Finite Group Databasebeta

← #517384

Subgroups — #517384

Order 512.

2930 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#14697C212yesmaximal
3#14682C211yesmaximal, characteristic
4#14286C212yesmaximal, complemented
5#14284C212yesmaximal, complemented
6#14053C212yesmaximal, complemented
7#14052C214yesmaximal
8#14029C212yesmaximal, complemented
9#14028C214yesmaximal
10#14006C212yesmaximal, complemented
11#13982C212yesmaximal, complemented
12#13793C212yesmaximal, complemented
13#13792C212yesmaximal, complemented
14C4×#380C212yesmaximal
15C2×C4×#95C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C42C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2857C4116yescomplemented
22C2×#391C2214yes—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#385C2214yes—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41#1446C2214yes—
42#1445C2214yes—
43#1445C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2214yes—
61#1434C2212yes—
62#1434C2212yes—
63#1434C2214yes—
64#1434C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C4×#95C2212yes—
70C4×#95C2211yescharacteristic
71C4×#95C2212yes—
72C4×#95C2211yescharacteristic
73C4×#95C2211yescharacteristic
74C4×#95C2211yescharacteristic
75C4×#95C2214yes—
76C23×Q8C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8—42no—
84C2×C4⋊Q8C2×C418yescomplemented
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8C2×C4116yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42—24no—
100C22×C42—42no—
101C22×C42—42no—
102C22×C42—42no—
103C22×C42—42no—
104C22×C42—42no—
105C22×C42—42no—
106C22×C42C2×C418yes—
107C22×C42Q818yescomplemented
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#391—216no—
121#391C2×C4116yescomplemented
122#389—44no—
123#389—44no—
124#389C2×C4116yescomplemented
125#389C2×C4116yescomplemented
126#387—216no—
127#385—48no—
128#384C2×C418yescomplemented
129#382—42no—
130#382—42no—
131#380—28no—
132#380C2×C4116yes—
133#380C2×C4116yes—
134C4×C4⋊C4—44no—
135C4×C4⋊C4—44no—
136C4×C22⋊C4—48no—
137C2×#95—28no—
138C2×#95—28no—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25C2411yescharacteristic
169C22×Q8—28no—
170C22×Q8—44no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8C22×C418yes—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—41no—
194C23×C4—41no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4C22×C414yes—
200C23×C4C22×C414yes—
201C23×C4C22×C414yes—
202C23×C4C2×Q818yes—
203C23×C4C2×Q814yes—
204C23×C4#4114yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C4⋊Q8—416no—
220C22⋊Q8—432no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C22×C4116yes—
248C2×C4⋊C4C22×C418yes—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4C22×C4116yes—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—416no—
269C2×C42—416no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—41no—
282C2×C42—416no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42C22×C414yes—
288C2×C42C22×C414yes—
289C2×C42C22×C414yes—
290C2×C42C22×C414yes—
291C2×C42C2×Q818yes—
292C2×C42C2×Q814yes—
293C2×C42C2×Q814yes—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95C22×C4116yes—
320#95C22×C4116yes—
321#95#4118yes—
322#95#4118yes—
323C24—22no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24C23×C412yes—
330C24C22×Q812yes—
331C24C2×#4112yes—
332C24H(5,𝔽2)14yes—
333C242−1+414yes—
334C24C2511yescharacteristic
335C2×Q8—216no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—88no—
339C2×Q8—88no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—216no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—216no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4C4×Q8116yes—
437C22×C4C23×C412yes—
438C22×C4C23×C418yes—
439C22×C4C23×C418yes—
440C22×C4C23×C412yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C23×C412yes—
444C22×C4C23×C412yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q812yes—
447C22×C4C22×Q814yes—
448C22×C4C22×Q812yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C22⋊C4—232no—
472C22⋊C4—416no—
473C42—28no—
474C42—28no—
475C42—216no—
476C42—44no—
477C42—44no—
478C42—44no—
479C42—44no—
480C42—44no—
481C42—44no—
482C42—48no—
483C42—88no—
484C42—88no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23C2×C4×Q814yes—
504C23#51814yes—
505C23#51914yes—
506C23#54218yes—
507C23#54418yes—
508C23C24×C411yescharacteristic
509C23C23×Q811yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4112yes—
512C23C22×#4112yes—
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)12yes—
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×2−1+412yes—
518C23C2×2−1+411yescharacteristic
519Q8—816no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—232no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—432no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—432no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4C2×C4×Q814yes—
605C2×C4C2×C4×Q814yes—
606C2×C4C4×#4114yes—
607C2×C4C4×#4114yes—
608C2×C4#51814yes—
609C2×C4#52014yes—
610C2×C4#53318yes—
611C2×C4#53814yes—
612C2×C4#53914yes—
613C2×C4#54314yes—
614C2×C4#54414yes—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—24no—
619C22—28no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22#233118yes—
626C22C22×C4×Q811yescharacteristic
627C22C2×C4×#4111yescharacteristic
628C22C2×C4×#4111yescharacteristic
629C22C2×#51811yescharacteristic
630C22C2×#51811yescharacteristic
631C22C2×#51911yescharacteristic
632C22C2×#52011yescharacteristic
633C22C2×#53312yes—
634C22#344412yes—
635C22C2×#53811yescharacteristic
636C22C2×#53911yescharacteristic
637C22C2×#54212yes—
638C22C2×#54311yescharacteristic
639C22C2×#54412yes—
640C22C2×#54411yescharacteristic
641C22#351912yes—
642C22#351912yes—
643C22#351912yes—
644C22#352012yes—
645C22#352012yes—
646C22#352012yes—
647C22#352212yes—
648C22#352312yes—
649C4—28no—
650C4—24no—
651C4—24no—
652C4—24no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—48no—
657C4—42no—
658C4—42no—
659C4—48no—
660C4—84no—
661C4—84no—
662C4#232918yes—
663C2—28no—
664C2C4×#53311yescharacteristic
665C2C2×#232911yescharacteristic
666C2C2×#233111yescharacteristic
667C2#3423511yescharacteristic
668C2#3423611yescharacteristic
669C2#3424511yescharacteristic
670C2#3424611yescharacteristic
671C2#3507912yes—
672C2#3508012yes—
673C2#3508312yes—
674C2#3523912yes—
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. #14286C2semidirect
5. #14284C2semidirect
6. #14053C2semidirect
8. #14029C2semidirect
10. #14006C2semidirect
11. #13982C2semidirect
12. #13793C2semidirect
13. #13792C2semidirect
21. #2857C4semidirect
84. C2×C4⋊Q8C2×C4semidirect
107. C22×C42Q8semidirect
121. #391C2×C4semidirect
124. #389C2×C4semidirect
125. #389C2×C4semidirect
128. #384C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#146972
#146822
#142862
#142842
#140532
#140522
#140292
#140282
#140062
#139822
#137932
#137922
C4×#3802
C2×C4×#952