Finite Group Databasebeta

← #224926

Subgroups — #224926

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36595C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic, complemented
3#22823C211yesmaximal, characteristic, complemented
4#9710C211yesmaximal, characteristic, complemented
5#9706C211yesmaximal, characteristic
6#9706C211yesmaximal, characteristic
7#9704C211yesmaximal, characteristic, complemented
8#9698C211yesmaximal, characteristic, complemented
9#9696C211yesmaximal, characteristic, complemented
10#8593C211yesmaximal, characteristic, complemented
11#8588C211yesmaximal, characteristic, complemented
12#8588C211yesmaximal, characteristic, complemented
13#8233C211yesmaximal, characteristic, complemented
14#8231C211yesmaximal, characteristic, complemented
15C4×#331C211yesmaximal, characteristic, complemented
16C2×#559—21no—
17#2867C2211yescharacteristic
18#2858C2211yescharacteristic
19#2858C2211yescharacteristic
20#2548—21no—
21#2548—21no—
22#2548C412yes—
23#2544—21no—
24#2544—21no—
25#2445—21no—
26#2445—21no—
27#2445C412yes—
28#2445C412yes—
29#2366—21no—
30#2366C412yes—
31C4×C4⋊Q8C2211yescharacteristic
32C4×#127C2211yescharacteristic
33C4×#127C2211yescharacteristic
34C42×Q8C2211yescharacteristic
35#1985—21no—
36#1984—21no—
37#1917—21no—
38#1917—21no—
39#1917—21no—
40#1916—21no—
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1831C2211yescharacteristic
44#1831C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1771—21no—
48#1770—21no—
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63C2×#331C2211yescharacteristic
64C2×#331C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#559—44no—
72#559—44no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×#127—21no—
75C2×#127—21no—
76C2×#127—21no—
77C2×#127—21no—
78C2×#127C2×C412yes—
79C2×#127C2×C412yes—
80C2×#127C2311yescharacteristic
81C2×#127C2311yescharacteristic
82C2×#124—22no—
83C2×#124—21no—
84C2×#124—21no—
85C2×#124—21no—
86C2×#124—21no—
87C2×#123—22no—
88C2×#123—22no—
89C2×C4×Q8—21no—
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8D411yescharacteristic
92C2×C4×Q8D411yescharacteristic
93C2×C4×D4—22no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4C2×C412yes—
99C2×C4×D4C2×C412yes—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118#394—22no—
119#394—22no—
120#394—22no—
121#394—22no—
122#394—22no—
123#394—22no—
124#391—21no—
125#390—22no—
126#390—22no—
127#390—21no—
128#390—21no—
129#390—21no—
130#390—21no—
131#390—21no—
132#390—21no—
133#390—21no—
134#390—21no—
135#390C2×C412yes—
136#390C2×C412yes—
137#390C2×C412yes—
138#390C2×C412yes—
139#390C2×C412yes—
140#390C2×C412yes—
141#389—21no—
142#389C2×C412yes—
143#386—22no—
144#386—22no—
145#386—22no—
146#386—22no—
147#384C2311yescharacteristic
148C4×C4⋊C4—21no—
149C4×C4⋊C4C2×C412yes—
150C4×C4⋊C4C2×C411yescharacteristic
151C4×C4⋊C4C2×C411yescharacteristic
152C4×C4⋊C4C2×C411yescharacteristic
153C4×C4⋊C4D411yescharacteristic
154C4×C4⋊C4D411yescharacteristic
155C4×C4⋊C4D411yescharacteristic
156C4×C4⋊C4D411yescharacteristic
157C4×C4⋊C4D411yescharacteristic
158C4×C4⋊C4D411yescharacteristic
159C43C2311yescharacteristic
160C43C2311yescharacteristic
161C43C2311yescharacteristic
162#336—21no—
163#336—21no—
164#336—21no—
165#336—21no—
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#332—22no—
177#332—22no—
178#332—41no—
179#332—41no—
180#331—22no—
181#331—22no—
182#331—22no—
183#331—22no—
184#331—41no—
185#331—41no—
186#331—41no—
187#331—41no—
188C22×Q8—21no—
189C22×Q8C22⋊C411yescharacteristic
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4C22×C412yes—
199C22×D4C22×C412yes—
200C23×C4—22no—
201C23×C4—22no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214#127—21no—
215#127—21no—
216#127—21no—
217#127—21no—
218#127—21no—
219#127—21no—
220#127—21no—
221#127—21no—
222#127—44no—
223#127—42no—
224#127—42no—
225#127—42no—
226#127—42no—
227#127—81no—
228#127—81no—
229#127—81no—
230#127—81no—
231#127—82no—
232#124—42no—
233#124—42no—
234#124—48no—
235#124—42no—
236#124—42no—
237#124—81no—
238#124—81no—
239#124—81no—
240#124—81no—
241#123—48no—
242#123—48no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—41no—
246C4×Q8—41no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—48no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—81no—
256C4×D4—81no—
257C4×D4—81no—
258C4×D4—81no—
259C4×D4—82no—
260C4×D4—82no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22⋊C411yescharacteristic
272C2×C4⋊C4C22⋊C411yescharacteristic
273C2×C4⋊C4C22⋊C411yescharacteristic
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C412yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C42—22no—
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2411yescharacteristic
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—42no—
325C4⋊C8—42no—
326C4⋊C8—81no—
327C4⋊C8—81no—
328C4⋊C8#4112yes—
329C4⋊C8#4112yes—
330#97—42no—
331#97—81no—
332#97—81no—
333#97—81no—
334C4×C8—42no—
335C4×C8—81no—
336#95—22no—
337#95—22no—
338#95C22⋊C411yescharacteristic
339#95C22⋊C411yescharacteristic
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344C24—22no—
345C24—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—44no—
393C2×D4—82no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4—82no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—22no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—44no—
508C42AGL(1,ℤ/8)12yes—
509C42AGL(1,ℤ/8)12yes—
510C42C2×#4111yescharacteristic
511C42C2×#4111yescharacteristic
512C42H(5,𝔽2)11yescharacteristic
513C422−1+411yescharacteristic
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C22×C22⋊C411yescharacteristic
529Q8—44no—
530Q8—81no—
531Q8—81no—
532Q8—81no—
533Q8—81no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—82no—
538D4—88no—
539D4—81no—
540D4—81no—
541D4—81no—
542D4—81no—
543D4—81no—
544D4—81no—
545D4—81no—
546D4—81no—
547D4—81no—
548D4—81no—
549D4—81no—
550D4—81no—
551D4—81no—
552D4—81no—
553D4—81no—
554D4—81no—
555D4—164no—
556D4—164no—
557D4—164no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—164no—
600C2×C4—164no—
601C2×C4#41312yes—
602C2×C4#41812yes—
603C2×C4#41812yes—
604C2×C4C4×#4111yescharacteristic
605C2×C4#52011yescharacteristic
606C2×C4#54811yescharacteristic
607C2×C4#54811yescharacteristic
608C2×C4#55011yescharacteristic
609C2×C4#55011yescharacteristic
610C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
611C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
612C2×C4#57511yescharacteristic
613C2×C4#57511yescharacteristic
614C2×C4#57611yescharacteristic
615C2×C4#57611yescharacteristic
616C2×C4#57711yescharacteristic
617C2×C4#57711yescharacteristic
618C8—42no—
619C8—42no—
620C8—81no—
621C8—81no—
622C8—81no—
623C8—81no—
624C8—81no—
625C8—81no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22#234111yescharacteristic
641C22C2×#41311yescharacteristic
642C22#288811yescharacteristic
643C22C2×#41811yescharacteristic
644C22#289611yescharacteristic
645C22#289711yescharacteristic
646C22#289711yescharacteristic
647C4—21no—
648C4—21no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—82no—
662C4—82no—
663C4#336611yescharacteristic
664C4#336811yescharacteristic
665C4#337011yescharacteristic
666C4#337111yescharacteristic
667C4#339711yescharacteristic
668C4#339911yescharacteristic
669C4#340111yescharacteristic
670C4#340311yescharacteristic
671C2—162no—
672C2—162no—
673C2#1773111yescharacteristic
674C2#1775811yescharacteristic
675C2#1776611yescharacteristic
676C2#1777611yescharacteristic
677C2#2253911yescharacteristic
678C2#2254511yescharacteristic
679C2#2254911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2867C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22825C2semidirect
3. #22823C2semidirect
4. #9710C2semidirect
7. #9704C2semidirect
8. #9698C2semidirect
9. #9696C2semidirect
10. #8593C2semidirect
11. #8588C2semidirect
12. #8588C2semidirect
13. #8233C2semidirect
14. #8231C2semidirect
15. C4×#331C2semidirect

Maximal subgroups

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

subgroupindex
#365952
#228252
#228232
#97102
#97062
#97042
#96982
#96962
#85932
#85882
#82332
#82312
C4×#3312