Finite Group Databasebeta

← #366822

Subgroups — #366822

Order 512.

1756 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#1895C211yesmaximal, characteristic
3C2×#1875C211yesmaximal, characteristic
4C2×#1851C211yesmaximal, characteristic, complemented
5C2×#1823C211yesmaximal, characteristic, complemented
6C2×#1736C211yesmaximal, characteristic, complemented
7#13801C211yesmaximal, characteristic, complemented
8#11358C212yesmaximal, complemented
9#11076C212yesmaximal, complemented
10#10931C212yesmaximal, complemented
11#10850C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2504—24no—
22#2504C418yescomplemented
23C2×#397—21no—
24C2×#397—21no—
25C2×#390C2212yes—
26C2×#386—22no—
27C2×#382C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1895—24no—
32#1875—24no—
33#1851—24no—
34#1823—24no—
35#1736—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1437—22no—
45C4⋊#95C2212yes—
46#1434—21no—
47C4×#95—21no—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—24no—
56C22×M16—42no—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2312yes—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121C2×C414yes—
66C2×#117—22no—
67C2×#117—24no—
68C2×#117—42no—
69C2×#117D414yes—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2×C414yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4D412yes—
88C22×C42—21no—
89C22×C42—22no—
90C22×C42C2312yes—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—42no—
94C2×#102—24no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104#397—28no—
105#397—28no—
106#390—24no—
107#390C2×C418yes—
108#386—216no—
109#382—24no—
110#382C2×C418yes—
111#380—22no—
112#380D414yes—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—42no—
115C4×C22⋊C4C2×C414yes—
116C4×C22⋊C4D414yes—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129#340—24no—
130#340—24no—
131#340—24no—
132#336—24no—
133C25—21no—
134C25C2×D411yescharacteristic
135C22×D4—24no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4C22×C412yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—81no—
163C23×C4—81no—
164C23×C4—81no—
165C23×C4—81no—
166C23×C4C22×C412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C2×M16—28no—
177C2×M16—48no—
178C2×M16—48no—
179C2×M16—44no—
180C2×M16—44no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8#4114yes—
198#121—28no—
199#121—28no—
200#121—44no—
201#121—44no—
202#117—28no—
203#117—44no—
204#117—48no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4#4114yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4—82no—
252C2×C22⋊C4—81no—
253C2×C22⋊C4—81no—
254C2×C22⋊C4C22⋊C418yes—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—82no—
264C2×C42C22×C414yes—
265C2×C42C2×D414yes—
266C4⋊C8—48no—
267#102—48no—
268#102—44no—
269#102—44no—
270C22⋊C8—44no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—44no—
275#95—44no—
276#95—82no—
277#95C22⋊C414yes—
278#95C22⋊C414yes—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—41no—
289C24—41no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24C2×C22⋊C412yes—
299C24#12112yes—
300C24#12112yes—
301C24C22×D411yescharacteristic
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—416no—
307C2×D4—48no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—84no—
323C2×D4—84no—
324C2×D4—88no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
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—22no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—48no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C414yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C4×D414yes—
393C22×C4C4×D414yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4#12414yes—
397C22×C4#12714yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408M16—416no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—48no—
444C4⋊C4#12314yes—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—84no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—48no—
456C42—88no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—81no—
489C23—81no—
490C23—82no—
491C23—81no—
492C23—81no—
493C23—82no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23#39014yes—
503C23C22×C22⋊C411yescharacteristic
504C23C2×#12111yescharacteristic
505C23C2×#12111yescharacteristic
506C23#54612yes—
507C23#54712yes—
508C23#57512yes—
509C23#57512yes—
510C23#57611yescharacteristic
511C23#57611yescharacteristic
512C23#57711yescharacteristic
513C23#57711yescharacteristic
514D4—42no—
515D4—44no—
516D4—42no—
517D4—816no—
518D4—82no—
519D4—81no—
520D4—81no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—816no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4#39014yes—
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C4×#4112yes—
579C2×C4#52012yes—
580C2×C4C2×#12311yescharacteristic
581C2×C4#52511yescharacteristic
582C2×C4C2×#12411yescharacteristic
583C2×C4#52912yes—
584C2×C4C2×#12711yescharacteristic
585C2×C4#53411yescharacteristic
586C2×C4#53611yescharacteristic
587C2×C4#53912yes—
588C2×C4#54612yes—
589C2×C4#54612yes—
590C2×C4#54712yes—
591C2×C4#54712yes—
592C2×C4#55112yes—
593C2×C4#55212yes—
594C8—44no—
595C8—81no—
596C8—81no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—44no—
607C22—42no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—84no—
612C22—84no—
613C22—161no—
614C22—161no—
615C22—162no—
616C22—162no—
617C22—161no—
618C22—161no—
619C22C2×#39011yescharacteristic
620C22#233912yes—
621C22#241211yescharacteristic
622C22#241311yescharacteristic
623C22#242512yes—
624C22#289612yes—
625C22#289711yescharacteristic
626C22#289711yescharacteristic
627C22#304011yescharacteristic
628C22#304011yescharacteristic
629C22#304012yes—
630C22#304111yescharacteristic
631C22#304311yescharacteristic
632C22#304511yescharacteristic
633C22#304511yescharacteristic
634C22#330712yes—
635C22#330912yes—
636C22#332212yes—
637C22#332612yes—
638C22#334712yes—
639C22#335012yes—
640C22#335112yes—
641C22#335512yes—
642C4—24no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—82no—
647C4—84no—
648C4—84no—
649C4—82no—
650C4C4×#12311yescharacteristic
651C4#232211yescharacteristic
652C4#235912yes—
653C4#248812yes—
654C4#250111yescharacteristic
655C4#253911yescharacteristic
656C2—44no—
657C2—162no—
658C2#1399611yescharacteristic
659C2#2126111yescharacteristic
660C2#2126311yescharacteristic
661C2#2249812yes—
662C2#2250112yes—
663C2#2454711yescharacteristic
664C2#2454911yescharacteristic
665C2#2459711yescharacteristic
666C2#2459911yescharacteristic
667C2#2522712yes—
668C2#2523112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1851C2semidirect
5. C2×#1823C2semidirect
6. C2×#1736C2semidirect
7. #13801C2semidirect
8. #11358C2semidirect
9. #11076C2semidirect
10. #10931C2semidirect
11. #10850C2semidirect
22. #2504C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#18952
C2×#18752
C2×#18512
C2×#18232
C2×#17362
#138012
#113582
#110762
#109312
#108502