Finite Group Databasebeta

← #357378

Subgroups — #357378

Order 512.

1079 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21237C211yesmaximal, characteristic
2C2×#1905C211yesmaximal, characteristic
3C2×#1902C211yesmaximal, characteristic
4#10959C211yesmaximal, characteristic
5#10957C211yesmaximal, characteristic, complemented
6#10954C211yesmaximal, characteristic
7#10952C211yesmaximal, characteristic, complemented
8#10904C211yesmaximal, characteristic, complemented
9#10903C211yesmaximal, characteristic, complemented
10#10813C211yesmaximal, characteristic, complemented
11#10805C211yesmaximal, characteristic, complemented
12#10786C211yesmaximal, characteristic, complemented
13#10784C211yesmaximal, characteristic
14#10751C211yesmaximal, characteristic
15#10750C211yesmaximal, characteristic, complemented
16C22×#137C2211yescharacteristic
17C2×#568C2211yescharacteristic
18C2×#519C2211yescharacteristic
19C2×#428C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C2×#421C2211yescharacteristic
22C2×#417C2211yescharacteristic
23C2×#419C2211yescharacteristic
24C2×#419C2211yescharacteristic
25C2×#419C2211yescharacteristic
26C22×#101C2211yescharacteristic
27C2×#410C2211yescharacteristic
28#2873C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30#2297C2211yescharacteristic
31C2×#391C2211yescharacteristic
32#2274C2211yescharacteristic
33#1973C412yescomplemented
34#1971C412yescomplemented
35#1969C412yescomplemented
36#1968C412yescomplemented
37#1905C418yescomplemented
38#1902C418yescomplemented
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41C2×#344C2211yescharacteristic
42C2×#344C2211yescharacteristic
43C2×#343C2211yescharacteristic
44#1740C2211yescharacteristic
45#1736C2211yescharacteristic
46#1736C2211yescharacteristic
47#1733C2211yescharacteristic
48#1733C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#338C2211yescharacteristic
54C2×#338C2211yescharacteristic
55C2×H(3,ℤ/4)C2211yescharacteristic
56C2×H(3,ℤ/4)C2211yescharacteristic
57C22×#41C2311yescharacteristic
58C23×Q8C2311yescharacteristic
59C2×#137—41no—
60C2×#137—41no—
61C2×#137—41no—
62C2×#137—41no—
63C2×#137C2×C414yes—
64C2×#137C2×C414yes—
65C22×Q16C2×C412yes—
66C22×SD16C2×C412yes—
67#568—41no—
68#568—41no—
69#568—41no—
70#568C2×C414yes—
71C2×#131—41no—
72C2×#131—41no—
73C22×M16—41no—
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80#519—41no—
81#519—41no—
82#519—41no—
83#519C2×C414yes—
84C2×C4×Q8—41no—
85C2×C4×Q8—41no—
86C2×#117—41no—
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#428—41no—
95#428—41no—
96C2×#107C2×C412yes—
97C2×#106C2×C412yes—
98#421—41no—
99#421—41no—
100#421C2×C414yes—
101#421C2×C414yes—
102C2×#104—41no—
103C2×#104—41no—
104#419—41no—
105#419—41no—
106#419—41no—
107#419—41no—
108#419C2×C414yes—
109#419C2×C414yes—
110#417—41no—
111#417—41no—
112C2×#103—41no—
113C2×#103—41no—
114C2×#103C2×C412yes—
115C2×#103C2×C412yes—
116C2×#103C2×C412yes—
117C2×#102C2×C412yes—
118C2×#101—41no—
119C2×#101—41no—
120C2×#101C2×C414yes—
121#410—41no—
122#410C2×C414yes—
123C2×#99—41no—
124#405—41no—
125#405—41no—
126C4×M16—41no—
127C4×M16—41no—
128C4×M16—41no—
129C2×#97C2×C412yes—
130C2×C4×C8C2×C412yes—
131#391—41no—
132#391C2×C412yes—
133#391C2×C411yescharacteristic
134#391C2×C411yescharacteristic
135#391C2×C412yes—
136#390C2×C411yescharacteristic
137#390C2×C411yescharacteristic
138#386—41no—
139C4×C4⋊C4—41no—
140C4×C4⋊C4—41no—
141C4×C22⋊C4—41no—
142C4×C22⋊C4—41no—
143C2×#95—41no—
144#344—41no—
145#344—41no—
146#344C2×C414yes—
147#344C2×C414yes—
148#343—41no—
149#342—41no—
150#340—41no—
151#340—41no—
152#339—41no—
153#339—41no—
154#339—41no—
155#339—41no—
156#339—41no—
157#339—41no—
158#338—41no—
159#338C2×C414yes—
160H(3,ℤ/4)—41no—
161H(3,ℤ/4)C2×C414yes—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41C22×C412yes—
173C2×#41C22×C412yes—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8C22×C412yes—
186C22×Q8C22×C412yes—
187C22×Q8C22×C411yescharacteristic
188C22×Q8C22×C412yes—
189C22×D4—21no—
190C22×D4C22×C411yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4C2411yescharacteristic
196#137—44no—
197#137—44no—
198#137—44no—
199#137—44no—
200C2×Q16—42no—
201C2×Q16—42no—
202C2×Q16C4218yescomplemented
203C2×SD16—42no—
204C2×SD16—42no—
205C2×SD16C4218yescomplemented
206#131—44no—
207#131—44no—
208#131—44no—
209#131—44no—
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—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—44no—
241C2×M16—42no—
242C2×M16—44no—
243C2×M16C22×C412yes—
244C2×M16C22×C412yes—
245C2×M16C22×C412yes—
246C2×M16C22×C412yes—
247C2×M16C22×C412yes—
248C2×M16C22×C412yes—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—42no—
254C22×C8C22×C412yes—
255C22×C8C22×C412yes—
256C4×Q8—44no—
257C4×Q8—44no—
258C4×Q8—44no—
259C4×Q8—44no—
260#117—21no—
261#117—21no—
262#117—21no—
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—44no—
277#117—44no—
278#117C22×C412yes—
279#117C22×C412yes—
280#117C22×C412yes—
281#117C22×C412yes—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4C22×C412yes—
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C42—21no—
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—42no—
313C2×C42—42no—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316#107—42no—
317#106—42no—
318#104—44no—
319#104—44no—
320#103—44no—
321#103—44no—
322#103—42no—
323#103—42no—
324#103—42no—
325#102—42no—
326#101—44no—
327#99—44no—
328#97—42no—
329#97—42no—
330C4×C8—42no—
331C4×C8—42no—
332#95—42no—
333#95—42no—
334#95—41no—
335#95—41no—
336#95—42no—
337C24—21no—
338C24H(5,𝔽2)11yescharacteristic
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—22no—
346#41—22no—
347#41—41no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353#41—41no—
354#41—41no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—21no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8C2×C4214yes—
388C2×Q8C2×C4212yes—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4C2×C4212yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—81no—
430C22×C4—82no—
431C22×C4C23×C411yescharacteristic
432C22×C4C23×C411yescharacteristic
433C22×C4C23×C411yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C4H(5,𝔽2)11yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C42−1+411yescharacteristic
444C22×C42−1+411yescharacteristic
445Q16—48no—
446SD16—48no—
447M16—22no—
448M16—22no—
449M16—22no—
450M16—22no—
451M16—22no—
452M16—22no—
453M16—22no—
454M16—22no—
455M16—22no—
456M16—22no—
457M16—22no—
458M16—22no—
459M16—48no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—48no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8C2×C4214yes—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—48no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C42—21no—
521C42—21no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—22no—
528C42—48no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—162no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—81no—
544C23—81no—
545C23#51811yescharacteristic
546C23#51811yescharacteristic
547C23#52011yescharacteristic
548C23#56011yescharacteristic
549C23#56011yescharacteristic
550C23#56011yescharacteristic
551C23#56211yescharacteristic
552Q8—24no—
553Q8—22no—
554Q8—24no—
555Q8—22no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—41no—
562Q8—42no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—41no—
571D4—41no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—82no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—82no—
613C2×C4—161no—
614C2×C4C22×C4211yescharacteristic
615C2×C4#51811yescharacteristic
616C2×C4#51811yescharacteristic
617C2×C4#51811yescharacteristic
618C2×C4#51811yescharacteristic
619C2×C4#51811yescharacteristic
620C2×C4#51811yescharacteristic
621C2×C4#51811yescharacteristic
622C2×C4#51811yescharacteristic
623C2×C4#51811yescharacteristic
624C2×C4#51811yescharacteristic
625C2×C4#52011yescharacteristic
626C2×C4#52011yescharacteristic
627C2×C4#52011yescharacteristic
628C2×C4#52011yescharacteristic
629C2×C4#52011yescharacteristic
630C2×C4#56011yescharacteristic
631C2×C4#56011yescharacteristic
632C2×C4#56011yescharacteristic
633C2×C4#56011yescharacteristic
634C2×C4#56011yescharacteristic
635C2×C4H(3,𝔽4)11yescharacteristic
636C2×C4#56211yescharacteristic
637C2×C4#56211yescharacteristic
638C2×C4#56211yescharacteristic
639C2×C4#56211yescharacteristic
640C2×C4#56211yescharacteristic
641C2×C4#56311yescharacteristic
642C8—24no—
643C8—42no—
644C8—42no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—82no—
660C22—81no—
661C22—81no—
662C22—161no—
663C22#227511yescharacteristic
664C22#237511yescharacteristic
665C22#237611yescharacteristic
666C22#237711yescharacteristic
667C22#237711yescharacteristic
668C22#237711yescharacteristic
669C22#238011yescharacteristic
670C4—42no—
671C4—42no—
672C4—44no—
673C4—81no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—81no—
679C4—81no—
680C4—82no—
681C4#227311yescharacteristic
682C4#227511yescharacteristic
683C4#237511yescharacteristic
684C4#237611yescharacteristic
685C4#237711yescharacteristic
686C4#237711yescharacteristic
687C4#237711yescharacteristic
688C4#237911yescharacteristic
689C2—21no—
690C2—21no—
691C2—21no—
692C2—21no—
693C2—21no—
694C2—21no—
695C2—82no—
696C2#1383211yescharacteristic
697C2#2336011yescharacteristic
698C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10957C2semidirect
7. #10952C2semidirect
8. #10904C2semidirect
9. #10903C2semidirect
10. #10813C2semidirect
11. #10805C2semidirect
12. #10786C2semidirect
15. #10750C2semidirect
33. #1973C4semidirect
34. #1971C4semidirect
35. #1969C4semidirect
36. #1968C4semidirect
37. #1905C4semidirect
38. #1902C4semidirect
202. C2×Q16C42semidirect
205. C2×SD16C42semidirect

Maximal subgroups

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

subgroupindex
#212372
C2×#19052
C2×#19022
#109592
#109572
#109542
#109522
#109042
#109032
#108132
#108052
#107862
#107842
#107512
#107502