Finite Group Databasebeta

← #826160

Subgroups — #826160

Order 512.

2594 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C212yesmaximal
2C2×#2551C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic
4#16560C212yesmaximal
5#15437C214yesmaximal
6#15332C214yesmaximal
7#15156C211yesmaximal, characteristic
8#15153C211yesmaximal, characteristic
9#15152C211yesmaximal, characteristic
10#14884C212yesmaximal
11#14869C212yesmaximal
12#14053C212yesmaximal
13#14053C212yesmaximal
14#14053C212yesmaximal
15#13918C212yesmaximal
16C4×#400C211yesmaximal, characteristic
17#13821C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2568—216no—
23#2551—28no—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#400C2211yescharacteristic
29C2×#395C2214yes—
30#2365—28no—
31C2×#391C2214yes—
32C2×#391C2212yes—
33C2×#391C2214yes—
34C2×#391C2214yes—
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#384C2211yescharacteristic
43C2×#382C2214yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#1446C2214yes—
52#1446C2214yes—
53#1446C2214yes—
54#1445C2212yes—
55#1445C2211yescharacteristic
56#1445C2214yes—
57#1443C2212yes—
58#1443C2214yes—
59#1442C2211yescharacteristic
60#1442C2214yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2211yescharacteristic
68#1441C2212yes—
69#1441C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2214yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2314yes—
87C23×Q8C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8Q818yescomplemented
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8—44no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#400—28no—
128#400—28no—
129#400—28no—
130#400—44no—
131#400D418yes—
132#395—216no—
133#391—216no—
134#391—216no—
135#391—216no—
136#391—28no—
137#389—24no—
138#389—28no—
139#389—28no—
140#389—24no—
141#389—28no—
142#387—28no—
143#387—28no—
144#384—24no—
145#382—216no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4D418yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—28no—
192C22×Q8—28no—
193C22×Q8—24no—
194C22×Q8—28no—
195C22×Q8—28no—
196C22×Q8—24no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×Q8—44no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8C2×Q818yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C4⋊Q8—232no—
221C4⋊Q8—432no—
222C4⋊Q8—432no—
223C4×Q8—432no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×Q818yes—
270C2×C4⋊C4C2×Q814yes—
271C2×C4⋊C4C2×Q814yes—
272C2×C4⋊C4#4114yes—
273C2×C4⋊C4#4118yes—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42C2×D418yes—
297C2×C42C2×D414yes—
298C2×C42C2×Q814yes—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95C2×D414yes—
332#95C2×D418yes—
333#95C2×D418yes—
334#95C2×D418yes—
335C24C2511yescharacteristic
336C2×Q8—216no—
337C2×Q8—216no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—416no—
344C2×Q8—416no—
345C2×Q8—416no—
346C2×Q8—416no—
347C2×Q8—88no—
348C2×Q8—88no—
349C2×Q8—88no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
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—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q814yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q812yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+414yes—
451C22×C42−1+412yes—
452C22×C42−1+414yes—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—416no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—84no—
469C4⋊C4C22⋊Q8116yes—
470C4⋊C4#124116yes—
471C42—28no—
472C42—28no—
473C42—216no—
474C42—416no—
475C42—48no—
476C42—48no—
477C42—48no—
478C42—44no—
479C42—48no—
480C23C23×D411yescharacteristic
481C23C23×D411yescharacteristic
482C23C23×Q811yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4112yes—
485C23C2×H(5,𝔽2)11yescharacteristic
486C23C2×2−1+412yes—
487C23C2×2−1+411yescharacteristic
488C23C2×2−1+412yes—
489C23#58511yescharacteristic
490C23#58512yes—
491Q8—432no—
492Q8—832no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
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—44no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4C2×C22⋊Q818yes—
588C2×C4C2×C22⋊Q814yes—
589C2×C4C2×#12418yes—
590C2×C4C2×#12414yes—
591C2×C4#53614yes—
592C2×C4D4214yes—
593C2×C4D4×Q814yes—
594C2×C4#55618yes—
595C2×C4#55718yes—
596C22C22×C22⋊Q811yescharacteristic
597C22C22×#12411yescharacteristic
598C22C2×#53611yescharacteristic
599C22C2×D4211yescharacteristic
600C22C2×D4×Q811yescharacteristic
601C22D4×#4112yes—
602C22D4×#4111yescharacteristic
603C22C2×#55612yes—
604C22C2×#55712yes—
605C22#348611yescharacteristic
606C22#348911yescharacteristic
607C22#349912yes—
608C22#350011yescharacteristic
609C22#350111yescharacteristic
610C22#350212yes—
611C22#351412yes—
612C22#351612yes—
613C22#351812yes—
614C22#352011yescharacteristic
615C22#352312yes—
616C22#354412yes—
617C22#355512yes—
618C22#356512yes—
619C4—28no—
620C4—24no—
621C4—28no—
622C4—24no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—44no—
630C4—84no—
631C4—84no—
632C4—81no—
633C4—82no—
634C4—82no—
635C4#256818yescomplemented
636C2C2×#256811yescharacteristic
637C2#3571111yescharacteristic
638C2#3591011yescharacteristic
639C2D4×C22⋊Q811yescharacteristic
640C2D4×#12411yescharacteristic
641C2#3933711yescharacteristic
642C2#4291511yescharacteristic
643C2#4495412yes—
644C2#4500112yes—
645C2#5966712yes—
646C2#5971112yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
96. C2×C4⋊Q8Q8semidirect
635. C4#2568semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#25512
C2×#23652
#165602
#154372
#153322
#151562
#151532
#151522
#148842
#148692
#140532
#139182
C4×#4002
#138212