Finite Group Databasebeta

← #370326

Subgroups — #370326

Order 512.

1144 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1922C211yesmaximal, characteristic, complemented
2#20835C211yesmaximal, characteristic
3#20658C211yesmaximal, characteristic
4#20386C211yesmaximal, characteristic
5C2×#1729C211yesmaximal, characteristic
6#11376C211yesmaximal, characteristic, complemented
7#11317C211yesmaximal, characteristic, complemented
8#11291C211yesmaximal, characteristic, complemented
9#11221C211yesmaximal, characteristic, complemented
10#10909C211yesmaximal, characteristic, complemented
11#10887C211yesmaximal, characteristic, complemented
12#10809C211yesmaximal, characteristic, complemented
13#10723C211yesmaximal, characteristic
14#10720C211yesmaximal, characteristic
15#10720C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C24×C8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27#2280C2211yescharacteristic
28#2280C2211yescharacteristic
29#1922—24no—
30#1851—22no—
31#1851—22no—
32#1808C412yescomplemented
33#1808C412yescomplemented
34#1807C412yescomplemented
35#1807C412yescomplemented
36#1753—22no—
37#1753—22no—
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52#1729—24no—
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C24×C4C2311yescharacteristic
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—41no—
67C22×M16—41no—
68C22×M16D412yes—
69C22×M16D412yes—
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8—22no—
75C23×C8—22no—
76C23×C8—22no—
77C23×C8—22no—
78C23×C8—41no—
79C23×C8—41no—
80C23×C8C2×C412yes—
81C23×C8C2×C412yes—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—22no—
89C2×#117—22no—
90C2×#117—41no—
91C2×#117—41no—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#427—22no—
101#427—22no—
102#427—41no—
103#427—41no—
104C2×#107C2×C411yescharacteristic
105C2×#107C2×C411yescharacteristic
106C2×#107C2×C412yes—
107C2×#106C2×C411yescharacteristic
108C2×#106C2×C411yescharacteristic
109C2×#106C2×C412yes—
110#424—22no—
111#424—22no—
112#424—41no—
113#424—41no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117#407—22no—
118#407—22no—
119#407—22no—
120#407Q814yes—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123#405—22no—
124#405—22no—
125#405—41no—
126#405—41no—
127C2×#97—21no—
128C2×#97—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131#381—21no—
132#381—21no—
133#380C2×C412yes—
134#380C2×C412yes—
135#380C2×C412yes—
136#380C2×C412yes—
137#379—21no—
138#379—21no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141#341—22no—
142#341—22no—
143#341—41no—
144#341—41no—
145#340C2×C412yes—
146#340C2×C412yes—
147#340C2×C412yes—
148#340C2×C412yes—
149#339—22no—
150#339—22no—
151#339—22no—
152#339—22no—
153#339—41no—
154#339—41no—
155#339—41no—
156#339—41no—
157#336—22no—
158#336—21no—
159#336—21no—
160#336—22no—
161#336—22no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336D414yes—
166C25#4111yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4C22×C412yes—
178C23×C4C2×D412yes—
179C23×C4C2×Q812yes—
180C23×C4#4112yes—
181C23×C4#4112yes—
182C23×C4#4111yescharacteristic
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—42no—
212C2×M16—42no—
213C2×M16—44no—
214C2×M16—44no—
215C2×M16—44no—
216C2×M16—44no—
217C2×M16C2×Q812yes—
218C2×M16C2×Q812yes—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—24no—
228C22×C8—24no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8—44no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8C22×C412yes—
254C22×C8C22×C411yescharacteristic
255C22×C8C22×C411yescharacteristic
256C22×C8C2×D412yes—
257C22×C8C2×D412yes—
258C22×C8C2×D412yes—
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D411yescharacteristic
261C22×C8#4112yes—
262C22×C8#4112yes—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—44no—
280#117—44no—
281#117—44no—
282#117—44no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C412yes—
291C2×C4⋊C4C22×C412yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42C2×D412yes—
318C2×C42#4112yes—
319#107—22no—
320#107—22no—
321#107—42no—
322#106—22no—
323#106—22no—
324#106—42no—
325C4⋊C8—44no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C22⋊C8—28no—
329C22⋊C8—44no—
330#97—42no—
331#97—42no—
332C4×C8—42no—
333C4×C8—42no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—42no—
343C24—42no—
344C24C4×D412yes—
345C24#12512yes—
346C24#12612yes—
347C24C2×#4111yescharacteristic
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
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—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
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—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4#11714yes—
393C22×C4C4×D412yes—
394C22×C4C4×Q814yes—
395C22×C4C22⋊Q814yes—
396C22×C4#12314yes—
397C22×C4#12512yes—
398C22×C4#12612yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×Q811yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—48no—
418M16—48no—
419M16—41no—
420M16—41no—
421M16—41no—
422M16—41no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—48no—
456C2×C8—48no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8C4×D412yes—
476C2×C8C4×D412yes—
477C2×C8#12112yes—
478C2×C8#12112yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23#38714yes—
542C23C2×C4×D411yescharacteristic
543C23#52511yescharacteristic
544C23C2×#12511yescharacteristic
545C23C2×#12611yescharacteristic
546C23#52911yescharacteristic
547C23#53811yescharacteristic
548C23#53911yescharacteristic
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
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—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4#38714yes—
588C2×C4C2×#11711yescharacteristic
589C2×C4C2×C4×D411yescharacteristic
590C2×C4C2×C4×Q811yescharacteristic
591C2×C4C4×#4111yescharacteristic
592C2×C4#51811yescharacteristic
593C2×C4#52011yescharacteristic
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×C22⋊Q811yescharacteristic
596C2×C4C2×#12311yescharacteristic
597C2×C4#53211yescharacteristic
598C2×C4#53411yescharacteristic
599C2×C4#53711yescharacteristic
600C2×C4#54611yescharacteristic
601C2×C4#54611yescharacteristic
602C2×C4#54611yescharacteristic
603C2×C4D4×Q811yescharacteristic
604C2×C4#55111yescharacteristic
605C2×C4#55111yescharacteristic
606C2×C4#55111yescharacteristic
607C2×C4#55111yescharacteristic
608C2×C4#55111yescharacteristic
609C2×C4#55111yescharacteristic
610C2×C4#55211yescharacteristic
611C2×C4#55211yescharacteristic
612C2×C4#55211yescharacteristic
613C2×C4#55411yescharacteristic
614C2×C4#55411yescharacteristic
615C2×C4#55411yescharacteristic
616C8—22no—
617C8—22no—
618C8—22no—
619C8—22no—
620C8—42no—
621C8—42no—
622C8—42no—
623C8—42no—
624C8—81no—
625C8—81no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22C2×#38711yescharacteristic
646C22C4×#12111yescharacteristic
647C22#232511yescharacteristic
648C22#247311yescharacteristic
649C22#248011yescharacteristic
650C22#249311yescharacteristic
651C22#250511yescharacteristic
652C22#295712yes—
653C22#295712yes—
654C4—24no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—82no—
662C4—82no—
663C4#234211yescharacteristic
664C4#234311yescharacteristic
665C4#236011yescharacteristic
666C4#236811yescharacteristic
667C4#241211yescharacteristic
668C4#241711yescharacteristic
669C4#249811yescharacteristic
670C4#256411yescharacteristic
671C2—21no—
672C2—21no—
673C2—21no—
674C2—21no—
675C2—44no—
676C2#1404711yescharacteristic
677C2#2104312yes—
678C2#2154511yescharacteristic
679C2#2154711yescharacteristic
680C2#2220612yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×M16 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1922C2semidirect
6. #11376C2semidirect
7. #11317C2semidirect
8. #11291C2semidirect
9. #11221C2semidirect
10. #10909C2semidirect
11. #10887C2semidirect
12. #10809C2semidirect
32. #1808C4semidirect
33. #1808C4semidirect
34. #1807C4semidirect
35. #1807C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19222
#208352
#206582
#203862
C2×#17292
#113762
#113172
#112912
#112212
#109092
#108872
#108092
#107232
#107202