Finite Group Databasebeta

← #443675

Subgroups — #443675

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2095C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#2083C211yesmaximal, characteristic
4C2×#1928C211yesmaximal, characteristic
5C2×#1921C211yesmaximal, characteristic
6C2×#1845C211yesmaximal, characteristic
7#15184C211yesmaximal, characteristic
8#13910C211yesmaximal, characteristic
9#11377C211yesmaximal, characteristic
10#11324C211yesmaximal, characteristic
11#11324C211yesmaximal, characteristic
12#11124C211yesmaximal, characteristic
13#11050C211yesmaximal, characteristic
14#10926C211yesmaximal, characteristic
15#10874C211yesmaximal, characteristic
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389—22no—
31C2×#389C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#2095—24no—
39#2095—24no—
40#2083—24no—
41#1928—24no—
42#1921—24no—
43#1845—24no—
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55#1444—22no—
56#1441—22no—
57#1441—22no—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C23×Q8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—41no—
97C2×#107—41no—
98C2×#107—41no—
99C2×#107—41no—
100C2×#107—41no—
101C2×#107D414yes—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×#103—41no—
111C2×#103—41no—
112C2×#103—41no—
113C2×#103Q814yes—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#391—22no—
121#391—22no—
122#391—41no—
123#389—22no—
124#389—22no—
125#389—48no—
126#384—22no—
127#384—22no—
128#384—22no—
129#384—22no—
130#384—22no—
131#384—41no—
132#384—41no—
133#384—41no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—22no—
147C2×#95D412yes—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#336—22no—
155#336—22no—
156#336—22no—
157#336D414yes—
158#336D414yes—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
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×Q8C2×Q812yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
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×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×Q812yes—
212C22×C8#4112yes—
213C22×C8#4112yes—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
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—21no—
234C2×C4⋊C4—21no—
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—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—82no—
266C2×C4⋊C4—82no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4Q1618yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×Q812yes—
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—21no—
287C2×C42—21no—
288C2×C42—44no—
289C2×C42—42no—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42#4112yes—
293C2×C42#4112yes—
294#107—28no—
295#107—44no—
296#107—44no—
297#107—44no—
298#107—44no—
299#107—44no—
300#103—28no—
301#103—44no—
302#103—44no—
303#103—44no—
304C4×C8—28no—
305#95—24no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—44no—
314#95—82no—
315C24C22×D411yescharacteristic
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
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—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
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#12314yes—
404C22×C4#12712yes—
405C22×C4#12712yes—
406C22×C4C2×Q1614yes—
407C22×C4C2×Q1614yes—
408C22×C4C2×Q1614yes—
409C22×C4AGL(1,ℤ/8)14yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×Q811yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C4H(5,𝔽2)11yescharacteristic
420C22×C42−1+411yescharacteristic
421C22×C42−1+411yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
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—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8#12114yes—
451C2×C8C22⋊Q814yes—
452C4⋊C4—216no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
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—88no—
482C4⋊C4—88no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—88no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—84no—
500C23C2×#12311yescharacteristic
501C23C2×#12711yescharacteristic
502C23#53611yescharacteristic
503C23D4211yescharacteristic
504C23#54711yescharacteristic
505C23D4×Q811yescharacteristic
506C23#55011yescharacteristic
507C23C22×Q1612yes—
508C23C2×AGL(1,ℤ/8)12yes—
509C23#57612yes—
510C23#57812yes—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—81no—
516Q8—81no—
517Q8—81no—
518Q8—81no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4#48418yes—
584C2×C4#49414yes—
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4#52511yescharacteristic
588C2×C4#53311yescharacteristic
589C2×C4#53511yescharacteristic
590C2×C4#53611yescharacteristic
591C2×C4#53611yescharacteristic
592C2×C4#53911yescharacteristic
593C2×C4#53911yescharacteristic
594C2×C4#54111yescharacteristic
595C2×C4#54211yescharacteristic
596C2×C4#54411yescharacteristic
597C2×C4D4211yescharacteristic
598C2×C4#54711yescharacteristic
599C2×C4#54711yescharacteristic
600C2×C4#54711yescharacteristic
601C2×C4#54711yescharacteristic
602C2×C4#54711yescharacteristic
603C2×C4#54811yescharacteristic
604C2×C4D4×Q811yescharacteristic
605C2×C4D4×Q811yescharacteristic
606C2×C4#55011yescharacteristic
607C2×C4#55211yescharacteristic
608C2×C4#55211yescharacteristic
609C2×C4#55311yescharacteristic
610C2×C4#55411yescharacteristic
611C2×C4#56311yescharacteristic
612C2×C4#56311yescharacteristic
613C8—24no—
614C8—24no—
615C8—42no—
616C8—42no—
617C22#242311yescharacteristic
618C22#244311yescharacteristic
619C22#250111yescharacteristic
620C22#254111yescharacteristic
621C22#254211yescharacteristic
622C22#267711yescharacteristic
623C22#272111yescharacteristic
624C22C2×#48412yes—
625C22#309812yes—
626C22C2×#49411yescharacteristic
627C22#315111yescharacteristic
628C22#315511yescharacteristic
629C22#315611yescharacteristic
630C22#323012yes—
631C22#324312yes—
632C22#326412yes—
633C22D4×Q1612yes—
634C22#333812yes—
635C22#334312yes—
636C22Q8×Q1612yes—
637C22#338612yes—
638C22#339512yes—
639C22#339912yes—
640C4—24no—
641C4—28no—
642C4—42no—
643C4—44no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—161no—
649C4—161no—
650C4—161no—
651C4—161no—
652C4#241311yescharacteristic
653C4#242911yescharacteristic
654C4#244111yescharacteristic
655C4#244911yescharacteristic
656C4#267411yescharacteristic
657C4#271811yescharacteristic
658C4#272011yescharacteristic
659C4#272111yescharacteristic
660C2#1576511yescharacteristic
661C2#2506212yes—
662C2#2613212yes—
663C2#2791011yescharacteristic
664C2#2792111yescharacteristic
665C2#2846211yescharacteristic
666C2#2847511yescharacteristic
667C2#2853111yescharacteristic
668C2#2854511yescharacteristic
669C2#3095712yes—
670C2#3238012yes—
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×#20952
C2×#20832
C2×#19282
C2×#19212
C2×#18452
#151842
#139102
#113772
#113242
#111242
#110502
#109262
#108742