Finite Group Databasebeta

← #2736747

Subgroups — #2736747

Order 512.

2088 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2489C211yesmaximal, characteristic
3#17205C214yesmaximal, complemented
4#17166C214yesmaximal
5#16963C212yesmaximal, complemented
6#16682C212yesmaximal
7#16564C212yesmaximal
8#15240C211yesmaximal, characteristic, complemented
9#15228C212yesmaximal, complemented
10#15220C212yesmaximal
11#15192C211yesmaximal, characteristic, complemented
12#15189C212yesmaximal, complemented
13#15182C212yesmaximal
14#15135C211yesmaximal, characteristic
15#14664C212yesmaximal, complemented
16#13918C211yesmaximal, characteristic, complemented
17C4×#397C211yesmaximal, characteristic
18C22×#125C2212yes—
19C22×#125C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21#2552—28no—
22#2489—28no—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#398C2214yes—
26C2×#397C2212yes—
27C2×#397C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2212yes—
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×#382C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1445C2214yes—
49#1445C2212yes—
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1443C2212yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2211yescharacteristic
63#1442C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#125—24no—
88C2×#125—28no—
89C2×#125—28no—
90C2×#125—44no—
91C2×#125—44no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—44no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#400—28no—
124#399—44no—
125#398—216no—
126#397—24no—
127#397—28no—
128#391—24no—
129#391—24no—
130#391—44no—
131#391—44no—
132#389—28no—
133#389—28no—
134#389—44no—
135#389—44no—
136#388—28no—
137#384—28no—
138#384—24no—
139#382—28no—
140#382—44no—
141#382—44no—
142#380—24no—
143#380—44no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4D418yes—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—44no—
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209#125—432no—
210#125—416no—
211C22⋊Q8—416no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4#4114yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42C2×D418yes—
288C2×C42C2×D414yes—
289#95—22no—
290#95—22no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—44no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C2×D418yes—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—22no—
331C24—21no—
332C24—22no—
333C242−1+412yes—
334C24C2511yescharacteristic
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—48no—
339C2×Q8—48no—
340C22×C4—22no—
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—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4C22×D412yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C42−1+414yes—
437C22×C42−1+414yes—
438C22×C42−1+412yes—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—416no—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—88no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—88no—
463C42—28no—
464C42—216no—
465C42—44no—
466C42—48no—
467C42—48no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23C23×D411yescharacteristic
492C23C22×#4111yescharacteristic
493C23C2×H(5,𝔽2)12yes—
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×2−1+412yes—
497C23C2×2−1+411yescharacteristic
498C23C2×2−1+412yes—
499C23C2×2−1+411yescharacteristic
500C23#58511yescharacteristic
501C23#58512yes—
502Q8—88no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4#53514yes—
593C2×C4#53614yes—
594C2×C4#53714yes—
595C2×C4#54614yes—
596C2×C4#56218yes—
597C2×C4#56318yes—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—82no—
607C22—81no—
608C22—82no—
609C22C2×#53511yescharacteristic
610C22C2×#53611yescharacteristic
611C22C2×#53711yescharacteristic
612C22C2×#54611yescharacteristic
613C22#348611yescharacteristic
614C22#348911yescharacteristic
615C22#350011yescharacteristic
616C22#350912yes—
617C22#351511yescharacteristic
618C22#351712yes—
619C22#352011yescharacteristic
620C22C2×#56212yes—
621C22C2×#56312yes—
622C22#354112yes—
623C22#354412yes—
624C22#354612yes—
625C22#354611yescharacteristic
626C22#355211yescharacteristic
627C22#355212yes—
628C22#355412yes—
629C22#356312yes—
630C22#356312yes—
631C22#356512yes—
632C4—24no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—44no—
638C4—81no—
639C4—84no—
640C4—81no—
641C4—82no—
642C4—82no—
643C4—84no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4#282018yes—
648C2—82no—
649C2C2×#282011yescharacteristic
650C2#3890811yescharacteristic
651C2#3895611yescharacteristic
652C2#3946511yescharacteristic
653C2#4279011yescharacteristic
654C2#4281511yescharacteristic
655C2#4297811yescharacteristic
656C2#4567112yes—
657C2#4607912yes—
658C2#5304012yes—
659C2#5305612yes—
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
1. C2×#2552C2semidirect
3. #17205C2semidirect
5. #16963C2semidirect
8. #15240C2semidirect
9. #15228C2semidirect
11. #15192C2semidirect
12. #15189C2semidirect
15. #14664C2semidirect
16. #13918C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24892
#172052
#171662
#169632
#166822
#165642
#152402
#152282
#152202
#151922
#151892
#151822
#151352
#146642
#139182
C4×#3972