Finite Group Databasebeta

← #761644

Subgroups — #761644

Order 512.

2194 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17012C212yesmaximal, complemented
2#16599C212yesmaximal
3#15196C211yesmaximal, characteristic, complemented
4#15181C212yesmaximal
5#15152C211yesmaximal, characteristic, complemented
6#15100C214yesmaximal, complemented
7#15095C214yesmaximal, complemented
8#14254C212yesmaximal, complemented
9#14157C212yesmaximal
10#13973C212yesmaximal, complemented
11#13973C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13#13876C212yesmaximal, complemented
14C4×#389C212yesmaximal, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C2×#400C2212yes—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#388C2214yes—
23C2×#388C2214yes—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2214yes—
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C43C2211yescharacteristic
37#1445C2212yes—
38#1444C2212yes—
39#1444C2212yes—
40#1442C2212yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2212yes—
47#1441C2212yes—
48#1441C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2212yes—
59#1437C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2211yescharacteristic
72#1434C2214yes—
73#1434C2214yes—
74C4×#95C2214yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C2×C4×D4—24no—
78C2×C4×D4—42no—
79C2×C4×D4—42no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111#400—28no—
112#390—24no—
113#390Q818yes—
114#389—28no—
115#389—28no—
116#388—48no—
117#388—48no—
118#384—42no—
119#384—42no—
120#380—28no—
121#380—28no—
122#380—44no—
123#380—44no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—42no—
126C4×C4⋊C4—48no—
127C4×C4⋊C4—44no—
128C4×C22⋊C4—28no—
129C4×C22⋊C4—28no—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C43—42no—
157C25C2412yes—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4C2×Q814yes—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C4×D4—416no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C2×Q814yes—
221C2×C4⋊C4#4118yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×Q818yes—
249C2×C22⋊C4C2×Q818yes—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—28no—
263C2×C42—41no—
264C2×C42—42no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42—44no—
268C2×C42—42no—
269C2×C42—41no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42C2×Q814yes—
273C2×C42#4114yes—
274C2×C42#4118yes—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—44no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—44no—
293#95—41no—
294#95—41no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24C22×Q814yes—
316C24C2511yescharacteristic
317C2×D4—28no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
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—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4C22×Q812yes—
400C22×C4C22×Q812yes—
401C22×C4C22×Q812yes—
402C22×C4C22×Q814yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4118yes—
405C22×C4C2×#4114yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4114yes—
414C22×C4H(5,𝔽2)14yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C4⋊C4—216no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—416no—
421C4⋊C4—416no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C22⋊C4—216no—
431C22⋊C4—216no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C42—28no—
440C42—216no—
441C42—44no—
442C42—416no—
443C42—44no—
444C42—48no—
445C42—44no—
446C42—44no—
447C42—88no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
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—
483C23C23×Q811yescharacteristic
484C23C22×#4111yescharacteristic
485C23C22×#4112yes—
486C23C22×#4112yes—
487C23C22×#4111yescharacteristic
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×H(5,𝔽2)12yes—
491C23#58511yescharacteristic
492C23#58511yescharacteristic
493C23#58512yes—
494D4—416no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
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—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
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—816no—
575C2×C4#52918yes—
576C2×C4#53314yes—
577C2×C4#53814yes—
578C2×C4#55318yes—
579C2×C4#55418yes—
580C2×C4#55514yes—
581C2×C4#55914yes—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×#52912yes—
598C22C2×#53311yescharacteristic
599C22C2×#53811yescharacteristic
600C22#345511yescharacteristic
601C22#346012yes—
602C22C2×#55312yes—
603C22C2×#55412yes—
604C22C2×#55511yescharacteristic
605C22#347512yes—
606C22Q8×#4112yes—
607C22C2×#55911yescharacteristic
608C22#348312yes—
609C22#350111yescharacteristic
610C22#350111yescharacteristic
611C22#350511yescharacteristic
612C22#350612yes—
613C22#350611yescharacteristic
614C22#351212yes—
615C22#351312yes—
616C22#352112yes—
617C22#354212yes—
618C22#354911yescharacteristic
619C22#355111yescharacteristic
620C4—24no—
621C4—28no—
622C4—42no—
623C4—48no—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—82no—
630C4—84no—
631C4—84no—
632C4—84no—
633C4#259318yes—
634C2—84no—
635C2C2×#259311yescharacteristic
636C2#3494112yes—
637C2#3501411yescharacteristic
638C2#3521812yes—
639C2#3751311yescharacteristic
640C2#5985612yes—
641C2#5985812yes—
642C2#5986411yescharacteristic
643C2#5986511yescharacteristic
644C2#5993511yescharacteristic
645C2#5995211yescharacteristic
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. #17012C2semidirect
3. #15196C2semidirect
5. #15152C2semidirect
6. #15100C2semidirect
7. #15095C2semidirect
8. #14254C2semidirect
10. #13973C2semidirect
11. #13973C2semidirect
12. #13876C2semidirect
13. #13876C2semidirect
14. C4×#389C2semidirect

Maximal subgroups

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

subgroupindex
#170122
#165992
#151962
#151812
#151522
#151002
#150952
#142542
#141572
#139732
#138762
C4×#3892
C4×#3902