Finite Group Databasebeta

← #556356

Subgroups — #556356

Order 512.

2786 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C22×#387C211yesmaximal, characteristic
4#14634C214yesmaximal, complemented
5#14339C211yesmaximal, characteristic
6#14327C212yesmaximal, complemented
7#14326C212yesmaximal, complemented
8#14307C212yesmaximal, complemented
9#14302C212yesmaximal, complemented
10#14281C211yesmaximal, characteristic
11#14275C211yesmaximal, characteristic
12#14203C214yesmaximal, complemented
13#14044C212yesmaximal
14#14039C212yesmaximal
15#13992C212yesmaximal
16#13972C212yesmaximal
17C4×#380C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2627C4116yescomplemented
24C2×#400C2212yes—
25C2×#398C2214yes—
26C2×#395C2212yes—
27C4×C22⋊Q8—28no—
28C2×#391C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#387—28no—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52C22×#95C2211yescharacteristic
53C22×#95C2211yescharacteristic
54#1446C2212yes—
55#1445C2212yes—
56#1444C2212yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2212yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77#1434C2214yes—
78#1434C2214yes—
79#1434C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8C2×C418yes—
97C2×C4×Q8—24no—
98C2×C4×Q8—44no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4C2×C418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42—44no—
123C22×C42D418yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#400C2×C4116yescomplemented
132#398C2×C4132yescomplemented
133#395C2×C4116yescomplemented
134#391—24no—
135#389—24no—
136#387—28no—
137#387—28no—
138#387—28no—
139#387—28no—
140#387—28no—
141#387—48no—
142#385—28no—
143#385—44no—
144#385C2×C4116yes—
145#381—44no—
146#381C2×C4116yes—
147#380—24no—
148#380—24no—
149#380C2×C418yes—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—44no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—44no—
155C2×#95—24no—
156C2×#95—24no—
157C2×#95—24no—
158C2×#95—24no—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8C22×C414yes—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4C22×C414yes—
228C23×C4C22×C414yes—
229C23×C4C22×C414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4#4114yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C22⋊Q8—232no—
251C4×Q8—416no—
252C2×C4⋊C4—232no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
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—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—416no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C22×C418yes—
285C2×C4⋊C4C22×C418yes—
286C2×C4⋊C4C22×C418yes—
287C2×C4⋊C4C22×C414yes—
288C2×C4⋊C4C22×C418yes—
289C2×C4⋊C4C22×C4116yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
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⋊C4C22×C418yes—
301C2×C22⋊C4C22×C418yes—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—416no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95C22×C418yes—
347#95C22×C4116yes—
348#95C22×C4116yes—
349#95C22×C4116yes—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24C23×C412yes—
355C24C22×D412yes—
356C24C2×#4112yes—
357C24H(5,𝔽2)12yes—
358C242−1+412yes—
359C242−1+412yes—
360C242−1+412yes—
361C24C2511yescharacteristic
362C2×Q8—28no—
363C2×Q8—48no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—48no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4C4×D4116yes—
473C22×C4C23×C414yes—
474C22×C4C23×C418yes—
475C22×C4C23×C414yes—
476C22×C4C23×C414yes—
477C22×C4C23×C414yes—
478C22×C4C23×C412yes—
479C22×C4C23×C412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C42−1+414yes—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—48no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—88no—
504C22⋊C4—216no—
505C22⋊C4—216no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C42—28no—
509C42—28no—
510C42—44no—
511C42—44no—
512C42—48no—
513C42—48no—
514C42—88no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23C2×C4×D414yes—
532C23#51914yes—
533C23#52014yes—
534C23#53614yes—
535C23#53714yes—
536C23#54114yes—
537C23#54214yes—
538C23C24×C411yescharacteristic
539C23C23×D411yescharacteristic
540C23C22×#4111yescharacteristic
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×H(5,𝔽2)12yes—
543C23C2×2−1+411yescharacteristic
544C23C2×2−1+411yescharacteristic
545C23C2×2−1+411yescharacteristic
546C23C2×2−1+412yes—
547C23#58512yes—
548C23#58512yes—
549Q8—416no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—416no—
587C2×C4—416no—
588C2×C4—416no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
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—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—88no—
638C2×C4—88no—
639C2×C4C2×C4×D414yes—
640C2×C4C2×C4×D414yes—
641C2×C4#53414yes—
642C2×C4#53614yes—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—28no—
648C22—28no—
649C22—28no—
650C22—28no—
651C22—24no—
652C22—24no—
653C22#232318yes—
654C22C22×C4×D411yescharacteristic
655C22C2×#51911yescharacteristic
656C22C2×#52011yescharacteristic
657C22C4×H(5,𝔽2)12yes—
658C22C4×2−1+412yes—
659C22C2×#53411yescharacteristic
660C22C2×#53611yescharacteristic
661C22C2×#53611yescharacteristic
662C22C2×#53711yescharacteristic
663C22#344911yescharacteristic
664C22#344911yescharacteristic
665C22C2×#54111yescharacteristic
666C22C2×#54211yescharacteristic
667C22#345912yes—
668C22#346112yes—
669C22#353512yes—
670C22#353512yes—
671C22#353612yes—
672C22#353812yes—
673C22#356012yes—
674C22#356312yes—
675C22#356312yes—
676C22#356412yes—
677C4—24no—
678C4—24no—
679C4—24no—
680C4—24no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—48no—
688C4—82no—
689C4—82no—
690C4—84no—
691C4—82no—
692C4—82no—
693C2—28no—
694C2C2×#232311yescharacteristic
695C2C4×#53411yescharacteristic
696C2C4×#53611yescharacteristic
697C2#3442712yes—
698C2#3443212yes—
699C2#3744111yescharacteristic
700C2#3744211yescharacteristic
701C2#3746412yes—
702C2#3746612yes—
703C2#3753811yescharacteristic
704C2#3754011yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #14634C2semidirect
6. #14327C2semidirect
7. #14326C2semidirect
8. #14307C2semidirect
9. #14302C2semidirect
12. #14203C2semidirect
23. #2627C4semidirect
131. #400C2×C4semidirect
132. #398C2×C4semidirect
133. #395C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C2×C4×C22⋊Q82
C22×#3872
#146342
#143392
#143272
#143262
#143072
#143022
#142812
#142752
#142032
#140442
#140392
#139922
#139722
C4×#3802