Finite Group Databasebeta

← #382701

Subgroups — #382701

Order 512.

2086 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2C2×#1780C211yesmaximal, characteristic
3#14337C211yesmaximal, characteristic
4#11200C212yesmaximal, complemented
5#11200C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10731C212yesmaximal, complemented
9#10730C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2311—24no—
18#2311C418yescomplemented
19C2×#391—21no—
20C2×#390—21no—
21C2×#386—21no—
22C2×#386—21no—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29#1780—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38#1446—21no—
39#1438—21no—
40#1438—21no—
41#1438—21no—
42C22×#41—24no—
43C22×#41—24no—
44C22×#41—41no—
45C22×#41—41no—
46C22×#41—41no—
47C22×#41—41no—
48C22×#41—42no—
49C22×#41—42no—
50C22×#41C2×C414yes—
51C22×#41D414yes—
52C23×Q8D411yescharacteristic
53C23×D4D411yescharacteristic
54C23×D4D411yescharacteristic
55C23×D4D411yescharacteristic
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2312yes—
62C2×#117—22no—
63C2×#117—42no—
64C2×#117—42no—
65C2×#117C2×C414yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—41no—
69C22×C4⋊C4—41no—
70C22×C4⋊C4—41no—
71C22×C4⋊C4—41no—
72C22×C4⋊C4C2×C414yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C42—21no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—41no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—41no—
96C2×C22⋊C8—41no—
97#391—28no—
98#390—28no—
99#386—28no—
100#386—28no—
101#380—24no—
102#380—24no—
103#380—24no—
104#380—24no—
105#380—41no—
106#380—41no—
107#380—42no—
108#380—42no—
109#380—41no—
110#380—41no—
111#380C2×C418yes—
112#380C2×C418yes—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119#340—42no—
120#340—42no—
121#340—42no—
122#340—42no—
123#340—42no—
124#340—42no—
125#340—42no—
126#340—42no—
127C25C2×D411yescharacteristic
128C25C2×D411yescharacteristic
129C25C2×D411yescharacteristic
130C2×#41—28no—
131C2×#41—28no—
132C2×#41—216no—
133C2×#41—44no—
134C2×#41—44no—
135C2×#41—44no—
136C2×#41—44no—
137C2×#41—44no—
138C2×#41—44no—
139C2×#41—48no—
140C2×#41—48no—
141C2×#41—48no—
142C2×#41—48no—
143C2×#41—48no—
144C2×#41—48no—
145C2×#41—48no—
146C2×#41—48no—
147C2×#41—48no—
148C22×Q8—24no—
149C22×Q8—42no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8C22⋊C414yes—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—42no—
157C22×D4—41no—
158C22×D4—41no—
159C22×D4—42no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—42no—
163C22×D4—41no—
164C22×D4—41no—
165C22×D4C22⋊C414yes—
166C22×D4C22⋊C414yes—
167C22×D4C22⋊C414yes—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—82no—
198C23×C4—82no—
199C23×C4C22×C412yes—
200C23×C4C22×C412yes—
201C23×C4C22×C412yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4#4114yes—
209C23×C4#4114yes—
210C23×C4C2411yescharacteristic
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225#117—44no—
226#117—44no—
227#117—48no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C22×C414yes—
258C2×C4⋊C4C22×C414yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—82no—
268C2×C22⋊C4—82no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4—82no—
277C2×C42—24no—
278C2×C42—44no—
279C22⋊C8—44no—
280C22⋊C8—48no—
281C22⋊C8—44no—
282C22⋊C8—48no—
283#95—28no—
284#95—28no—
285#95—42no—
286#95—44no—
287#95—42no—
288#95—44no—
289C24—22no—
290C24—21no—
291C24—21no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C2×C22⋊C412yes—
313C24C2×C22⋊C412yes—
314C24C2×C22⋊C412yes—
315C24#12314yes—
316C24C22×D411yescharacteristic
317#41—416no—
318#41—416no—
319#41—416no—
320#41—416no—
321#41—432no—
322#41—432no—
323C2×Q8—28no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
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—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C412yes—
440C22×C4C2×C22⋊C412yes—
441C22×C4#11714yes—
442C22×C4#11714yes—
443C22×C4#12318yes—
444C22×C4#12314yes—
445C22×C4C23×C411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C4H(5,𝔽2)11yescharacteristic
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C22⋊C4—48no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C42—48no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23#38114yes—
557C23C22×C22⋊C411yescharacteristic
558C23C2×#12312yes—
559C23#53411yescharacteristic
560C23#53412yes—
561C23#53411yescharacteristic
562C23#57612yes—
563C23#57712yes—
564C23#57712yes—
565C23#57812yes—
566Q8—416no—
567D4—416no—
568D4—416no—
569D4—416no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—816no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4#38114yes—
632C2×C4C2×#11711yescharacteristic
633C2×C4C2×#11711yescharacteristic
634C2×C4#51811yescharacteristic
635C2×C4#51811yescharacteristic
636C2×C4#51811yescharacteristic
637C2×C4#51811yescharacteristic
638C2×C4C2×#12312yes—
639C2×C4#53411yescharacteristic
640C2×C4#53411yescharacteristic
641C2×C4#53412yes—
642C2×C4#53812yes—
643C2×C4#53812yes—
644C2×C4#53812yes—
645C2×C4#53812yes—
646C2×C4#53812yes—
647C2×C4#53812yes—
648C2×C4#53812yes—
649C2×C4#53812yes—
650C8—82no—
651C8—82no—
652C22—24no—
653C22—28no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22C2×#38111yescharacteristic
680C22#230811yescharacteristic
681C22#230811yescharacteristic
682C22#262212yes—
683C22#262212yes—
684C22#289712yes—
685C22#289812yes—
686C22#309612yes—
687C22#309712yes—
688C22#309812yes—
689C22#309912yes—
690C22#319812yes—
691C22#319812yes—
692C22#319911yescharacteristic
693C22#320011yescharacteristic
694C22#320011yescharacteristic
695C22#320112yes—
696C22#320112yes—
697C22#320211yescharacteristic
698C22#320312yes—
699C22#320412yes—
700C4—24no—
701C4—42no—
702C4—42no—
703C4—84no—
704C4—84no—
705C4—162no—
706C4—162no—
707C4#231211yescharacteristic
708C4#231211yescharacteristic
709C4#231211yescharacteristic
710C4#231211yescharacteristic
711C4#262212yes—
712C4#262212yes—
713C2—44no—
714C2—82no—
715C2—82no—
716C2#1434111yescharacteristic
717C2#2070312yes—
718C2#2177011yescharacteristic
719C2#2177111yescharacteristic
720C2#2177212yes—
721C2#2989812yes—
722C2#2989912yes—
723C2#2990012yes—
724C2#2990112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×#41 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11200C2semidirect
5. #11200C2semidirect
6. #11199C2semidirect
7. #11199C2semidirect
8. #10731C2semidirect
9. #10730C2semidirect
18. #2311C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#17802
#143372
#112002
#111992
#107312
#107302