Finite Group Databasebeta

← #374192

Subgroups — #374192

Order 512.

1639 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3091C211yesmaximal, characteristic
2C2×#2924C211yesmaximal, characteristic, complemented
3C2×#1896C211yesmaximal, characteristic
4#20897C211yesmaximal, characteristic
5C22×#339C211yesmaximal, characteristic, complemented
6#11221C212yesmaximal, complemented
7#11068C212yesmaximal, complemented
8#10934C212yesmaximal, complemented
9#10909C212yesmaximal, complemented
10#10734C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C2×#525C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16#3091—24no—
17#3091—24no—
18C2×#484C411yescharacteristic, complemented
19C2×#481C411yescharacteristic, complemented
20C2×#485C411yescharacteristic, complemented
21C2×#485C411yescharacteristic, complemented
22C2×#482C411yescharacteristic, complemented
23C2×#482C411yescharacteristic, complemented
24C2×#483C411yescharacteristic, complemented
25C2×#480C411yescharacteristic, complemented
26#2924—24no—
27#2924—24no—
28C2×#433—22no—
29C2×#432—21no—
30C2×#431—21no—
31C2×#427C2211yescharacteristic
32C2×#427C2211yescharacteristic
33C2×#424C2212yes—
34C22×#104C2212yes—
35C2×#416C2211yescharacteristic
36C2×#416C2211yescharacteristic
37C2×#408C2212yes—
38C2×#407C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41C2×#405C2212yes—
42#1896—24no—
43#1870—22no—
44#1869—21no—
45#1869—21no—
46C2×#341C2212yes—
47C2×#341C2212yes—
48C2×#339—24no—
49C2×#339C2212yes—
50C2×#339C2212yes—
51C2×#339C2212yes—
52#1729C2212yes—
53C2×#336C2212yes—
54C2×#336C2212yes—
55C22×#41C2311yescharacteristic
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C22×M16—21no—
59C22×M16—22no—
60C22×M16—24no—
61C22×M16—22no—
62C22×M16—42no—
63C22×M16C2312yes—
64C22×M16C2312yes—
65C23×C8—21no—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2312yes—
69#525—22no—
70#525—22no—
71#525—22no—
72#525—22no—
73C2×#123—21no—
74C2×C22⋊Q8C2×C411yescharacteristic
75C2×C22⋊Q8C2×C411yescharacteristic
76C2×#121C2×C411yescharacteristic
77C2×#121C2×C411yescharacteristic
78C2×#120—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×#117—21no—
82C2×#117—22no—
83C2×#117—21no—
84C2×#117—22no—
85C2×#117C2×C411yescharacteristic
86C2×#117C2×C411yescharacteristic
87C2×#117D414yes—
88C2×#117D414yes—
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C4⋊C4C2×C411yescharacteristic
94C22×C22⋊C4—21no—
95C22×C42—21no—
96C22×C42—22no—
97C22×C42—24no—
98C22×C42—22no—
99C22×C42—21no—
100C22×C42—42no—
101C22×C42C2312yes—
102#485—24no—
103#485—24no—
104#484—24no—
105#483—24no—
106#482—24no—
107#482—24no—
108#481—24no—
109#480—24no—
110#433—216no—
111#432—28no—
112#431—28no—
113#427—22no—
114#427—22no—
115#427—22no—
116#427—22no—
117C2×#107C2×C411yescharacteristic, complemented
118C2×#107C2×C411yescharacteristic, complemented
119C2×#107C2×C411yescharacteristic, complemented
120C2×#107C2×C411yescharacteristic, complemented
121C2×#106C2×C411yescharacteristic, complemented
122C2×#106C2×C411yescharacteristic, complemented
123C2×#106C2×C411yescharacteristic, complemented
124C2×#106C2×C411yescharacteristic, complemented
125#424—24no—
126#424—24no—
127C2×C4⋊C8—22no—
128C2×C4⋊C8—22no—
129C2×#104—24no—
130C2×#104—24no—
131C2×#104—42no—
132#416—22no—
133#416—22no—
134#416—22no—
135#416—22no—
136C2×#103C2×C411yescharacteristic, complemented
137C2×#103C2×C411yescharacteristic, complemented
138C2×#103C2×C411yescharacteristic, complemented
139C2×#103C2×C411yescharacteristic, complemented
140C2×#102C2×C411yescharacteristic, complemented
141C2×#102C2×C411yescharacteristic, complemented
142C2×#102C2×C411yescharacteristic, complemented
143C2×#102C2×C411yescharacteristic, complemented
144#408—24no—
145#408—42no—
146#407—22no—
147#407D414yes—
148C2×C22⋊C8—22no—
149C2×C22⋊C8—22no—
150C2×C22⋊C8—22no—
151C2×C22⋊C8—22no—
152C2×C22⋊C8—21no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—22no—
155C2×C22⋊C8—22no—
156C2×C22⋊C8C2×C411yescharacteristic
157C2×C22⋊C8C2×C411yescharacteristic
158C2×C22⋊C8C2×C411yescharacteristic
159C2×C22⋊C8C2×C411yescharacteristic
160C2×C22⋊C8D411yescharacteristic
161C2×C22⋊C8D411yescharacteristic
162C2×C22⋊C8D412yes—
163#405—24no—
164#405—24no—
165C2×#97—22no—
166C2×C4×C8—22no—
167#341—24no—
168#341—42no—
169#339—24no—
170#339—24no—
171#339—24no—
172#336—22no—
173#336—22no—
174#336—24no—
175#336—42no—
176C25#4111yescharacteristic
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—22no—
182C2×#41—22no—
183C2×#41—22no—
184C2×#41—22no—
185C22×Q8C22×C411yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4C22×C411yescharacteristic
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—21no—
197C23×C4—24no—
198C23×C4—21no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4#4111yescharacteristic
211C23×C4#4112yes—
212C23×C4C2411yescharacteristic
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—28no—
216C2×M16—22no—
217C2×M16—22no—
218C2×M16—22no—
219C2×M16—22no—
220C2×M16—22no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—22no—
224C2×M16—44no—
225C2×M16—48no—
226C2×M16—44no—
227C2×M16—42no—
228C2×M16C2×D414yes—
229C2×M16#4114yes—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—44no—
247C22×C8—42no—
248C22×C8C22×C411yescharacteristic
249C22×C8C22×C411yescharacteristic
250C22×C8C22×C411yescharacteristic
251C22×C8C22×C411yescharacteristic
252C22×C8C2×D412yes—
253C22×C8C2×D412yes—
254#123—42no—
255#123—42no—
256C22⋊Q8—22no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260#121—22no—
261#121—22no—
262#121—22no—
263#121—22no—
264#120—42no—
265#120—42no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—42no—
269C4×D4—42no—
270#117—22no—
271#117—22no—
272#117—24no—
273#117—24no—
274#117—24no—
275#117—24no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117C22⋊C414yes—
285#117C22⋊C414yes—
286#117C2×D412yes—
287#117C2×D412yes—
288#117C2×D412yes—
289#117C2×D412yes—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4C22⋊C414yes—
297C2×C4⋊C4C22⋊C414yes—
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—28no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—48no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42—42no—
332C2×C42#4114yes—
333#107—22no—
334#107—22no—
335#107—22no—
336#107—22no—
337#106—22no—
338#106—22no—
339#106—22no—
340#106—22no—
341C4⋊C8—28no—
342C4⋊C8—28no—
343#104—48no—
344#103—22no—
345#103—22no—
346#103—22no—
347#103—22no—
348#102—22no—
349#102—22no—
350#102—22no—
351#102—22no—
352C22⋊C8—22no—
353C22⋊C8—22no—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C22⋊C8—22no—
357C22⋊C8—22no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8—44no—
362C22⋊C8—44no—
363#97—28no—
364C4×C8—28no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—42no—
371C24C4×D411yescharacteristic
372C24C4×D411yescharacteristic
373C24#12112yes—
374C24#12312yes—
375C24C2×#4111yescharacteristic
376#41—41no—
377#41—41no—
378#41—42no—
379#41—41no—
380#41—41no—
381#41—42no—
382#41—42no—
383#41—42no—
384#41—42no—
385#41—42no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—82no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4C2×C22⋊C412yes—
452C22×C4C2×C22⋊C412yes—
453C22×C4C4×D412yes—
454C22×C4C4×D412yes—
455C22×C4C4×D411yescharacteristic
456C22×C4C4×D411yescharacteristic
457C22×C4#12012yes—
458C22×C4#12012yes—
459C22×C4#12112yes—
460C22×C4#12112yes—
461C22×C4#12112yes—
462C22×C4#12312yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473M16—24no—
474M16—24no—
475M16—24no—
476M16—24no—
477M16—48no—
478M16—42no—
479M16—42no—
480M16—42no—
481M16—42no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—48no—
499C2×C8—42no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—42no—
503C2×C8—42no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—42no—
507C2×C8—42no—
508C2×C8—42no—
509C2×C8—42no—
510C2×C8—42no—
511C2×C8—42no—
512C2×C8—42no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4C2×C22⋊C412yes—
526C4⋊C4C2×C22⋊C412yes—
527C4⋊C4C2×C22⋊C412yes—
528C4⋊C4C2×C22⋊C412yes—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—24no—
546C42—24no—
547C42—24no—
548C42—24no—
549C42—48no—
550C42—41no—
551C42—41no—
552C42—42no—
553C42—42no—
554C23—22no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—24no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—42no—
575C23—41no—
576C23—41no—
577C23—42no—
578C23—41no—
579C23#38612yes—
580C23#38612yes—
581C23#44312yes—
582C23#44412yes—
583C23C2×C4×D411yescharacteristic
584C23C2×#12111yescharacteristic
585C23C2×#12311yescharacteristic
586C23#55112yes—
587C23#55212yes—
588Q8—41no—
589Q8—42no—
590Q8—41no—
591D4—41no—
592D4—42no—
593D4—41no—
594D4—84no—
595D4—84no—
596D4—81no—
597D4—81no—
598D4—82no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—432no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—84no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—82no—
646C2×C4#38612yes—
647C2×C4#38612yes—
648C2×C4C22×C22⋊C411yescharacteristic
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C4×#4112yes—
651C2×C4#52012yes—
652C2×C4C2×#12011yescharacteristic
653C2×C4C2×#12111yescharacteristic
654C2×C4#52511yescharacteristic
655C2×C4#52511yescharacteristic
656C2×C4#53212yes—
657C2×C4#53511yescharacteristic
658C2×C4#53611yescharacteristic
659C2×C4#53712yes—
660C2×C4#54612yes—
661C2×C4#54612yes—
662C2×C4#54712yes—
663C2×C4#54712yes—
664C2×C4#55112yes—
665C2×C4#55212yes—
666C8—41no—
667C8—41no—
668C8—44no—
669C8—42no—
670C8—42no—
671C8—44no—
672C8—41no—
673C8—41no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—22no—
679C22—22no—
680C22—24no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—22no—
685C22—21no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—44no—
690C22—44no—
691C22—48no—
692C22—42no—
693C22—42no—
694C22—81no—
695C22—82no—
696C22—82no—
697C22—81no—
698C22—81no—
699C22#187614yes—
700C22C2×#38611yescharacteristic
701C22#235912yes—
702C22#241111yescharacteristic
703C22#241311yescharacteristic
704C22#249812yes—
705C22C2×#44311yescharacteristic
706C22C2×#44411yescharacteristic
707C22#295612yes—
708C4—22no—
709C4—22no—
710C4—48no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—81no—
716C4—81no—
717C4—81no—
718C4#229711yescharacteristic
719C4#231111yescharacteristic
720C4#235912yes—
721C4#241111yescharacteristic
722C4#241311yescharacteristic
723C4#243012yes—
724C2—21no—
725C2—24no—
726C2—21no—
727C2—22no—
728C2—42no—
729C2—161no—
730C2#1413511yescharacteristic
731C2C2×#187611yescharacteristic
732C2#2091811yescharacteristic
733C2#2294812yes—
734C2#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
2. C2×#2924C2semidirect
5. C22×#339C2semidirect
6. #11221C2semidirect
7. #11068C2semidirect
8. #10934C2semidirect
9. #10909C2semidirect
10. #10734C2semidirect
11. #10712C2semidirect
18. C2×#484C4semidirect
19. C2×#481C4semidirect
20. C2×#485C4semidirect
21. C2×#485C4semidirect
22. C2×#482C4semidirect
23. C2×#482C4semidirect
24. C2×#483C4semidirect
25. C2×#480C4semidirect
117. C2×#107C2×C4semidirect
118. C2×#107C2×C4semidirect
119. C2×#107C2×C4semidirect
120. C2×#107C2×C4semidirect
121. C2×#106C2×C4semidirect
122. C2×#106C2×C4semidirect
123. C2×#106C2×C4semidirect
124. C2×#106C2×C4semidirect
136. C2×#103C2×C4semidirect
137. C2×#103C2×C4semidirect
138. C2×#103C2×C4semidirect
139. C2×#103C2×C4semidirect
140. C2×#102C2×C4semidirect
141. C2×#102C2×C4semidirect
142. C2×#102C2×C4semidirect
143. 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×#29242
C2×#18962
#208972
C22×#3392
#112212
#110682
#109342
#109092
#107342
#107122