Finite Group Databasebeta

← #6752315

Subgroups — #6752315

Order 512.

1431 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic
2#36189C211yesmaximal, characteristic, complemented
3#33048C211yesmaximal, characteristic, complemented
4#32211C214yesmaximal, complemented
5#32018C214yesmaximal, complemented
6#30307C211yesmaximal, characteristic, complemented
7#29843C211yesmaximal, characteristic, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29192C211yesmaximal, characteristic, complemented
10#27973C211yesmaximal, characteristic, complemented
11#27891C211yesmaximal, characteristic, complemented
12#27878C211yesmaximal, characteristic, complemented
13#27877C211yesmaximal, characteristic, complemented
14#27388C214yesmaximal, complemented
15#27348C214yesmaximal, complemented
16#22220C211yesmaximal, characteristic, complemented
17#21982C211yesmaximal, characteristic, complemented
18#21102C211yesmaximal, characteristic, complemented
19#20824C211yesmaximal, characteristic, complemented
20C2×#486C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2856C2211yescharacteristic
29#2570—21no—
30#2535C2212yes—
31#2513—21no—
32#2513—21no—
33#2511—21no—
34#2507C2212yes—
35#2497C2212yes—
36#2490C2212yes—
37#2490C2212yes—
38#2487C2212yes—
39#2442—21no—
40#2441—21no—
41#2423—21no—
42#2420—21no—
43#2420—21no—
44#2417—21no—
45#2414—21no—
46#2414—21no—
47#2363—21no—
48#2363—21no—
49#2360—21no—
50#2357C2212yes—
51#2352—21no—
52#2345C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C42×D4C2211yescharacteristic
60#2101C2214yes—
61#2085C2214yes—
62#2076C2214yes—
63#2075C2214yes—
64#2071C2214yes—
65#2067C2214yes—
66#2065C2214yes—
67#2059C2214yes—
68#2034C2214yes—
69#2033C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1989C2214yes—
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1947C2214yes—
78#1936C2214yes—
79#1925C2214yes—
80#1925C2214yes—
81#1924C2212yes—
82#1921C2214yes—
83#1918C2214yes—
84#1917C2212yes—
85#1895C2214yes—
86#1874C2214yes—
87#1870C2212yes—
88#1862C2214yes—
89#1862C2214yes—
90#1852C2214yes—
91#1851C2212yes—
92#1845C2211yescharacteristic
93#1845C2211yescharacteristic
94#1845C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1834C2211yescharacteristic
101#1809C2214yes—
102#1776C2214yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106#1744C2211yescharacteristic
107C22×M16C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#125—21no—
111C2×#125C2312yes—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×#121C2312yes—
120C2×C4×Q8—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—21no—
128C2×#117—21no—
129C2×#117—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#486—24no—
136#431—24no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C4×M16—24no—
151C2×#97C2312yes—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156#400—22no—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#397—22no—
162#397—22no—
163#397—22no—
164#396—22no—
165#394—22no—
166#394—22no—
167#392—22no—
168#391—21no—
169#391—21no—
170#391—21no—
171#390—22no—
172#390—22no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
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#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#388—22no—
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#386—22no—
207#386—22no—
208#386—22no—
209#385—22no—
210#384—21no—
211#384—21no—
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2312yes—
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#382—21no—
220#382—21no—
221#382—21no—
222#382—21no—
223#382C2312yes—
224#381—22no—
225#380—21no—
226#380—21no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—22no—
231#380C2312yes—
232#380C2312yes—
233#380C2312yes—
234#380C2312yes—
235#380C2312yes—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2311yescharacteristic
247C4×C22⋊C4—22no—
248C4×C22⋊C4—22no—
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4C2311yescharacteristic
254C4×C22⋊C4C2311yescharacteristic
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—
269C22×Q8—21no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4—21no—
273C22×D4—21no—
274C22×D4C2×D411yescharacteristic
275C22×D4C2412yes—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C23×C4C2411yescharacteristic
284C2×M16—24no—
285C2×M16—42no—
286C2×M16—42no—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293#127—22no—
294#127—22no—
295#125—44no—
296#125—82no—
297C22⋊Q8—44no—
298C22⋊Q8—82no—
299#121—44no—
300#121—44no—
301#121—44no—
302#121—44no—
303#121—82no—
304#121—82no—
305C4×Q8—82no—
306C4×D4—24no—
307C4×D4—44no—
308C4×D4—42no—
309C4×D4—42no—
310C4×D4—82no—
311C4×D4—82no—
312#117—82no—
313#117—82no—
314#117—82no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
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—44no—
339C2×C4⋊C4—82no—
340C2×C4⋊C4—82no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C22⋊C4C2×D411yescharacteristic
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C42—22no—
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×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—44no—
393#107—44no—
394#107—44no—
395#106—44no—
396C4⋊C8—24no—
397C4⋊C8—42no—
398#102—28no—
399#102—44no—
400#102—44no—
401#102—44no—
402C22⋊C8—24no—
403C22⋊C8—42no—
404C22⋊C8—42no—
405#97—24no—
406C4×C8—22no—
407C4×C8—22no—
408C4×C8—22no—
409C4×C8#4114yes—
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#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432C24—22no—
433C24C22×D411yescharacteristic
434C24C22×D411yescharacteristic
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4—82no—
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—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
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—22no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4#13714yes—
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×D411yescharacteristic
502C22×C4C2511yescharacteristic
503M16—48no—
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—
514C2×C8—22no—
515C2×C8C2×#4114yes—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—44no—
528C4⋊C4—44no—
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—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—21no—
579C42—21no—
580C42—21no—
581C42—21no—
582C42—22no—
583C42—22no—
584C42—82no—
585C42—82no—
586C42—82no—
587C42—82no—
588C42C2×#4112yes—
589C422−1+412yes—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23C2×#13712yes—
611C23C23×D411yescharacteristic
612Q8—84no—
613D4—44no—
614D4—44no—
615D4—44no—
616D4—84no—
617D4—84no—
618D4—164no—
619D4—164no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—28no—
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—21no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
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—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—164no—
679C2×C4—164no—
680C2×C4#54712yes—
681C2×C4#54712yes—
682C2×C4C2×#13712yes—
683C2×C4C2×#13712yes—
684C2×C4#57512yes—
685C2×C4#57612yes—
686C2×C4#57612yes—
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×2−1+411yescharacteristic
689C2×C4#58512yes—
690C2×C4#58512yes—
691C8—24no—
692C8—42no—
693C8—42no—
694C8—42no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22#344911yescharacteristic
717C22C2×#54711yescharacteristic
718C22#350011yescharacteristic
719C22C22×#13711yescharacteristic
720C22C2×#57511yescharacteristic
721C22C2×#57611yescharacteristic
722C22C2×#57611yescharacteristic
723C4—22no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4#334614yes—
741C4#335514yes—
742C4#349312yes—
743C4#356512yes—
744C2—42no—
745C2—81no—
746C2—162no—
747C2#3787111yescharacteristic
748C2#6096611yescharacteristic
749C2#6101111yescharacteristic
750C2C2×#334611yescharacteristic
751C2C2×#335511yescharacteristic
752C2#6253711yescharacteristic
753C2#6254611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36211C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36189C2semidirect
3. #33048C2semidirect
4. #32211C2semidirect
5. #32018C2semidirect
6. #30307C2semidirect
7. #29843C2semidirect
8. #29220C2semidirect
9. #29192C2semidirect
10. #27973C2semidirect
11. #27891C2semidirect
12. #27878C2semidirect
13. #27877C2semidirect
14. #27388C2semidirect
15. #27348C2semidirect
16. #22220C2semidirect
17. #21982C2semidirect
18. #21102C2semidirect
19. #20824C2semidirect

Maximal subgroups

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

subgroupindex
#362112
#361892
#330482
#322112
#320182
#303072
#298432
#292202
#291922
#279732
#278912
#278782
#278772
#273882
#273482
#222202
#219822
#211022
#208242