Finite Group Databasebeta

← #412815

Subgroups — #412815

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2097C211yesmaximal, characteristic
2C2×#2097C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic
4C2×#1847C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6C2×#1763C211yesmaximal, characteristic
7#13942C211yesmaximal, characteristic
8#13911C211yesmaximal, characteristic
9#11378C211yesmaximal, characteristic
10#11377C211yesmaximal, characteristic
11#11325C211yesmaximal, characteristic
12#11325C211yesmaximal, characteristic
13#11124C211yesmaximal, characteristic
14#11122C211yesmaximal, characteristic
15#10835C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#2097—24no—
38#2097—24no—
39#1949—24no—
40#1847—24no—
41#1803—24no—
42#1763—24no—
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53#1442—22no—
54#1441—22no—
55C4⋊#95—22no—
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60#1434—22no—
61C23×Q8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8Q814yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—41no—
95C2×#107—41no—
96C2×#107—41no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—41no—
104C2×#106—41no—
105C2×#106—41no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8D414yes—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—22no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—22no—
117C2×#103—41no—
118C2×#103—41no—
119C2×#103D414yes—
120C2×#103Q814yes—
121C2×#97—22no—
122C2×#97—22no—
123C2×#97D414yes—
124#391—22no—
125#391—22no—
126#391—41no—
127#391—41no—
128#384—22no—
129#384—22no—
130#382—22no—
131#382—22no—
132#382—22no—
133#382—22no—
134#382—41no—
135#382—41no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95D412yes—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#336—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8C2×D412yes—
174C22×Q8C2×Q812yes—
175C22×Q8C2×Q812yes—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8C2×Q812yes—
215C22×C8#4112yes—
216C22×C8#4112yes—
217C4×Q8—28no—
218C4×Q8—28no—
219C4×Q8—44no—
220C4×Q8—44no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×Q812yes—
272C2×C4⋊C4C2×Q812yes—
273C2×C42—24no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—42no—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×Q812yes—
293C2×C42#4112yes—
294C2×C42#4112yes—
295C2×C42#4112yes—
296#107—44no—
297#107—44no—
298#107—44no—
299#106—44no—
300#106—44no—
301#106—44no—
302C4⋊C8—28no—
303#103—28no—
304#103—28no—
305#103—44no—
306#103—44no—
307#97—28no—
308#95—24no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—82no—
319#95—82no—
320C24C22×D411yescharacteristic
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8C4⋊Q814yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
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—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4C22⋊Q812yes—
410C22×C4C22⋊Q812yes—
411C22×C4#12312yes—
412C22×C4#12312yes—
413C22×C4#12712yes—
414C22×C4#12712yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×Q811yescharacteristic
419C22×C4C22×Q811yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C4H(5,𝔽2)11yescharacteristic
427C22×C42−1+411yescharacteristic
428C22×C42−1+411yescharacteristic
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8#12114yes—
456C2×C8C22⋊Q814yes—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—48no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—48no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—48no—
498C42—84no—
499C42C22⋊Q814yes—
500C42C22⋊Q814yes—
501C42#12314yes—
502C42#12314yes—
503C23C2×C22⋊Q811yescharacteristic
504C23C2×#12311yescharacteristic
505C23C2×#12711yescharacteristic
506C23D4211yescharacteristic
507C23#54711yescharacteristic
508C23D4×Q811yescharacteristic
509C23#55011yescharacteristic
510C23#57512yes—
511C23#57512yes—
512C23#57712yes—
513C23#57812yes—
514Q8—24no—
515Q8—24no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—42no—
521Q8—42no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
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#39914yes—
587C2×C4#39914yes—
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×C22⋊Q811yescharacteristic
590C2×C4C2×C22⋊Q811yescharacteristic
591C2×C4C2×C22⋊Q811yescharacteristic
592C2×C4C2×#12311yescharacteristic
593C2×C4C2×#12311yescharacteristic
594C2×C4C2×C4⋊Q811yescharacteristic
595C2×C4#53211yescharacteristic
596C2×C4#53311yescharacteristic
597C2×C4#53311yescharacteristic
598C2×C4#53511yescharacteristic
599C2×C4#53611yescharacteristic
600C2×C4#54211yescharacteristic
601C2×C4#54211yescharacteristic
602C2×C4#54411yescharacteristic
603C2×C4#54411yescharacteristic
604C2×C4D4211yescharacteristic
605C2×C4#54711yescharacteristic
606C2×C4#54711yescharacteristic
607C2×C4#54711yescharacteristic
608C2×C4D4×Q811yescharacteristic
609C2×C4#55011yescharacteristic
610C2×C4#55211yescharacteristic
611C2×C4#55211yescharacteristic
612C2×C4#55311yescharacteristic
613C2×C4#55311yescharacteristic
614C2×C4#55411yescharacteristic
615C2×C4#55411yescharacteristic
616C8—24no—
617C8—24no—
618C8—42no—
619C8—42no—
620C22C2×#39911yescharacteristic
621C22#242211yescharacteristic
622C22#242311yescharacteristic
623C22#244011yescharacteristic
624C22#244111yescharacteristic
625C22#254111yescharacteristic
626C22#254211yescharacteristic
627C22#308012yes—
628C22#308212yes—
629C22#309312yes—
630C22#309712yes—
631C22#315311yescharacteristic
632C22#315311yescharacteristic
633C22#315411yescharacteristic
634C22#315611yescharacteristic
635C22#326912yes—
636C22#327112yes—
637C22#334812yes—
638C22#334912yes—
639C22#338812yes—
640C22#339012yes—
641C22#340112yes—
642C22#340212yes—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—161no—
652C4—161no—
653C4—161no—
654C4—161no—
655C4#242211yescharacteristic
656C4#242311yescharacteristic
657C4#244011yescharacteristic
658C4#244111yescharacteristic
659C4#248611yescharacteristic
660C4#248711yescharacteristic
661C4#262811yescharacteristic
662C4#285711yescharacteristic
663C2#1513511yescharacteristic
664C2#2373311yescharacteristic
665C2#2373611yescharacteristic
666C2#2502512yes—
667C2#2507112yes—
668C2#2605712yes—
669C2#2610512yes—
670C2#2847611yescharacteristic
671C2#2848111yescharacteristic
672C2#2854611yescharacteristic
673C2#2854911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20972
C2×#19492
C2×#18472
C2×#18032
C2×#17632
#139422
#139112
#113782
#113772
#113252
#111242
#111222
#108352