Finite Group Databasebeta

← #805542

Subgroups — #805542

Order 512.

2916 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2427C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#17058C212yesmaximal, complemented
4#15554C214yesmaximal, complemented
5#15544C214yesmaximal, complemented
6#15192C211yesmaximal, characteristic, complemented
7#15185C211yesmaximal, characteristic, complemented
8#15150C211yesmaximal, characteristic, complemented
9#14842C212yesmaximal, complemented
10#14776C212yesmaximal, complemented
11#14027C212yesmaximal, complemented
12#13997C212yesmaximal, complemented
13#13970C212yesmaximal
14#13933C212yesmaximal, complemented
15#13924C212yesmaximal, complemented
16C4×#397C211yesmaximal, characteristic, complemented
17#13820C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2212yes—
23C23×C4⋊C4C2212yes—
24#2427—28no—
25C2×#397C2211yescharacteristic
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#394C2212yes—
30D4×C4⋊C4—28no—
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1445C2211yescharacteristic
50#1445C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1445C2214yes—
56#1443C2212yes—
57#1443C2214yes—
58#1442C2211yescharacteristic
59#1442C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1440C2212yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2211yescharacteristic
68#1439C2214yes—
69#1438C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77C4⋊#95C2214yes—
78#1434C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C22⋊#95C2212yes—
82C22⋊#95C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2314yes—
91C2×#124—24no—
92C2×#124—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×#121—24no—
96C2×#121—28no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—44no—
100C2×C4×D4—44no—
101C2×C4×D4D418yes—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—44no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134#397—24no—
135#397—28no—
136#397—44no—
137#397—44no—
138#394—28no—
139#391—24no—
140#391—44no—
141#391—44no—
142#390—28no—
143#389—28no—
144#389—24no—
145#386—28no—
146#386—48no—
147#385—216no—
148#384—24no—
149#380—28no—
150#380—28no—
151#380—44no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—44no—
156C4×C22⋊C4—44no—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C25C2411yescharacteristic
188C25C2412yes—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—24no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4C2×D414yes—
203C22×D4#4118yes—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—28no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D418yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236#124—416no—
237C22⋊Q8—416no—
238#121—416no—
239C4×D4—232no—
240C4×D4—432no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—432no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4#4114yes—
269C2×C4⋊C4#4118yes—
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—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4C2×D418yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C2×D414yes—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—42no—
335#95—42no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—42no—
345#95—44no—
346#95—44no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—22no—
366C24C22×D414yes—
367C24C2×#4118yes—
368C24H(5,𝔽2)12yes—
369C24C2511yescharacteristic
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—48no—
380C2×D4—416no—
381C2×D4—48no—
382C2×D4—84no—
383C2×D4—88no—
384C2×D4#124116yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4118yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+414yes—
489C4⋊C4—28no—
490C4⋊C4—416no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4#123116yes—
500C22⋊C4—216no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C42—28no—
514C42—44no—
515C42—48no—
516C42—48no—
517C42—88no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—42no—
563C23C2×#12418yes—
564C23C23×D411yescharacteristic
565C23C22×#4112yes—
566C23C22×#4112yes—
567C23C22×#4111yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×2−1+412yes—
573C23#58512yes—
574C23#58511yescharacteristic
575Q8—88no—
576D4—232no—
577D4—816no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—432no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—832no—
660C2×C4—88no—
661C2×C4C2×#12318yes—
662C2×C4C2×#12314yes—
663C2×C4C2×#12414yes—
664C2×C4#53514yes—
665C2×C4#54614yes—
666C2×C4#55318yes—
667C2×C4#55318yes—
668C22—28no—
669C22—28no—
670C22—28no—
671C22—48no—
672C22—48no—
673C22—44no—
674C22—48no—
675C22—44no—
676C22—44no—
677C22—48no—
678C22—44no—
679C22—44no—
680C22—42no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—81no—
685C22—82no—
686C22—81no—
687C22—82no—
688C22—81no—
689C22—81no—
690C22C22×#12311yescharacteristic
691C22C22×#12411yescharacteristic
692C22C2×#53511yescharacteristic
693C22C2×#54611yescharacteristic
694C22C2×#55312yes—
695C22C2×#55312yes—
696C22#347512yes—
697C22#348212yes—
698C22#348511yescharacteristic
699C22#348811yescharacteristic
700C22#349911yescharacteristic
701C22#351312yes—
702C22#351312yes—
703C22#351411yescharacteristic
704C22#351612yes—
705C22#351911yescharacteristic
706C22#352012yes—
707C22#352112yes—
708C22#352212yes—
709C22#354112yes—
710C22#354311yescharacteristic
711C22#354611yescharacteristic
712C22#355412yes—
713C4—24no—
714C4—28no—
715C4—44no—
716C4—42no—
717C4—44no—
718C4—82no—
719C4—81no—
720C4—82no—
721C4—84no—
722C4—82no—
723C4—81no—
724C4—81no—
725C4—82no—
726C4—84no—
727C4#250418yes—
728C2—48no—
729C2—82no—
730C2C2×#250411yescharacteristic
731C2#3584711yescharacteristic
732C2#3599611yescharacteristic
733C2#3615511yescharacteristic
734C2#3630311yescharacteristic
735C2#3891011yescharacteristic
736C2#4264811yescharacteristic
737C2#4506912yes—
738C2#4520312yes—
739C2#5939812yes—
740C2#5957512yes—
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
2. C2×D4×C4⋊C4C2semidirect
3. #17058C2semidirect
4. #15554C2semidirect
5. #15544C2semidirect
6. #15192C2semidirect
7. #15185C2semidirect
8. #15150C2semidirect
9. #14842C2semidirect
10. #14776C2semidirect
11. #14027C2semidirect
12. #13997C2semidirect
14. #13933C2semidirect
15. #13924C2semidirect
16. C4×#397C2semidirect
17. #13820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24272
C2×D4×C4⋊C42
#170582
#155542
#155442
#151922
#151852
#151502
#148422
#147762
#140272
#139972
#139702
#139332
#139242
C4×#3972
#138202