Finite Group Databasebeta

← #364658

Subgroups — #364658

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic
3C2×#1894C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic
6#11219C212yesmaximal, complemented
7#11190C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#484—24no—
18C2×#484—24no—
19C2×#484C418yescomplemented
20C22×#107C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24#2362—24no—
25#2362—24no—
26C2×#391C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#388—21no—
29C2×#388—21no—
30C2×#387—21no—
31C2×#381—21no—
32C2×#381—21no—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#1894—24no—
38#1736—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1445—21no—
47#1445—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×Q8C2311yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—42no—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8C2×C414yes—
64C2×C22⋊Q8D414yes—
65C2×C22⋊Q8D414yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—24no—
69C2×C4×Q8—42no—
70C2×#117—22no—
71C2×#117—24no—
72C2×#117—42no—
73C2×#117D414yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4C2×C414yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42—21no—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#484—232no—
100C2×#107—24no—
101C2×#107—24no—
102C2×#107C2×C418yes—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—42no—
108C2×#103C2×C418yes—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8C2×C414yes—
112#391—22no—
113#391—22no—
114#389—22no—
115#389—22no—
116#388—28no—
117#388—28no—
118#387—28no—
119#381—44no—
120#381—44no—
121#380—22no—
122#380—22no—
123C4×C4⋊C4—28no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95D411yescharacteristic
138C2×#95D411yescharacteristic
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—42no—
144#336—24no—
145#336—42no—
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22×C412yes—
155C22×Q8C2×D412yes—
156C22×Q8C2×D412yes—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4C22×C412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C2×M16—28no—
187C2×M16—48no—
188C2×M16—48no—
189C2×M16—44no—
190C2×M16—44no—
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×C8C22×C414yes—
204C22×C8#4114yes—
205C22⋊Q8—28no—
206C22⋊Q8—28no—
207C22⋊Q8—28no—
208C22⋊Q8C22⋊C4116yes—
209C4×Q8—48no—
210C4×Q8—44no—
211C4×Q8—44no—
212#117—28no—
213#117—44no—
214#117—48no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4#4114yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—42no—
284C2×C42—42no—
285#107—216no—
286#103—216no—
287#103—48no—
288C22⋊C8—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—22no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95—44no—
300C24—22no—
301C24—22no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—22no—
308C24C4×D412yes—
309C24#12012yes—
310C24#12112yes—
311C24C22×D411yescharacteristic
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8C2×C22⋊C414yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—48no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4C2×C22⋊C414yes—
408C22×C4C4×D414yes—
409C22×C4C4×D412yes—
410C22×C4#12012yes—
411C22×C4#12114yes—
412C22×C4#12112yes—
413C22×C4#12314yes—
414C22×C4AGL(1,ℤ/8)12yes—
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4111yescharacteristic
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+412yes—
426M16—416no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—216no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—48no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4C2×C22⋊C414yes—
462C4⋊C4#12314yes—
463C22⋊C4—216no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—84no—
468C42—48no—
469C42—48no—
470C42—48no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—42no—
476C42—42no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23#38614yes—
500C23C2×C4×D411yescharacteristic
501C23C2×#12011yescharacteristic
502C23C2×#12111yescharacteristic
503C23#54612yes—
504C23#54712yes—
505C23C2×AGL(1,ℤ/8)11yescharacteristic
506C23C2×AGL(1,ℤ/8)11yescharacteristic
507C23#57511yescharacteristic
508C23#57511yescharacteristic
509C23#57812yes—
510C23#57812yes—
511Q8—28no—
512Q8—42no—
513Q8—44no—
514Q8—42no—
515Q8—42no—
516Q8—42no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—44no—
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—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—816no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4#38614yes—
585C2×C4C22×C22⋊C411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C4×#4112yes—
588C2×C4#52012yes—
589C2×C4C2×#12111yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4C2×#12311yescharacteristic
592C2×C4#52511yescharacteristic
593C2×C4#52912yes—
594C2×C4#53411yescharacteristic
595C2×C4#53611yescharacteristic
596C2×C4#53912yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4#54712yes—
600C2×C4#54712yes—
601C2×C4#55112yes—
602C2×C4#55212yes—
603C8—44no—
604C8—44no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22C2×#38611yescharacteristic
617C22#235912yes—
618C22#240511yescharacteristic
619C22#240611yescharacteristic
620C22#242512yes—
621C22C4×AGL(1,ℤ/8)11yescharacteristic
622C22C4×AGL(1,ℤ/8)11yescharacteristic
623C22#295012yes—
624C22#300611yescharacteristic
625C22#300611yescharacteristic
626C22#301111yescharacteristic
627C22#301111yescharacteristic
628C22#303811yescharacteristic
629C22#304011yescharacteristic
630C22#304412yes—
631C22#330812yes—
632C22#331512yes—
633C22#332712yes—
634C22#333012yes—
635C22#334312yes—
636C22#334912yes—
637C22#335212yes—
638C22#335312yes—
639C4—24no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—42no—
644C4—44no—
645C4—81no—
646C4—84no—
647C4—82no—
648C4—81no—
649C4—162no—
650C4C4×#12311yescharacteristic
651C4#232211yescharacteristic
652C4#233912yes—
653C4#241912yes—
654C4#248611yescharacteristic
655C4#253411yescharacteristic
656C2—44no—
657C2#1396811yescharacteristic
658C2#2092811yescharacteristic
659C2#2093511yescharacteristic
660C2#2292512yes—
661C2#2294712yes—
662C2#2405111yescharacteristic
663C2#2405411yescharacteristic
664C2#2410711yescharacteristic
665C2#2411011yescharacteristic
666C2#2521712yes—
667C2#2522912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
6. #11219C2semidirect
7. #11190C2semidirect
8. #10907C2semidirect
9. #10789C2semidirect
19. C2×#484C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#23622
C2×#18942
C2×#17362
#139042
#112192
#111902
#109072
#107892
#107312
#107152