Finite Group Databasebeta

← #163929

Subgroups — #163929

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34253C211yesmaximal, characteristic
2#22592C211yesmaximal, characteristic, complemented
3#22590C211yesmaximal, characteristic, complemented
4#7645C211yesmaximal, characteristic
5#7644C211yesmaximal, characteristic, complemented
6#7631C211yesmaximal, characteristic
7#7630C211yesmaximal, characteristic, complemented
8#7614C211yesmaximal, characteristic, complemented
9#7610C211yesmaximal, characteristic, complemented
10#7608C211yesmaximal, characteristic, complemented
11#7607C211yesmaximal, characteristic, complemented
12#7596C211yesmaximal, characteristic, complemented
13#7580C211yesmaximal, characteristic, complemented
14#7574C211yesmaximal, characteristic, complemented
15#7562C211yesmaximal, characteristic, complemented
16C2×#520C411yescharacteristic
17#2353—21no—
18#2353—21no—
19#2352—21no—
20#2352—21no—
21#2347—21no—
22#2346—21no—
23#2345—21no—
24#2344—21no—
25#2320C411yescharacteristic
26#2320C411yescharacteristic
27#2319C411yescharacteristic
28#2319C411yescharacteristic
29#2316C411yescharacteristic
30#2311C411yescharacteristic
31#2297C2211yescharacteristic
32#2295C2211yescharacteristic
33#2295C2211yescharacteristic
34#2293C2211yescharacteristic
35#2293C2211yescharacteristic
36C2×#389C2211yescharacteristic
37#2274C411yescharacteristic
38C4×#117C2211yescharacteristic
39#1917—21no—
40#1917—21no—
41#1916—21no—
42#1916—21no—
43#1851—21no—
44#1851—21no—
45#1839C2211yescharacteristic
46#1839C2211yescharacteristic
47#1831C2211yescharacteristic
48#1831C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1753—21no—
54#1753—21no—
55#1745C2211yescharacteristic
56#1744C2211yescharacteristic
57#1728C2211yescharacteristic
58#1728C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×#41C2×C411yescharacteristic
76C22×M16—21no—
77C22×M16—21no—
78#520—24no—
79#520—24no—
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×#117—21no—
87C2×#117—21no—
88C2×#117—21no—
89C2×#117C2×C411yescharacteristic
90C2×#117C2×C411yescharacteristic
91C2×#117C2×C411yescharacteristic
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2×C411yescharacteristic
95C22×C4⋊C4C2×C411yescharacteristic
96C22×C42C2311yescharacteristic
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107#391C2×C411yescharacteristic
108#391C2×C411yescharacteristic
109#391C2×C411yescharacteristic
110#391C2311yescharacteristic
111#390C2×C411yescharacteristic
112#390C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#390C2311yescharacteristic
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#389C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#389C2×C411yescharacteristic
124#389C2×C411yescharacteristic
125#389C2311yescharacteristic
126#389C2311yescharacteristic
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133#388—21no—
134#388—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#386—21no—
144#386—21no—
145#386—21no—
146#386—21no—
147#383—21no—
148#383—21no—
149#382—21no—
150#382—21no—
151#381—21no—
152#381—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157#380—21no—
158#380—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4C2×C411yescharacteristic
164C4×C4⋊C4C2×C411yescharacteristic
165C4×C4⋊C4C2×C411yescharacteristic
166C4×C4⋊C4C2×C411yescharacteristic
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4D411yescharacteristic
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4D411yescharacteristic
178C4×C22⋊C4D411yescharacteristic
179C4×C22⋊C4D411yescharacteristic
180C4×C22⋊C4D411yescharacteristic
181#376—21no—
182#376—21no—
183#376C2×C411yescharacteristic
184#376C2×C411yescharacteristic
185C2×#95—21no—
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#336—21no—
199#336—21no—
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208C2×#41—22no—
209C2×#41—22no—
210C2×#41—42no—
211C2×#41—42no—
212C22×Q8C22×C411yescharacteristic
213C22×D4—21no—
214C22×D4C22×C411yescharacteristic
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C2×M16—42no—
224C2×M16—42no—
225C2×M16—42no—
226C2×M16—42no—
227C2×M16—81no—
228C2×M16—81no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8M1614yescomplemented
241C4×D4—28no—
242C4×D4—28no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4M1614yescomplemented
247#117—28no—
248#117—22no—
249#117—22no—
250#117—42no—
251#117—42no—
252#117—81no—
253#117—81no—
254#117—81no—
255#117—81no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—81no—
273C2×C4⋊C4—81no—
274C2×C4⋊C4C22⋊C411yescharacteristic
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22⋊C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—42no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2411yescharacteristic
323C4⋊C8—42no—
324C4⋊C8—42no—
325C4⋊C8—81no—
326C4⋊C8—81no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329#97—42no—
330#97—81no—
331C4×C8—42no—
332C4×C8—81no—
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#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356#95C22⋊C411yescharacteristic
357#95C22⋊C411yescharacteristic
358#95C22⋊C411yescharacteristic
359#95C22⋊C411yescharacteristic
360#95C22⋊C411yescharacteristic
361C24—21no—
362C24C2×C22⋊C411yescharacteristic
363#41—48no—
364#41—48no—
365C2×Q8—21no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8C2×M1612yes—
369C2×D4—24no—
370C2×D4—21no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4C2×M1612yes—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424M16—84no—
425M16—84no—
426C2×C8—22no—
427C2×C8—22no—
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—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4C2×M1612yes—
475C4⋊C4C2×M1612yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—42no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C42—24no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—24no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—42no—
516C42—81no—
517C42—81no—
518C42—81no—
519C42—81no—
520C42C2×M1612yes—
521C42C2×M1612yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23#41012yes—
536C23C22×C22⋊C411yescharacteristic
537Q8—42no—
538Q8—42no—
539D4—42no—
540D4—42no—
541D4—48no—
542C2×C4—24no—
543C2×C4—24no—
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—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
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—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—48no—
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—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—162no—
610C2×C4#40712yes—
611C2×C4#40712yes—
612C2×C4#41312yes—
613C2×C4C22×M1611yescharacteristic
614C2×C4C22×M1611yescharacteristic
615C2×C4#57611yescharacteristic
616C2×C4#57711yescharacteristic
617C2×C4#57711yescharacteristic
618C2×C4#57811yescharacteristic
619C8—42no—
620C8—42no—
621C8—81no—
622C8—81no—
623C8—81no—
624C8—81no—
625C8—81no—
626C8—81no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22C2×#40711yescharacteristic
642C22C2×#41011yescharacteristic
643C22C2×#41311yescharacteristic
644C22#289711yescharacteristic
645C22#289811yescharacteristic
646C22#290111yescharacteristic
647C22#290111yescharacteristic
648C4—22no—
649C4—22no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—22no—
655C4—21no—
656C4—21no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—42no—
668C4—81no—
669C4—81no—
670C4—81no—
671C4—81no—
672C2—42no—
673C2—82no—
674C2#1735711yescharacteristic
675C2#1745311yescharacteristic
676C2#1745411yescharacteristic
677C2#1760411yescharacteristic
678C2#1760911yescharacteristic
679C2#1768711yescharacteristic
680C2#1778111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
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. #22592C2semidirect
3. #22590C2semidirect
5. #7644C2semidirect
7. #7630C2semidirect
8. #7614C2semidirect
9. #7610C2semidirect
10. #7608C2semidirect
11. #7607C2semidirect
12. #7596C2semidirect
13. #7580C2semidirect
14. #7574C2semidirect
15. #7562C2semidirect
240. C4×Q8M16semidirect
246. C4×D4M16semidirect

Maximal subgroups

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

subgroupindex
#342532
#225922
#225902
#76452
#76442
#76312
#76302
#76142
#76102
#76082
#76072
#75962
#75802
#75742
#75622