Finite Group Databasebeta

← #950059

Subgroups — #950059

Order 512.

2394 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17306C212yesmaximal, complemented
2#17304C212yesmaximal, complemented
3#16520C214yesmaximal, complemented
4#16404C214yesmaximal, complemented
5#15506C212yesmaximal
6#15429C212yesmaximal
7#15152C211yesmaximal, characteristic, complemented
8#15131C212yesmaximal, complemented
9#15128C211yesmaximal, characteristic, complemented
10#15125C212yesmaximal, complemented
11#14266C212yesmaximal, complemented
12#14130C212yesmaximal, complemented
13#13907C212yesmaximal, complemented
14#13872C212yesmaximal
15#13847C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C2×#400C2212yes—
20C2×#396C2214yes—
21C2×#395C2212yes—
22C2×#393C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#386C2214yes—
27C2×#385C2214yes—
28C2×#384C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2214yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×#376C2211yescharacteristic
37#1446C2214yes—
38#1446C2212yes—
39#1445C2214yes—
40#1445C2211yescharacteristic
41#1445C2211yescharacteristic
42#1445C2212yes—
43#1444C2212yes—
44#1444C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1438C2211yescharacteristic
64#1438C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C23×Q8C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2314yes—
79C2×C4⋊Q8—44no—
80C2×C4⋊Q8—42no—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C2×C4×D4Q818yescomplemented
84C22×C4⋊C4—28no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#400—44no—
114#396—48no—
115#395—44no—
116#393—48no—
117#389—28no—
118#389—44no—
119#389—44no—
120#386—48no—
121#385—48no—
122#384—42no—
123#380—28no—
124#380—48no—
125#380—44no—
126#380—44no—
127C4×C4⋊C4—42no—
128C4×C22⋊C4—44no—
129C4×C22⋊C4—44no—
130#376—42no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C25C2412yes—
161C22×Q8—24no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—44no—
165C22×Q8—42no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4C2×Q814yes—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4C2×Q818yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C4⋊Q8—416no—
200C4×D4—416no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—432no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×Q814yes—
241C2×C4⋊C4#4118yes—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4C2×Q818yes—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42C2×Q814yes—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—42no—
282#95—41no—
283#95—41no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—42no—
288#95—41no—
289#95—41no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—42no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24C22×Q814yes—
318C24H(5,𝔽2)14yes—
319C242−1+414yes—
320C24C2511yescharacteristic
321C2×Q8—48no—
322C2×Q8—88no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8—88no—
326C2×D4—216no—
327C2×D4—44no—
328C2×D4—48no—
329C2×D4—44no—
330C2×D4—44no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—21no—
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—24no—
368C22×C4—22no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
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—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4C22×Q812yes—
419C22×C4C22×Q812yes—
420C22×C4C22×Q814yes—
421C22×C4C22×Q812yes—
422C22×C4C2×#4118yes—
423C22×C4C2×#4114yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+412yes—
430C22×C42−1+412yes—
431C4⋊C4—216no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—88no—
441C4⋊C4—84no—
442C4⋊C4—88no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C22⋊C4—48no—
446C22⋊C4—416no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C42—28no—
454C42—48no—
455C42—44no—
456C42—44no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23#54418yes—
492C23C23×Q811yescharacteristic
493C23C22×#4112yes—
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)12yes—
496C23C2×H(5,𝔽2)11yescharacteristic
497C23C2×H(5,𝔽2)12yes—
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×2−1+412yes—
501C23C2×2−1+411yescharacteristic
502C23C2×2−1+411yescharacteristic
503Q8—816no—
504D4—416no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—432no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—832no—
588C2×C4#53818yes—
589C2×C4#54314yes—
590C2×C4#55418yes—
591C2×C4#55514yes—
592C2×C4#56014yes—
593C22—28no—
594C22—28no—
595C22—28no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22C2×#53812yes—
609C22C2×#54311yescharacteristic
610C22C2×#54412yes—
611C22C2×#55412yes—
612C22C2×#55511yescharacteristic
613C22#350311yescharacteristic
614C22#350311yescharacteristic
615C22#351912yes—
616C22#352012yes—
617C22#352212yes—
618C22#352312yes—
619C22C2×#56011yescharacteristic
620C22#353611yescharacteristic
621C22#353612yes—
622C22#353611yescharacteristic
623C22#353612yes—
624C22#353612yes—
625C22#353711yescharacteristic
626C22#353712yes—
627C22#353811yescharacteristic
628C22#353812yes—
629C22#354911yescharacteristic
630C22#355011yescharacteristic
631C4—42no—
632C4—42no—
633C4—44no—
634C4—42no—
635C4—48no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—84no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4—84no—
644C4#278418yes—
645C2—48no—
646C2#3730611yescharacteristic
647C2#3747711yescharacteristic
648C2#3809812yes—
649C2#3809912yes—
650C2#4260612yes—
651C2#4286212yes—
652C2#4537811yescharacteristic
653C2#4538411yescharacteristic
654C2C2×#278411yescharacteristic
655C2#5298811yescharacteristic
656C2#5299311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17306C2semidirect
2. #17304C2semidirect
3. #16520C2semidirect
4. #16404C2semidirect
7. #15152C2semidirect
8. #15131C2semidirect
9. #15128C2semidirect
10. #15125C2semidirect
11. #14266C2semidirect
12. #14130C2semidirect
13. #13907C2semidirect
83. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#173062
#173042
#165202
#164042
#155062
#154292
#151522
#151312
#151282
#151252
#142662
#141302
#139072
#138722
#138472