Finite Group Databasebeta

← #373825

Subgroups — #373825

Order 512.

1482 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
1#34183C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic, complemented
3C2×#1860C212yesmaximal
4C2×#1806C211yesmaximal, characteristic
5C4×#417C211yesmaximal, characteristic
6#13911C211yesmaximal, characteristic, complemented
7#11339C212yesmaximal, complemented
8#11276C212yesmaximal, complemented
9#10857C211yesmaximal, characteristic, complemented
10#10851C211yesmaximal, characteristic, complemented
11#10781C212yesmaximal
12C2×#520C2212yes—
13C2×C4×#41C2211yescharacteristic
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C22×#104C2212yes—
17C2×#417C2211yescharacteristic
18C2×#417C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20Q8×C4⋊C4—21no—
21Q8×C4⋊C4—21no—
22D4×C4⋊C4—21no—
23D4×C4⋊C4—21no—
24D4×C4⋊C4—22no—
25D4×C4⋊C4—22no—
26#2317—22no—
27#2317C2212yes—
28#2285—21no—
29#2285C2211yescharacteristic
30C2×#384—21no—
31C2×#384C2211yescharacteristic
32C2×#382C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1949—22no—
36#1948—21no—
37#1947—21no—
38#1860—28no—
39#1806C418yescomplemented
40#1763—21no—
41#1762—21no—
42C4×#103—21no—
43C4×#102—21no—
44C2×#343C2212yes—
45C2×#343C2212yes—
46#1736C2211yescharacteristic
47#1736C2212yes—
48#1736C2211yescharacteristic
49C2×#339C2212yes—
50C2×#339C2212yes—
51C2×#339C2212yes—
52#1442—22no—
53C4⋊#95—22no—
54C4⋊#95C2212yes—
55C22×#41C2311yescharacteristic
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59#520—24no—
60#520—24no—
61#520—42no—
62#520—42no—
63C4×#41—22no—
64C4×#41—22no—
65C4×#41C2×C414yes—
66C4×#41Q814yes—
67C2×C4×Q8—22no—
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—24no—
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×#117—22no—
77C2×#117—21no—
78C2×#117C2312yes—
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C42—21no—
90C22×C42—21no—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93#428—24no—
94#428—42no—
95C2×#107—22no—
96C2×#106—22no—
97#423—22no—
98#423C2×C414yescomplemented
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×#104—24no—
102C2×#104—24no—
103C2×#104C2×C418yes—
104#417—22no—
105#417—22no—
106#417D414yes—
107#417Q814yes—
108C2×#103—21no—
109C2×#103—21no—
110C2×#102—21no—
111C2×#102—21no—
112C4×M16—22no—
113C4×M16—22no—
114C2×#97—21no—
115C2×C4×C8—21no—
116#389—24no—
117#389—24no—
118#389—22no—
119#389—22no—
120#385—24no—
121#385—24no—
122#385—24no—
123#384—22no—
124#384—22no—
125#384—22no—
126#384—21no—
127#384—21no—
128#384—21no—
129#384—21no—
130#384D411yescharacteristic
131#384D411yescharacteristic
132#382—22no—
133#382—24no—
134#382—42no—
135#382D412yes—
136#380—22no—
137#380—22no—
138#380—21no—
139#380—21no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4C2×C414yescomplemented
150C2×#95—21no—
151C2×#95—24no—
152C2×#95—21no—
153C2×#95—24no—
154#343—24no—
155#343—42no—
156#340—22no—
157#340—22no—
158#340—21no—
159#340—21no—
160#340—21no—
161#340—21no—
162#339—24no—
163#339—42no—
164#339C2×C418yescomplemented
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41C22×C412yes—
172C2×#41C2×D412yes—
173C2×#41C2×Q812yes—
174C2×#41C2×Q812yes—
175C2×#41#4112yes—
176C2×#41#4112yes—
177C22×Q8C2×D411yescharacteristic
178C22×D4—21no—
179C22×D4C2×D411yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—24no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D414yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—22no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—22no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—22no—
206C2×M16C22×C414yes—
207C2×M16C2×D412yes—
208C2×M16C2×Q812yes—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—44no—
215C4×Q8—44no—
216C4×Q8C4⋊C414yes—
217C4×Q8C4⋊C414yes—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—48no—
225C4×D4—48no—
226C4×D4—44no—
227C4×D4—44no—
228C4×D4C4⋊C414yes—
229C4×D4C4⋊C414yes—
230#117—21no—
231#117—21no—
232#117—24no—
233#117—24no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117—44no—
239#117—44no—
240#117C22×C412yes—
241#117C2×Q812yes—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—84no—
278C2×C4⋊C4—84no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×Q812yes—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—44no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42C22×C414yes—
307C2×C42C22×C412yes—
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×Q812yes—
313C2×C42#4112yes—
314C2×C42#4112yes—
315#107—44no—
316#106—44no—
317C4⋊C8—24no—
318C4⋊C8—24no—
319#104—216no—
320#103—24no—
321#103—24no—
322#102—24no—
323#102—24no—
324#97—42no—
325C4×C8—42no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—48no—
332#95—41no—
333#95—41no—
334#95—84no—
335C24—21no—
336C24C22×D411yescharacteristic
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—22no—
346#41—22no—
347#41—22no—
348#41—22no—
349#41C4×D414yes—
350#41C4×Q814yes—
351#41#12514yes—
352#41C4⋊Q814yes—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8C2×C4⋊C412yes—
357C2×Q8#12112yes—
358C2×Q8C22⋊Q812yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C2×C4⋊C412yes—
369C2×D4#12112yes—
370C2×D4C22⋊Q812yes—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4C4×D412yes—
418C22×C4C4×D412yes—
419C22×C4#12011yescharacteristic
420C22×C4#12011yescharacteristic
421C22×C4#12011yescharacteristic
422C22×C4#12011yescharacteristic
423C22×C4C22⋊Q812yes—
424C22×C4C22⋊Q812yes—
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×Q811yescharacteristic
430C22×C4C22×Q811yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4H(5,𝔽2)11yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C42−1+411yescharacteristic
438C22×C42−1+411yescharacteristic
439M16—24no—
440M16—24no—
441M16—22no—
442M16—22no—
443M16—22no—
444M16—22no—
445M16—42no—
446M16—41no—
447M16—41no—
448M16—42no—
449M16—41no—
450M16—41no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—42no—
459C2×C8—42no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4C2×C4⋊C412yes—
493C4⋊C4C2×C4⋊C412yes—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—48no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—24no—
507C42—48no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—82no—
513C42—82no—
514C42C2×C4⋊C412yes—
515C42C2×C4⋊C412yes—
516C42C4×D412yes—
517C42C4×D412yes—
518C42C22⋊Q812yes—
519C42C22⋊Q812yes—
520C42#12312yes—
521C42#12312yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C2×C4×D411yescharacteristic
534C23C2×#12011yescharacteristic
535C23C2×C22⋊Q811yescharacteristic
536C23D4211yescharacteristic
537C23#54611yescharacteristic
538C23#54811yescharacteristic
539C23D4×Q811yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—22no—
543Q8#38914yes—
544D4—24no—
545D4—24no—
546D4—24no—
547D4—24no—
548D4—22no—
549D4—22no—
550D4—22no—
551D4#38914yes—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—168no—
606C2×C4#38512yes—
607C2×C4#38512yes—
608C2×C4#38512yes—
609C2×C4#38512yes—
610C2×C4#45412yes—
611C2×C4#45512yes—
612C2×C4C22×C4⋊C411yescharacteristic
613C2×C4C2×C4×D411yescharacteristic
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×C4×Q811yescharacteristic
616C2×C4#52011yescharacteristic
617C2×C4#52011yescharacteristic
618C2×C4C2×#12111yescharacteristic
619C2×C4C2×C22⋊Q811yescharacteristic
620C2×C4C2×C22⋊Q811yescharacteristic
621C2×C4C2×#12311yescharacteristic
622C2×C4C2×#12511yescharacteristic
623C2×C4C2×C4⋊Q811yescharacteristic
624C2×C4#53511yescharacteristic
625C2×C4#53611yescharacteristic
626C2×C4#54411yescharacteristic
627C2×C4#54411yescharacteristic
628C2×C4D4211yescharacteristic
629C2×C4#54611yescharacteristic
630C2×C4#54711yescharacteristic
631C2×C4#54711yescharacteristic
632C2×C4#54811yescharacteristic
633C2×C4D4×Q811yescharacteristic
634C2×C4#55211yescharacteristic
635C2×C4#55211yescharacteristic
636C2×C4#55311yescharacteristic
637C2×C4#55311yescharacteristic
638C2×C4#55411yescharacteristic
639C2×C4#55411yescharacteristic
640C8—44no—
641C8—42no—
642C8—42no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—24no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22C2×#38511yescharacteristic
663C22#236011yescharacteristic
664C22#236111yescharacteristic
665C22#240811yescharacteristic
666C22#240911yescharacteristic
667C22#244011yescharacteristic
668C22#244211yescharacteristic
669C22C2×#45511yescharacteristic
670C22C2×#45411yescharacteristic
671C22#301911yescharacteristic
672C22#302011yescharacteristic
673C4—24no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4#186014yescomplemented
686C4C2×#38911yescharacteristic
687C4#231711yescharacteristic
688C4#236011yescharacteristic
689C4#236111yescharacteristic
690C4#244011yescharacteristic
691C4#244211yescharacteristic
692C4#248611yescharacteristic
693C4#248711yescharacteristic
694C2—21no—
695C2—21no—
696C2—21no—
697C2—21no—
698C2—44no—
699C2#1413011yescharacteristic
700C2C2×#186011yescharacteristic
701C2#2086411yescharacteristic
702C2#2423111yescharacteristic
703C2#2423211yescharacteristic
704C2#2432111yescharacteristic
705C2#2432211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21170C2semidirect
6. #13911C2semidirect
7. #11339C2semidirect
8. #11276C2semidirect
9. #10857C2semidirect
10. #10851C2semidirect
39. #1806C4semidirect
98. #423C2×C4semidirect
149. C4×C4⋊C4C2×C4semidirect
164. #339C2×C4semidirect
685. C4#1860semidirect

Maximal subgroups

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

subgroupindex
#341832
#211702
C2×#18602
C2×#18062
C4×#4172
#139112
#113392
#112762
#108572
#108512
#107812