Finite Group Databasebeta

← #378575

Subgroups — #378575

Order 512.

823 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23666C211yesmaximal, characteristic
2#21148C211yesmaximal, characteristic
3#20439C211yesmaximal, characteristic
4#11291C211yesmaximal, characteristic
5#11221C211yesmaximal, characteristic
6#11209C211yesmaximal, characteristic
7#10909C211yesmaximal, characteristic
8#10887C211yesmaximal, characteristic
9#10887C211yesmaximal, characteristic
10#10887C211yesmaximal, characteristic
11#10886C211yesmaximal, characteristic
12#10882C211yesmaximal, characteristic
13#10876C211yesmaximal, characteristic
14#10853C211yesmaximal, characteristic
15#10706C211yesmaximal, characteristic
16C2×#427C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#405C2211yescharacteristic
23C2×#405C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2296C2211yescharacteristic
28#2296C2211yescharacteristic
29#2088C411yescharacteristic, complemented
30#2088C411yescharacteristic, complemented
31#2088C411yescharacteristic, complemented
32#2088C411yescharacteristic, complemented
33#2087C411yescharacteristic, complemented
34#2087C411yescharacteristic, complemented
35#2086C411yescharacteristic, complemented
36#2086C411yescharacteristic, complemented
37#1940—21no—
38#1940—21no—
39#1939—21no—
40#1939—21no—
41#1771—21no—
42#1771—21no—
43#1770—21no—
44C8×C4⋊C4—21no—
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49#1735C2211yescharacteristic
50#1735C2211yescharacteristic
51#1735C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#427—41no—
83#427—41no—
84C2×#107C2×C411yescharacteristic
85C2×#107C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88#424—22no—
89#424—22no—
90#424—22no—
91#424—41no—
92#424—41no—
93#424Q814yes—
94#423—22no—
95#423—41no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—41no—
105C2×C4⋊C8—41no—
106C2×C4⋊C8C2×C411yescharacteristic
107C2×C4⋊C8C2×C411yescharacteristic
108C2×C4⋊C8C2×C411yescharacteristic
109C2×C4⋊C8C2×C411yescharacteristic
110#405—22no—
111#405—22no—
112#405—22no—
113#405—22no—
114#405—41no—
115#405—41no—
116C4×M16—22no—
117C4×M16—41no—
118C2×#97—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8—22no—
126C2×C4×C8—21no—
127C2×C4×C8—41no—
128C2×C4×C8—41no—
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C2×#95—21no—
146C2×#95—21no—
147#341—22no—
148#341—22no—
149#341—41no—
150#341—41no—
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#340C2×C411yescharacteristic
162#340C2×C411yescharacteristic
163#339—22no—
164#339—22no—
165#339—22no—
166#339—22no—
167#339—22no—
168#339—41no—
169#339—41no—
170#339Q814yes—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—22no—
176#336—41no—
177#336—41no—
178#336D411yescharacteristic
179#336D411yescharacteristic
180#336D411yescharacteristic
181#336D411yescharacteristic
182C23×C4—21no—
183C23×C4—21no—
184C23×C4#4111yescharacteristic
185C23×C4#4111yescharacteristic
186C23×C4C2411yescharacteristic
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16C2×Q812yes—
207C2×M16C2×Q812yes—
208C2×M16#4112yes—
209C2×M16#4112yes—
210C2×M16#4112yes—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8C22×C411yescharacteristic
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C22×C411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2×Q812yes—
246C22×C8#4112yes—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117C2×Q812yes—
260#117C2×Q812yes—
261#117#4112yes—
262#117#4112yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×Q812yes—
307#107—41no—
308#107—41no—
309#106—41no—
310#106—41no—
311C4⋊C8—24no—
312C4⋊C8—24no—
313C4⋊C8—41no—
314C4⋊C8—41no—
315C4⋊C8—41no—
316C4⋊C8—41no—
317C4⋊C8—42no—
318C4⋊C8—42no—
319C4⋊C8—42no—
320C4⋊C8—42no—
321C4⋊C8—42no—
322C4⋊C8—42no—
323#97—24no—
324#97—42no—
325#97—42no—
326#97—42no—
327C4×C8—24no—
328C4×C8—42no—
329C4×C8—42no—
330C4×C8—42no—
331C4×C8—44no—
332#95—24no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—42no—
342C24C2×#4111yescharacteristic
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×Q811yescharacteristic
380C22×C4C22×Q811yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C42−1+411yescharacteristic
391M16—22no—
392M16—22no—
393M16—22no—
394M16—22no—
395M16—22no—
396M16—22no—
397M16—22no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16#12514yes—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—22no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8C22⋊Q812yes—
458C2×C8C22⋊Q812yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4C4×Q812yes—
480C4⋊C4C4×Q812yes—
481C4⋊C4C4×Q812yes—
482C4⋊C4C4×Q812yes—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—22no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23C2×C4×D411yescharacteristic
520C23#52911yescharacteristic
521C23#53311yescharacteristic
522C23#55211yescharacteristic
523C23#55311yescharacteristic
524C23#55311yescharacteristic
525C23#55411yescharacteristic
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4#44312yes—
580C2×C4C2×C4×Q811yescharacteristic
581C2×C4C2×C4×Q811yescharacteristic
582C2×C4C4×#4111yescharacteristic
583C2×C4C4×#4111yescharacteristic
584C2×C4#51811yescharacteristic
585C2×C4#52011yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4#52511yescharacteristic
588C2×C4C2×#12511yescharacteristic
589C2×C4#52911yescharacteristic
590C2×C4#52911yescharacteristic
591C2×C4#53311yescharacteristic
592C2×C4#53811yescharacteristic
593C2×C4#53911yescharacteristic
594C2×C4#54011yescharacteristic
595C2×C4#54211yescharacteristic
596C2×C4#54611yescharacteristic
597C2×C4#54611yescharacteristic
598C2×C4#54611yescharacteristic
599C2×C4D4×Q811yescharacteristic
600C2×C4#55211yescharacteristic
601C2×C4#55311yescharacteristic
602C2×C4#55411yescharacteristic
603C2×C4#55411yescharacteristic
604C2×C4#55411yescharacteristic
605C2×C4#55511yescharacteristic
606C2×C4#55711yescharacteristic
607C2×C4Q8211yescharacteristic
608C8—22no—
609C8—22no—
610C8—24no—
611C8—41no—
612C8—41no—
613C8—41no—
614C8—41no—
615C8—42no—
616C8—41no—
617C8—41no—
618C8—42no—
619C8—81no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22C4×C22⋊Q811yescharacteristic
633C22#236011yescharacteristic
634C22#236411yescharacteristic
635C22#250711yescharacteristic
636C22#252911yescharacteristic
637C22#258711yescharacteristic
638C22#259011yescharacteristic
639C22C2×#44311yescharacteristic
640C22#295511yescharacteristic
641C22#295611yescharacteristic
642C22#295711yescharacteristic
643C4—22no—
644C4—44no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4C4×#12511yescharacteristic
658C4#233511yescharacteristic
659C4#236911yescharacteristic
660C4#237011yescharacteristic
661C4#245011yescharacteristic
662C4#245111yescharacteristic
663C4#247311yescharacteristic
664C4#247611yescharacteristic
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2#1421711yescharacteristic
670C2#2162711yescharacteristic
671C2#2162911yescharacteristic
672C2#2298211yescharacteristic
673C2#2298411yescharacteristic
674C2#2314611yescharacteristic
675C2#2314811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #2088C4semidirect
30. #2088C4semidirect
31. #2088C4semidirect
32. #2088C4semidirect
33. #2087C4semidirect
34. #2087C4semidirect
35. #2086C4semidirect
36. #2086C4semidirect

Maximal subgroups

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

subgroupindex
#236662
#211482
#204392
#112912
#112212
#112092
#109092
#108872
#108862
#108822
#108762
#108532
#107062