Finite Group Databasebeta

← #793577

Subgroups — #793577

Order 512.

2140 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#16979C212yesmaximal, complemented
4#15700C214yesmaximal
5#15693C214yesmaximal, complemented
6#15230C211yesmaximal, characteristic, complemented
7#15175C212yesmaximal
8#15156C211yesmaximal, characteristic, complemented
9#15125C211yesmaximal, characteristic
10#14972C212yesmaximal
11#14250C212yesmaximal, complemented
12#14150C212yesmaximal, complemented
13#13994C212yesmaximal
14#13885C212yesmaximal, complemented
15#13875C212yesmaximal
16C4×#399C211yesmaximal, characteristic
17#13812C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19#2494—28no—
20C2×#401C2212yes—
21C2×#400C2212yes—
22C2×#399C2211yescharacteristic
23C2×#397C2212yes—
24C2×#393C2212yes—
25#2369—28no—
26C2×#391C2214yes—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#388C2212yes—
32C2×#388C2214yes—
33C2×#387C2212yes—
34C2×#382C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47#1446C2212yes—
48#1446C2214yes—
49#1445C2212yes—
50#1445C2211yescharacteristic
51#1445C2214yes—
52#1444C2211yescharacteristic
53#1444C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2211yescharacteristic
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78#1434C2214yes—
79#1434C2214yes—
80#1434C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—24no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2314yes—
116C22×C42C2312yes—
117#401—28no—
118#400—28no—
119#399—24no—
120#397—28no—
121#393—44no—
122#391—216no—
123#389—28no—
124#389—24no—
125#389—28no—
126#389—24no—
127#388—28no—
128#388—48no—
129#387—28no—
130#382—24no—
131#382—28no—
132#380—24no—
133#380—24no—
134#380—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4D418yes—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—28no—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C25C2411yescharacteristic
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—44no—
182C22×Q8—42no—
183C22×Q8—42no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C4⋊Q8—232no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
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—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4#4118yes—
242C2×C4⋊C4#4114yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—44no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42C2×D414yes—
279C2×C42C2×D418yes—
280C2×C42#4118yes—
281#95—22no—
282#95—22no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—22no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—22no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95C2×D418yes—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—22no—
326C24H(5,𝔽2)12yes—
327C24C2511yescharacteristic
328C2×Q8—216no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8—88no—
332C2×Q8—88no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4C22×D412yes—
418C22×C4C22×D414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4118yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C42−1+412yes—
430C22×C42−1+412yes—
431C22×C42−1+412yes—
432C4⋊C4—216no—
433C4⋊C4—216no—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—28no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—84no—
445C4⋊C4—88no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—44no—
449C22⋊C4—48no—
450C22⋊C4—416no—
451C22⋊C4—44no—
452C22⋊C4—84no—
453C42—28no—
454C42—216no—
455C42—48no—
456C42—48no—
457C42—44no—
458C42—88no—
459C42—88no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23C23×D411yescharacteristic
484C23C22×#4112yes—
485C23C22×#4112yes—
486C23C22×#4111yescharacteristic
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×2−1+411yescharacteristic
489C23C2×2−1+411yescharacteristic
490C23C2×2−1+411yescharacteristic
491C23#58512yes—
492C23#58512yes—
493C23#58511yescharacteristic
494Q8—816no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—88no—
584C2×C4#52514yes—
585C2×C4#53214yes—
586C2×C4#53614yes—
587C2×C4#54814yes—
588C2×C4#55218yes—
589C2×C4#55618yes—
590C22—44no—
591C22—42no—
592C22—44no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—82no—
597C22—81no—
598C22—82no—
599C22—81no—
600C22—81no—
601C22C2×#52511yescharacteristic
602C22C2×#53211yescharacteristic
603C22C2×#53611yescharacteristic
604C22C2×#54811yescharacteristic
605C22C2×#55212yes—
606C22C2×#55612yes—
607C22#347512yes—
608C22#348012yes—
609C22#348511yescharacteristic
610C22#348811yescharacteristic
611C22#349312yes—
612C22#349412yes—
613C22#350011yescharacteristic
614C22#350712yes—
615C22#351312yes—
616C22#351612yes—
617C22#351811yescharacteristic
618C22#352112yes—
619C22#352211yescharacteristic
620C22#354411yescharacteristic
621C22#355511yescharacteristic
622C22#355712yes—
623C22#356512yes—
624C4—24no—
625C4—48no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—44no—
632C4—81no—
633C4—84no—
634C4—81no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4#260418yes—
640C2—82no—
641C2C2×#260411yescharacteristic
642C2#3598211yescharacteristic
643C2#3606211yescharacteristic
644C2#3653411yescharacteristic
645C2#3658211yescharacteristic
646C2#3912511yescharacteristic
647C2#4264711yescharacteristic
648C2#4500512yes—
649C2#4512412yes—
650C2#5925012yes—
651C2#5929612yes—
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
2. C2×#2369C2semidirect
3. #16979C2semidirect
5. #15693C2semidirect
6. #15230C2semidirect
8. #15156C2semidirect
11. #14250C2semidirect
12. #14150C2semidirect
14. #13885C2semidirect
17. #13812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#23692
#169792
#157002
#156932
#152302
#151752
#151562
#151252
#149722
#142502
#141502
#139942
#138852
#138752
C4×#3992
#138122