no—
649C4—84no—
650C4#236018yes—
651C2—48no—
652C2#3406511yescharacteristic
653C2C2×#236011yescharacteristic
654C2#3426911yescharacteristic
655C2#3433312yes—
656C2#3450012yes—
657C2#3890212yes—
658C2#3896911yescharacteristic
659C2#3897111yescharacteristic
660C2#3959012yes—
661C2#3962811yescharacteristic
662C2#3963011yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2295C2semidirect
4. #14707C2semidirect
5. #14553C2semidirect
6. #14551C2semidirect
7. #14266C2semidirect
8. #14131C2semidirect
9. #14046C2semidirect
11. #14011C2semidirect
12. #13925C2semidirect
13. #13804C2semidirect
19. #2580C4semidirect
84. C2×C4×Q8C2×C4semidirect
119. #399C2×C4semidirect
120. #393C2×C4semidirect
124. #390Q8semidirect
129. #389C2×C4semidirect
134. #385C2×C4semidirect
136. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#23492
C2×#22952
#147072
#145532
#145512
#142662
#141312
#140462
#140182
#140112
#139252
#138042
#137852
C4×#3902