Finite Group Databasebeta

← #613842

Subgroups — #613842

Order 512.

3302 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3C2×#2580C211yesmaximal, characteristic
4C2×C42×D4C211yesmaximal, characteristic
5#15204C214yesmaximal, complemented
6#14771C214yesmaximal, complemented
7#14770C214yesmaximal, complemented
8#14709C211yesmaximal, characteristic, complemented
9#14697C214yesmaximal, complemented
10#14006C212yesmaximal, complemented
11#14005C212yesmaximal, complemented
12#13954C211yesmaximal, characteristic, complemented
13#13920C212yesmaximal
14#13920C212yesmaximal
15C4×#391C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C42C2212yes—
21#2584—28no—
22#2580—28no—
23#2566—28no—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#396C2214yes—
27C2×#395C2212yes—
28C2×#393C2212yes—
29C2×#391C2212yes—
30C2×#390C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#380C2214yes—
46C2×#380C2214yes—
47C42×D4—28no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2214yes—
53C2×C43C2211yescharacteristic
54#1444C2214yes—
55#1444C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1436C2212yes—
64#1436C2212yes—
65#1436C2212yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
73C4⋊#95C2214yes—
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2314yes—
82C24×C4C2312yes—
83C2×#125—24no—
84C2×#125—24no—
85C2×#125—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4D418yes—
94C2×C4×D4Q8116yescomplemented
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C42—28no—
118C22×C42—48no—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#399—28no—
130#398—28no—
131#396—216no—
132#395—28no—
133#393—28no—
134#391—28no—
135#390—24no—
136#390—24no—
137#390—28no—
138#389—28no—
139#389—28no—
140#389—28no—
141#389—28no—
142#389—28no—
143#389—216no—
144#389—48no—
145#385—216no—
146#385—216no—
147#384—24no—
148#382—24no—
149#382—24no—
150#380—216no—
151#380—48no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4D418yes—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—216no—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C43—24no—
181C25C2412yes—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—42no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4C2×D414yes—
191C22×D4C2×Q818yes—
192C23×C4—28no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C2×D418yes—
203C23×C4C2×Q8116yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218#125—416no—
219C4×Q8—416no—
220C4×D4—264no—
221C4×D4—232no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×Q818yes—
269C2×C4⋊C4#4118yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×D418yes—
287C2×C22⋊C4C2×Q8116yes—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—28no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—432no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42C2×D418yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×Q818yes—
313C2×C42#4114yes—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—42no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95C2×D418yes—
336#95#4118yes—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24C22×D414yes—
345C24C22×Q818yes—
346C24C2511yescharacteristic
347C2×Q8—28no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×D4—28no—
353C2×D4—216no—
354C2×D4—28no—
355C2×D4—216no—
356C2×D4C4⋊Q8116yes—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—216no—
392C22×C4—232no—
393C22×C4—232no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
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—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D418yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q818yes—
444C22×C4C22×Q814yes—
445C22×C4C22×Q814yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)14yes—
455C4⋊C4—28no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C4⋊C4C22⋊Q8132yes—
471C22⋊C4—216no—
472C22⋊C4—232no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C42—28no—
478C42—216no—
479C42—48no—
480C42—48no—
481C42—416no—
482C42—44no—
483C42—48no—
484C42—44no—
485C42—44no—
486C42#121116yes—
487C23—24no—
488C23—28no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23C2×C4⋊Q818yes—
508C23C23×D411yescharacteristic
509C23C23×D411yescharacteristic
510C23C23×Q812yes—
511C23C22×#4111yescharacteristic
512C23C22×#4112yes—
513C23C22×#4112yes—
514C23C2×H(5,𝔽2)12yes—
515C23#58512yes—
516C23#58512yes—
517Q8—416no—
518D4—232no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—464no—
551C2×C4—432no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4C2×#12114yes—
599C2×C4C2×#12114yes—
600C2×C4C2×#12114yes—
601C2×C4C2×C22⋊Q8116yes—
602C2×C4C2×C22⋊Q818yes—
603C2×C4C2×C4⋊Q814yes—
604C2×C4#53214yes—
605C2×C4#53318yes—
606C2×C4#55418yes—
607C22—28no—
608C22—216no—
609C22—48no—
610C22—48no—
611C22—44no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22C22×#12111yescharacteristic
617C22C22×C22⋊Q812yes—
618C22C22×C4⋊Q811yescharacteristic
619C22C2×#53211yescharacteristic
620C22C2×#53312yes—
621C22#344411yescharacteristic
622C22#345511yescharacteristic
623C22#346012yes—
624C22D4×#4112yes—
625C22D4×#4112yes—
626C22C2×#55412yes—
627C22Q8×#4112yes—
628C22#348012yes—
629C22#348112yes—
630C22#348411yescharacteristic
631C22#348711yescharacteristic
632C22#349612yes—
633C22#350112yes—
634C22#350412yes—
635C22#351312yes—
636C22#352412yes—
637C4—24no—
638C4—28no—
639C4—216no—
640C4—24no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—84no—
649C4—82no—
650C4#285718yes—
651C2—48no—
652C2#3474311yescharacteristic
653C2#3489312yes—
654C2#3516811yescharacteristic
655C2#3517512yes—
656C2#3621512yes—
657C2#3626112yes—
658C2#3643112yes—
659C2#3647112yes—
660C2C2×#285711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2584C2semidirect
5. #15204C2semidirect
6. #14771C2semidirect
7. #14770C2semidirect
8. #14709C2semidirect
9. #14697C2semidirect
10. #14006C2semidirect
11. #14005C2semidirect
12. #13954C2semidirect
15. C4×#391C2semidirect
94. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
C2×#25662
C2×#25802
C2×C42×D42
#152042
#147712
#147702
#147092
#146972
#140062
#140052
#139542
#139202
C4×#3912