Finite Group Databasebeta

← #2736730

Subgroups — #2736730

Order 512.

2434 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2447C211yesmaximal, characteristic
3#17190C214yesmaximal, complemented
4#17108C214yesmaximal, complemented
5#16606C212yesmaximal, complemented
6#16364C212yesmaximal, complemented
7#16264C212yesmaximal, complemented
8#15226C212yesmaximal
9#15192C211yesmaximal, characteristic, complemented
10#15191C211yesmaximal, characteristic, complemented
11#15184C212yesmaximal, complemented
12#15147C212yesmaximal
13#15137C212yesmaximal, complemented
14#15128C211yesmaximal, characteristic, complemented
15#14660C212yesmaximal, complemented
16#13920C211yesmaximal, characteristic
17C4×#398C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2542—28no—
24#2447—28no—
25C2×#399C2214yes—
26C2×#398C2211yescharacteristic
27C2×#398C2214yes—
28C2×#397C2212yes—
29C2×#397C2212yes—
30C2×#397C2214yes—
31C2×#395C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#385C2212yes—
43C2×#384C2212yes—
44C2×#381C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49#1446C2212yes—
50#1446C2212yes—
51#1446C2212yes—
52#1445C2211yescharacteristic
53#1445C2212yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2211yescharacteristic
57#1443C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2211yescharacteristic
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—44no—
91C2×#124—24no—
92C2×#124—44no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—48no—
97C2×C22⋊Q8—44no—
98C2×C4×D4—44no—
99C2×C4×D4D418yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
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×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#399—216no—
131#398—24no—
132#398—48no—
133#397—28no—
134#397—28no—
135#397—48no—
136#395—44no—
137#391—24no—
138#391—24no—
139#391—44no—
140#391—44no—
141#390—24no—
142#390—28no—
143#390—44no—
144#389—24no—
145#389—44no—
146#389—44no—
147#385—28no—
148#384—28no—
149#381—28no—
150#380—44no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—44no—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C25C2412yes—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—44no—
195C22×Q8—44no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4C2×D414yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4C2×D418yes—
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C4⋊Q8—416no—
223#124—416no—
224C22⋊Q8—432no—
225C22⋊Q8—432no—
226C4×D4—416no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
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⋊C4—44no—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4#4114yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—42no—
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⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×D418yes—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42C2×D414yes—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—24no—
316#95—22no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24C22×D414yes—
348C24C2511yescharacteristic
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—88no—
358C2×Q8—88no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—48no—
363C2×D4—48no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—216no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+414yes—
460C22×C42−1+412yes—
461C22×C42−1+414yes—
462C22×C42−1+412yes—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—84no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—88no—
486C22⋊C4—84no—
487C22⋊C4—88no—
488C42—28no—
489C42—44no—
490C42—48no—
491C42—44no—
492C42—48no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23C23×D411yescharacteristic
517C23C22×#4111yescharacteristic
518C23C2×H(5,𝔽2)12yes—
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)12yes—
522C23C2×2−1+412yes—
523C23C2×2−1+411yescharacteristic
524C23C2×2−1+411yescharacteristic
525C23C2×2−1+412yes—
526C23#58511yescharacteristic
527Q8—816no—
528Q8—88no—
529D4—416no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
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—22no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—816no—
619C2×C4#53414yes—
620C2×C4#53614yes—
621C2×C4#53714yes—
622C2×C4#54614yes—
623C2×C4#56218yes—
624C2×C4#56318yes—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22C2×#53411yescharacteristic
637C22C2×#53611yescharacteristic
638C22C2×#53711yescharacteristic
639C22C2×#54611yescharacteristic
640C22#348411yescharacteristic
641C22#348911yescharacteristic
642C22#349911yescharacteristic
643C22#350812yes—
644C22#351511yescharacteristic
645C22#352011yescharacteristic
646C22#352312yes—
647C22C2×#56212yes—
648C22C2×#56312yes—
649C22#353612yes—
650C22#354312yes—
651C22#354412yes—
652C22#354611yescharacteristic
653C22#355012yes—
654C22#355012yes—
655C22#355212yes—
656C22#355211yescharacteristic
657C22#355412yes—
658C22#355412yes—
659C4—24no—
660C4—44no—
661C4—44no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—42no—
666C4—81no—
667C4—84no—
668C4—82no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4#281418yes—
675C2—84no—
676C2C2×#281411yescharacteristic
677C2#3890811yescharacteristic
678C2#3895411yescharacteristic
679C2#3946411yescharacteristic
680C2#4242811yescharacteristic
681C2#4248111yescharacteristic
682C2#4291711yescharacteristic
683C2#4567512yes—
684C2#4610512yes—
685C2#5296412yes—
686C2#5311812yes—
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. #17190C2semidirect
4. #17108C2semidirect
5. #16606C2semidirect
6. #16364C2semidirect
7. #16264C2semidirect
9. #15192C2semidirect
10. #15191C2semidirect
11. #15184C2semidirect
13. #15137C2semidirect
14. #15128C2semidirect
15. #14660C2semidirect
17. C4×#398C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24472
#171902
#171082
#166062
#163642
#162642
#152262
#151922
#151912
#151842
#151472
#151372
#151282
#146602
#139202
C4×#3982