Finite Group Databasebeta

← #790828

Subgroups — #790828

Order 512.

2422 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16933C212yesmaximal, complemented
2#16229C212yesmaximal, complemented
3#15586C214yesmaximal, complemented
4#15506C212yesmaximal
5#15152C211yesmaximal, characteristic, complemented
6#15125C212yesmaximal, complemented
7#14832C214yesmaximal, complemented
8#14224C212yesmaximal
9#14161C211yesmaximal, characteristic, complemented
10#13936C212yesmaximal, complemented
11#13924C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13#13872C212yesmaximal
14C4×#394C211yesmaximal, characteristic
15C22⋊C4×C4⋊C4C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C2×#400C2212yes—
20C2×#394C2211yescharacteristic
21C2×#393C2214yes—
22C2×#388C2214yes—
23C2×#386C2214yes—
24C2×#384C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2214yes—
34#1446C2214yes—
35#1446C2212yes—
36#1446C2212yes—
37#1445C2214yes—
38#1445C2212yes—
39#1445C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2211yescharacteristic
44#1442C2214yes—
45#1442C2212yes—
46#1441C2211yescharacteristic
47#1441C2214yes—
48#1441C2212yes—
49#1440C2212yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2211yescharacteristic
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69#1434C2214yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C4×#95C2211yescharacteristic
75C23×Q8C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2314yes—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×C4×D4—28no—
83C2×C4×D4—42no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111C22×C42C2312yes—
112#400—44no—
113#394Q818yescomplemented
114#393—48no—
115#388—48no—
116#386—48no—
117#384—42no—
118#380—28no—
119#380—44no—
120#380—44no—
121C4×C4⋊C4—28no—
122C4×C4⋊C4—42no—
123C4×C4⋊C4—42no—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—28no—
126C4×C22⋊C4—48no—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C25C2412yes—
158C22×Q8—24no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8—44no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4C2×Q814yes—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—28no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4#4118yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2411yescharacteristic
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C4⋊Q8—416no—
197C4×D4—416no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
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—24no—
216C2×C4⋊C4—432no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C2×Q814yes—
229C2×C4⋊C4#4118yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4C2×Q818yes—
253C2×C22⋊C4C2×Q818yes—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—28no—
264C2×C42—24no—
265C2×C42—44no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42—42no—
269C2×C42#4114yes—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—42no—
278#95—44no—
279#95—44no—
280#95—42no—
281#95—44no—
282#95—44no—
283#95—41no—
284#95—44no—
285#95—41no—
286#95—41no—
287#95—42no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95C2×Q814yes—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C22×Q814yes—
313C24C2×#4114yes—
314C24H(5,𝔽2)14yes—
315C24C2511yescharacteristic
316C2×Q8—48no—
317C2×Q8—88no—
318C2×Q8—84no—
319C2×Q8—84no—
320C2×Q8—88no—
321C2×D4—216no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—48no—
325C2×D4—44no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—416no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—416no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4C22×Q814yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q812yes—
414C22×C4C22×Q812yes—
415C22×C4C2×#4118yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4114yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+412yes—
426C4⋊C4—216no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—416no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C22⋊C4—216no—
438C22⋊C4—216no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—48no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—28no—
446C42—48no—
447C42—44no—
448C42—44no—
449C42—84no—
450C42—88no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
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—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23#55418yes—
486C23C23×Q811yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4112yes—
489C23C22×#4112yes—
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)12yes—
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×2−1+411yescharacteristic
495C23#58512yes—
496C23#58511yescharacteristic
497Q8—816no—
498D4—416no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
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—22no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—432no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—832no—
581C2×C4#52918yes—
582C2×C4#53814yes—
583C2×C4#55114yes—
584C2×C4#55118yes—
585C2×C4#55414yes—
586C2×C4#55514yes—
587C22—28no—
588C22—28no—
589C22—28no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22C2×#52912yes—
603C22C2×#53811yescharacteristic
604C22C2×#55111yescharacteristic
605C22C2×#55112yes—
606C22C2×#55412yes—
607C22C2×#55411yescharacteristic
608C22C2×#55511yescharacteristic
609C22Q8×#4112yes—
610C22#349412yes—
611C22#349412yes—
612C22#349512yes—
613C22#350111yescharacteristic
614C22#350311yescharacteristic
615C22#350612yes—
616C22#350711yescharacteristic
617C22#351912yes—
618C22#352112yes—
619C22#352212yes—
620C22#353611yescharacteristic
621C22#353711yescharacteristic
622C22#355111yescharacteristic
623C22#355811yescharacteristic
624C22#355812yes—
625C4—24no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—48no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—84no—
634C4—84no—
635C4—81no—
636C4—84no—
637C4—82no—
638C4#257218yes—
639C2—48no—
640C2C2×#257211yescharacteristic
641C2#3583312yes—
642C2#3655112yes—
643C2#3963211yescharacteristic
644C2#4298211yescharacteristic
645C2#4518812yes—
646C2#4525111yescharacteristic
647C2#4537811yescharacteristic
648C2#5919911yescharacteristic
649C2#5927412yes—
650C2#5993511yescharacteristic
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. #16933C2semidirect
2. #16229C2semidirect
3. #15586C2semidirect
5. #15152C2semidirect
6. #15125C2semidirect
7. #14832C2semidirect
9. #14161C2semidirect
10. #13936C2semidirect
11. #13924C2semidirect
12. #13876C2semidirect
15. C22⋊C4×C4⋊C4C2semidirect
113. #394Q8semidirect

Maximal subgroups

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

subgroupindex
#169332
#162292
#155862
#155062
#151522
#151252
#148322
#142242
#141612
#139362
#139242
#138762
#138722
C4×#3942
C22⋊C4×C4⋊C42