Finite Group Databasebeta

← #536514

Subgroups — #536514

Order 512.

3154 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3C2×#2331C211yesmaximal, characteristic, complemented
4#14584C214yesmaximal, complemented
5#14362C214yesmaximal, complemented
6#14230C214yesmaximal, complemented
7#14161C211yesmaximal, characteristic, complemented
8#14151C212yesmaximal, complemented
9#14052C214yesmaximal, complemented
10#14029C212yesmaximal, complemented
11#14018C212yesmaximal
12#13848C211yesmaximal, characteristic, complemented
13#13820C212yesmaximal
14#13804C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2542C4116yescomplemented
21C2×#399C2212yes—
22C2×#399C2212yes—
23#2349—28no—
24#2331—28no—
25C2×#391C2214yes—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#384C2212yes—
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2214yes—
46C2×#376C2211yescharacteristic
47C22×#95C2212yes—
48#1446C2212yes—
49#1446C2212yes—
50#1445C2214yes—
51#1444C2212yes—
52#1442C2212yes—
53#1441C2212yes—
54#1441C2212yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C4×#95C2212yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8C2×C418yescomplemented
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8—48no—
87C2×C22⋊Q8C2×C4132yescomplemented
88C2×C4×D4—28no—
89C2×C4×D4—28no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—44no—
93C2×C4×D4C2×C418yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2314yes—
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122#399C2×C4116yescomplemented
123#399C2×C4116yescomplemented
124#391—216no—
125#390—24no—
126#390C2×C4116yes—
127#390Q818yescomplemented
128#389—28no—
129#389—28no—
130#389—48no—
131#387—216no—
132#387—48no—
133#385—216no—
134#385—48no—
135#384C2×C4116yescomplemented
136#382D418yes—
137#380—28no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—44no—
142C4×C4⋊C4—44no—
143C4×C4⋊C4C2×C418yescomplemented
144C4×C22⋊C4—48no—
145#376—24no—
146C2×#95—28no—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C25C2412yes—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8C22×C418yes—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4C22×C414yes—
185C22×D4C2×Q814yes—
186C22×D4#4114yes—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C22×C418yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C4⋊Q8—232no—
214C22⋊Q8—464no—
215C4×D4—232no—
216C4×D4—432no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C22×C414yes—
252C2×C4⋊C4C22×C418yes—
253C2×C4⋊C4C22×C4116yes—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C4116yes—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×Q814yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C22×C418yes—
274C2×C22⋊C4C22×C4116yes—
275C2×C22⋊C4C22×C4116yes—
276C2×C22⋊C4C2×Q8116yes—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42C22×C418yes—
296C2×C42C22×C414yes—
297C2×C42C2×D414yes—
298C2×C42C2×Q814yes—
299C2×C42#4114yes—
300#95—22no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C22×C414yes—
324#95C22×C414yes—
325#95C2×D418yes—
326#95C2×D418yes—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—24no—
335C24C23×C414yes—
336C24C22×Q814yes—
337C24C2×#4114yes—
338C24C2511yescharacteristic
339C2×Q8—216no—
340C2×Q8—416no—
341C2×Q8—88no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4C4×Q8116yes—
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—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—216no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—88no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4C23×C414yes—
429C22×C4C23×C418yes—
430C22×C4C23×C414yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C23×C414yes—
434C22×C4C23×C412yes—
435C22×C4C22×D418yes—
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)14yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+414yes—
450C22×C42−1+412yes—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—232no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—28no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C22⋊C4—216no—
466C22⋊C4—232no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—88no—
471C42—216no—
472C42—28no—
473C42—44no—
474C42—48no—
475C42—84no—
476C42—88no—
477C42—88no—
478C42C4×D4116yes—
479C23—24no—
480C23—28no—
481C23—24no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23C2×C4×Q818yes—
502C23C24×C411yescharacteristic
503C23C23×D411yescharacteristic
504C23C23×Q811yescharacteristic
505C23C22×#4111yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×H(5,𝔽2)12yes—
508C23C2×H(5,𝔽2)12yes—
509C23C2×H(5,𝔽2)11yescharacteristic
510C23C2×2−1+411yescharacteristic
511C23C2×2−1+412yes—
512C23#58512yes—
513Q8—816no—
514D4—232no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—816no—
594C2×C4—832no—
595C2×C4C2×C4×D414yes—
596C2×C4C2×C4×D414yes—
597C2×C4C2×C4×D414yes—
598C2×C4C2×C4×Q814yes—
599C2×C4#51818yes—
600C2×C4#52014yes—
601C2×C4#54614yes—
602C2×C4D4×Q814yes—
603C2×C4#55214yes—
604C2×C4#55414yes—
605C22—216no—
606C22—28no—
607C22—48no—
608C22—44no—
609C22—44no—
610C22—48no—
611C22—44no—
612C22—44no—
613C22—48no—
614C22—44no—
615C22—44no—
616C22C22×C4×D411yescharacteristic
617C22C22×C4×Q811yescharacteristic
618C22C2×#51812yes—
619C22C2×#52011yescharacteristic
620C22C4×2−1+412yes—
621C22C2×#54611yescharacteristic
622C22C2×D4×Q811yescharacteristic
623C22C2×#55211yescharacteristic
624C22C2×#55411yescharacteristic
625C22#348912yes—
626C22#349812yes—
627C22#350112yes—
628C22#350312yes—
629C22#350811yescharacteristic
630C22#350811yescharacteristic
631C22#350911yescharacteristic
632C22#351011yescharacteristic
633C22#351412yes—
634C22#352012yes—
635C22#353612yes—
636C22#354212yes—
637C22#354412yes—
638C22#355912yes—
639C4—28no—
640C4—24no—
641C4—24no—
642C4—48no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—48no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4#236018yes—
653C2—48no—
654C2C2×#236011yescharacteristic
655C2#3421011yescharacteristic
656C2#3426811yescharacteristic
657C2#3433412yes—
658C2#3446012yes—
659C2#4494112yes—
660C2#4494711yescharacteristic
661C2#4507912yes—
662C2#4508311yescharacteristic
663C2#4521111yescharacteristic
664C2#4543411yescharacteristic
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×#2331C2semidirect
4. #14584C2semidirect
5. #14362C2semidirect
6. #14230C2semidirect
7. #14161C2semidirect
8. #14151C2semidirect
9. #14052C2semidirect
10. #14029C2semidirect
12. #13848C2semidirect
14. #13804C2semidirect
20. #2542C4semidirect
84. C2×C4⋊Q8C2×C4semidirect
87. C2×C22⋊Q8C2×C4semidirect
122. #399C2×C4semidirect
123. #399C2×C4semidirect
127. #390Q8semidirect
135. #384C2×C4semidirect
143. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#23492
C2×#23312
#145842
#143622
#142302
#141612
#141512
#140522
#140292
#140182
#138482
#138202
#138042
#137852