Finite Group Databasebeta

← #729376

Subgroups — #729376

Order 512.

2920 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2504C211yesmaximal, characteristic
3C22×#398C211yesmaximal, characteristic, complemented
4C2×#2345C211yesmaximal, characteristic, complemented
5#16800C212yesmaximal, complemented
6#16705C212yesmaximal, complemented
7#15621C214yesmaximal, complemented
8#15583C214yesmaximal, complemented
9#15163C212yesmaximal, complemented
10#15085C212yesmaximal, complemented
11#14921C212yesmaximal, complemented
12#14729C212yesmaximal, complemented
13#14271C211yesmaximal, characteristic
14#14020C212yesmaximal, complemented
15#14019C212yesmaximal, complemented
16#13924C211yesmaximal, characteristic, complemented
17#13782C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2591—28no—
23#2504—28no—
24C2×#400C2211yescharacteristic
25C2×#400C2211yescharacteristic
26C2×#398—28no—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#398C2212yes—
31C2×#397C2214yes—
32C2×#396C2214yes—
33C2×#394C2211yescharacteristic
34C2×#394C2211yescharacteristic
35#2345—28no—
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#388C2212yes—
39C2×#388C2212yes—
40C2×#388C2214yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#386C2212yes—
48C2×#386C2212yes—
49C2×#385C2212yes—
50C2×#382C2212yes—
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2211yescharacteristic
59C22×#95C2212yes—
60C22×#95C2211yescharacteristic
61#1445C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1443C2214yes—
65#1443C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1438C2212yes—
77#1437C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2214yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×D4C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—48no—
104C22×C4⋊C4D418yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136#400—24no—
137#400—24no—
138#398—28no—
139#398—28no—
140#398—28no—
141#398—28no—
142#397—216no—
143#396—48no—
144#394—24no—
145#394—24no—
146#390—28no—
147#389—28no—
148#388—28no—
149#388—28no—
150#388—48no—
151#387—28no—
152#387—28no—
153#386—28no—
154#386—28no—
155#386—28no—
156#386—28no—
157#386—28no—
158#386—28no—
159#385—28no—
160#382—28no—
161#380—24no—
162#380—24no—
163#380—24no—
164C4×C4⋊C4—28no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4D418yes—
169C2×#95—24no—
170C2×#95—28no—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C25C2412yes—
199C25C2411yescharacteristic
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—44no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—42no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4#4118yes—
233C23×C4#4114yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2414yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249#121—416no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—432no—
281C2×C4⋊C4—416no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D414yes—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D418yes—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95C2×D418yes—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24C22×D412yes—
375C24C22×D412yes—
376C24C2×#4112yes—
377C24C2×#4114yes—
378C242−1+414yes—
379C24C2511yescharacteristic
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—84no—
390C2×D4—84no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—216no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4#121116yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C2×#4118yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—416no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—88no—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C42—48no—
519C42—48no—
520C42—84no—
521C42—84no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23C2×#12114yes—
557C23#53214yes—
558C23#53614yes—
559C23#54818yes—
560C23#55618yes—
561C23C23×D411yescharacteristic
562C23C23×D411yescharacteristic
563C23C22×#4111yescharacteristic
564C23C22×#4112yes—
565C23C2×H(5,𝔽2)12yes—
566C23C2×2−1+412yes—
567C23#58512yes—
568C23#58514yes—
569D4—88no—
570D4—816no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—432no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—88no—
649C2×C4—816no—
650C2×C4C2×#12114yes—
651C2×C4C2×#12114yes—
652C2×C4#53514yes—
653C22—28no—
654C22—28no—
655C22—216no—
656C22—28no—
657C22—24no—
658C22—24no—
659C22—28no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—84no—
665C22—84no—
666C22—82no—
667C22—84no—
668C22—82no—
669C22—82no—
670C22—84no—
671C22#254518yes—
672C22C22×#12111yescharacteristic
673C22C2×#53211yescharacteristic
674C22C2×#53511yescharacteristic
675C22C2×#53611yescharacteristic
676C22#344911yescharacteristic
677C22C2×#54812yes—
678C22C2×#55612yes—
679C22#348512yes—
680C22#348512yes—
681C22#348812yes—
682C22#349112yes—
683C22#349412yes—
684C22#350514yes—
685C22#351214yes—
686C22#354114yes—
687C22#355314yes—
688C4—24no—
689C4—24no—
690C4—24no—
691C4—48no—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—84no—
703C2—28no—
704C2—162no—
705C2C2×#254511yescharacteristic
706C2#3525211yescharacteristic
707C2#3544711yescharacteristic
708C2#3595312yes—
709C2#3971712yes—
710C2#3972212yes—
711C2#4140612yes—
712C2#4357614yes—
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
3. C22×#398C2semidirect
4. C2×#2345C2semidirect
5. #16800C2semidirect
6. #16705C2semidirect
7. #15621C2semidirect
8. #15583C2semidirect
9. #15163C2semidirect
10. #15085C2semidirect
11. #14921C2semidirect
12. #14729C2semidirect
14. #14020C2semidirect
15. #14019C2semidirect
16. #13924C2semidirect
17. #13782C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#25042
C22×#3982
C2×#23452
#168002
#167052
#156212
#155832
#151632
#150852
#149212
#147292
#142712
#140202
#140192
#139242
#137822