Finite Group Databasebeta

← #790823

Subgroups — #790823

Order 512.

2606 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2572C212yesmaximal
2#16653C212yesmaximal, complemented
3#16116C212yesmaximal, complemented
4#16021C214yesmaximal, complemented
5#15952C212yesmaximal
6#15152C211yesmaximal, characteristic, complemented
7#15125C212yesmaximal, complemented
8#14768C214yesmaximal, complemented
9#14155C211yesmaximal, characteristic, complemented
10#13921C212yesmaximal, complemented
11#13904C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13#13872C212yesmaximal
14C4×#394C211yesmaximal, characteristic
15C22⋊C4×C4⋊C4C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19#2572—216no—
20C2×#400C2212yes—
21C2×#394C2211yescharacteristic
22C2×#393C2214yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#388C2214yes—
26C2×#386C2214yes—
27C2×#384C2211yescharacteristic
28C2×#383C2214yes—
29C2×#381C2214yes—
30C2×#381C2214yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×C4×C22⋊C4C2212yes—
40#1446C2214yes—
41#1445C2214yes—
42#1445C2212yes—
43#1445C2212yes—
44#1445C2212yes—
45#1445C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2211yescharacteristic
51#1442C2212yes—
52#1441C2211yescharacteristic
53#1441C2214yes—
54#1441C2212yes—
55#1440C2212yes—
56#1439C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2211yescharacteristic
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2211yescharacteristic
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2314yes—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4×D4—28no—
84C2×C4×D4—42no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113#400—28no—
114#394D418yes—
115#393—216no—
116#391—28no—
117#391—44no—
118#388—216no—
119#386—216no—
120#384—42no—
121#383—216no—
122#381—216no—
123#381—48no—
124#380—28no—
125#380—44no—
126#380—44no—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4—42no—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—216no—
132C4×C22⋊C4—28no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C25C2412yes—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4C2×D414yes—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4#4118yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C4⋊Q8—416no—
205C4×D4—416no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
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—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—432no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4#4118yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4C2×D418yes—
261C2×C22⋊C4C2×D418yes—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—28no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—44no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42#4114yes—
278#95—28no—
279#95—28no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—28no—
294#95—44no—
295#95—42no—
296#95—41no—
297#95—44no—
298#95—44no—
299#95—41no—
300#95—41no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95C2×D414yes—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24C22×D414yes—
321C24C2×#4114yes—
322C242−1+414yes—
323C24C2511yescharacteristic
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—84no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×D4—216no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—48no—
335C2×D4—44no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—88no—
421C22×C4C22×D412yes—
422C22×C4C22×D414yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C2×#4118yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+412yes—
436C4⋊C4—216no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—416no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C22⋊C4—216no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C42—28no—
456C42—416no—
457C42—48no—
458C42—44no—
459C42—44no—
460C42—84no—
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—24no—
476C23—24no—
477C23—24no—
478C23—24no—
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—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23#54718yes—
496C23C23×D411yescharacteristic
497C23C22×#4111yescharacteristic
498C23C22×#4112yes—
499C23C22×#4112yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×2−1+412yes—
504C23C2×2−1+411yescharacteristic
505C23#58512yes—
506C23#58511yescharacteristic
507Q8—816no—
508D4—416no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—432no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—44no—
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—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—832no—
591C2×C4#52918yes—
592C2×C4#53814yes—
593C2×C4#54714yes—
594C2×C4#55114yes—
595C2×C4#55118yes—
596C2×C4#55514yes—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22C2×#52912yes—
613C22C2×#53811yescharacteristic
614C22C2×#54711yescharacteristic
615C22C2×#54712yes—
616C22D4×#4112yes—
617C22C2×#55111yescharacteristic
618C22C2×#55112yes—
619C22C2×#55511yescharacteristic
620C22#349412yes—
621C22#349512yes—
622C22#349512yes—
623C22#349611yescharacteristic
624C22#349811yescharacteristic
625C22#350611yescharacteristic
626C22#350712yes—
627C22#352012yes—
628C22#352312yes—
629C22#352412yes—
630C22#353611yescharacteristic
631C22#353811yescharacteristic
632C22#355111yescharacteristic
633C22#356111yescharacteristic
634C22#356112yes—
635C4—24no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—48no—
642C4—42no—
643C4—81no—
644C4—84no—
645C4—84no—
646C4—84no—
647C4—82no—
648C4#257218yescomplemented
649C2—48no—
650C2C2×#257211yescharacteristic
651C2#3582812yes—
652C2#3611612yes—
653C2#3937511yescharacteristic
654C2#4295811yescharacteristic
655C2#4511511yescharacteristic
656C2#4511612yes—
657C2#4538411yescharacteristic
658C2#5919811yescharacteristic
659C2#5927312yes—
660C2#5984111yescharacteristic
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
2. #16653C2semidirect
3. #16116C2semidirect
4. #16021C2semidirect
6. #15152C2semidirect
7. #15125C2semidirect
8. #14768C2semidirect
9. #14155C2semidirect
10. #13921C2semidirect
11. #13904C2semidirect
12. #13876C2semidirect
15. C22⋊C4×C4⋊C4C2semidirect
648. C4#2572semidirect

Maximal subgroups

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

subgroupindex
C2×#25722
#166532
#161162
#160212
#159522
#151522
#151252
#147682
#141552
#139212
#139042
#138762
#138722
C4×#3942
C22⋊C4×C4⋊C42