Finite Group Databasebeta

← #515360

Subgroups — #515360

Order 512.

2910 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2320C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#2285C211yesmaximal, characteristic
4#14078C214yesmaximal, complemented
5#14072C214yesmaximal
6#14005C212yesmaximal, complemented
7#14000C212yesmaximal, complemented
8#13997C212yesmaximal
9#13994C212yesmaximal
10#13919C212yesmaximal
11#13918C212yesmaximal, complemented
12#13915C212yesmaximal
13#13914C212yesmaximal, complemented
14#13786C211yesmaximal, characteristic, complemented
15#13783C211yesmaximal, characteristic, complemented
16#13753C211yesmaximal, characteristic
17#13750C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2212yes—
21C23×C42C2211yescharacteristic
22#2320—28no—
23C4×C22⋊Q8C4116yescomplemented
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#389C2211yescharacteristic
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#387C2212yes—
34C2×#385C2212yes—
35#2285—28no—
36C2×#384C2212yes—
37C2×#383C2212yes—
38C2×#382C2212yes—
39C2×#382C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53C2×#376C2211yescharacteristic
54#1445C2214yes—
55#1444C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2211yescharacteristic
73#1434C2212yes—
74#1434C2212yes—
75#1434C2214yes—
76#1434C2211yescharacteristic
77#1434C2212yes—
78#1434C2212yes—
79C22⋊#95C2211yescharacteristic
80C22⋊#95C2211yescharacteristic
81C22⋊#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2214yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8C2×C418yes—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8C2×C418yes—
95C2×#117—28no—
96C2×#117—44no—
97C2×#117—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42—24no—
116C22×C42—24no—
117C22×C42C2×C418yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#391—28no—
131#391—28no—
132#391C2×C418yes—
133#389—28no—
134#389—28no—
135#389—48no—
136#389C2×C418yes—
137#388—48no—
138#387—216no—
139#387C2×C4116yes—
140#385C2×C4116yes—
141#384—28no—
142#383—28no—
143#382—28no—
144#382—28no—
145#380—28no—
146#380—28no—
147#380C2×C418yes—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4C2×C4116yes—
153C4×C4⋊C4C2×C418yes—
154C4×C4⋊C4D418yes—
155C4×C22⋊C4—28no—
156C4×C22⋊C4C2×C4116yes—
157#376—24no—
158#376—24no—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
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×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—24no—
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—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—42no—
211C23×C4C22×C414yes—
212C23×C4C22×C414yes—
213C23×C4C22×C414yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C22⋊Q8—232no—
231C4×Q8—232no—
232#117—432no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4#4118yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C418yes—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—232no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42C22×C418yes—
307C2×C42C22×C414yes—
308C2×C42C22×C414yes—
309C2×C42C22×C414yes—
310C2×C42C22×C418yes—
311C2×C42C22×C414yes—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42#4118yes—
315C2×C42#4114yes—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—28no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95C22×C418yes—
343#95C22×C418yes—
344#95C22×C418yes—
345#95C22×C418yes—
346#95C22×C418yes—
347#95C2×D418yes—
348C24—21no—
349C24—22no—
350C24—22no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—21no—
355C24—22no—
356C24C23×C412yes—
357C24H(5,𝔽2)12yes—
358C242−1+412yes—
359C24C2511yescharacteristic
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—48no—
364C2×Q8—48no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4C23×C414yes—
456C22×C4C23×C412yes—
457C22×C4C23×C414yes—
458C22×C4C23×C414yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C22×D418yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4118yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4114yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+414yes—
479C22×C42−1+412yes—
480C4⋊C4—216no—
481C4⋊C4—28no—
482C4⋊C4—216no—
483C4⋊C4—28no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4#117116yescomplemented
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C42—28no—
498C42—28no—
499C42—28no—
500C42—216no—
501C42—216no—
502C42—48no—
503C42—48no—
504C42—48no—
505C42—48no—
506C42—88no—
507C42—88no—
508C42C4×D4116yes—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23#52014yes—
532C23C24×C411yescharacteristic
533C23C23×D411yescharacteristic
534C23C22×#4112yes—
535C23C22×#4111yescharacteristic
536C23C22×#4112yes—
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×2−1+412yes—
540C23C2×2−1+411yescharacteristic
541C23C2×2−1+411yescharacteristic
542C23#58512yes—
543Q8—416no—
544C2×C4—232no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—416no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4C2×#11718yes—
636C2×C4C2×#11714yes—
637C2×C4C2×C4×D414yes—
638C2×C4C2×C4×D414yes—
639C2×C4C2×C4×D414yes—
640C2×C4C4×#4114yes—
641C2×C4#51914yes—
642C2×C4#52014yes—
643C2×C4#52918yes—
644C2×C4#53218yes—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22C22×#11711yescharacteristic
657C22C22×C4×D411yescharacteristic
658C22C2×C4×#4111yescharacteristic
659C22C2×#51911yescharacteristic
660C22C2×#52011yescharacteristic
661C22C2×#52011yescharacteristic
662C22#342811yescharacteristic
663C22C2×#52912yes—
664C22C2×#53212yes—
665C22#348411yescharacteristic
666C22#348611yescharacteristic
667C22#348912yes—
668C22#349211yescharacteristic
669C22#349211yescharacteristic
670C22#349412yes—
671C22#349512yes—
672C22#349512yes—
673C22#350912yes—
674C22#351112yes—
675C22#351412yes—
676C22#351512yes—
677C22#352212yes—
678C22#352312yes—
679C4—28no—
680C4—24no—
681C4—24no—
682C4—24no—
683C4—24no—
684C4—48no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—44no—
691C4—82no—
692C4—84no—
693C4—82no—
694C4#229318yes—
695C2—44no—
696C2C2×#229311yescharacteristic
697C2#3419111yescharacteristic
698C2#3419611yescharacteristic
699C2#3420011yescharacteristic
700C2#3420311yescharacteristic
701C2#3422011yescharacteristic
702C2#3422511yescharacteristic
703C2#3560712yes—
704C2#3560812yes—
705C2#3561312yes—
706C2#3563712yes—
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×#2320C2semidirect
4. #14078C2semidirect
6. #14005C2semidirect
7. #14000C2semidirect
11. #13918C2semidirect
13. #13914C2semidirect
14. #13786C2semidirect
15. #13783C2semidirect
17. #13750C2semidirect
23. C4×C22⋊Q8C4semidirect
491. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23202
C2×C4×C22⋊Q82
C2×#22852
#140782
#140722
#140052
#140002
#139972
#139942
#139192
#139182
#139152
#139142
#137862
#137832
#137532
#137502