Finite Group Databasebeta

← #366574

Subgroups — #366574

Order 512.

2610 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
1C23×#103C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic
3C2×#1956C212yesmaximal, complemented
4C2×#1892C211yesmaximal, characteristic
5C2×#1872C211yesmaximal, characteristic
6C2×#1787C211yesmaximal, characteristic
7C2×C8×C22⋊C4C211yesmaximal, characteristic
8#13880C211yesmaximal, characteristic
9#11050C212yesmaximal, complemented
10#11010C212yesmaximal, complemented
11#10926C212yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4⋊Q8C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#103—28no—
18C22×#103—24no—
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#103C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C4×C8C2212yes—
28#2434—24no—
29#2434C418yescomplemented
30C2×#395—22no—
31C2×#391C2212yes—
32C2×#385—22no—
33C2×#385—22no—
34C2×#384C2212yes—
35C2×#380C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#1956—28no—
38#1892—24no—
39#1872—24no—
40#1787—24no—
41C8×C22⋊C4—24no—
42C2×#340C2211yescharacteristic
43C2×#340C2212yes—
44C2×#340C2211yescharacteristic
45C2×#336C2212yes—
46#1444—22no—
47C4⋊#95C2212yes—
48C23×Q8—21no—
49C23×Q8—22no—
50C23×Q8—24no—
51C23×Q8—24no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8C2×C414yes—
59C23×Q8C2311yescharacteristic
60C23×Q8C2312yes—
61C24×C4—21no—
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C23×C8—22no—
65C23×C8—22no—
66C23×C8—24no—
67C23×C8—42no—
68C23×C8D414yes—
69C23×C8C2312yes—
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—42no—
76C2×C4⋊Q8—42no—
77C2×C4⋊Q8C2×C418yescomplemented
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8C2×C414yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4D412yes—
99C22×C42C2312yes—
100C2×#103—232no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—42no—
116C2×#103—42no—
117C2×#103—42no—
118C2×#103—42no—
119C2×#103D418yes—
120C2×C22⋊C8—22no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8D414yes—
123C2×C4×C8—24no—
124C2×C4×C8—24no—
125C2×C4×C8—42no—
126#395—216no—
127#391—24no—
128#391C2×C418yescomplemented
129#385—216no—
130#385—48no—
131#384—24no—
132#384—42no—
133#380—22no—
134#380—22no—
135C4×C22⋊C4C2×C414yes—
136C4×C22⋊C4D414yes—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95—21no—
140C2×#95D412yes—
141#340—24no—
142#340—22no—
143#340—22no—
144#336—24no—
145C25C2×D411yescharacteristic
146C22×Q8—24no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—28no—
152C22×Q8—216no—
153C22×Q8—28no—
154C22×Q8—216no—
155C22×Q8—28no—
156C22×Q8—28no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—48no—
178C22×Q8—48no—
179C22×Q8—48no—
180C22×Q8—48no—
181C22×Q8—48no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—48no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8C22×C412yes—
190C22×Q8C22×C414yes—
191C22×Q8C2×D414yes—
192C22×Q8#4114yes—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—82no—
212C23×C4C22×C412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D414yes—
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4#4112yes—
219C23×C4#4112yes—
220C23×C4C2411yescharacteristic
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—22no—
224C22×C8—28no—
225C22×C8—21no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—21no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—28no—
236C22×C8—44no—
237C22×C8—48no—
238C22×C8—48no—
239C22×C8C2×D412yes—
240C22×C8C2×D414yes—
241C22×C8C2×D414yes—
242C22×C8C2×D412yes—
243C4⋊Q8—216no—
244C4⋊Q8—48no—
245C4⋊Q8—48no—
246C4⋊Q8—48no—
247C22⋊Q8—28no—
248C22⋊Q8—28no—
249C22⋊Q8—44no—
250C22⋊Q8—44no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4—84no—
283C2×C4⋊C4C22×C412yes—
284C2×C4⋊C4C2×D414yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C22⋊C418yes—
291C2×C22⋊C4SD1618yes—
292C2×C22⋊C4Q1618yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42C22×C414yes—
298C2×C42C2×D414yes—
299#103—216no—
300#103—216no—
301#103—48no—
302#103—48no—
303#103—48no—
304#103—44no—
305#103—44no—
306C22⋊C8—28no—
307C4×C8—48no—
308#95—216no—
309#95—42no—
310#95—48no—
311#95C22⋊C418yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24C2×C22⋊C412yes—
317C24#12412yes—
318C24#12712yes—
319C24C2×SD1614yes—
320C24C2×Q1614yes—
321C24C22×D411yescharacteristic
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—232no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—416no—
343C2×Q8—432no—
344C2×Q8—48no—
345C2×Q8—416no—
346C2×Q8—48no—
347C2×Q8—416no—
348C2×Q8—416no—
349C2×Q8—416no—
350C2×Q8—432no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—48no—
368C2×Q8—41no—
369C2×Q8—42no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8C4×D418yes—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—416no—
406C22×C4—48no—
407C22×C4—416no—
408C22×C4—48no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—816no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C418yes—
440C22×C4C4×D414yes—
441C22×C4C4×D414yes—
442C22×C4#12114yes—
443C22×C4#12114yes—
444C22×C4#12412yes—
445C22×C4#12712yes—
446C22×C4C2×SD1618yes—
447C22×C4C2×Q1618yes—
448C22×C4C23×C411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—416no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8#12014yes—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—432no—
481C4⋊C4—48no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—816no—
494C22⋊C4—48no—
495C22⋊C4—84no—
496C22⋊C4#103132yescomplemented
497C42—24no—
498C42—24no—
499C42—24no—
500C42—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—22no—
513C23#39014yes—
514C23C2×#10318yes—
515C23#48718yes—
516C23#49214yes—
517C23#49414yes—
518C23C22×C22⋊C411yescharacteristic
519C23C2×#12411yescharacteristic
520C23C2×#12711yescharacteristic
521C23D4212yes—
522C23#54612yes—
523C23C22×SD1612yes—
524C23C22×Q1612yes—
525C23#57712yes—
526C23#57812yes—
527Q8—28no—
528Q8—28no—
529Q8—432no—
530Q8—44no—
531Q8—44no—
532Q8—44no—
533Q8—44no—
534Q8—44no—
535Q8—44no—
536Q8—42no—
537Q8—44no—
538Q8—42no—
539Q8—81no—
540Q8—81no—
541Q8—82no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—432no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—816no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—81no—
590C2×C4#39014yes—
591C2×C4C2×#103116yescomplemented
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C2×C4×D412yes—
595C2×C4#51812yes—
596C2×C4C2×#12011yescharacteristic
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4#52511yescharacteristic
600C2×C4#52512yes—
601C2×C4#53411yescharacteristic
602C2×C4#53511yescharacteristic
603C2×C4#53812yes—
604C2×C4D4212yes—
605C2×C4D4212yes—
606C2×C4#54612yes—
607C2×C4#54612yes—
608C2×C4#54612yes—
609C2×C4#55112yes—
610C8—24no—
611C8—42no—
612C8—44no—
613C22—216no—
614C22—216no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22#195618yescomplemented
622C22C2×#39011yescharacteristic
623C22D4×C22⋊C412yes—
624C22#243311yescharacteristic
625C22#247212yes—
626C22#253911yescharacteristic
627C22C22×#10312yes—
628C22#289812yes—
629C22C2×#48712yes—
630C22#313712yes—
631C22C2×#49211yescharacteristic
632C22C2×#49411yescharacteristic
633C22#315411yescharacteristic
634C22#315611yescharacteristic
635C22D4×SD1614yes—
636C22D4×Q1614yes—
637C22#329814yes—
638C22#329914yes—
639C4—216no—
640C4—24no—
641C4—48no—
642C4—82no—
643C4—84no—
644C4—82no—
645C4—161no—
646C4—161no—
647C4C4×#12011yescharacteristic
648C4#232111yescharacteristic
649C4#235812yes—
650C4#241011yescharacteristic
651C4#241111yescharacteristic
652C4#242412yes—
653C2—28no—
654C2#1399511yescharacteristic
655C2C2×#195611yescharacteristic
656C2#2120511yescharacteristic
657C2D4×#10314yes—
658C2#2577411yescharacteristic
659C2#2578011yescharacteristic
660C2#2725814yes—
661C2#2833311yescharacteristic
662C2#2833611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1956C2semidirect
9. #11050C2semidirect
10. #11010C2semidirect
11. #10926C2semidirect
29. #2434C4semidirect
77. C2×C4⋊Q8C2×C4semidirect
128. #391C2×C4semidirect
496. C22⋊C4#103semidirect
591. C2×C4C2×#103semidirect
621. C22#1956semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×#24342
C2×#19562
C2×#18922
C2×#18722
C2×#17872
C2×C8×C22⋊C42
#138802
#110502
#110102
#109262