Finite Group Databasebeta

← #6468359

Subgroups — #6468359

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic, complemented
2#34226C211yesmaximal, characteristic
3#29220C211yesmaximal, characteristic, complemented
4#29206C211yesmaximal, characteristic, complemented
5#29190C211yesmaximal, characteristic, complemented
6#29190C211yesmaximal, characteristic, complemented
7#29178C211yesmaximal, characteristic, complemented
8#29177C211yesmaximal, characteristic, complemented
9#29174C214yesmaximal, complemented
10#29139C211yesmaximal, characteristic, complemented
11#28514C211yesmaximal, characteristic, complemented
12#24852C214yesmaximal, complemented
13#23786C214yesmaximal, complemented
14#21619C214yesmaximal, complemented
15#21410C211yesmaximal, characteristic
16#21401C211yesmaximal, characteristic
17#20824C211yesmaximal, characteristic, complemented
18#20816C211yesmaximal, characteristic, complemented
19C4×#424C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×#478C2214yes—
22C2×#424C2212yes—
23C2×C4×M16C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2629C2212yes—
30#2628C2212yes—
31#2570—21no—
32#2570—21no—
33#2550—21no—
34#2550—21no—
35#2441—21no—
36#2441—21no—
37#2439—21no—
38#2439—21no—
39#2367—21no—
40#2367—21no—
41#2363—21no—
42#2363—21no—
43#2357—21no—
44#2349—21no—
45#2341—21no—
46#2339—21no—
47#2331C2212yes—
48#2330C2212yes—
49#2324C2212yes—
50#2324C2212yes—
51#2319C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55C4×#127C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2275C2212yes—
60C4×#117C2211yescharacteristic
61#2101C2214yes—
62#2092C2214yes—
63#2088C2214yes—
64#2086C2214yes—
65#2065C2214yes—
66#2065C2214yes—
67#2058C2214yes—
68#2023C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1989C2214yes—
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1948C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1895C2214yes—
82#1893C2214yes—
83#1864C2214yes—
84#1864C2214yes—
85#1845C2211yescharacteristic
86#1844C2211yescharacteristic
87#1844C2211yescharacteristic
88#1844C2211yescharacteristic
89#1834C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1809C2214yes—
94#1807C2214yes—
95#1802C2214yes—
96#1776C2214yes—
97C4×#106C2214yes—
98#1768C2212yes—
99#1768C2212yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1762C2214yes—
105C4×#102C2214yes—
106#1729C2212yes—
107C22×#41—21no—
108C22×M16C2312yes—
109C23×C8C2312yes—
110#532—24no—
111#532Q818yescomplemented
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×#127C2311yescharacteristic
117C2×#125—21no—
118C2×#125—21no—
119C2×#125C2312yes—
120C2×#124—21no—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#478—216no—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144#424—28no—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C4×M16—24no—
154C2×#97C2312yes—
155C2×C4×C8—24no—
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161#400—22no—
162#399—22no—
163#398—22no—
164#398—22no—
165#396—22no—
166#396—22no—
167#395—22no—
168#393—22no—
169#391—21no—
170#391—21no—
171#391—22no—
172#391—22no—
173#390—22no—
174#390—22no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#389—21no—
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—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#384—21no—
212#384—21no—
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2312yes—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#382—21no—
221#382—21no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382C2312yes—
226#381—22no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—22no—
231#380C2312yes—
232#380C2312yes—
233#380C2312yes—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C22⋊C4—22no—
249C4×C22⋊C4—22no—
250C4×C22⋊C4—22no—
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255#376C2311yescharacteristic
256#376C2311yescharacteristic
257C2×#95—22no—
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C2×#41—28no—
273C2×#41—82no—
274C2×#41—82no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×D4—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2412yes—
281C23×C4—22no—
282C23×C4—22no—
283C23×C4—22no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C2×M16—24no—
288C2×M16—24no—
289C2×M16—42no—
290C22×C8—24no—
291C22×C8—24no—
292C22×C8—42no—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C4⋊Q8—22no—
298C4⋊Q8—44no—
299C4⋊Q8—82no—
300C4⋊Q8—82no—
301C4⋊Q8C2×Q814yes—
302#127—22no—
303#127C2×Q814yes—
304#125—28no—
305#125—82no—
306#125—82no—
307#124—28no—
308#121—28no—
309#121—24no—
310#121C2×Q818yes—
311C4×Q8—82no—
312C4×Q8—82no—
313C4×D4—28no—
314C4×D4—24no—
315C4×D4—28no—
316C4×D4—44no—
317C4×D4—82no—
318C4×D4—82no—
319C4×D4—82no—
320C4×D4—82no—
321C4×D4C2×Q818yes—
322#117—24no—
323#117—42no—
324#117—82no—
325#117—82no—
326#117—82no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
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—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
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⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
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×C42C2×Q814yes—
393C2×C42#4114yes—
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—28no—
401#106—28no—
402#106—44no—
403#106—44no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8#4118yescomplemented
408#102—28no—
409#102—28no—
410#102—44no—
411#102—44no—
412#97—24no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—24no—
416C4×C8—22no—
417C4×C8—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
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442C24—22no—
443C24C22×D411yescharacteristic
444#41—816no—
445C2×Q8—24no—
446C2×Q8—44no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—82no—
450C2×Q8—82no—
451C2×D4—28no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—82no—
462C2×D4—82no—
463C2×D4C22×Q814yes—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—28no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4C22⋊Q814yes—
487C22×C4C22⋊Q814yes—
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×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×Q812yes—
502C22×C4C2×#4112yes—
503C22×C4C2511yescharacteristic
504M16—48no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—48no—
516C2×C8C2×#4114yes—
517C2×C8C2×#4114yes—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
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—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4C22×Q814yes—
554C22⋊C4—28no—
555C22⋊C4—28no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—82no—
581C42—82no—
582C42—82no—
583C42—82no—
584C42—82no—
585C42C22⋊Q814yes—
586C42C22⋊Q814yes—
587C42C22×Q812yes—
588C42C22×Q812yes—
589C42C2×#4112yes—
590C42C2×#4112yes—
591C42C2×#4114yes—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—24no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23C2×C22⋊Q812yes—
606C23C23×D411yescharacteristic
607Q8—48no—
608Q8—164no—
609D4—28no—
610D4—48no—
611D4—48no—
612D4—44no—
613D4—88no—
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—28no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
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—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
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—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—88no—
668C2×C4—164no—
669C2×C4—164no—
670C2×C4C2×C22⋊Q812yes—
671C2×C4C2×C22⋊Q812yes—
672C2×C4C2×C22⋊Q812yes—
673C2×C4C2×C22⋊Q812yes—
674C2×C4C2×C22⋊Q812yes—
675C2×C4#53212yes—
676C2×C4#53212yes—
677C2×C4#57612yes—
678C2×C4#57712yes—
679C2×C4#57712yes—
680C2×C4#57712yes—
681C2×C4C23×Q811yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C22×#4112yes—
684C2×C4#58512yes—
685C8—24no—
686C8—24no—
687C8—42no—
688C8—42no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22C22×C22⋊Q811yescharacteristic
704C22C2×#53211yescharacteristic
705C22#344911yescharacteristic
706C22C2×#57611yescharacteristic
707C22C2×#57711yescharacteristic
708C22C2×#57711yescharacteristic
709C22C2×#57711yescharacteristic
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—41no—
716C4—41no—
717C4—44no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4#308114yes—
729C4#308414yes—
730C4#347512yes—
731C4Q8×#4112yes—
732C2—42no—
733C2—84no—
734C2#3494711yescharacteristic
735C2C2×#308111yescharacteristic
736C2C2×#308411yescharacteristic
737C2#6071611yescharacteristic
738C2#6072211yescharacteristic
739C2#6100611yescharacteristic
740C2#6101211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
1. #35294C2semidirect
3. #29220C2semidirect
4. #29206C2semidirect
5. #29190C2semidirect
6. #29190C2semidirect
7. #29178C2semidirect
8. #29177C2semidirect
9. #29174C2semidirect
10. #29139C2semidirect
11. #28514C2semidirect
12. #24852C2semidirect
13. #23786C2semidirect
14. #21619C2semidirect
17. #20824C2semidirect
18. #20816C2semidirect
19. C4×#424C2semidirect
111. #532Q8semidirect
407. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#352942
#342262
#292202
#292062
#291902
#291782
#291772
#291742
#291392
#285142
#248522
#237862
#216192
#214102
#214012
#208242
#208162
C4×#4242