Finite Group Databasebeta

← #530484

Subgroups — #530484

Order 512.

2714 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#2283C211yesmaximal, characteristic
4#14682C211yesmaximal, characteristic
5#14394C214yesmaximal, complemented
6#14392C214yesmaximal
7#14266C212yesmaximal, complemented
8#13973C212yesmaximal
9#13969C212yesmaximal
10#13955C211yesmaximal, characteristic, complemented
11#13942C211yesmaximal, characteristic, complemented
12#13917C212yesmaximal
13#13914C212yesmaximal, complemented
14#13893C212yesmaximal, complemented
15#13887C212yesmaximal
16#13804C211yesmaximal, characteristic, complemented
17#13786C212yesmaximal, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2351—28no—
23C4×C22⋊Q8C4116yescomplemented
24C2×#391C2212yes—
25C2×#391C2211yescharacteristic
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#387C2212yes—
30C2×#385C2212yes—
31#2283—28no—
32C2×#383C2211yescharacteristic
33C2×#383C2211yescharacteristic
34C2×#382C2212yes—
35C2×#382C2212yes—
36C2×#382C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#381C2211yescharacteristic
39C2×#381C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×#376C2212yes—
52#1445C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2212yes—
75#1434C2214yes—
76#1434C2214yes—
77#1434C2212yes—
78#1434C2212yes—
79#1434C2212yes—
80#1434C2212yes—
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8C2×C418yes—
93C2×C4×Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—28no—
96C2×C4×Q8—24no—
97C2×C4×Q8—44no—
98C2×C4×Q8C2×C418yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C42—24no—
117C22×C42—24no—
118C22×C42C2×C418yes—
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#391—28no—
130#391C2×C418yes—
131#389—216no—
132#389—28no—
133#389C2×C418yes—
134#387C2×C4116yes—
135#385C2×C4116yes—
136#383—24no—
137#383—24no—
138#382—28no—
139#382—24no—
140#382—24no—
141#382—28no—
142#381—24no—
143#381—24no—
144#380—28no—
145#380—44no—
146#380C2×C418yes—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—44no—
150C4×C4⋊C4C2×C4116yes—
151C4×C4⋊C4C2×C418yes—
152C4×C4⋊C4D418yes—
153C4×C22⋊C4—28no—
154C4×C22⋊C4C2×C4116yes—
155#376—28no—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C25C2411yescharacteristic
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8C22×C414yes—
196C22×Q8#4114yes—
197C22×Q8#4114yes—
198C23×C4—24no—
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—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4C22×C414yes—
213C23×C4C22×C414yes—
214C23×C4C22×C414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C22⋊Q8—232no—
232C4×Q8—232no—
233C4×Q8—432no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C22×C418yes—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—232no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C22×C418yes—
305C2×C42C22×C414yes—
306C2×C42C22×C414yes—
307C2×C42C22×C414yes—
308C2×C42C22×C418yes—
309C2×C42C22×C414yes—
310C2×C42C2×D418yes—
311C2×C42C2×D414yes—
312C2×C42#4114yes—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C22×C418yes—
341#95C22×C418yes—
342#95C22×C418yes—
343#95C22×C418yes—
344#95C22×C418yes—
345#95C2×D414yes—
346#95C2×D414yes—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24C23×C412yes—
357C242−1+414yes—
358C24C2511yescharacteristic
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8#117116yes—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
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—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4C23×C412yes—
458C22×C4C23×C414yes—
459C22×C4C23×C412yes—
460C22×C4C23×C412yes—
461C22×C4C23×C414yes—
462C22×C4C23×C414yes—
463C22×C4C23×C414yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C42−1+412yes—
481C4⋊C4—28no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—88no—
498C42—28no—
499C42—28no—
500C42—28no—
501C42—216no—
502C42—216no—
503C42—48no—
504C42—416no—
505C42—48no—
506C42—88no—
507C42C4×D4116yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23#51914yes—
531C23C24×C411yescharacteristic
532C23C23×D411yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C22×#4111yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+412yes—
538C23C2×2−1+411yescharacteristic
539C23#58512yes—
540C23#58512yes—
541C23#58512yes—
542Q8—232no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4C2×#11718yes—
635C2×C4C2×#11714yes—
636C2×C4C2×C4×D414yes—
637C2×C4C2×C4×D414yes—
638C2×C4C2×C4×D414yes—
639C2×C4#52014yes—
640C2×C4#54614yes—
641C2×C4#54814yes—
642C2×C4#55214yes—
643C2×C4#55314yes—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22C22×#11711yescharacteristic
656C22C22×C4×D411yescharacteristic
657C22C2×#51911yescharacteristic
658C22C2×#52011yescharacteristic
659C22#342811yescharacteristic
660C22#342811yescharacteristic
661C22#342811yescharacteristic
662C22#344912yes—
663C22#345512yes—
664C22C2×#54611yescharacteristic
665C22C2×#54811yescharacteristic
666C22C2×#55211yescharacteristic
667C22C2×#55311yescharacteristic
668C22#348912yes—
669C22#349512yes—
670C22#351612yes—
671C22#351612yes—
672C22#352412yes—
673C22#352412yes—
674C22#356112yes—
675C22#356212yes—
676C22#356312yes—
677C22#356512yes—
678C4—24no—
679C4—28no—
680C4—24no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—44no—
686C4—42no—
687C4—44no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4—84no—
693C4#235218yes—
694C2—44no—
695C2#3406511yescharacteristic
696C2C2×#235211yescharacteristic
697C2#3419211yescharacteristic
698C2#3419711yescharacteristic
699C2#3420311yescharacteristic
700C2#3439211yescharacteristic
701C2#3439911yescharacteristic
702C2#3890312yes—
703C2#3892712yes—
704C2#3911912yes—
705C2#3914312yes—
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
1. C2×#2351C2semidirect
5. #14394C2semidirect
7. #14266C2semidirect
10. #13955C2semidirect
11. #13942C2semidirect
13. #13914C2semidirect
14. #13893C2semidirect
16. #13804C2semidirect
17. #13786C2semidirect
23. C4×C22⋊Q8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
C2×C4×C22⋊Q82
C2×#22832
#146822
#143942
#143922
#142662
#139732
#139692
#139552
#139422
#139172
#139142
#138932
#138872
#138042
#137862