Finite Group Databasebeta

← #163833

Subgroups — #163833

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35268C211yesmaximal, characteristic
2#22634C211yesmaximal, characteristic, complemented
3#22590C211yesmaximal, characteristic, complemented
4#7732C211yesmaximal, characteristic, complemented
5#7719C211yesmaximal, characteristic, complemented
6#7706C211yesmaximal, characteristic, complemented
7#7697C211yesmaximal, characteristic, complemented
8#7692C211yesmaximal, characteristic, complemented
9#7679C211yesmaximal, characteristic, complemented
10#7666C211yesmaximal, characteristic, complemented
11#7657C211yesmaximal, characteristic, complemented
12#7647C211yesmaximal, characteristic, complemented
13#7633C211yesmaximal, characteristic, complemented
14#7620C211yesmaximal, characteristic, complemented
15#7616C211yesmaximal, characteristic, complemented
16#2864C2211yescharacteristic
17#2860C2211yescharacteristic
18#2860C2211yescharacteristic
19#2858C2211yescharacteristic
20#2628C411yescharacteristic
21#2624C411yescharacteristic
22#2624C411yescharacteristic
23#2619C411yescharacteristic
24#2521—21no—
25#2521—21no—
26#2516—21no—
27#2516—21no—
28#2467—21no—
29#2467—21no—
30#2459—21no—
31#2459—21no—
32#2331C411yescharacteristic
33#2328C411yescharacteristic
34#2319C411yescharacteristic
35#2319C411yescharacteristic
36C4×#121C2211yescharacteristic
37C4×#121C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1940—21no—
40#1924—21no—
41#1923—21no—
42#1917—21no—
43#1916—21no—
44#1851—21no—
45#1839C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1831C2211yescharacteristic
49#1817C2211yescharacteristic
50#1797C2211yescharacteristic
51#1771—21no—
52#1767C2211yescharacteristic
53#1753—21no—
54#1749C2211yescharacteristic
55#1745C2211yescharacteristic
56#1744C2211yescharacteristic
57#1744C2211yescharacteristic
58#1728C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1270C2211yescharacteristic
68#1270C2211yescharacteristic
69#1270C2211yescharacteristic
70#1270C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×M16—21no—
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×#127C2×C411yescharacteristic
78C2×#126—21no—
79C2×#126—21no—
80C2×#124—21no—
81C2×#124—21no—
82C2×#121C2×C411yescharacteristic
83C2×#121C2×C411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×#117C2×C411yescharacteristic
89C2×#117C2×C411yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104#400—21no—
105#399—21no—
106#399—21no—
107#399—21no—
108#397—21no—
109#397—21no—
110#395—21no—
111#394—21no—
112#394—21no—
113#394—21no—
114#394—21no—
115#394—21no—
116#394—21no—
117#393—21no—
118#392—21no—
119#392—21no—
120#391—21no—
121#391—21no—
122#390—21no—
123#390—21no—
124#390C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#390C2311yescharacteristic
131#390C2311yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#389C2311yescharacteristic
139#389C2311yescharacteristic
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#388—21no—
150#388—21no—
151#388—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#386—21no—
159#386—21no—
160#386—21no—
161#386—21no—
162#386—21no—
163#386—21no—
164#382C2×C411yescharacteristic
165#382C2×C411yescharacteristic
166#381—21no—
167#381—21no—
168#380—21no—
169#380—21no—
170#380C2×C411yescharacteristic
171#380C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4D411yescharacteristic
175C4×C4⋊C4D411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4D411yescharacteristic
181C4×C22⋊C4D411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184#376C2×C411yescharacteristic
185#376C2×C411yescharacteristic
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C22×Q8—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C22×C411yescharacteristic
212C22×D4C22×C411yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C23×C4C22×C411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C2×M16—42no—
219C2×M16—42no—
220C2×M16—81no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C4⋊Q8—42no—
230#127—42no—
231#126—81no—
232#126—81no—
233#126—81no—
234#126—81no—
235#124—81no—
236#124—81no—
237#124—81no—
238#124—81no—
239#121—42no—
240#121—42no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—81no—
244C4×D4—81no—
245C4×D4—81no—
246C4×D4—81no—
247#117—42no—
248#117—42no—
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⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C22⋊C4C22⋊C411yescharacteristic
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—42no—
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2411yescharacteristic
313C4⋊C8—42no—
314C4⋊C8—42no—
315C4⋊C8—81no—
316C4⋊C8—81no—
317C4⋊C8—81no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320C22⋊C8—81no—
321#97—42no—
322#97—42no—
323#97—81no—
324C4×C8—42no—
325C4×C8—42no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350C24—21no—
351C24—21no—
352C24C2×C22⋊C411yescharacteristic
353C2×Q8—42no—
354C2×Q8—81no—
355C2×Q8—81no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—81no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411M16—84no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
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—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C22⋊C4—22no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C42—22no—
488C42—22no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—81no—
508C42—81no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23C22×C22⋊C411yescharacteristic
531Q8—162no—
532D4—82no—
533D4—82no—
534D4—84no—
535D4—84no—
536D4—162no—
537D4—162no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
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—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—162no—
599C2×C4—162no—
600C2×C4#41012yes—
601C2×C4#41312yes—
602C2×C4#56812yes—
603C2×C4#57611yescharacteristic
604C2×C4#57611yescharacteristic
605C2×C4#57711yescharacteristic
606C2×C4#57711yescharacteristic
607C8—42no—
608C8—42no—
609C8—81no—
610C8—81no—
611C8—81no—
612C8—81no—
613C8—81no—
614C8—81no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22#287911yescharacteristic
637C22C2×#41011yescharacteristic
638C22C2×#41311yescharacteristic
639C22#289711yescharacteristic
640C22#289711yescharacteristic
641C22#290111yescharacteristic
642C22#290111yescharacteristic
643C4—22no—
644C4—21no—
645C4—21no—
646C4—22no—
647C4—21no—
648C4—21no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C2—81no—
666C2—161no—
667C2—161no—
668C2#1740111yescharacteristic
669C2#1750511yescharacteristic
670C2#1750511yescharacteristic
671C2#1760411yescharacteristic
672C2#1761011yescharacteristic
673C2#1768611yescharacteristic
674C2#1777611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2619C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22634C2semidirect
3. #22590C2semidirect
4. #7732C2semidirect
5. #7719C2semidirect
6. #7706C2semidirect
7. #7697C2semidirect
8. #7692C2semidirect
9. #7679C2semidirect
10. #7666C2semidirect
11. #7657C2semidirect
12. #7647C2semidirect
13. #7633C2semidirect
14. #7620C2semidirect
15. #7616C2semidirect

Maximal subgroups

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

subgroupindex
#352682
#226342
#225902
#77322
#77192
#77062
#76972
#76922
#76792
#76662
#76572
#76472
#76332
#76202
#76162