Finite Group Databasebeta

← #2736678

Subgroups — #2736678

Order 512.

2298 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2454C211yesmaximal, characteristic
3#17124C214yesmaximal, complemented
4#17110C214yesmaximal, complemented
5#17059C212yesmaximal, complemented
6#16601C212yesmaximal, complemented
7#16394C212yesmaximal, complemented
8#15218C212yesmaximal, complemented
9#15202C211yesmaximal, characteristic, complemented
10#15187C212yesmaximal, complemented
11#15153C211yesmaximal, characteristic, complemented
12#15152C211yesmaximal, characteristic, complemented
13#15147C212yesmaximal
14#15138C212yesmaximal
15#14660C212yesmaximal, complemented
16#13916C211yesmaximal, characteristic
17C4×#395C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2538—28no—
23#2454—28no—
24C2×#400C2212yes—
25C2×#400C2214yes—
26C2×#399C2212yes—
27C2×#397C2214yes—
28C2×#397C2214yes—
29C2×#396C2214yes—
30C2×#396C2212yes—
31C2×#395C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#385C2212yes—
41C2×#384C2211yescharacteristic
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47#1446C2211yescharacteristic
48#1446C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1444C2212yes—
54#1444C2211yescharacteristic
55#1444C2212yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2214yes—
59#1443C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1441C2211yescharacteristic
63#1441C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#125—24no—
89C2×#125—44no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×C4×Q8—24no—
94C2×C4×Q8—44no—
95C2×C4×D4—44no—
96C2×C4×D4D418yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#400—48no—
131#400—44no—
132#399—28no—
133#397—48no—
134#397—48no—
135#396—28no—
136#396—216no—
137#395—24no—
138#390—24no—
139#390—24no—
140#390—28no—
141#390—44no—
142#389—24no—
143#389—28no—
144#389—24no—
145#389—44no—
146#385—28no—
147#384—24no—
148#381—44no—
149#380—28no—
150#380—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—44no—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C25C2412yes—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8#4114yes—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4C2×D414yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C2×D418yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214#125—416no—
215C22⋊Q8—432no—
216C4×Q8—416no—
217C4×D4—416no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D414yes—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D418yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42C2×D414yes—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24C22×D414yes—
343C24C2511yescharacteristic
344C2×Q8—28no—
345C2×Q8—48no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—48no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—48no—
353C2×D4—48no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C42−1+412yes—
450C22×C42−1+412yes—
451C22×C42−1+412yes—
452C22×C42−1+414yes—
453C4⋊C4—28no—
454C4⋊C4—44no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—88no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—88no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C22⋊C4—84no—
480C22⋊C4—88no—
481C42—28no—
482C42—48no—
483C42—44no—
484C42—44no—
485C42—48no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23C23×D411yescharacteristic
510C23C22×#4111yescharacteristic
511C23C2×H(5,𝔽2)12yes—
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+412yes—
516C23C2×2−1+411yescharacteristic
517C23C2×2−1+411yescharacteristic
518C23#58512yes—
519C23#58511yescharacteristic
520Q8—416no—
521D4—416no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—816no—
611C2×C4#53414yes—
612C2×C4#53614yes—
613C2×C4#53614yes—
614C2×C4#54614yes—
615C2×C4#56318yes—
616C2×C4#56318yes—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22C2×#53411yescharacteristic
629C22C2×#53611yescharacteristic
630C22C2×#53611yescharacteristic
631C22C2×#54611yescharacteristic
632C22#348511yescharacteristic
633C22#348911yescharacteristic
634C22#349811yescharacteristic
635C22#350512yes—
636C22#351811yescharacteristic
637C22#352212yes—
638C22#352311yescharacteristic
639C22C2×#56312yes—
640C22C2×#56312yes—
641C22#354212yes—
642C22#354212yes—
643C22#354812yes—
644C22#354812yes—
645C22#355211yescharacteristic
646C22#355412yes—
647C22#355411yescharacteristic
648C22#355512yes—
649C22#355712yes—
650C22#355912yes—
651C4—24no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—84no—
658C4—81no—
659C4—84no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4—82no—
666C4#281718yes—
667C2—84no—
668C2C2×#281711yescharacteristic
669C2#3891111yescharacteristic
670C2#3895211yescharacteristic
671C2#3946611yescharacteristic
672C2#4265111yescharacteristic
673C2#4267611yescharacteristic
674C2#4290911yescharacteristic
675C2#4570212yes—
676C2#4634712yes—
677C2#5311212yes—
678C2#5312312yes—
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
3. #17124C2semidirect
4. #17110C2semidirect
5. #17059C2semidirect
6. #16601C2semidirect
7. #16394C2semidirect
8. #15218C2semidirect
9. #15202C2semidirect
10. #15187C2semidirect
11. #15153C2semidirect
12. #15152C2semidirect
15. #14660C2semidirect
17. C4×#395C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24542
#171242
#171102
#170592
#166012
#163942
#152182
#152022
#151872
#151532
#151522
#151472
#151382
#146602
#139162
C4×#3952