Finite Group Databasebeta

← #121529

Subgroups — #121529

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1852C211yesmaximal, characteristic
2#14275C211yesmaximal, characteristic
3C2×#1442C211yesmaximal, characteristic
4#10730C211yesmaximal, characteristic
5C2×#1294C211yesmaximal, characteristic
6C2×#1294C211yesmaximal, characteristic
7C2×#1280C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×#107—21no—
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C2×#385—21no—
13C2×#380C2211yescharacteristic
14C2×#380C2211yescharacteristic
15C22×#95—21no—
16C22×#95—21no—
17C22×#95C2211yescharacteristic
18C22×#95C2211yescharacteristic
19#1852—22no—
20#1852—22no—
21C2×#340—21no—
22C2×#340—22no—
23#1444—21no—
24#1442—22no—
25#1442—22no—
26#1442—21no—
27#1441—22no—
28C22⋊#95—21no—
29#1294—22no—
30#1294C414yescomplemented
31#1280—22no—
32C24×C4—21no—
33C24×C4—21no—
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C23×C8—21no—
38C23×C8—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—44no—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D412yes—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C2×#107—24no—
53C2×#107—44no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8C2×C412yes—
63C2×C22⋊C8D412yes—
64C2×C22⋊C8Q812yes—
65C2×C22⋊C8Q812yes—
66#385—42no—
67#385—42no—
68#380—21no—
69#380—21no—
70#380—21no—
71#380—21no—
72#380—21no—
73#380—21no—
74#380C2×C412yes—
75#380Q812yes—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—24no—
82C2×#95—24no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—44no—
93C2×#95—42no—
94C2×#95—42no—
95C2×#95C2×C412yes—
96C2×#95D412yes—
97C2×#95Q812yes—
98#340—24no—
99#340—44no—
100C25C2×D411yescharacteristic
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4D812yes—
145C23×C4Q1612yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×Q811yescharacteristic
153C23×C4C2×Q811yescharacteristic
154C23×C4C2×Q811yescharacteristic
155C23×C4C2×Q811yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4#4111yescharacteristic
158C23×C4#4111yescharacteristic
159C23×C4#4111yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—42no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—41no—
169C22×C8—41no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C4⋊C414yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207#107—48no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8C4⋊C414yescomplemented
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—48no—
222#95—48no—
223#95—44no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—82no—
231#95—82no—
232#95C4⋊C414yescomplemented
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24C4×D411yescharacteristic
238C24#12011yescharacteristic
239C24#12011yescharacteristic
240C24#12111yescharacteristic
241C24C22⋊Q811yescharacteristic
242C24C22⋊Q811yescharacteristic
243C24C4⋊Q811yescharacteristic
244C24C2×D811yescharacteristic
245C24C2×Q1611yescharacteristic
246C24#13511yescharacteristic
247C24#13511yescharacteristic
248C24AGL(1,ℤ/8)11yescharacteristic
249C24AGL(1,ℤ/8)11yescharacteristic
250C24#13711yescharacteristic
251C24#13711yescharacteristic
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—82no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4C2×C4⋊C411yescharacteristic
370C22×C4C2×C4⋊C411yescharacteristic
371C22×C4C2×C4⋊C411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×Q811yescharacteristic
375C22×C4#12011yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4C22⋊Q811yescharacteristic
379C22×C4C22⋊Q811yescharacteristic
380C22×C4C22⋊Q811yescharacteristic
381C22×C4C22⋊Q811yescharacteristic
382C22×C4C22⋊Q811yescharacteristic
383C22×C4C22⋊Q811yescharacteristic
384C22×C4C22⋊Q811yescharacteristic
385C22×C4#12311yescharacteristic
386C22×C4#12311yescharacteristic
387C22×C4#12311yescharacteristic
388C22×C4#12511yescharacteristic
389C22×C4#12511yescharacteristic
390C22×C4#12511yescharacteristic
391C22×C4#12711yescharacteristic
392C22×C4C4⋊Q811yescharacteristic
393C22×C4C4⋊Q811yescharacteristic
394C22×C4C4⋊Q811yescharacteristic
395C22×C4C4⋊Q811yescharacteristic
396C22×C4C4⋊Q811yescharacteristic
397C2×C8—24no—
398C2×C8—42no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—42no—
402C2×C8—44no—
403C2×C8—42no—
404C2×C8—82no—
405C2×C8—81no—
406C2×C8—81no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—82no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—88no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—81no—
425C22⋊C4—81no—
426C22⋊C4—81no—
427C22⋊C4—81no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23#38511yescharacteristic
461C23#38511yescharacteristic
462C23#38911yescharacteristic
463C23#39211yescharacteristic
464C23#39511yescharacteristic
465C23#39511yescharacteristic
466C23#39811yescharacteristic
467C23C4×D811yescharacteristic
468C23C4×Q1611yescharacteristic
469C23#44011yescharacteristic
470C23#44011yescharacteristic
471C23#44711yescharacteristic
472C23#44711yescharacteristic
473C23#44911yescharacteristic
474C23#44911yescharacteristic
475C23#45111yescharacteristic
476C23#45111yescharacteristic
477C23#45211yescharacteristic
478C23#45211yescharacteristic
479C23#45412yes—
480C23#45512yes—
481C23B(4,𝔽2)12yes—
482C23#45812yes—
483C23#46611yescharacteristic
484C23#46711yescharacteristic
485C23#46911yescharacteristic
486C23#47011yescharacteristic
487C23#47411yescharacteristic
488C23#47411yescharacteristic
489C23#47711yescharacteristic
490C23#47711yescharacteristic
491C23#47811yescharacteristic
492C23#47811yescharacteristic
493C23#47911yescharacteristic
494C23#47911yescharacteristic
495C23#49911yescharacteristic
496C23#50011yescharacteristic
497C23#50111yescharacteristic
498C23#50111yescharacteristic
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
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—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—84no—
563C2×C4—82no—
564C2×C4—84no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4#38411yescharacteristic
589C2×C4#38511yescharacteristic
590C2×C4#38911yescharacteristic
591C2×C4#38911yescharacteristic
592C2×C4#39511yescharacteristic
593C2×C4#39811yescharacteristic
594C2×C4#39811yescharacteristic
595C2×C4#39911yescharacteristic
596C8—44no—
597C8—82no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
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#144411yescharacteristic
639C22#159712yes—
640C22#159812yes—
641C22#160212yes—
642C22#160612yes—
643C22#160712yes—
644C22#161212yes—
645C22#161512yes—
646C22#161712yes—
647C22#186012yes—
648C22#186211yescharacteristic
649C22#186211yescharacteristic
650C22#186311yescharacteristic
651C22#186311yescharacteristic
652C22#186712yes—
653C22#194211yescharacteristic
654C22#194411yescharacteristic
655C22#199911yescharacteristic
656C22#200111yescharacteristic
657C22#200311yescharacteristic
658C22#200411yescharacteristic
659C22#202111yescharacteristic
660C22#202211yescharacteristic
661C22#205711yescharacteristic
662C22#205711yescharacteristic
663C22#205911yescharacteristic
664C22#205911yescharacteristic
665C22#206112yes—
666C22#206212yes—
667C22#208611yescharacteristic
668C22#208711yescharacteristic
669C4—21no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—84no—
678C4—84no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4—161no—
683C4—161no—
684C2—21no—
685C2—21no—
686C2—21no—
687C2—21no—
688C2—21no—
689C2—21no—
690C2—21no—
691C2—21no—
692C2#886712yes—
693C2#886912yes—
694C2#991511yescharacteristic
695C2#991811yescharacteristic
696C2#992411yescharacteristic
697C2#992911yescharacteristic
698C2#1056912yes—
699C2#1057112yes—
700C2#1132811yescharacteristic
701C2#1133911yescharacteristic
702C2#1134411yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #1294C4semidirect
215. C22⋊C8C4⋊C4semidirect
232. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18522
#142752
C2×#14422
#107302
C2×#12942
C2×#12802