Finite Group Databasebeta

← #6484832

Subgroups — #6484832

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#33155C214yesmaximal
4#29979C214yesmaximal
5#29292C214yesmaximal, complemented
6#29285C214yesmaximal, complemented
7#29219C211yesmaximal, characteristic, complemented
8#29218C211yesmaximal, characteristic, complemented
9#29207C211yesmaximal, characteristic, complemented
10#29207C211yesmaximal, characteristic, complemented
11#29179C211yesmaximal, characteristic, complemented
12#29178C211yesmaximal, characteristic, complemented
13#29177C211yesmaximal, characteristic, complemented
14#29176C211yesmaximal, characteristic, complemented
15#21982C211yesmaximal, characteristic
16#21981C211yesmaximal, characteristic
17#21835C211yesmaximal, characteristic
18#21835C211yesmaximal, characteristic
19#20809C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2629C2212yes—
28#2628C2212yes—
29#2570—21no—
30#2570—21no—
31#2550—21no—
32#2550—21no—
33#2441—21no—
34#2441—21no—
35#2439—21no—
36#2439—21no—
37#2360—21no—
38#2360—21no—
39#2360—21no—
40#2360—21no—
41#2355—21no—
42#2355—21no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4—21no—
45#2331C2212yes—
46#2330C2212yes—
47#2330C2212yes—
48#2330C2212yes—
49#2317C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2274C2212yes—
58C4×#117C2211yescharacteristic
59#2100C2214yes—
60#2099C2214yes—
61#2088C2214yes—
62#2088C2214yes—
63#2087C2214yes—
64#2086C2214yes—
65#2059C2214yes—
66#2059C2214yes—
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1989C2214yes—
76#1989C2214yes—
77#1988C2214yes—
78#1987C2214yes—
79#1949C2214yes—
80#1949C2214yes—
81#1948C2214yes—
82#1947C2214yes—
83#1921C2214yes—
84#1920C2214yes—
85#1919C2214yes—
86#1918C2214yes—
87#1853C2214yes—
88#1852C2214yes—
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1844C2211yescharacteristic
92#1844C2211yescharacteristic
93#1843C2212yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1832C2212yes—
99#1825C2214yes—
100#1825C2214yes—
101#1798C2212yes—
102#1798C2212yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106C4×C4⋊C8C2211yescharacteristic
107#533—24no—
108#533Q818yescomplemented
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×#125—21no—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×#117—21no—
127C2×#117—21no—
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135#431—24no—
136#431—24no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153#400—22no—
154#399—22no—
155#398—22no—
156#398—22no—
157#396—22no—
158#396—22no—
159#395—22no—
160#393—22no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—22no—
166#391—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2312yes—
211#384C2312yes—
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#382—22no—
217#382—21no—
218#382—21no—
219#382—21no—
220#382—21no—
221#382C2312yes—
222#380—21no—
223#380—21no—
224#380C2312yes—
225#380C2312yes—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4D411yescharacteristic
235C4×C4⋊C4D411yescharacteristic
236C4×C4⋊C4D411yescharacteristic
237C4×C4⋊C4D411yescharacteristic
238C4×C4⋊C4C2312yes—
239C4×C4⋊C4C2312yes—
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—22no—
246C4×C22⋊C4D411yescharacteristic
247C4×C22⋊C4D411yescharacteristic
248C4×C22⋊C4D411yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250#376C2311yescharacteristic
251#376C2311yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8—21no—
272C22×Q8—22no—
273C23×C4—22no—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—28no—
284C4⋊Q8—82no—
285C4⋊Q8—82no—
286C4⋊Q8C2×Q814yes—
287C4⋊Q8C2×Q814yes—
288#125—24no—
289#125—44no—
290#125—82no—
291#125—82no—
292#125C2×Q818yes—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—82no—
298C4×Q8—82no—
299C4×Q8—82no—
300C4×Q8—82no—
301#117—24no—
302#117—24no—
303#117—42no—
304#117—82no—
305#117—82no—
306#117C2×Q818yes—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
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—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4—82no—
341C2×C4⋊C4—82no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4C2×D411yescharacteristic
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×Q814yes—
379C2×C42#4114yes—
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—44no—
387#107—44no—
388#106—44no—
389#106—44no—
390C4⋊C8—24no—
391C4⋊C8—24no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394C4⋊C8—42no—
395C4⋊C8—42no—
396C22⋊C8—24no—
397C22⋊C8—24no—
398C4×C8—22no—
399C4×C8—22no—
400C4×C8—22no—
401C4×C8—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95C2×D411yescharacteristic
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428C24C22×D411yescharacteristic
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—44no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—82no—
437C2×Q8—82no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4C22⋊Q814yes—
456C22×C4C22⋊Q814yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×Q812yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4C22×Q814yes—
530C4⋊C4C22×Q814yes—
531C22⋊C4—28no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—28no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—82no—
551C42—82no—
552C42—82no—
553C42—82no—
554C42—82no—
555C42—82no—
556C42C22⋊Q814yes—
557C42C22⋊Q814yes—
558C42C22×Q812yes—
559C42C22×Q812yes—
560C42C2×#4112yes—
561C42C2×#4112yes—
562C422−1+414yes—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23C2×C22⋊Q812yes—
570C23C23×D411yescharacteristic
571Q8—48no—
572Q8—48no—
573Q8—164no—
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—21no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
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—21no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—164no—
631C2×C4C2×C22⋊Q812yes—
632C2×C4C2×C22⋊Q812yes—
633C2×C4C2×C22⋊Q812yes—
634C2×C4C2×C22⋊Q812yes—
635C2×C4C2×C22⋊Q812yes—
636C2×C4#53612yes—
637C2×C4#53612yes—
638C2×C4#57612yes—
639C2×C4#57712yes—
640C2×C4#57712yes—
641C2×C4#57812yes—
642C2×C4C23×Q811yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×2−1+412yes—
645C2×C4#58512yes—
646C8—42no—
647C8—42no—
648C8—42no—
649C8—42no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22C22×C22⋊Q811yescharacteristic
658C22C2×#53611yescharacteristic
659C22#344911yescharacteristic
660C22C2×#57611yescharacteristic
661C22C2×#57711yescharacteristic
662C22C2×#57711yescharacteristic
663C22C2×#57811yescharacteristic
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—44no—
670C4—44no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#308314yes—
684C4#308414yes—
685C4#350212yes—
686C4#350712yes—
687C2—42no—
688C2#3517911yescharacteristic
689C2C2×#308311yescharacteristic
690C2C2×#308411yescharacteristic
691C2#6087711yescharacteristic
692C2#6087811yescharacteristic
693C2#6100611yescharacteristic
694C2#6100811yescharacteristic
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
5. #29292C2semidirect
6. #29285C2semidirect
7. #29219C2semidirect
8. #29218C2semidirect
9. #29207C2semidirect
10. #29207C2semidirect
11. #29179C2semidirect
12. #29178C2semidirect
13. #29177C2semidirect
14. #29176C2semidirect
108. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#342142
#331552
#299792
#292922
#292852
#292192
#292182
#292072
#291792
#291782
#291772
#291762
#219822
#219812
#218352
#208092