Finite Group Databasebeta

← #6490470

Subgroups — #6490470

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#33103C214yesmaximal, complemented
4#30158C214yesmaximal, complemented
5#29401C214yesmaximal, complemented
6#29246C214yesmaximal, complemented
7#29179C211yesmaximal, characteristic, complemented
8#29177C211yesmaximal, characteristic, complemented
9#28655C211yesmaximal, characteristic, complemented
10#28655C211yesmaximal, characteristic, complemented
11#27974C211yesmaximal, characteristic
12#27949C211yesmaximal, characteristic, complemented
13#25903C211yesmaximal, characteristic
14#25813C211yesmaximal, characteristic, complemented
15#22248C211yesmaximal, characteristic, complemented
16#22247C211yesmaximal, characteristic, complemented
17#22220C211yesmaximal, characteristic, complemented
18#22219C211yesmaximal, characteristic, complemented
19#20782C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#486C2211yescharacteristic
25#2859C2211yescharacteristic
26#2858C2211yescharacteristic
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2644C2212yes—
30#2624C2212yes—
31#2513—21no—
32#2490—21no—
33#2454—21no—
34#2448—21no—
35#2447—21no—
36#2427—21no—
37#2426—21no—
38#2420—21no—
39#2369—21no—
40#2369—21no—
41#2365—21no—
42#2365—21no—
43#2345—21no—
44#2345—21no—
45Q8×C22⋊C4—21no—
46Q8×C22⋊C4—21no—
47#2331C2212yes—
48#2331C2212yes—
49#2324C2212yes—
50#2319C2212yes—
51#2316C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×#127C2211yescharacteristic
58#2285C2211yescharacteristic
59#2274C2212yes—
60C4×#117C2211yescharacteristic
61#2088C2214yes—
62#2087C2214yes—
63#2085C2214yes—
64#2073C2214yes—
65#2067C2214yes—
66#2067C2214yes—
67#2057C2214yes—
68#2023C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1926C2214yes—
78#1925C2214yes—
79#1925C2214yes—
80#1925C2214yes—
81#1921C2214yes—
82#1921C2214yes—
83#1921C2214yes—
84#1919C2214yes—
85#1918C2214yes—
86#1918C2214yes—
87#1875C2214yes—
88#1874C2214yes—
89#1864C2214yes—
90#1862C2214yes—
91#1854C2214yes—
92#1852C2214yes—
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1840C2212yes—
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1832C2212yes—
99#1818C2212yes—
100#1798C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1744C2211yescharacteristic
106#1744C2211yescharacteristic
107C23×Q8—21no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×#127C2311yescharacteristic
117C2×#125—21no—
118C2×#124—21no—
119C2×#124—21no—
120C2×C22⋊Q8—21no—
121C2×#121—21no—
122C2×#121—21no—
123C2×#121C2312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4C2312yes—
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C42C2311yescharacteristic
138#500—24no—
139#500—24no—
140#486—24no—
141#486—24no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158#399—22no—
159#397—22no—
160#397—22no—
161#397—22no—
162#397—22no—
163#396—22no—
164#394—22no—
165#394—22no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—22no—
176#390—22no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#390C2312yes—
187#389—22no—
188#389—22no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#388—22no—
206#388—22no—
207#388—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#386—22no—
213#385—22no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#382—21no—
219#382C2312yes—
220#381—22no—
221#380—21no—
222#380—21no—
223#380—21no—
224#380—22no—
225#380—22no—
226#380—22no—
227#380C2312yes—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4D411yescharacteristic
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248C4×C22⋊C4D411yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252#376C2311yescharacteristic
253#376C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—28no—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—22no—
276C22×Q8—21no—
277C22×Q8—82no—
278C22×Q8—82no—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4—22no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—28no—
293C4⋊Q8—28no—
294C4⋊Q8—24no—
295C4⋊Q8—24no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8#4114yes—
301C4⋊Q8#4114yes—
302#127—22no—
303#127—22no—
304#125—82no—
305#124—82no—
306#124—82no—
307C22⋊Q8—82no—
308#121—28no—
309#121—82no—
310#121—82no—
311C4×Q8—82no—
312C4×Q8—82no—
313C4×Q8—82no—
314C4×Q8—82no—
315C4×Q8—82no—
316C4×D4—44no—
317C4×D4—82no—
318C4×D4—82no—
319#117—24no—
320#117—42no—
321#117—44no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
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—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
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×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—28no—
398#107—28no—
399#107—44no—
400#106—44no—
401C4⋊C8—42no—
402C4⋊C8—42no—
403#102—28no—
404#102—28no—
405#102—44no—
406#102—44no—
407C22⋊C8—24no—
408C22⋊C8—24no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8—22no—
413#95—22no—
414#95—22no—
415#95—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#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437C24—22no—
438C24C22×D411yescharacteristic
439C2×Q8—28no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—82no—
450C2×Q8—82no—
451C2×Q8—816no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—82no—
464C2×D4—82no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—44no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—82no—
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×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C42−1+414yes—
501C22×C4C2511yescharacteristic
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—
513C2×C8—22no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—41no—
529C4⋊C4—41no—
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⋊C4C2×#4114yes—
548C4⋊C4C2×#4114yes—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—21no—
576C42—21no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—44no—
582C42—82no—
583C42—82no—
584C42—82no—
585C42—82no—
586C42—82no—
587C42C2×#4112yes—
588C42C2×#4112yes—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23#53612yes—
603C23C23×D411yescharacteristic
604Q8—48no—
605Q8—48no—
606Q8—88no—
607Q8—84no—
608D4—44no—
609D4—44no—
610D4—84no—
611D4—84no—
612D4—164no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—24no—
620C2×C4—24no—
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—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—21no—
648C2×C4—21no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
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—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—88no—
671C2×C4—164no—
672C2×C4#52512yes—
673C2×C4#52512yes—
674C2×C4#53612yes—
675C2×C4#57612yes—
676C2×C4#57612yes—
677C2×C4#57712yes—
678C2×C4#57812yes—
679C2×C4C22×#4111yescharacteristic
680C2×C4C22×#4111yescharacteristic
681C2×C4C2×2−1+412yes—
682C2×C4#58512yes—
683C8—42no—
684C8—42no—
685C8—42no—
686C8—42no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22C2×#52511yescharacteristic
702C22C2×#53611yescharacteristic
703C22#344911yescharacteristic
704C22C2×#57611yescharacteristic
705C22C2×#57611yescharacteristic
706C22C2×#57711yescharacteristic
707C22C2×#57811yescharacteristic
708C4—22no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—44no—
716C4—44no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4#312114yes—
727C4#312214yes—
728C4#351612yes—
729C4#351612yes—
730C2—42no—
731C2—162no—
732C2#3524611yescharacteristic
733C2C2×#312111yescharacteristic
734C2C2×#312211yescharacteristic
735C2#6087711yescharacteristic
736C2#6088011yescharacteristic
737C2#6100611yescharacteristic
738C2#6101411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. #34216C2semidirect
3. #33103C2semidirect
4. #30158C2semidirect
5. #29401C2semidirect
6. #29246C2semidirect
7. #29179C2semidirect
8. #29177C2semidirect
9. #28655C2semidirect
10. #28655C2semidirect
12. #27949C2semidirect
14. #25813C2semidirect
15. #22248C2semidirect
16. #22247C2semidirect
17. #22220C2semidirect
18. #22219C2semidirect
19. #20782C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#342162
#331032
#301582
#294012
#292462
#291792
#291772
#286552
#279742
#279492
#259032
#258132
#222482
#222472
#222202
#222192
#207822