Finite Group Databasebeta

← #766199

Subgroups — #766199

Order 512.

2710 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2396C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic
3#15930C212yesmaximal
4#15922C214yesmaximal
5#15916C212yesmaximal
6#15884C212yesmaximal, complemented
7#15857C214yesmaximal, complemented
8#15854C212yesmaximal, complemented
9#15201C212yesmaximal, complemented
10#15192C212yesmaximal, complemented
11#14021C212yesmaximal, complemented
12#14021C212yesmaximal, complemented
13#13939C212yesmaximal
14#13922C212yesmaximal
15#13777C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2396—28no—
21C2×#400C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#398C2212yes—
24C2×#396C2212yes—
25C2×#395C2211yescharacteristic
26C2×#393C2211yescharacteristic
27#2310—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44#1446C2214yes—
45#1446C2214yes—
46#1446C2214yes—
47#1445C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1437C2212yes—
61#1437C2212yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—28no—
80C23×Q8—42no—
81C23×Q8—42no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×Q8C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C2×C4×Q8—24no—
88C2×C4×Q8—42no—
89C2×C4×Q8—42no—
90C2×#117—24no—
91C2×#117—28no—
92C2×#117—42no—
93C2×#117—42no—
94C2×#117D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118#400—24no—
119#399—24no—
120#398—28no—
121#396—28no—
122#395—24no—
123#393—24no—
124#391—28no—
125#391—28no—
126#391—28no—
127#391—28no—
128#391—28no—
129#391—28no—
130#391—44no—
131#391—44no—
132#381—28no—
133#381—28no—
134#380—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—42no—
138C4×C4⋊C4—42no—
139C4×C22⋊C4—28no—
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142C2×#95C2312yes—
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C25C2411yescharacteristic
172C22×Q8—232no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—28no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—432no—
190C22×Q8—432no—
191C22×Q8—416no—
192C22×Q8—416no—
193C22×Q8#4118yes—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4C2×D414yes—
203C23×C4#4118yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4×Q8—416no—
219#117—416no—
220#117—416no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4C2×D414yes—
267C2×C22⋊C4C2×D414yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42C2×D414yes—
285C2×C42C2×D414yes—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24C22×D412yes—
325C24C2×#4114yes—
326C24C2511yescharacteristic
327C2×Q8—216no—
328C2×Q8—216no—
329C2×Q8—216no—
330C2×Q8—216no—
331C2×Q8—48no—
332C2×Q8—44no—
333C2×Q8—416no—
334C2×Q8—416no—
335C2×Q8—44no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—4128no—
339C2×Q8—44no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—48no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4#124116yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C2×#4118yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C42−1+414yes—
440C22×C42−1+414yes—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—88no—
457C42—28no—
458C42—28no—
459C42—48no—
460C42—48no—
461C42—48no—
462C42—84no—
463C42—84no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23C2×#12414yes—
485C23C23×D411yescharacteristic
486C23C22×#4112yes—
487C23C22×#4112yes—
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×2−1+412yes—
491C23C2×2−1+412yes—
492C23#58512yes—
493C23#58512yes—
494Q8—432no—
495Q8—416no—
496Q8—432no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—232no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—432no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4#52514yes—
580C2×C4C2×#12414yes—
581C2×C4C2×#12414yes—
582C2×C4#53414yes—
583C2×C4#55118yes—
584C2×C4#55118yes—
585C22—24no—
586C22—28no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22C2×#52511yescharacteristic
595C22C22×#12411yescharacteristic
596C22C2×#53411yescharacteristic
597C22C2×#55112yes—
598C22C2×#55112yes—
599C22#348912yes—
600C22#348912yes—
601C22#350712yes—
602C22#350712yes—
603C22#350912yes—
604C22#351012yes—
605C22#351612yes—
606C22#351812yes—
607C22#352312yes—
608C22#352412yes—
609C22#353811yescharacteristic
610C22#354011yescharacteristic
611C22#354112yes—
612C22#354712yes—
613C22#355011yescharacteristic
614C22#355111yescharacteristic
615C4—24no—
616C4—42no—
617C4—42no—
618C4—48no—
619C4—44no—
620C4—42no—
621C4—42no—
622C4—44no—
623C4—82no—
624C4—84no—
625C4—82no—
626C4—84no—
627C4—84no—
628C4#256118yes—
629C2—44no—
630C2C2×#256111yescharacteristic
631C2#3562812yes—
632C2#3566312yes—
633C2#3737411yescharacteristic
634C2#3737611yescharacteristic
635C2#4514812yes—
636C2#4514912yes—
637C2#4539012yes—
638C2#4539212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #15884C2semidirect
7. #15857C2semidirect
8. #15854C2semidirect
9. #15201C2semidirect
10. #15192C2semidirect
11. #14021C2semidirect
12. #14021C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
C2×#23102
#159302
#159222
#159162
#158842
#158572
#158542
#152012
#151922
#140212
#139392
#139222
#137772