Finite Group Databasebeta

← #636440

Subgroups — #636440

Order 512.

2954 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×#2440C211yesmaximal, characteristic
2C2×#2627C212yesmaximal
3C22×#389C211yesmaximal, characteristic
4#16554C214yesmaximal, complemented
5#16553C214yesmaximal, complemented
6#15149C212yesmaximal, complemented
7#15139C212yesmaximal, complemented
8#15137C212yesmaximal, complemented
9#15128C212yesmaximal, complemented
10#14935C212yesmaximal
11#14934C212yesmaximal
12#14697C211yesmaximal, characteristic
13#14263C212yesmaximal
14#13907C211yesmaximal, characteristic
15#13904C211yesmaximal, characteristic
16#13904C211yesmaximal, characteristic
17C4×#385C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2627—216no—
25#2440—28no—
26C2×#400C2214yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#398C2212yes—
30C2×#395C2214yes—
31C2×#395C2212yes—
32C2×#389—28no—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2214yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#385C2211yescharacteristic
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95C2212yes—
56#1446C2214yes—
57#1445C2212yes—
58#1445C2214yes—
59#1445C2212yes—
60#1444C2214yes—
61#1444C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1436C2214yes—
73#1436C2212yes—
74#1436C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2212yes—
84C22⋊#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—44no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—44no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4D418yes—
104C22×C4⋊C4Q818yescomplemented
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C42—24no—
131C22×C42—44no—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137#400—216no—
138#398—216no—
139#398—28no—
140#398—216no—
141#395—216no—
142#395—28no—
143#389—24no—
144#389—24no—
145#389—28no—
146#389—28no—
147#389—216no—
148#389—44no—
149#389—44no—
150#385—28no—
151#385—28no—
152#385—24no—
153#385D418yes—
154#384—24no—
155#381—28no—
156#381—28no—
157#381—28no—
158#381—44no—
159#380—24no—
160#380—24no—
161#380—28no—
162C4×C4⋊C4—44no—
163C4×C22⋊C4—24no—
164C4×C22⋊C4—24no—
165C2×#95—28no—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C25C2411yescharacteristic
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—42no—
199C22×Q8—44no—
200C22×Q8—44no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D418yes—
225C23×C4C2×Q818yes—
226C23×C4C2×Q814yes—
227C23×C4#4114yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C4⋊Q8—416no—
245C22⋊Q8—432no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—416no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—432no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D418yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×Q818yes—
298C2×C4⋊C4C2×Q818yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4#4114yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—416no—
331#95—28no—
332#95—28no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—42no—
352#95—44no—
353#95—42no—
354#95—42no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95C2×D418yes—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C22×Q812yes—
367C24C2×#4112yes—
368C24C2×#4112yes—
369C24H(5,𝔽2)12yes—
370C242−1+412yes—
371C24C2511yescharacteristic
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—84no—
375C2×Q8—88no—
376C2×Q8—88no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4C22⋊Q8116yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×Q818yes—
487C22×C4C22×Q812yes—
488C22×C4C22×Q812yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4H(5,𝔽2)14yes—
494C22×C42−1+414yes—
495C22×C42−1+414yes—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—416no—
504C4⋊C4—416no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—84no—
513C4⋊C4—88no—
514C4⋊C4—88no—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C42—48no—
522C42—48no—
523C42—44no—
524C42—44no—
525C42—48no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23C2×C22⋊Q814yes—
543C23#53214yes—
544C23#54214yes—
545C23D4214yes—
546C23#54814yes—
547C23D4×Q814yes—
548C23#55414yes—
549C23C23×D411yescharacteristic
550C23C23×D411yescharacteristic
551C23C23×Q811yescharacteristic
552C23C22×#4111yescharacteristic
553C23C22×#4111yescharacteristic
554C23C2×H(5,𝔽2)12yes—
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×2−1+412yes—
557C23C2×2−1+411yescharacteristic
558C23C2×2−1+412yes—
559C23#58512yes—
560Q8—816no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—232no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—432no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
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—816no—
652C2×C4C2×C22⋊Q814yes—
653C2×C4C2×C22⋊Q814yes—
654C2×C4#53414yes—
655C2×C4#53614yes—
656C2×C4#53918yes—
657C2×C4#54418yes—
658C22—28no—
659C22—24no—
660C22—28no—
661C22—24no—
662C22—24no—
663C22—28no—
664C22—28no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22#245218yes—
669C22C22×C22⋊Q811yescharacteristic
670C22C2×#53211yescharacteristic
671C22C2×#53411yescharacteristic
672C22C2×#53611yescharacteristic
673C22C2×#53912yes—
674C22C2×#54211yescharacteristic
675C22C2×#54412yes—
676C22#346112yes—
677C22C2×D4211yescharacteristic
678C22C2×#54811yescharacteristic
679C22C2×D4×Q811yescharacteristic
680C22C2×#55411yescharacteristic
681C22#348511yescharacteristic
682C22#348611yescharacteristic
683C22#349612yes—
684C22#349912yes—
685C22#350212yes—
686C22#351512yes—
687C22#352012yes—
688C22#353812yes—
689C22#354212yes—
690C22#354412yes—
691C22#356312yes—
692C4—24no—
693C4—48no—
694C4—42no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—42no—
699C4—44no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—84no—
707C4#262718yescomplemented
708C2—28no—
709C2C2×#262711yescharacteristic
710C2C2×#245211yescharacteristic
711C2#3500812yes—
712C2#3515411yescharacteristic
713C2#3750312yes—
714C2#3896312yes—
715C2#3912111yescharacteristic
716C2#3912211yescharacteristic
717C2#3933711yescharacteristic
718C2#3933811yescharacteristic
719C2#3936812yes—
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
4. #16554C2semidirect
5. #16553C2semidirect
6. #15149C2semidirect
7. #15139C2semidirect
8. #15137C2semidirect
9. #15128C2semidirect
104. C22×C4⋊C4Q8semidirect
707. C4#2627semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#26272
C22×#3892
#165542
#165532
#151492
#151392
#151372
#151282
#149352
#149342
#146972
#142632
#139072
#139042
C4×#3852