Finite Group Databasebeta

← #770342

Subgroups — #770342

Order 512.

2554 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C211yesmaximal, characteristic, complemented
2C2×#2326C211yesmaximal, characteristic
3#15826C214yesmaximal, complemented
4#15772C214yesmaximal, complemented
5#15475C214yesmaximal, complemented
6#15431C214yesmaximal, complemented
7#15201C212yesmaximal, complemented
8#15157C212yesmaximal
9#14153C212yesmaximal
10#13942C211yesmaximal, characteristic, complemented
11#13941C211yesmaximal, characteristic
12#13926C211yesmaximal, characteristic, complemented
13#13911C212yesmaximal, complemented
14#13906C211yesmaximal, characteristic
15#13779C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2399—28no—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#395C2212yes—
24C2×#395C2212yes—
25C2×#393C2214yes—
26#2326—28no—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#388C2214yes—
34C2×#385C2214yes—
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#382C2214yes—
39C2×#380C2214yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×#376C2211yescharacteristic
45#1446C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2214yes—
73#1434C2212yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#124—24no—
82C2×#124—24no—
83C2×#124Q818yescomplemented
84C2×C4×Q8—24no—
85C2×C4×Q8—28no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—44no—
89C2×C4×Q8—44no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#398—28no—
123#398—28no—
124#395—28no—
125#395—28no—
126#393—48no—
127#391—24no—
128#391—44no—
129#391—44no—
130#390—24no—
131#390—44no—
132#390—44no—
133#388—216no—
134#385—216no—
135#384—24no—
136#384—24no—
137#382—24no—
138#382—48no—
139#380—48no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—28no—
144#376D418yes—
145C2×#95C2314yes—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C25C2412yes—
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8C2×Q814yes—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4C2×Q814yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212#124—232no—
213C4×Q8—432no—
214C4×Q8—416no—
215C4×D4—416no—
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—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
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—44no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×Q8116yes—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×Q814yes—
298C2×C42#4114yes—
299C2×C42#4114yes—
300C2×C42#4114yes—
301#95—24no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
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#95—44no—
323#95—44no—
324#95—44no—
325#95C2×D418yes—
326#95C2×D418yes—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24C22×Q814yes—
338C24C2511yescharacteristic
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
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—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—88no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q814yes—
442C22×C4C22×Q812yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+414yes—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—88no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C22⋊C4—232no—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C42—28no—
478C42—416no—
479C42—48no—
480C42—48no—
481C42—48no—
482C42C22⋊Q8116yes—
483C42#124116yes—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23C23×D411yescharacteristic
508C23C23×Q811yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)12yes—
515C23C2×2−1+412yes—
516C23#58512yes—
517C23#58512yes—
518Q8—432no—
519Q8—416no—
520D4—416no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—816no—
602C2×C4C2×C22⋊Q818yes—
603C2×C4C2×C22⋊Q814yes—
604C2×C4C2×#12414yes—
605C2×C4C2×#12414yes—
606C2×C4C2×#12414yes—
607C2×C4#53514yes—
608C2×C4#55414yes—
609C2×C4#55414yes—
610C2×C4#55514yes—
611C2×C4#55514yes—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—48no—
616C22—48no—
617C22—82no—
618C22—84no—
619C22—82no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22C22×C22⋊Q811yescharacteristic
624C22C22×#12411yescharacteristic
625C22C2×#53511yescharacteristic
626C22C2×#55411yescharacteristic
627C22C2×#55411yescharacteristic
628C22C2×#55511yescharacteristic
629C22C2×#55511yescharacteristic
630C22#348812yes—
631C22#348912yes—
632C22#350312yes—
633C22#350412yes—
634C22#350612yes—
635C22#350612yes—
636C22#350712yes—
637C22#350712yes—
638C22#350811yescharacteristic
639C22#350911yescharacteristic
640C22#351011yescharacteristic
641C22#351011yescharacteristic
642C22#353612yes—
643C22#354012yes—
644C22#354112yes—
645C22#354212yes—
646C4—24no—
647C4—28no—
648C4—24no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—84no—
654C4—82no—
655C4—82no—
656C4—84no—
657C4—82no—
658C4—82no—
659C4#256318yes—
660C2—84no—
661C2C2×#256311yescharacteristic
662C2#3551912yes—
663C2#3555611yescharacteristic
664C2#3562811yescharacteristic
665C2#3727112yes—
666C2#4496311yescharacteristic
667C2#4496612yes—
668C2#4515111yescharacteristic
669C2#4536912yes—
670C2#4542511yescharacteristic
671C2#4543411yescharacteristic
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. C2×#2399C2semidirect
3. #15826C2semidirect
4. #15772C2semidirect
5. #15475C2semidirect
6. #15431C2semidirect
7. #15201C2semidirect
10. #13942C2semidirect
12. #13926C2semidirect
13. #13911C2semidirect
15. #13779C2semidirect
83. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
C2×#23262
#158262
#157722
#154752
#154312
#152012
#151572
#141532
#139422
#139412
#139262
#139112
#139062
#137792