Finite Group Databasebeta

← #364663

Subgroups — #364663

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3091C211yesmaximal, characteristic
2C2×#2933C211yesmaximal, characteristic
3C2×#1896C211yesmaximal, characteristic
4#20685C211yesmaximal, characteristic, complemented
5C22×#339C211yesmaximal, characteristic, complemented
6#11221C212yesmaximal, complemented
7#11191C212yesmaximal, complemented
8#10909C212yesmaximal, complemented
9#10791C212yesmaximal, complemented
10#10734C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C22×#131C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C2×#525C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17#3091—24no—
18#3091—24no—
19C2×#484C411yescharacteristic, complemented
20C2×#481C411yescharacteristic, complemented
21C2×#485C412yescomplemented
22C2×#482C412yescomplemented
23C2×#483C411yescharacteristic, complemented
24C2×#480C411yescharacteristic, complemented
25#2933—24no—
26#2933—24no—
27C2×#436—21no—
28C2×#435—21no—
29C2×#435—21no—
30C2×C8×D4—21no—
31C2×#427C2212yes—
32C2×#424C2211yescharacteristic
33C22×#104C2212yes—
34C2×#416C2212yes—
35C2×#408C2211yescharacteristic
36C2×#407C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39C2×#405C2211yescharacteristic
40#1896—24no—
41#1818—21no—
42#1818—21no—
43#1817—21no—
44#1817—21no—
45C2×#341C2212yes—
46C2×#341C2212yes—
47C2×#339—24no—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1729C2212yes—
52#1729C2212yes—
53C2×#336C2212yes—
54C2×#336C2212yes—
55C22×#41C2311yescharacteristic
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C2×#131—22no—
59C2×#131—22no—
60C2×#131—24no—
61C2×#131—42no—
62C22×M16—21no—
63C22×M16—22no—
64C22×M16—21no—
65C22×M16—24no—
66C22×M16—42no—
67C22×M16D414yes—
68C22×M16C2312yes—
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8—21no—
72C23×C8—21no—
73C23×C8C2312yes—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76#525—22no—
77#525—22no—
78#525D414yes—
79#525D414yes—
80C2×#123—21no—
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×#121C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×#120—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×#117—21no—
89C2×#117—21no—
90C2×#117—22no—
91C2×#117—22no—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117C2×C412yes—
95C2×#117C2312yes—
96C2×#117C2311yescharacteristic
97C22×C4⋊C4C2×C411yescharacteristic
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C22⋊C4—21no—
100C22×C42—22no—
101C22×C42—24no—
102C22×C42—21no—
103C22×C42—22no—
104C22×C42—21no—
105C22×C42—42no—
106C22×C42C2312yes—
107#485—28no—
108#484—24no—
109#483—24no—
110#482—28no—
111#481—24no—
112#480—24no—
113#436—28no—
114#435—28no—
115#435—28no—
116C8×D4—28no—
117#427—24no—
118#427—24no—
119C2×#107C2×C412yescomplemented
120C2×#107C2×C412yescomplemented
121C2×#106C2×C412yescomplemented
122C2×#106C2×C412yescomplemented
123#424—22no—
124#424—22no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×#104—24no—
128C2×#104—24no—
129C2×#104—42no—
130#416—24no—
131#416—24no—
132C2×#103C2×C412yescomplemented
133C2×#103C2×C412yescomplemented
134C2×#102C2×C412yescomplemented
135C2×#102C2×C412yescomplemented
136#408—22no—
137#408—22no—
138#407—22no—
139#407—22no—
140C2×C22⋊C8—22no—
141C2×C22⋊C8—22no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8—22no—
144C2×C22⋊C8—21no—
145C2×C22⋊C8—21no—
146C2×C22⋊C8—21no—
147C2×C22⋊C8—21no—
148C2×C22⋊C8C2×C411yescharacteristic
149C2×C22⋊C8C2×C411yescharacteristic
150C2×C22⋊C8C2×C412yes—
151C2×C22⋊C8D411yescharacteristic
152C2×C22⋊C8D411yescharacteristic
153C2×C22⋊C8D411yescharacteristic
154C2×C22⋊C8D411yescharacteristic
155#405—22no—
156#405—22no—
157C2×#97—21no—
158C2×C4×C8—21no—
159#341—24no—
160#341—42no—
161#339—24no—
162#339—24no—
163#339—24no—
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336—24no—
169#336—42no—
170C25#4111yescharacteristic
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41C2×D412yes—
178C2×#41C2×D412yes—
179C22×Q8C22×C411yescharacteristic
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C22×C411yescharacteristic
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—21no—
190C23×C4—24no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4#4111yescharacteristic
205C23×C4#4112yes—
206C23×C4C2411yescharacteristic
207#131—48no—
208#131—44no—
209#131—44no—
210C2×M16—21no—
211C2×M16—22no—
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—21no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—28no—
218C2×M16—22no—
219C2×M16—22no—
220C2×M16—22no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—22no—
224C2×M16—44no—
225C2×M16—44no—
226C2×M16—44no—
227C2×M16—48no—
228C2×M16C2×D412yes—
229C2×M16C2×D412yes—
230C2×M16#4114yes—
231C22×C8—21no—
232C22×C8—24no—
233C22×C8—24no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—21no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C22×C412yes—
253C22×C8C22×C412yes—
254C22×C8C2×D411yescharacteristic
255C22×C8C2×D411yescharacteristic
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D411yescharacteristic
258#123—24no—
259#123—24no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C22⋊Q8C22⋊C414yes—
263C22⋊Q8C22⋊C414yes—
264#121—22no—
265#121—22no—
266#121C22⋊C414yes—
267#121C22⋊C414yes—
268#120—24no—
269#120—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274#117—21no—
275#117—21no—
276#117—24no—
277#117—24no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117—22no—
282#117—42no—
283#117—42no—
284#117—42no—
285#117—42no—
286#117C2×D412yes—
287#117C2×D412yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4C22×C412yes—
297C2×C4⋊C4C22×C412yes—
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C42—21no—
310C2×C42—28no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—48no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42#4114yes—
325#107—24no—
326#107—24no—
327#106—24no—
328#106—24no—
329C4⋊C8—24no—
330C4⋊C8—24no—
331#104—48no—
332#103—24no—
333#103—24no—
334#102—24no—
335#102—24no—
336C22⋊C8—24no—
337C22⋊C8—24no—
338C22⋊C8—22no—
339C22⋊C8—22no—
340C22⋊C8—24no—
341C22⋊C8—22no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344C22⋊C8—22no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347#97—24no—
348C4×C8—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—42no—
355C24C4×D411yescharacteristic
356C24C4×D411yescharacteristic
357C24#12112yes—
358C24#12312yes—
359C24C2×#4111yescharacteristic
360#41—24no—
361#41—24no—
362#41—24no—
363#41—24no—
364#41—41no—
365#41—41no—
366#41—42no—
367#41—41no—
368#41—41no—
369#41—42no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8C2×C22⋊C412yes—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—82no—
386C2×D4C2×C22⋊C412yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—82no—
433C22×C4C2×C22⋊C412yes—
434C22×C4C2×C22⋊C412yes—
435C22×C4C4×D412yes—
436C22×C4C4×D412yes—
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4#12012yes—
440C22×C4#12012yes—
441C22×C4#12112yes—
442C22×C4#12112yes—
443C22×C4#12112yes—
444C22×C4#12312yes—
445C22×C4C23×C411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455M16—22no—
456M16—24no—
457M16—22no—
458M16—22no—
459M16—48no—
460M16—48no—
461M16—42no—
462M16—42no—
463M16—42no—
464M16—42no—
465M16—42no—
466M16#12314yes—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—22no—
479C2×C8—21no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—48no—
485C2×C8—48no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8—42no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4C2×C22⋊C412yes—
511C4⋊C4C2×C22⋊C412yes—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—24no—
525C42—48no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—44no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23#38612yes—
557C23#38612yes—
558C23#44312yes—
559C23#44412yes—
560C23C2×C4×D411yescharacteristic
561C23C2×#12111yescharacteristic
562C23C2×#12311yescharacteristic
563C23#55112yes—
564C23#55212yes—
565Q8—24no—
566Q8—41no—
567Q8—41no—
568D4—24no—
569D4—48no—
570D4—44no—
571D4—42no—
572D4—42no—
573D4—41no—
574D4—41no—
575D4—84no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—432no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4#38612yes—
619C2×C4#38612yes—
620C2×C4C22×C22⋊C411yescharacteristic
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C4×#4112yes—
623C2×C4#52012yes—
624C2×C4C2×#12011yescharacteristic
625C2×C4C2×#12111yescharacteristic
626C2×C4C2×#12311yescharacteristic
627C2×C4#52511yescharacteristic
628C2×C4#52912yes—
629C2×C4#53411yescharacteristic
630C2×C4#53611yescharacteristic
631C2×C4#53912yes—
632C2×C4#54612yes—
633C2×C4#54612yes—
634C2×C4#54612yes—
635C2×C4#54712yes—
636C2×C4#54712yes—
637C2×C4#54712yes—
638C8—24no—
639C8—42no—
640C8—42no—
641C8—41no—
642C8—41no—
643C8—42no—
644C8—41no—
645C8—41no—
646C8—82no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—22no—
652C22—22no—
653C22—24no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—22no—
658C22—21no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—44no—
663C22—48no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—82no—
671C22—82no—
672C22#187614yes—
673C22C2×#38611yescharacteristic
674C22#235912yes—
675C22#241912yes—
676C22#248611yescharacteristic
677C22#253411yescharacteristic
678C22C2×#44311yescharacteristic
679C22C2×#44411yescharacteristic
680C22#295612yes—
681C4—22no—
682C4—22no—
683C4—48no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4C4×#12311yescharacteristic
690C4#232211yescharacteristic
691C4#233912yes—
692C4#240511yescharacteristic
693C4#240611yescharacteristic
694C4#242512yes—
695C2—21no—
696C2—24no—
697C2—22no—
698C2—21no—
699C2—42no—
700C2—82no—
701C2#1396811yescharacteristic
702C2C2×#187611yescharacteristic
703C2#2091811yescharacteristic
704C2#2294812yes—
705C2#2295012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20685C2semidirect
5. C22×#339C2semidirect
6. #11221C2semidirect
7. #11191C2semidirect
8. #10909C2semidirect
9. #10791C2semidirect
10. #10734C2semidirect
11. #10712C2semidirect
19. C2×#484C4semidirect
20. C2×#481C4semidirect
21. C2×#485C4semidirect
22. C2×#482C4semidirect
23. C2×#483C4semidirect
24. C2×#480C4semidirect
119. C2×#107C2×C4semidirect
120. C2×#107C2×C4semidirect
121. C2×#106C2×C4semidirect
122. C2×#106C2×C4semidirect
132. C2×#103C2×C4semidirect
133. C2×#103C2×C4semidirect
134. C2×#102C2×C4semidirect
135. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30912
C2×#29332
C2×#18962
#206852
C22×#3392
#112212
#111912
#109092
#107912
#107342
#107122