Finite Group Databasebeta

← #6494880

Subgroups — #6494880

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic
3#33206C214yesmaximal, complemented
4#29992C214yesmaximal, complemented
5#29220C211yesmaximal, characteristic, complemented
6#29219C211yesmaximal, characteristic, complemented
7#29208C211yesmaximal, characteristic, complemented
8#29207C211yesmaximal, characteristic, complemented
9#29144C211yesmaximal, characteristic, complemented
10#29141C211yesmaximal, characteristic, complemented
11#28515C211yesmaximal, characteristic, complemented
12#28511C211yesmaximal, characteristic, complemented
13#27690C214yesmaximal
14#24311C214yesmaximal
15#21984C211yesmaximal, characteristic
16#21982C211yesmaximal, characteristic
17#21864C211yesmaximal, characteristic
18#21860C211yesmaximal, characteristic
19#20809C211yesmaximal, characteristic
20C2×#544C2212yes—
21C2×#533C2211yescharacteristic
22C2×#479C2214yes—
23C2×#477C2214yes—
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2660C2212yes—
31#2660C2212yes—
32#2552—21no—
33#2552—21no—
34#2551—21no—
35#2551—21no—
36#2440—21no—
37#2440—21no—
38#2440—21no—
39#2440—21no—
40#2370—21no—
41#2370—21no—
42#2362—21no—
43#2362—21no—
44#2356—21no—
45#2351—21no—
46#2347—21no—
47#2346—21no—
48#2323C2212yes—
49#2323C2212yes—
50#2320C2212yes—
51#2317C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#384C2211yescharacteristic
59#2276C2212yes—
60C4×#117C2211yescharacteristic
61#2101C2214yes—
62#2099C2214yes—
63#2097C2214yes—
64#2095C2214yes—
65#2059C2214yes—
66#2057C2214yes—
67#2026C2214yes—
68#2024C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1989C2214yes—
74#1988C2214yes—
75#1956C2211yescharacteristic
76#1956C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1949C2214yes—
80#1949C2214yes—
81#1947C2214yes—
82#1947C2214yes—
83#1873C2214yes—
84#1872C2214yes—
85#1854C2214yes—
86#1852C2214yes—
87#1845C2211yescharacteristic
88#1845C2211yescharacteristic
89#1845C2211yescharacteristic
90#1844C2211yescharacteristic
91#1843C2212yes—
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1832C2212yes—
97#1805C2214yes—
98#1805C2214yes—
99#1803C2214yes—
100#1803C2214yes—
101#1798C2212yes—
102#1798C2212yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106C4×C4⋊C8C2211yescharacteristic
107#544—48no—
108#533—24no—
109#533Q818yescomplemented
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119C2×#125—22no—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#125C2312yes—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8C2312yes—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×#117—21no—
134C2×#117—21no—
135C2×#117C2312yes—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2312yes—
141C22×C42C2311yescharacteristic
142#479—48no—
143#477—48no—
144#431—24no—
145#431—24no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#395—22no—
171#395—22no—
172#395—22no—
173#395—22no—
174#391—21no—
175#391—21no—
176#391C2312yes—
177#391C2312yes—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389C2312yes—
197#389C2312yes—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#385—22no—
207#385—22no—
208#384—21no—
209#384—21no—
210#384—21no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2312yes—
217#384C2312yes—
218#384C2312yes—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#383—22no—
224#382—21no—
225#382—22no—
226#382C2312yes—
227#382C2312yes—
228#382C2312yes—
229#380—21no—
230#380—21no—
231#380—21no—
232#380C2312yes—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—22no—
244C4×C4⋊C4—22no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256#376—22no—
257#376C2311yescharacteristic
258#376C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×Q8—22no—
276C22×Q8C2412yes—
277C23×C4—22no—
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—28no—
288C4⋊Q8—44no—
289C4⋊Q8—48no—
290C4⋊Q8—82no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293C4⋊Q8—82no—
294C4⋊Q8C2×Q814yes—
295C4⋊Q8C2×Q814yes—
296#125—28no—
297#125—28no—
298#125—44no—
299#125—48no—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8—48no—
303C22⋊Q8—48no—
304C22⋊Q8C2×Q818yes—
305C4×Q8—28no—
306C4×Q8—24no—
307C4×Q8—28no—
308C4×Q8—28no—
309C4×Q8—82no—
310C4×Q8—82no—
311C4×Q8C2×Q818yes—
312#117—28no—
313#117—24no—
314#117—24no—
315#117—44no—
316#117—82no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4—82no—
353C2×C4⋊C4—82no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42C2×D411yescharacteristic
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×Q814yes—
390C2×C42#4114yes—
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—44no—
398#107—44no—
399#107—44no—
400#106—44no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407#103—44no—
408#103—44no—
409#103—44no—
410#103—44no—
411C22⋊C8—24no—
412C22⋊C8—24no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—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#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441C24C22×D411yescharacteristic
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—82no—
451C2×Q8—82no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×Q8C22×Q814yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4C22⋊Q814yes—
473C22×C4C22⋊Q814yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×Q812yes—
488C22×C4C2×#4112yes—
489C22×C4C2511yescharacteristic
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4C22×Q814yes—
548C22⋊C4—28no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C42—28no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—44no—
567C42—44no—
568C42—44no—
569C42—82no—
570C42—82no—
571C42—82no—
572C42C22⋊Q814yes—
573C42C22⋊Q814yes—
574C42C22×Q812yes—
575C42C22×Q812yes—
576C42C2×#4112yes—
577C42C2×#4112yes—
578C42H(5,𝔽2)14yes—
579C422−1+414yes—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23C2×C22⋊Q812yes—
587C23C23×D411yescharacteristic
588Q8—28no—
589Q8—48no—
590Q8—44no—
591Q8—164no—
592Q8—164no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
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—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
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—164no—
648C2×C4C2×C22⋊Q812yes—
649C2×C4C2×C22⋊Q812yes—
650C2×C4C2×C22⋊Q812yes—
651C2×C4C2×C22⋊Q812yes—
652C2×C4C2×C22⋊Q812yes—
653C2×C4#53512yes—
654C2×C4#53512yes—
655C2×C4#53612yes—
656C2×C4#53612yes—
657C2×C4#57612yes—
658C2×C4#57712yes—
659C2×C4#57812yes—
660C2×C4#57812yes—
661C2×C4C23×Q811yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)12yes—
664C2×C4C2×2−1+412yes—
665C8—42no—
666C8—42no—
667C8—42no—
668C8—42no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22C22×C22⋊Q811yescharacteristic
677C22C2×#53511yescharacteristic
678C22C2×#53611yescharacteristic
679C22C2×#57611yescharacteristic
680C22C2×#57711yescharacteristic
681C22C2×#57811yescharacteristic
682C22C2×#57811yescharacteristic
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—44no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—44no—
695C4—41no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#308214yes—
703C4#308314yes—
704C4#350312yes—
705C4#351112yes—
706C2—42no—
707C2#3527911yescharacteristic
708C2C2×#308211yescharacteristic
709C2C2×#308311yescharacteristic
710C2#6082111yescharacteristic
711C2#6082211yescharacteristic
712C2#6087811yescharacteristic
713C2#6088011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33206C2semidirect
4. #29992C2semidirect
5. #29220C2semidirect
6. #29219C2semidirect
7. #29208C2semidirect
8. #29207C2semidirect
9. #29144C2semidirect
10. #29141C2semidirect
11. #28515C2semidirect
12. #28511C2semidirect
109. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#342252
#332062
#299922
#292202
#292192
#292082
#292072
#291442
#291412
#285152
#285112
#276902
#243112
#219842
#219822
#218642
#218602
#208092