Finite Group Databasebeta

← #2736932

Subgroups — #2736932

Order 512.

2356 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2492C211yesmaximal, characteristic
3#17222C214yesmaximal, complemented
4#17153C214yesmaximal
5#17080C212yesmaximal, complemented
6#16704C212yesmaximal
7#16694C212yesmaximal, complemented
8#15201C211yesmaximal, characteristic, complemented
9#15191C212yesmaximal, complemented
10#15190C211yesmaximal, characteristic, complemented
11#15188C212yesmaximal, complemented
12#15179C212yesmaximal
13#15178C212yesmaximal
14#15178C211yesmaximal, characteristic
15#14625C212yesmaximal
16#13933C211yesmaximal, characteristic, complemented
17C4×#393C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C22×C4×Q8C2211yescharacteristic
24#2568—28no—
25#2492—28no—
26C2×#400C2214yes—
27C2×#400C2212yes—
28C2×#393C2212yes—
29C2×#393C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#388C2212yes—
41C2×#382C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46#1446C2214yes—
47#1446C2214yes—
48#1446C2212yes—
49#1446C2211yescharacteristic
50#1444C2214yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2211yescharacteristic
63#1442C2212yes—
64#1442C2212yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1439C2212yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1439C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8—44no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—44no—
99C2×C22⋊Q8—44no—
100C2×C22⋊Q8—44no—
101C2×C22⋊Q8—44no—
102C2×C4×Q8—24no—
103C2×C4×Q8—44no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#400—216no—
131#400—28no—
132#393—28no—
133#393Q818yescomplemented
134#391—28no—
135#391—28no—
136#391—24no—
137#391—28no—
138#391—28no—
139#391—24no—
140#391—44no—
141#391—44no—
142#391—44no—
143#389—44no—
144#388—28no—
145#382—28no—
146#380—24no—
147C4×C4⋊C4D418yes—
148C4×C22⋊C4—24no—
149C4×C22⋊C4—44no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—44no—
204C22×Q8—42no—
205C22×Q8—44no—
206C22×Q8C2×Q814yes—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C4⋊Q8—416no—
229C4⋊Q8—416no—
230C4⋊Q8—432no—
231C22⋊Q8—416no—
232C22⋊Q8—432no—
233C4×Q8—416no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
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⋊C4—44no—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D414yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×Q814yes—
287C2×C22⋊C4C2×Q818yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—44no—
298C2×C42C2×D414yes—
299C2×C42C2×D418yes—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—42no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95C2×D418yes—
332#95C2×Q818yes—
333#95C2×Q814yes—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—21no—
342C24—22no—
343C24—21no—
344C24C22×Q812yes—
345C24C2511yescharacteristic
346C2×Q8—28no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—84no—
366C2×Q8—88no—
367C2×Q8—88no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
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—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×Q812yes—
455C22×C4C22×Q814yes—
456C22×C4C22×Q814yes—
457C22×C4C22×Q812yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+414yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C22×C42−1+412yes—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—88no—
479C22⋊C4—28no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—84no—
486C42—28no—
487C42—48no—
488C42—48no—
489C42—44no—
490C42—48no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23C23×D411yescharacteristic
515C23C23×Q811yescharacteristic
516C23C2×H(5,𝔽2)12yes—
517C23C2×H(5,𝔽2)12yes—
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×2−1+411yescharacteristic
520C23C2×2−1+411yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23C2×2−1+412yes—
523C23#58512yes—
524C23#58511yescharacteristic
525Q8—416no—
526Q8—816no—
527Q8—88no—
528Q8—816no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—88no—
616C2×C4#53514yes—
617C2×C4#53514yes—
618C2×C4#53714yes—
619C2×C4D4×Q814yes—
620C2×C4H(3,𝔽4)18yes—
621C2×C4#56218yes—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—81no—
628C22—82no—
629C22—82no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×#53511yescharacteristic
634C22C2×#53511yescharacteristic
635C22C2×#53711yescharacteristic
636C22C2×D4×Q811yescharacteristic
637C22#348511yescharacteristic
638C22#348911yescharacteristic
639C22#349911yescharacteristic
640C22#350112yes—
641C22#350211yescharacteristic
642C22#350311yescharacteristic
643C22#350312yes—
644C22C2×H(3,𝔽4)12yes—
645C22C2×#56212yes—
646C22#354112yes—
647C22#354112yes—
648C22#355012yes—
649C22#355212yes—
650C22#355411yescharacteristic
651C22#355512yes—
652C22#355611yescharacteristic
653C22#356112yes—
654C22#356112yes—
655C22#356212yes—
656C4—24no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—81no—
666C4—82no—
667C4—84no—
668C4—84no—
669C4—82no—
670C4—82no—
671C4#281918yes—
672C2—82no—
673C2C2×#281911yescharacteristic
674C2#3959411yescharacteristic
675C2#3959911yescharacteristic
676C2#3961611yescharacteristic
677C2#4264911yescharacteristic
678C2#4268011yescharacteristic
679C2#4291111yescharacteristic
680C2#4550012yes—
681C2#4600412yes—
682C2#5314012yes—
683C2#5314412yes—
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×#2568C2semidirect
3. #17222C2semidirect
5. #17080C2semidirect
7. #16694C2semidirect
8. #15201C2semidirect
9. #15191C2semidirect
10. #15190C2semidirect
11. #15188C2semidirect
16. #13933C2semidirect
133. #393Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#24922
#172222
#171532
#170802
#167042
#166942
#152012
#151912
#151902
#151882
#151792
#151782
#146252
#139332
C4×#3932