Finite Group Databasebeta

← #4308097

Subgroups — #4308097

Order 512.

2546 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic
3#14852C214yesmaximal, complemented
4#14797C214yesmaximal
5#14787C214yesmaximal, complemented
6#14742C214yesmaximal
7#14714C211yesmaximal, characteristic, complemented
8#14707C211yesmaximal, characteristic, complemented
9#14699C212yesmaximal, complemented
10#14697C212yesmaximal
11#14343C212yesmaximal
12#14340C211yesmaximal, characteristic
13#14282C211yesmaximal, characteristic, complemented
14#14276C211yesmaximal, characteristic, complemented
15#14266C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2628—28no—
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C2×#391C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#387C2214yes—
30C2×#385C2214yes—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C4×#117—28no—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×#376C2211yescharacteristic
42C2×#376C2211yescharacteristic
43C2×C43C2211yescharacteristic
44C2×C43C2211yescharacteristic
45#1446C2214yes—
46#1445C2214yes—
47#1445C2214yes—
48#1445C2214yes—
49#1445C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2212yes—
60#1439C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C23×Q8C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117D418yes—
88C2×#117Q818yescomplemented
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C42—24no—
108C22×C42—24no—
109C22×C42—24no—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#400—28no—
122#399—28no—
123#395—28no—
124#393—28no—
125#391—48no—
126#389—28no—
127#389—28no—
128#389—28no—
129#387—48no—
130#385—48no—
131#384—24no—
132#382—24no—
133#382—24no—
134#380—28no—
135#380—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—28no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—28no—
140#376—24no—
141#376—24no—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C43—24no—
170C43—24no—
171C25C2411yescharacteristic
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—44no—
176C22×Q8—44no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4C2×D414yes—
189C23×C4C2×Q814yes—
190C23×C4#4114yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C4⋊Q8—416no—
206#117—232no—
207#117—232no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×Q818yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4C2×D414yes—
258C2×C22⋊C4C2×D414yes—
259C2×C22⋊C4C2×Q818yes—
260C2×C42—232no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42C2×D414yes—
287C2×C42C2×D414yes—
288C2×C42C2×Q814yes—
289C2×C42C2×Q814yes—
290C2×C42#4114yes—
291C2×C42#4114yes—
292C2×C42#4114yes—
293C2×C42#4114yes—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24C22×D412yes—
330C24C22×Q812yes—
331C24C2×#4112yes—
332C24C2511yescharacteristic
333C2×Q8—416no—
334C2×Q8—48no—
335C2×Q8—88no—
336C2×Q8—84no—
337C2×Q8—84no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—216no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
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—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
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—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4C22⋊Q8116yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×Q814yes—
425C22×C4C22×Q814yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q812yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C42−1+414yes—
442C4⋊C4—216no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C4⋊C4—88no—
454C4⋊C4—88no—
455C22⋊C4—216no—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—416no—
459C22⋊C4—88no—
460C42—216no—
461C42—28no—
462C42—28no—
463C42—28no—
464C42—28no—
465C42—28no—
466C42—48no—
467C42—48no—
468C42—48no—
469C42—48no—
470C42#123116yes—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C2×C22⋊Q814yes—
494C23C23×D411yescharacteristic
495C23C23×Q811yescharacteristic
496C23C22×#4111yescharacteristic
497C23C22×#4111yescharacteristic
498C23C22×#4111yescharacteristic
499C23C22×#4111yescharacteristic
500C23C22×#4111yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×2−1+412yes—
503C23#58512yes—
504C23#58512yes—
505Q8—816no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—232no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—432no—
541C2×C4—42no—
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—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4C2×C22⋊Q814yes—
588C2×C4C2×C22⋊Q814yes—
589C2×C4C2×#12314yes—
590C2×C4C2×#12314yes—
591C2×C4C2×#12314yes—
592C2×C4#52514yes—
593C2×C4#52914yes—
594C2×C4#52914yes—
595C2×C4#53314yes—
596C2×C4#53314yes—
597C2×C4#54218yes—
598C22—24no—
599C22—28no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22C22×C22⋊Q811yescharacteristic
610C22C22×#12311yescharacteristic
611C22C2×#52511yescharacteristic
612C22C2×#52911yescharacteristic
613C22C2×#52911yescharacteristic
614C22C2×#53311yescharacteristic
615C22C2×#53311yescharacteristic
616C22C2×#54212yes—
617C22#345512yes—
618C22#346012yes—
619C22#346012yes—
620C22#346112yes—
621C22#346112yes—
622C22#348812yes—
623C22#348912yes—
624C22#349412yes—
625C22#349512yes—
626C22#350312yes—
627C22#350412yes—
628C22#350811yescharacteristic
629C22#350911yescharacteristic
630C22#351011yescharacteristic
631C22#351011yescharacteristic
632C4—24no—
633C4—24no—
634C4—28no—
635C4—24no—
636C4—42no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—42no—
641C4—84no—
642C4—84no—
643C4—84no—
644C4—84no—
645C4#285518yes—
646C2—44no—
647C2#3461912yes—
648C2#3525411yescharacteristic
649C2#3525712yes—
650C2#3526012yes—
651C2#3526811yescharacteristic
652C2#3528212yes—
653C2#3528311yescharacteristic
654C2#3530711yescharacteristic
655C2#3531411yescharacteristic
656C2#3533711yescharacteristic
657C2C2×#285511yescharacteristic
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
3. #14852C2semidirect
5. #14787C2semidirect
7. #14714C2semidirect
8. #14707C2semidirect
9. #14699C2semidirect
13. #14282C2semidirect
14. #14276C2semidirect
15. #14266C2semidirect
88. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×C4×#1172
#148522
#147972
#147872
#147422
#147142
#147072
#146992
#146972
#143432
#143402
#142822
#142762
#142662