Finite Group Databasebeta

← #2737673

Subgroups — #2737673

Order 512.

2306 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2490C211yesmaximal, characteristic
3#17126C214yesmaximal, complemented
4#17110C214yesmaximal, complemented
5#17071C212yesmaximal, complemented
6#16701C212yesmaximal, complemented
7#16692C212yesmaximal, complemented
8#15228C212yesmaximal, complemented
9#15219C212yesmaximal
10#15155C211yesmaximal, characteristic, complemented
11#15147C211yesmaximal, characteristic, complemented
12#15141C212yesmaximal, complemented
13#15135C211yesmaximal, characteristic, complemented
14#15131C212yesmaximal, complemented
15#14628C212yesmaximal, complemented
16#13933C211yesmaximal, characteristic, complemented
17C4×#397C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#121C2211yescharacteristic
20#2571—28no—
21#2490—28no—
22C2×#400C2214yes—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#399C2214yes—
27C2×#397C2211yescharacteristic
28C2×#397C2212yes—
29C2×#397C2212yes—
30C2×#396C2214yes—
31C2×#396C2214yes—
32C2×#394C2212yes—
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45C2×#382C2212yes—
46C2×#382C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53#1445C2212yes—
54#1445C2211yescharacteristic
55#1444C2212yes—
56#1444C2214yes—
57#1444C2214yes—
58#1443C2212yes—
59#1442C2211yescharacteristic
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2212yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2211yescharacteristic
74#1439C2214yes—
75#1439C2214yes—
76#1438C2214yes—
77#1438C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2311yescharacteristic
88C2×#125—24no—
89C2×#125—44no—
90C2×#121—24no—
91C2×#121—44no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#400—216no—
129#399—28no—
130#399—44no—
131#399—44no—
132#399—48no—
133#397—28no—
134#397—24no—
135#397—44no—
136#396—48no—
137#396—48no—
138#394—28no—
139#390—24no—
140#390—24no—
141#390—28no—
142#390—44no—
143#390—44no—
144#389—28no—
145#389—28no—
146#389—44no—
147#389—44no—
148#388—28no—
149#384—24no—
150#384—44no—
151#382—28no—
152#382—28no—
153#380—24no—
154#380—44no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4D418yes—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—44no—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C25C2412yes—
186C25C2411yescharacteristic
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—44no—
195C22×D4—42no—
196C22×D4—44no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217#125—416no—
218#121—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
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—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—42no—
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⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
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—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
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⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—44no—
311C2×C42C2×D414yes—
312C2×C42C2×D418yes—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—44no—
326#95—44no—
327#95—42no—
328#95—42no—
329#95—42no—
330#95—42no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C2×D418yes—
338#95#4114yes—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—22no—
360C242−1+412yes—
361C24C2511yescharacteristic
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—84no—
370C2×D4—88no—
371C2×D4—88no—
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—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+414yes—
463C22×C42−1+412yes—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—84no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C42—28no—
496C42—48no—
497C42—44no—
498C42—48no—
499C42—48no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23C23×D411yescharacteristic
547C23C22×#4111yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×H(5,𝔽2)12yes—
552C23C2×2−1+412yes—
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+411yescharacteristic
555C23#58512yes—
556C23#58511yescharacteristic
557D4—816no—
558D4—88no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—88no—
641C2×C4#53514yes—
642C2×C4#53614yes—
643C2×C4#53714yes—
644C2×C4#55014yes—
645C2×C4#56218yes—
646C2×C4#56318yes—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—82no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—84no—
669C22C2×#53511yescharacteristic
670C22C2×#53611yescharacteristic
671C22C2×#53711yescharacteristic
672C22C2×#55011yescharacteristic
673C22#348511yescharacteristic
674C22#348911yescharacteristic
675C22#349811yescharacteristic
676C22#350812yes—
677C22#351512yes—
678C22#351811yescharacteristic
679C22#352011yescharacteristic
680C22C2×#56212yes—
681C22C2×#56312yes—
682C22#354212yes—
683C22#354312yes—
684C22#354611yescharacteristic
685C22#354712yes—
686C22#355211yescharacteristic
687C22#355212yes—
688C22#355912yes—
689C22#355912yes—
690C22#356112yes—
691C22#356212yes—
692C4—24no—
693C4—44no—
694C4—44no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—81no—
699C4—82no—
700C4—84no—
701C4—81no—
702C4—82no—
703C4—81no—
704C4—82no—
705C4—84no—
706C4#282018yes—
707C2—82no—
708C2—162no—
709C2C2×#282011yescharacteristic
710C2#3913411yescharacteristic
711C2#3913611yescharacteristic
712C2#3974511yescharacteristic
713C2#4264611yescharacteristic
714C2#4268011yescharacteristic
715C2#4291411yescharacteristic
716C2#4562512yes—
717C2#4610912yes—
718C2#5297612yes—
719C2#5300112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2571C2semidirect
3. #17126C2semidirect
4. #17110C2semidirect
5. #17071C2semidirect
6. #16701C2semidirect
7. #16692C2semidirect
8. #15228C2semidirect
10. #15155C2semidirect
11. #15147C2semidirect
12. #15141C2semidirect
13. #15135C2semidirect
14. #15131C2semidirect
15. #14628C2semidirect
16. #13933C2semidirect
17. C4×#397C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#24902
#171262
#171102
#170712
#167012
#166922
#152282
#152192
#151552
#151472
#151412
#151352
#151312
#146282
#139332
C4×#3972