Finite Group Databasebeta

← #370327

Subgroups — #370327

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1922C211yesmaximal, characteristic, complemented
2#20835C211yesmaximal, characteristic
3#20660C211yesmaximal, characteristic
4#20386C211yesmaximal, characteristic
5C2×#1728C211yesmaximal, 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#10724C211yesmaximal, characteristic
14#10719C211yesmaximal, characteristic
15#10719C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C23×M16C2211yescharacteristic
18C23×C42C2211yescharacteristic
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#1809C412yescomplemented
33#1809C412yescomplemented
34#1809C412yescomplemented
35#1809C412yescomplemented
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#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1728—24no—
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—22no—
67C22×M16—22no—
68C22×M16—22no—
69C22×M16—22no—
70C22×M16—22no—
71C22×M16—41no—
72C22×M16—41no—
73C22×M16—41no—
74C22×M16—41no—
75C22×M16C2×C412yes—
76C22×M16C2×C412yes—
77C22×M16D412yes—
78C22×M16D412yes—
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C22×M16C2311yescharacteristic
82C22×M16C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C22×C42—21no—
92C22×C42—22no—
93C22×C42—22no—
94C22×C42—22no—
95C22×C42—22no—
96C22×C42—41no—
97C22×C42—41no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#427—22no—
101#427—22no—
102#427—41no—
103#427—41no—
104C2×#107C2×C411yescharacteristic, complemented
105C2×#107C2×C411yescharacteristic, complemented
106C2×#107C2×C412yescomplemented
107C2×#106C2×C411yescharacteristic, complemented
108C2×#106C2×C411yescharacteristic, complemented
109C2×#106C2×C412yescomplemented
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×C412yescomplemented
146#340C2×C412yescomplemented
147#340C2×C412yescomplemented
148#340C2×C412yescomplemented
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×C4—24no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4C22×C412yes—
183C23×C4C2×D412yes—
184C23×C4C2×Q812yes—
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4#4111yescharacteristic
188C23×C4#4112yes—
189C23×C4C2411yescharacteristic
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
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—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—42no—
221C2×M16—42no—
222C2×M16—42no—
223C2×M16—42no—
224C2×M16—44no—
225C2×M16—44no—
226C2×M16—44no—
227C2×M16—44no—
228C2×M16—44no—
229C2×M16—44no—
230C2×M16—44no—
231C2×M16—44no—
232C2×M16C2×Q812yes—
233C2×M16C2×Q812yes—
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—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8C22×C412yes—
259C22×C8C22×C411yescharacteristic
260C22×C8C22×C411yescharacteristic
261C22×C8C2×D412yes—
262C22×C8C2×D412yes—
263C22×C8C2×D412yes—
264C22×C8C2×D411yescharacteristic
265C22×C8C2×D411yescharacteristic
266C22×C8#4112yes—
267C22×C8#4112yes—
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—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C412yes—
289C2×C4⋊C4C22×C412yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—24no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42C2×D412yes—
320C2×C42#4112yes—
321#107—22no—
322#107—22no—
323#107—42no—
324#106—22no—
325#106—22no—
326#106—42no—
327C4⋊C8—44no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C22⋊C8—28no—
331C22⋊C8—44no—
332#97—42no—
333#97—42no—
334C4×C8—42no—
335C4×C8—42no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—42no—
345C24—42no—
346C24C4×D412yes—
347C24#12512yes—
348C24#12612yes—
349C24C2×#4111yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
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—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4#11714yes—
400C22×C4C4×D412yes—
401C22×C4C4×Q814yes—
402C22×C4C22⋊Q814yes—
403C22×C4#12314yes—
404C22×C4#12512yes—
405C22×C4#12612yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×Q811yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C42−1+411yescharacteristic
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—48no—
425M16—48no—
426M16—48no—
427M16—48no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
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×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8C4×D412yes—
483C2×C8C4×D412yes—
484C2×C8#12112yes—
485C2×C8#12112yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—48no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23#38714yes—
548C23C2×C4×D411yescharacteristic
549C23#52511yescharacteristic
550C23C2×#12511yescharacteristic
551C23C2×#12611yescharacteristic
552C23#52911yescharacteristic
553C23#53811yescharacteristic
554C23#53911yescharacteristic
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—432no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4#38714yes—
595C2×C4C2×#11711yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×Q811yescharacteristic
598C2×C4C4×#4111yescharacteristic
599C2×C4#51811yescharacteristic
600C2×C4#52011yescharacteristic
601C2×C4C2×#12111yescharacteristic
602C2×C4C2×C22⋊Q811yescharacteristic
603C2×C4C2×#12311yescharacteristic
604C2×C4#53211yescharacteristic
605C2×C4#53411yescharacteristic
606C2×C4#53711yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54611yescharacteristic
610C2×C4D4×Q811yescharacteristic
611C2×C4#55111yescharacteristic
612C2×C4#55111yescharacteristic
613C2×C4#55111yescharacteristic
614C2×C4#55111yescharacteristic
615C2×C4#55111yescharacteristic
616C2×C4#55111yescharacteristic
617C2×C4#55211yescharacteristic
618C2×C4#55211yescharacteristic
619C2×C4#55211yescharacteristic
620C2×C4#55411yescharacteristic
621C2×C4#55411yescharacteristic
622C2×C4#55411yescharacteristic
623C8—22no—
624C8—22no—
625C8—22no—
626C8—22no—
627C8—42no—
628C8—42no—
629C8—42no—
630C8—42no—
631C8—81no—
632C8—81no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22C2×#38711yescharacteristic
653C22C4×#12111yescharacteristic
654C22#232511yescharacteristic
655C22#247311yescharacteristic
656C22#248011yescharacteristic
657C22#249311yescharacteristic
658C22#250511yescharacteristic
659C22#295612yes—
660C22#295612yes—
661C4—24no—
662C4—44no—
663C4—44no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—82no—
669C4—82no—
670C4#234211yescharacteristic
671C4#234311yescharacteristic
672C4#236011yescharacteristic
673C4#236811yescharacteristic
674C4#241211yescharacteristic
675C4#241711yescharacteristic
676C4#249811yescharacteristic
677C4#256411yescharacteristic
678C2—21no—
679C2—21no—
680C2—21no—
681C2—21no—
682C2—44no—
683C2#1404711yescharacteristic
684C2#2104212yes—
685C2#2154911yescharacteristic
686C2#2155111yescharacteristic
687C2#2220712yes—
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. #1809C4semidirect
33. #1809C4semidirect
34. #1809C4semidirect
35. #1809C4semidirect
104. C2×#107C2×C4semidirect
105. C2×#107C2×C4semidirect
106. C2×#107C2×C4semidirect
107. C2×#106C2×C4semidirect
108. C2×#106C2×C4semidirect
109. C2×#106C2×C4semidirect
145. #340C2×C4semidirect
146. #340C2×C4semidirect
147. #340C2×C4semidirect
148. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19222
#208352
#206602
#203862
C2×#17282
#113762
#113172
#112912
#112212
#109092
#108872
#108092
#107242
#107192