Finite Group Databasebeta

← #647029

Subgroups — #647029

Order 512.

2578 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic
3#16991C212yesmaximal, complemented
4#16889C212yesmaximal, complemented
5#16688C214yesmaximal, complemented
6#15083C212yesmaximal, complemented
7#15019C211yesmaximal, characteristic
8#15014C211yesmaximal, characteristic
9#14935C212yesmaximal
10#14934C211yesmaximal, characteristic
11#14914C211yesmaximal, characteristic
12#14878C212yesmaximal, complemented
13#14728C212yesmaximal
14#14682C211yesmaximal, characteristic
15#14270C214yesmaximal, complemented
16#14039C212yesmaximal
17#13992C212yesmaximal
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#398C2214yes—
24C4×C22⋊Q8—28no—
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#387—28no—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#382C2212yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1446C2212yes—
50#1446C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2212yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75#1434C2212yes—
76#1434C2212yes—
77#1434C2214yes—
78C22⋊#95C2211yescharacteristic
79C22⋊#95C2212yes—
80C22⋊#95C2211yescharacteristic
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42—44no—
119C22×C42D418yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#398—48no—
127#391—24no—
128#389—24no—
129#387—216no—
130#387—28no—
131#387—28no—
132#387—28no—
133#387—28no—
134#387—28no—
135#385—28no—
136#385—28no—
137#385—44no—
138#382—44no—
139#381—28no—
140#381—44no—
141#380—24no—
142#380—44no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C22⋊C4—28no—
146C2×#95—24no—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—42no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C22⋊Q8—232no—
237C4×Q8—416no—
238C2×C4⋊C4—232no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4#4118yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4#4118yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—28no—
289C2×C42—416no—
290C2×C42—44no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24C22×D412yes—
336C24C2×#4114yes—
337C24H(5,𝔽2)12yes—
338C242−1+414yes—
339C242−1+412yes—
340C24C2511yescharacteristic
341C2×Q8—28no—
342C2×Q8—416no—
343C2×Q8—44no—
344C2×Q8—44no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C42−1+414yes—
456C4⋊C4—216no—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—416no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—216no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C42—28no—
474C42—28no—
475C42—44no—
476C42—44no—
477C42—48no—
478C42—48no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23#52514yes—
492C23#53614yes—
493C23#53714yes—
494C23#54118yes—
495C23#54218yes—
496C23C23×D411yescharacteristic
497C23C22×#4112yes—
498C23C22×#4112yes—
499C23C2×H(5,𝔽2)12yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×2−1+412yes—
502C23C2×2−1+412yes—
503C23C2×2−1+411yescharacteristic
504C23#58512yes—
505Q8—416no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—416no—
534C2×C4—416no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—416no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
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—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—88no—
589C2×C4—88no—
590C2×C4#52514yes—
591C2×C4#53414yes—
592C2×C4#53614yes—
593C22—24no—
594C22—24no—
595C22—216no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—216no—
601C22#263718yes—
602C22C2×#52511yescharacteristic
603C22C2×#52511yescharacteristic
604C22C2×#53411yescharacteristic
605C22C2×#53611yescharacteristic
606C22C2×#53611yescharacteristic
607C22C2×#53711yescharacteristic
608C22#344911yescharacteristic
609C22C2×#54112yes—
610C22C2×#54212yes—
611C22#347514yes—
612C22#352014yes—
613C22#352014yes—
614C22#352212yes—
615C22#352312yes—
616C22#353412yes—
617C22#353512yes—
618C22#356314yes—
619C4—24no—
620C4—24no—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—84no—
630C4—84no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C2—28no—
636C2C2×#263711yescharacteristic
637C2#3487511yescharacteristic
638C2#3508411yescharacteristic
639C2#3508411yescharacteristic
640C2#3524711yescharacteristic
641C2#3745211yescharacteristic
642C2#3754011yescharacteristic
643C2#3814314yes—
644C2#3817314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16991C2semidirect
4. #16889C2semidirect
5. #16688C2semidirect
6. #15083C2semidirect
12. #14878C2semidirect
15. #14270C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C22×#3872
#169912
#168892
#166882
#150832
#150192
#150142
#149352
#149342
#149142
#148782
#147282
#146822
#142702
#140392
#139922