Finite Group Databasebeta

← #361434

Subgroups — #361434

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2290C211yesmaximal, characteristic
2C2×#1870C212yesmaximal
3C2×#1851C212yesmaximal, complemented
4C2×#1798C211yesmaximal, characteristic, complemented
5C2×#1728C211yesmaximal, characteristic, complemented
6#11202C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10833C211yesmaximal, characteristic, complemented
9#10796C211yesmaximal, characteristic, complemented
10#10713C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C23×M16C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21#2290—28no—
22C2×#388C412yes—
23C2×#387C412yes—
24C2×#386C412yes—
25C2×#385C412yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1870—28no—
31#1851—28no—
32#1798—24no—
33#1728—24no—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C412yes—
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—24no—
49C22×M16—24no—
50C22×M16—24no—
51C22×M16—24no—
52C22×M16—24no—
53C22×M16—42no—
54C22×M16—42no—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4D414yes—
62C2×C4×D4Q814yes—
63C2×#117—22no—
64C2×#117—24no—
65C2×#117—22no—
66C2×#117—42no—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C412yes—
75C22×C42—24no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—42no—
79C22×C42C2×C412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—42no—
90C2×C4⋊C8—42no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100#388—28no—
101#387—28no—
102#386—28no—
103#385—28no—
104C4×C4⋊C4—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4D418yes—
107#376—24no—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113#336—24no—
114#336—22no—
115#336—24no—
116#336—24no—
117#336—24no—
118#336—24no—
119#336—22no—
120#336—22no—
121#336—42no—
122#336D414yescomplemented
123C25C22×C411yescharacteristic
124C25C22×C411yescharacteristic
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4C4⋊C414yes—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D414yes—
161C23×C4C2×D412yes—
162C23×C4C2×Q812yes—
163C23×C4C2×Q812yes—
164C23×C4#4114yes—
165C23×C4C2411yescharacteristic
166C2×M16—216no—
167C2×M16—48no—
168C2×M16—48no—
169C2×M16—48no—
170C2×M16—48no—
171C2×M16—48no—
172C2×M16—48no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
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×C8C2×D414yes—
200C22×C8#4114yes—
201C4×D4—216no—
202C4×D4—216no—
203#117—48no—
204#117—48no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C4⋊C414yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4C22⋊C418yes—
227C2×C22⋊C4C4⋊C414yes—
228C2×C22⋊C4C4⋊C414yes—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—48no—
246C2×C42—48no—
247C2×C42—42no—
248C2×C42C2×D414yes—
249C2×C42C2×D414yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×Q812yes—
253C2×C42#4112yes—
254C4⋊C8—48no—
255C4⋊C8—48no—
256C22⋊C8—48no—
257C22⋊C8—48no—
258C22⋊C8—48no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95C22⋊C418yes—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24C2×C22⋊C412yes—
283C24C2×C4⋊C412yes—
284C24C2×C4⋊C412yes—
285C24#11712yes—
286C24C23×C411yescharacteristic
287C2×D4—28no—
288C2×D4—48no—
289C2×D4—48no—
290C2×D4—44no—
291C2×D4—44no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
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—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4C2×C22⋊C414yes—
372C22×C4C2×C22⋊C414yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C4⋊C412yes—
375C22×C4C2×C4⋊C412yes—
376C22×C4C2×C4⋊C412yes—
377C22×C4C2×C4⋊C412yes—
378C22×C4#11712yes—
379C22×C4C4×D414yes—
380C22×C4#12118yes—
381C22×C4C22⋊Q818yes—
382C22×C4#13114yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×Q811yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)12yes—
392C22×C42−1+411yescharacteristic
393M16—432no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—28no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C42—24no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—48no—
439C42—42no—
440C42—42no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23#38014yes—
487C23C22×C22⋊C411yescharacteristic
488C23C22×C4⋊C411yescharacteristic
489C23C2×#11711yescharacteristic
490C23C2×C4×D411yescharacteristic
491C23C4×#4111yescharacteristic
492C23#51811yescharacteristic
493C23#52011yescharacteristic
494C23C2×#13112yes—
495C23#56812yes—
496C23#56812yes—
497C23#56812yes—
498D4—416no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—432no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4#38014yes—
560C2×C4C2×#12112yes—
561C2×C4C2×C22⋊Q812yes—
562C2×C4#52512yes—
563C2×C4#52912yes—
564C2×C4#53812yes—
565C2×C4#53912yes—
566C2×C4D4211yescharacteristic
567C2×C4D4212yes—
568C2×C4#54612yes—
569C2×C4#54612yes—
570C2×C4#54612yes—
571C2×C4#54711yescharacteristic
572C2×C4#54712yes—
573C2×C4D4×Q811yescharacteristic
574C2×C4#55412yes—
575C2×C4#55411yescharacteristic
576C8—44no—
577C8—44no—
578C8—82no—
579C8—82no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—44no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—81no—
597C22—81no—
598C22—82no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22C2×#38011yescharacteristic
603C22#229011yescharacteristic
604C22#231811yescharacteristic
605C22D4×C22⋊C411yescharacteristic
606C22#233911yescharacteristic
607C22D4×C4⋊C411yescharacteristic
608C22#234911yescharacteristic
609C22#287912yes—
610C22#288012yes—
611C22#290512yes—
612C22#290612yes—
613C22#292111yescharacteristic
614C22#292111yescharacteristic
615C22#292211yescharacteristic
616C22#292311yescharacteristic
617C22#293012yes—
618C22#293312yes—
619C22#296712yes—
620C22#296912yes—
621C22#297612yes—
622C22#297812yes—
623C22#299312yes—
624C22#299512yes—
625C4—24no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4#242412yes—
634C4#242512yes—
635C4#244912yes—
636C4#245012yes—
637C2—44no—
638C2—82no—
639C2#1389911yescharacteristic
640C2#2062711yescharacteristic
641C2#2063011yescharacteristic
642C2#2110411yescharacteristic
643C2#2110611yescharacteristic
644C2#2199911yescharacteristic
645C2#2200311yescharacteristic
646C2#2240012yes—
647C2#2247212yes—
648C2#2264912yes—
649C2#2269912yes—
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
3. C2×#1851C2semidirect
4. C2×#1798C2semidirect
5. C2×#1728C2semidirect
6. #11202C2semidirect
7. #11042C2semidirect
8. #10833C2semidirect
9. #10796C2semidirect
10. #10713C2semidirect
11. #10712C2semidirect
122. #336D4semidirect

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×#18512
C2×#17982
C2×#17282
#112022
#110422
#108332
#107962
#107132
#107122