Finite Group Databasebeta

← #108398

Subgroups — #108398

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#95C211yesmaximal, characteristic
2#10712C212yesmaximal
3C2×#1280C212yesmaximal
4C2×#1280C212yesmaximal
5C23×C42C411yescharacteristic
6C23×C42C411yescharacteristic
7C23×C42C2211yescharacteristic
8C22×C22⋊C8C2214yes—
9C22×#95C411yescharacteristic
10C22×#95C411yescharacteristic
11C22×#95C2211yescharacteristic
12C22×#95C2211yescharacteristic
13C2×#336—22no—
14C2×#336—22no—
15C4×#95—22no—
16C4×#95—22no—
17#1280—24no—
18#1280C418yescomplemented
19C24×C4C2×C411yescharacteristic
20C24×C4C2×C411yescharacteristic
21C24×C4C2×C411yescharacteristic
22C24×C4C2×C411yescharacteristic
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C23×C8—24no—
27C22×C42—21no—
28C22×C42—21no—
29C22×C42—21no—
30C22×C42—21no—
31C22×C42—22no—
32C22×C42—22no—
33C22×C42—22no—
34C22×C42—22no—
35C22×C42—22no—
36C22×C42—42no—
37C22×C42—42no—
38C22×C42—42no—
39C22×C42—42no—
40C22×C42—42no—
41C22×C42—42no—
42C22×C42C814yescomplemented
43C22×C42D412yes—
44C22×C42D412yes—
45C2×C22⋊C8—24no—
46C2×C22⋊C8—24no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8C2×C418yes—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95C814yescomplemented
61C2×#95C2×C414yes—
62C2×#95D412yes—
63C2×#95Q812yes—
64#336—44no—
65#336—44no—
66C25C22×C411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—41no—
113C23×C4—41no—
114C23×C4—41no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—41no—
119C23×C4C4212yes—
120C23×C4C4212yes—
121C23×C4C22⋊C412yes—
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C411yescharacteristic
124C23×C4C4⋊C412yes—
125C23×C4C4⋊C412yes—
126C23×C4C2×C812yes—
127C23×C4C2×C812yes—
128C23×C4C2×C812yes—
129C23×C4C2×C812yes—
130C23×C4C2×C812yes—
131C23×C4C2×C812yes—
132C23×C4M1612yes—
133C23×C4M1612yes—
134C23×C4M1612yes—
135C23×C4M1612yes—
136C23×C4M1612yes—
137C23×C4M1612yes—
138C23×C4C22×C412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×Q811yescharacteristic
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4#4112yes—
146C22×C8—28no—
147C22×C8—44no—
148C22×C8—44no—
149C22×C8—44no—
150C22×C8—44no—
151C22×C8—44no—
152C22×C8—44no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—48no—
164C2×C42—48no—
165C2×C42—48no—
166C2×C42—48no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—48no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—48no—
177C22⋊C8—28no—
178C22⋊C8—28no—
179C22⋊C8—44no—
180C22⋊C8—44no—
181#95—28no—
182#95—28no—
183#95—28no—
184#95—24no—
185#95—24no—
186C24—21no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24C2×C4211yescharacteristic
195C24C2×C22⋊C411yescharacteristic
196C24C2×C4⋊C411yescharacteristic
197C24#11711yescharacteristic
198C24#11711yescharacteristic
199C24#11711yescharacteristic
200C24#11711yescharacteristic
201C24C22×C811yescharacteristic
202C24C22×C811yescharacteristic
203C24C2×M1611yescharacteristic
204C24C2×M1611yescharacteristic
205C24C2×M1611yescharacteristic
206C24C2×M1611yescharacteristic
207C24C2×M1611yescharacteristic
208C24C2×M1611yescharacteristic
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4C4×C814yes—
362C22×C4#9714yes—
363C22×C4C22⋊C814yes—
364C22×C4C22⋊C812yes—
365C22×C4C22⋊C812yes—
366C22×C4#9912yes—
367C22×C4#9912yes—
368C22×C4#10012yes—
369C22×C4#10112yes—
370C22×C4C4⋊C814yes—
371C22×C4C4⋊C814yes—
372C22×C4#11714yes—
373C22×C4C4×D414yes—
374C22×C4C4×D412yes—
375C22×C4C4×Q812yes—
376C22×C4#12112yes—
377C22×C4C22⋊Q812yes—
378C22×C4#12314yes—
379C22×C4#12412yes—
380C22×C4#12512yes—
381C22×C4#12614yes—
382C2×C8—48no—
383C2×C8—48no—
384C2×C8—48no—
385C2×C8—84no—
386C2×C8—84no—
387C2×C8—84no—
388C2×C8—84no—
389C42—48no—
390C42—48no—
391C42—48no—
392C42—48no—
393C42—48no—
394C42—48no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23#37611yescharacteristic
452C23C4×C22⋊C411yescharacteristic
453C23C4×C4⋊C411yescharacteristic
454C23#38011yescharacteristic
455C23#38111yescharacteristic
456C23#38211yescharacteristic
457C23#38311yescharacteristic
458C23C2×C4×C811yescharacteristic
459C23C2×#9711yescharacteristic
460C23C4×M1611yescharacteristic
461C23C4×M1611yescharacteristic
462C23C2×C22⋊C811yescharacteristic
463C23C2×C22⋊C811yescharacteristic
464C23#40711yescharacteristic
465C23#40711yescharacteristic
466C23C2×#9911yescharacteristic
467C23C2×#9911yescharacteristic
468C23#41011yescharacteristic
469C23#41011yescharacteristic
470C23C2×#10011yescharacteristic
471C23C2×#10111yescharacteristic
472C23#41311yescharacteristic
473C23#41311yescharacteristic
474C23C2×C4⋊C811yescharacteristic
475C23C2×C4⋊C811yescharacteristic
476C23#42311yescharacteristic
477C23#42311yescharacteristic
478C23#43111yescharacteristic
479C23#43111yescharacteristic
480C23#43111yescharacteristic
481C23#43111yescharacteristic
482C23#43111yescharacteristic
483C23#43111yescharacteristic
484C23#43111yescharacteristic
485C23#43111yescharacteristic
486C23#43211yescharacteristic
487C23#43211yescharacteristic
488C23#43211yescharacteristic
489C23#43211yescharacteristic
490C23#43211yescharacteristic
491C23#43211yescharacteristic
492C23#43211yescharacteristic
493C23#43211yescharacteristic
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
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—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4#32314yes—
580C2×C4#32414yes—
581C2×C4#38714yes—
582C2×C4#38814yes—
583C8—88no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22#143411yescharacteristic
641C22C2×#32311yescharacteristic
642C22C2×#32411yescharacteristic
643C22#145811yescharacteristic
644C22#145811yescharacteristic
645C22#145911yescharacteristic
646C22#145911yescharacteristic
647C22#146011yescharacteristic
648C22#146111yescharacteristic
649C22#146211yescharacteristic
650C22#146311yescharacteristic
651C22#146411yescharacteristic
652C22#146411yescharacteristic
653C22#146511yescharacteristic
654C22#146511yescharacteristic
655C22#146611yescharacteristic
656C22#146711yescharacteristic
657C22#174411yescharacteristic
658C22#174511yescharacteristic
659C22C4×C22⋊C811yescharacteristic
660C22#174911yescharacteristic
661C22C4×#9911yescharacteristic
662C22C4×#9911yescharacteristic
663C22C4×#10011yescharacteristic
664C22C4×#10111yescharacteristic
665C22C4×C4⋊C811yescharacteristic
666C22#176711yescharacteristic
667C22#179711yescharacteristic
668C22#179711yescharacteristic
669C22#179911yescharacteristic
670C22#179911yescharacteristic
671C22#180011yescharacteristic
672C22#180111yescharacteristic
673C22#181711yescharacteristic
674C22#181711yescharacteristic
675C22#181911yescharacteristic
676C22#182011yescharacteristic
677C22#182111yescharacteristic
678C22#182211yescharacteristic
679C22#183111yescharacteristic
680C22#183111yescharacteristic
681C22#183911yescharacteristic
682C22#183911yescharacteristic
683C4—24no—
684C4—24no—
685C4—24no—
686C4—24no—
687C4—44no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—44no—
692C2—21no—
693C2—21no—
694C2—21no—
695C2—21no—
696C2—21no—
697C2—21no—
698C2—21no—
699C2—21no—
700C2C4×#32311yescharacteristic
701C2C4×#32411yescharacteristic
702C2#815211yescharacteristic
703C2#815311yescharacteristic
704C2#855711yescharacteristic
705C2#855811yescharacteristic
706C2#855911yescharacteristic
707C2#856011yescharacteristic
708C2#869811yescharacteristic
709C2#869911yescharacteristic
710C2#870011yescharacteristic
711C2#870111yescharacteristic
712C2#1079511yescharacteristic
713C2#1080111yescharacteristic
714C2#1080211yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23C2×C4×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1280C4semidirect
42. C22×C42C8semidirect
60. C2×#95C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#952
#107122
C2×#12802