Finite Group Databasebeta

← #722898

Subgroups — #722898

Order 512.

3078 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2549C211yesmaximal, characteristic, complemented
2C22×#399C211yesmaximal, characteristic
3#16794C214yesmaximal, complemented
4#15675C214yesmaximal, complemented
5#15499C214yesmaximal, complemented
6#15065C214yesmaximal, complemented
7#14843C212yesmaximal, complemented
8#14717C212yesmaximal
9#14338C211yesmaximal, characteristic
10#14139C212yesmaximal, complemented
11#13971C212yesmaximal
12#13924C211yesmaximal, characteristic, complemented
13#13924C211yesmaximal, characteristic, complemented
14#13802C211yesmaximal, characteristic, complemented
15#13775C211yesmaximal, characteristic
16C22×C22⋊Q8C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21#2549—28no—
22C2×#400C2214yescomplemented
23C2×#399—28no—
24C2×#399C2214yes—
25C2×#399C2212yes—
26C2×#399C2212yes—
27C2×#398C2212yescomplemented
28C2×#395C2212yescomplemented
29C2×#393C2214yes—
30C2×#391C2212yescomplemented
31C2×#391C2214yescomplemented
32C2×#389C2212yescomplemented
33C2×#388C2214yes—
34C2×#385C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#379C2211yescharacteristic
38C2×#379C2212yes—
39C2×#379C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C22×#95C2211yescharacteristic
44#1445C2212yescomplemented
45#1445C2212yescomplemented
46#1445C2214yescomplemented
47#1445C2214yescomplemented
48#1442C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2214yes—
53#1439C2212yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1436C2214yescomplemented
64C4⋊#95C2214yescomplemented
65C4⋊#95C2214yescomplemented
66C4⋊#95C2214yescomplemented
67#1434C2212yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C4×#95C2212yescomplemented
77C26C2311yescharacteristic
78C23×Q8C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—44no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—48no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—48no—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124#400—48no—
125#399—216no—
126#399—28no—
127#399—28no—
128#398—28no—
129#395—28no—
130#393—48no—
131#391—28no—
132#391—48no—
133#389—28no—
134#388—48no—
135#385—28no—
136#380—24no—
137#380—24no—
138#379—24no—
139#379—44no—
140#379Q818yescomplemented
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—44no—
144C2×#95D418yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2314yes—
154C2×#95C2314yes—
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×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C25—22no—
171C25—22no—
172C25—22no—
173C25—22no—
174C25—22no—
175C25—22no—
176C25—44no—
177C25—44no—
178C25C2×Q814yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—28no—
184C22×Q8—42no—
185C22×Q8—44no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C2×D418yes—
201C23×C4C2×D414yes—
202C23×C4#4114yes—
203C23×C4#4114yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C22⋊Q8—432no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
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—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—432no—
245C2×C4⋊C4—416no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—432no—
280C2×C22⋊C4—416no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×Q814yes—
290C2×C22⋊C4C2×Q814yes—
291C2×C22⋊C4C2×Q8116yes—
292C2×C22⋊C4#4114yes—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—44no—
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#95—44no—
321#95—44no—
322#95C2×D418yes—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—24no—
326C24—24no—
327C24—28no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—28no—
339C24—28no—
340C24—416no—
341C24—48no—
342C24—48no—
343C24—416no—
344C24—44no—
345C24—44no—
346C24—44no—
347C24—44no—
348C24—84no—
349C24—84no—
350C24—84no—
351C24—84no—
352C24—88no—
353C24—88no—
354C24C22×D412yes—
355C24C22×Q814yes—
356C24C22×Q812yes—
357C24C2×#4112yes—
358C24C2×#4112yes—
359C24C2×#4112yes—
360C24H(5,𝔽2)12yes—
361C242−1+412yes—
362C24C2511yescharacteristic
363C2×Q8—48no—
364C2×Q8—416no—
365C2×Q8—48no—
366C2×Q8—84no—
367C2×Q8—88no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—216no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4#124116yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q814yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)18yes—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—416no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—28no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C42—84no—
479C42—84no—
480C42—88no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—432no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—416no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—816no—
517C23—816no—
518C23—816no—
519C23—88no—
520C23—816no—
521C23—816no—
522C23—88no—
523C23C2×#12414yes—
524C23#53314yes—
525C23#54214yes—
526C23#54614yes—
527C23D4×Q814yes—
528C23#55414yes—
529C23#55614yes—
530C23C23×D411yescharacteristic
531C23C23×Q811yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C2×H(5,𝔽2)14yes—
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23#58514yes—
539Q8—816no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—416no—
560C2×C4—432no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—816no—
607C2×C4—88no—
608C2×C4C2×#12414yes—
609C2×C4C2×#12414yes—
610C2×C4#53518yes—
611C22—28no—
612C22—24no—
613C22—28no—
614C22—28no—
615C22—28no—
616C22—24no—
617C22—28no—
618C22—28no—
619C22—48no—
620C22—44no—
621C22—48no—
622C22—44no—
623C22—44no—
624C22—832no—
625C22—84no—
626C22—82no—
627C22—84no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22#249218yes—
632C22C22×#12411yescharacteristic
633C22C2×#53311yescharacteristic
634C22C2×#53512yes—
635C22C2×#54211yescharacteristic
636C22C2×#54611yescharacteristic
637C22C2×D4×Q811yescharacteristic
638C22C2×#55411yescharacteristic
639C22C2×#55611yescharacteristic
640C22#348512yes—
641C22#349112yes—
642C22#349112yes—
643C22#350114yes—
644C22#350812yes—
645C22#350812yes—
646C22#351314yes—
647C22#353414yes—
648C22#354114yes—
649C4—24no—
650C4—24no—
651C4—28no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—84no—
660C4—82no—
661C4—84no—
662C2—28no—
663C2—84no—
664C2C2×#249211yescharacteristic
665C2#3528511yescharacteristic
666C2#3545511yescharacteristic
667C2#3583912yes—
668C2#3802312yes—
669C2#3959412yes—
670C2#4139012yes—
671C2#4351212yes—
672C2#4410212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2549C2semidirect
3. #16794C2semidirect
4. #15675C2semidirect
5. #15499C2semidirect
6. #15065C2semidirect
7. #14843C2semidirect
10. #14139C2semidirect
12. #13924C2semidirect
13. #13924C2semidirect
14. #13802C2semidirect
22. C2×#400C22semidirect
27. C2×#398C22semidirect
28. C2×#395C22semidirect
30. C2×#391C22semidirect
31. C2×#391C22semidirect
32. C2×#389C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
63. #1436C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
76. C4×#95C22semidirect
140. #379Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25492
C22×#3992
#167942
#156752
#154992
#150652
#148432
#147172
#143382
#141392
#139712
#139242
#138022
#137752