Finite Group Databasebeta

← #364111

Subgroups — #364111

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2290C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1798C211yesmaximal, characteristic, complemented
5C2×#1729C211yesmaximal, characteristic, complemented
6#11202C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10796C212yesmaximal, complemented
9#10715C211yesmaximal, characteristic, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10712C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22#2290—24no—
23#2290—24no—
24C2×#388C412yes—
25C2×#387C412yes—
26C2×#386C412yes—
27C2×#385C412yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#1870—24no—
33#1869—24no—
34#1798—24no—
35#1729—24no—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—42no—
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—42no—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C23×C8C2312yes—
62C23×C8C2312yes—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4D414yes—
68C2×C4×D4Q814yes—
69C2×#117—42no—
70C2×#117—42no—
71C2×#117D414yes—
72C2×#117D414yes—
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C412yes—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—42no—
84C22×C42—42no—
85C22×C42C2×C412yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—24no—
92C2×C4⋊C8—42no—
93C2×C4⋊C8—42no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—42no—
105C2×C22⋊C8—42no—
106#388—28no—
107#387—28no—
108#386—28no—
109#385—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116#376—22no—
117#376—22no—
118C2×#95C2×C412yes—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123#336—24no—
124#336—24no—
125#336—24no—
126#336—24no—
127#336—24no—
128#336—24no—
129#336—24no—
130#336—42no—
131#336—42no—
132C25C22×C411yescharacteristic
133C25C22×C411yescharacteristic
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—42no—
138C22×D4C4⋊C414yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×Q812yes—
171C23×C4C2×Q812yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×M16—28no—
176C2×M16—48no—
177C2×M16—48no—
178C2×M16—44no—
179C2×M16—44no—
180C22×C8—28no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—48no—
208C22×C8—48no—
209C22×C8—44no—
210C22×C8—44no—
211C22×C8#4114yes—
212C22×C8#4114yes—
213C4×D4—28no—
214C4×D4—28no—
215C4×D4—28no—
216C4×D4—28no—
217#117—48no—
218#117—44no—
219#117—44no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C22⋊C418yes—
228C2×C4⋊C4C4⋊C414yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4C22⋊C418yes—
242C2×C22⋊C4C4⋊C414yes—
243C2×C22⋊C4C4⋊C414yes—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—48no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×Q812yes—
272C4⋊C8—48no—
273C22⋊C8—48no—
274C22⋊C8—48no—
275C22⋊C8—44no—
276C22⋊C8—44no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24C2×C22⋊C412yes—
296C24C2×C4⋊C412yes—
297C24C2×C4⋊C412yes—
298C24#11712yes—
299C24#13114yes—
300C24C23×C411yescharacteristic
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—44no—
304C2×D4—48no—
305C2×D4—48no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4C2×C22⋊C414yes—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C4⋊C412yes—
388C22×C4C2×C4⋊C412yes—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4#11712yes—
392C22×C4#12114yes—
393C22×C4#12114yes—
394C22×C4C22⋊Q814yes—
395C22×C4C22⋊Q814yes—
396C22×C4#13114yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×Q811yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4H(5,𝔽2)12yes—
406C22×C4H(5,𝔽2)12yes—
407M16—416no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—416no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C4⋊C4—28no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—48no—
456C42—41no—
457C42—41no—
458C42—42no—
459C42—42no—
460C42—41no—
461C42—41no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23#38014yes—
503C23#42414yes—
504C23C22×C22⋊C411yescharacteristic
505C23C22×C4⋊C411yescharacteristic
506C23C2×#11711yescharacteristic
507C23C4×#4112yes—
508C23#51812yes—
509C23C2×#13112yes—
510C23C2×#13112yes—
511C23#56812yes—
512C23#56812yes—
513D4—416no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—432no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4#38014yes—
575C2×C4C2×#12111yescharacteristic
576C2×C4C2×#12111yescharacteristic
577C2×C4C2×C22⋊Q811yescharacteristic
578C2×C4C2×C22⋊Q811yescharacteristic
579C2×C4#52511yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4#53212yes—
582C2×C4#53412yes—
583C2×C4#53411yescharacteristic
584C2×C4#53411yescharacteristic
585C2×C4#54612yes—
586C2×C4#54612yes—
587C2×C4#55112yes—
588C2×C4#55112yes—
589C2×C4#55312yes—
590C2×C4#55312yes—
591C2×C4#55412yes—
592C2×C4#55412yes—
593C8—44no—
594C8—44no—
595C8—82no—
596C8—82no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—28no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22C2×#38011yescharacteristic
616C22#229711yescharacteristic
617C22#230811yescharacteristic
618C22#234912yes—
619C22#235312yes—
620C22#288012yes—
621C22#288012yes—
622C22C2×#42411yescharacteristic
623C22#290511yescharacteristic
624C22#290612yes—
625C22#292211yescharacteristic
626C22#292311yescharacteristic
627C22#292412yes—
628C22#296612yes—
629C22#296712yes—
630C22#296812yes—
631C22#296912yes—
632C22#297612yes—
633C22#297612yes—
634C22#297712yes—
635C22#297812yes—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—82no—
644C4#253411yescharacteristic
645C4#253411yescharacteristic
646C4#254312yes—
647C4#256412yes—
648C4#257911yescharacteristic
649C4#257911yescharacteristic
650C2—44no—
651C2—44no—
652C2#1396011yescharacteristic
653C2#2062811yescharacteristic
654C2#2063011yescharacteristic
655C2#2138112yes—
656C2#2174412yes—
657C2#2269712yes—
658C2#2269912yes—
659C2#2279312yes—
660C2#2279412yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1798C2semidirect
5. C2×#1729C2semidirect
6. #11202C2semidirect
7. #11042C2semidirect
8. #10796C2semidirect
9. #10715C2semidirect
10. #10714C2semidirect
11. #10712C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22902
C2×#18702
C2×#18692
C2×#17982
C2×#17292
#112022
#110422
#107962
#107152
#107142
#107122