Finite Group Databasebeta

← #6806131

Subgroups — #6806131

Order 512.

1546 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33232C211yesmaximal, characteristic, complemented
4#31663C214yesmaximal, complemented
5#31660C214yesmaximal, complemented
6#30206C211yesmaximal, characteristic, complemented
7#30205C211yesmaximal, characteristic, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29144C211yesmaximal, characteristic, complemented
10#29141C211yesmaximal, characteristic, complemented
11#28613C211yesmaximal, characteristic
12#28613C211yesmaximal, characteristic
13#28438C211yesmaximal, characteristic
14#25945C214yesmaximal, complemented
15#25331C214yesmaximal, complemented
16#22268C211yesmaximal, characteristic
17#22268C211yesmaximal, characteristic
18#21646C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×#554C2212yes—
21C2×#492C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#491C2214yes—
24C2×#486C2211yescharacteristic
25C2×#461C2214yes—
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862C2211yescharacteristic
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2582—21no—
33#2571—21no—
34#2570—21no—
35#2570—21no—
36#2568C2212yes—
37#2566C2212yes—
38#2566C2212yes—
39#2563C2212yes—
40#2538—21no—
41#2538—21no—
42#2511—21no—
43#2511—21no—
44#2452—21no—
45#2452—21no—
46#2452—21no—
47#2452—21no—
48#2446C2212yes—
49#2446C2212yes—
50#2365C2212yes—
51#2355—21no—
52#2355—21no—
53D4×C4⋊C4—21no—
54D4×C4⋊C4—21no—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2101C2214yes—
63#2074C2214yes—
64#2072C2214yes—
65#2055C2214yes—
66#2055C2214yes—
67#2026C2214yes—
68#2026C2214yes—
69#2024C2214yes—
70#2024C2214yes—
71#2013C2214yes—
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1989C2214yes—
76#1985C2212yes—
77#1960C2214yes—
78#1957C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1940C2212yes—
86#1936C2214yes—
87#1917C2212yes—
88#1917C2212yes—
89#1880C2214yes—
90#1880C2214yes—
91#1877C2214yes—
92#1877C2214yes—
93#1877C2214yes—
94#1877C2214yes—
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1805C2214yes—
99#1803C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×SD16C2314yes—
108C22×SD16C2314yes—
109#554—48no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×#127C2311yescharacteristic
119C2×#125C2312yes—
120C2×#125C2312yes—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124C2×C4×Q8—21no—
125C2×C4×Q8D411yescharacteristic
126C2×C4×Q8D411yescharacteristic
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2311yescharacteristic
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2312yes—
139C22×C4⋊C4—22no—
140#492—24no—
141#492—24no—
142#491—48no—
143#486—24no—
144#461—48no—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#97C2312yes—
157C2×#97C2312yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162#400—22no—
163#400—22no—
164#400—22no—
165#400—22no—
166#398—22no—
167#398—22no—
168#396—22no—
169#396—22no—
170#396—22no—
171#396—22no—
172#393—22no—
173#393—22no—
174#391—22no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390C2312yes—
193#390C2312yes—
194#390C2312yes—
195#389—22no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389C2312yes—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#385—22no—
217#385—22no—
218#385—22no—
219#385—22no—
220#384—21no—
221#384—21no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2312yes—
231#382—22no—
232#382—21no—
233#382—21no—
234#382—21no—
235#382—21no—
236#382C2312yes—
237#382C2312yes—
238#380—22no—
239#380—22no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8C2×D411yescharacteristic
275C22×Q8C2412yes—
276C22×Q8C2412yes—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4C2412yes—
280C23×C4—22no—
281C23×C4—22no—
282C23×C4—22no—
283C2×SD16—28no—
284C2×SD16—28no—
285C2×SD16—44no—
286C2×SD16—44no—
287C2×SD16—44no—
288C2×SD16—44no—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—28no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—44no—
301C4⋊Q8—44no—
302C4⋊Q8—82no—
303C4⋊Q8—82no—
304#127—22no—
305#127#4114yes—
306#125—28no—
307#125—44no—
308C22⋊Q8—48no—
309C22⋊Q8—48no—
310C22⋊Q8—48no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—42no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×D4—28no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—82no—
323C4×D4—82no—
324C4×D4—82no—
325C4×D4—82no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—48no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C4⋊C8—24no—
400C4⋊C8—42no—
401C4⋊C8—42no—
402#103—44no—
403#103—44no—
404#103—44no—
405#103—44no—
406#103—44no—
407#103—44no—
408#102—28no—
409#102—44no—
410#97—24no—
411#97—42no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440C24—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—41no—
452C2×Q8—44no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—82no—
457C2×Q8—82no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4C2×#4114yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C2511yescharacteristic
499SD16—88no—
500SD16—88no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—44no—
581C42—82no—
582C42—82no—
583C42#13714yes—
584C42C2×#4112yes—
585C42H(5,𝔽2)14yes—
586C422−1+412yes—
587C23—24no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23C23×D411yescharacteristic
595Q8—44no—
596Q8—44no—
597Q8—44no—
598Q8—44no—
599Q8—44no—
600Q8—84no—
601Q8—164no—
602D4—28no—
603D4—48no—
604D4—48no—
605D4—44no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—88no—
660C2×C4—164no—
661C2×C4—164no—
662C2×C4#53512yes—
663C2×C4#53512yes—
664C2×C4#54812yes—
665C2×C4#54812yes—
666C2×C4C2×#13712yes—
667C2×C4C2×#13712yes—
668C2×C4C2×#13712yes—
669C2×C4#57512yes—
670C2×C4#57712yes—
671C2×C4#57712yes—
672C2×C4C22×#4111yescharacteristic
673C2×C4C2×H(5,𝔽2)12yes—
674C2×C4C2×2−1+411yescharacteristic
675C2×C4#58512yes—
676C8—42no—
677C8—42no—
678C8—42no—
679C8—42no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22C2×#53511yescharacteristic
688C22C2×#54811yescharacteristic
689C22#350011yescharacteristic
690C22C22×#13711yescharacteristic
691C22C2×#57511yescharacteristic
692C22C2×#57711yescharacteristic
693C22C2×#57711yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4#337214yes—
714C4#337614yes—
715C4#351412yes—
716C4#355412yes—
717C2—84no—
718C2#3917111yescharacteristic
719C2#6080411yescharacteristic
720C2#6081811yescharacteristic
721C2C2×#337211yescharacteristic
722C2C2×#337611yescharacteristic
723C2#6254011yescharacteristic
724C2#6254311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
3. #33232C2semidirect
4. #31663C2semidirect
5. #31660C2semidirect
6. #30206C2semidirect
7. #30205C2semidirect
8. #29220C2semidirect
9. #29144C2semidirect
10. #29141C2semidirect
14. #25945C2semidirect
15. #25331C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#365592
#332322
#316632
#316602
#302062
#302052
#292202
#291442
#291412
#286132
#284382
#259452
#253312
#222682
#216462
#213402