Finite Group Databasebeta

← #164178

Subgroups — #164178

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36385C211yesmaximal, characteristic
2#22589C211yesmaximal, characteristic, complemented
3#20468C211yesmaximal, characteristic, complemented
4#7690C211yesmaximal, characteristic
5#7682C211yesmaximal, characteristic, complemented
6#7676C211yesmaximal, characteristic, complemented
7#7670C211yesmaximal, characteristic
8#7662C211yesmaximal, characteristic, complemented
9#7656C211yesmaximal, characteristic, complemented
10#7623C211yesmaximal, characteristic, complemented
11#7614C211yesmaximal, characteristic, complemented
12#7611C211yesmaximal, characteristic, complemented
13#7566C211yesmaximal, characteristic, complemented
14#7562C211yesmaximal, characteristic, complemented
15#7560C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17#2862C2211yescharacteristic
18#2861C2211yescharacteristic
19#2850C2211yescharacteristic
20#2609—21no—
21#2574—21no—
22#2550C411yescharacteristic
23#2545C411yescharacteristic
24#2532—21no—
25#2529—21no—
26#2518—21no—
27#2491C411yescharacteristic
28#2489C411yescharacteristic
29#2474—21no—
30#2429C411yescharacteristic
31#2427C411yescharacteristic
32#2372—21no—
33#2367C411yescharacteristic
34#2356C411yescharacteristic
35#2346—21no—
36C4×C22⋊Q8C2211yescharacteristic
37#2295C2211yescharacteristic
38#2293C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1916—21no—
41#1916—21no—
42#1850—21no—
43#1839C2211yescharacteristic
44#1831C2211yescharacteristic
45#1817C2211yescharacteristic
46#1768—21no—
47#1767C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C8×C22⋊C4—21no—
51#1750—21no—
52#1749C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1745C2211yescharacteristic
55#1744C2211yescharacteristic
56#1744C2211yescharacteristic
57C2×#336C2211yescharacteristic
58#1726—21no—
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×C8—21no—
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×#125C2×C411yescharacteristic
79C2×#124—21no—
80C2×#124C2×C411yescharacteristic
81C2×#124C2×C411yescharacteristic
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×#117—21no—
89C2×#117D411yescharacteristic
90C2×#117D411yescharacteristic
91C22×C42C2311yescharacteristic
92#433—28no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#400—21no—
109#398—21no—
110#398—21no—
111#397—21no—
112#397—21no—
113#396—21no—
114#396—21no—
115#396—21no—
116#395—21no—
117#394—21no—
118#394—21no—
119#393—21no—
120#391—21no—
121#391—21no—
122#391—21no—
123#391C2×C411yescharacteristic
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#391C2311yescharacteristic
128#391C2311yescharacteristic
129#390C2×C411yescharacteristic
130#390C2×C411yescharacteristic
131#390C2311yescharacteristic
132#389—21no—
133#389—21no—
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2311yescharacteristic
137#388—21no—
138#388—21no—
139#388—21no—
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#386—21no—
158#386—21no—
159#386—21no—
160#385—21no—
161#385—21no—
162#383—21no—
163#382—21no—
164#382—21no—
165#382—21no—
166#382C2×C411yescharacteristic
167#380—21no—
168#380C2×C411yescharacteristic
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4C2×C411yescharacteristic
182C4×C22⋊C4C2×C411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185#376—21no—
186#376—21no—
187#376D411yescharacteristic
188#376D411yescharacteristic
189C43C2311yescharacteristic
190C43C2311yescharacteristic
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#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207C22×Q8—21no—
208C22×Q8C22×C411yescharacteristic
209C22×Q8C22×C411yescharacteristic
210C22×D4—21no—
211C22×D4C22×C411yescharacteristic
212C23×C4—21no—
213C23×C4C22×C411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—81no—
227C4⋊Q8—81no—
228C4⋊Q8—81no—
229#125—81no—
230#124—42no—
231#124—44no—
232#124—81no—
233C22⋊Q8—22no—
234C22⋊Q8—41no—
235C22⋊Q8—81no—
236#121—81no—
237C4×Q8—22no—
238C4×Q8—41no—
239C4×Q8—42no—
240C4×D4—22no—
241C4×D4—41no—
242#117—24no—
243#117—22no—
244#117—22no—
245#117—42no—
246#117—42no—
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⋊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⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C22⋊C4C22⋊C411yescharacteristic
286C2×C22⋊C4C22⋊C411yescharacteristic
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—42no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
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×C42C2411yescharacteristic
309C4⋊C8—24no—
310C4⋊C8—24no—
311C4⋊C8—42no—
312C4⋊C8—81no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—42no—
316#97—24no—
317#97—42no—
318#97—42no—
319C4×C8—24no—
320C4×C8—42no—
321C4×C8—42no—
322C4×C8—42no—
323C4×C8—81no—
324#95—21no—
325#95—21no—
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#95—21no—
343#95—21no—
344#95—21no—
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353C24—21no—
354C24C2×C22⋊C411yescharacteristic
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—81no—
365C2×Q8—81no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—81no—
372C2×D4—81no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—81no—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
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—84no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4#13112yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4#10414yes—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—81no—
503C42#13512yes—
504C42#13512yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×#10412yes—
519C23C22×C22⋊C411yescharacteristic
520Q8—81no—
521Q8—81no—
522Q8—81no—
523Q8—81no—
524Q8—162no—
525D4—81no—
526D4—81no—
527D4—162no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
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—22no—
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—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
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—81no—
588C2×C4—81no—
589C2×C4—84no—
590C2×C4#41012yes—
591C2×C4#41612yes—
592C2×C4#41612yes—
593C2×C4C2×#10412yes—
594C2×C4C2×#10412yes—
595C2×C4C2×#13111yescharacteristic
596C2×C4#56811yescharacteristic
597C2×C4C2×#13511yescharacteristic
598C2×C4C2×#13511yescharacteristic
599C2×C4#57711yescharacteristic
600C2×C4#57811yescharacteristic
601C8—42no—
602C8—42no—
603C8—42no—
604C8—42no—
605C8—42no—
606C8—81no—
607C8—81no—
608C8—81no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22#288011yescharacteristic
624C22C2×#41011yescharacteristic
625C22#288811yescharacteristic
626C22C2×#41611yescharacteristic
627C22#289811yescharacteristic
628C22C22×#10411yescharacteristic
629C22#290111yescharacteristic
630C4—21no—
631C4—21no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—42no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—81no—
654C2—42no—
655C2—161no—
656C2#1741811yescharacteristic
657C2#1750811yescharacteristic
658C2#1751911yescharacteristic
659C2#1753311yescharacteristic
660C2#1760911yescharacteristic
661C2#1765211yescharacteristic
662C2#1772611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×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. #22589C2semidirect
3. #20468C2semidirect
5. #7682C2semidirect
6. #7676C2semidirect
8. #7662C2semidirect
9. #7656C2semidirect
10. #7623C2semidirect
11. #7614C2semidirect
12. #7611C2semidirect
13. #7566C2semidirect
14. #7562C2semidirect
15. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#363852
#225892
#204682
#76902
#76822
#76762
#76702
#76622
#76562
#76232
#76142
#76112
#75662
#75622
#75602