Finite Group Databasebeta

← #770345

Subgroups — #770345

Order 512.

3106 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#395C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic
3#15627C214yesmaximal, complemented
4#15625C214yesmaximal, complemented
5#15592C214yesmaximal, complemented
6#15510C214yesmaximal, complemented
7#15092C212yesmaximal
8#14874C212yesmaximal
9#14159C211yesmaximal, characteristic
10#14149C212yesmaximal
11#14139C211yesmaximal, characteristic
12#14130C212yesmaximal
13#13965C211yesmaximal, characteristic
14#13880C211yesmaximal, characteristic
15#13778C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#395—28no—
22C2×#395C2214yes—
23C2×#395C2214yes—
24C2×#395C2214yes—
25C2×#395C2214yes—
26#2329—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#384C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2214yes—
48#1446C2214yes—
49#1446C2212yes—
50#1445C2212yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2212yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70C4⋊#95C2214yes—
71C4⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—44no—
79C23×Q8—44no—
80C23×Q8—44no—
81C23×Q8C2314yes—
82C23×Q8C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8Q818yescomplemented
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122#395—216no—
123#395—216no—
124#395—48no—
125#395—48no—
126#391—28no—
127#391—44no—
128#389—28no—
129#389—48no—
130#387—28no—
131#387—28no—
132#385—28no—
133#385—44no—
134#385—44no—
135#385—44no—
136#385—44no—
137#384—44no—
138#380—24no—
139#380Q8116yescomplemented
140C4×C22⋊C4—28no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143C2×#95—24no—
144C2×#95—24no—
145C2×#95D418yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C25C2411yescharacteristic
174C22×Q8—22no—
175C22×Q8—28no—
176C22×Q8—22no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—416no—
186C22×Q8—416no—
187C22×Q8—416no—
188C22×Q8—464no—
189C22×Q8C2×Q814yescomplemented
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×Q814yes—
217C23×C4C2×Q818yes—
218C23×C4#4114yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C22⋊Q8—232no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—416no—
260C2×C4⋊C4—416no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×Q814yescomplemented
276C2×C4⋊C4C2×Q818yescomplemented
277C2×C4⋊C4C2×Q8116yescomplemented
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4C2×Q818yes—
288C2×C22⋊C4C2×Q818yes—
289C2×C22⋊C4C2×Q818yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—42no—
300#95—24no—
301#95—24no—
302#95—28no—
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—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×D418yes—
325#95C2×D418yes—
326#95C2×Q8116yescomplemented
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24C22×D412yes—
332C24C22×Q812yes—
333C24C22×Q814yes—
334C24C2×#4112yes—
335C24H(5,𝔽2)14yes—
336C24C2511yescharacteristic
337C2×Q8—28no—
338C2×Q8—48no—
339C2×Q8—416no—
340C2×Q8—416no—
341C2×Q8—416no—
342C2×Q8—416no—
343C2×Q8—416no—
344C2×Q8—416no—
345C2×Q8—4128no—
346C2×Q8—416no—
347C2×Q8—82no—
348C2×Q8—82no—
349C2×Q8—84no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
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—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—416no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—88no—
434C22×C4C22⋊Q8116yes—
435C22×C4C4⋊Q8116yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C22×Q818yes—
441C22×C4C22×Q818yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C22×Q814yes—
447C22×C4C22×Q814yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)14yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+414yes—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—416no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—416no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—48no—
472C42—48no—
473C42—84no—
474C42—84no—
475C42—84no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23C2×C22⋊Q814yes—
489C23C2×C4⋊Q814yes—
490C23#53514yes—
491C23#55418yes—
492C23Q8218yes—
493C23C23×D411yescharacteristic
494C23C23×Q811yescharacteristic
495C23C23×Q812yes—
496C23C22×#4111yescharacteristic
497C23C2×H(5,𝔽2)12yes—
498C23C2×H(5,𝔽2)12yes—
499C23C2×H(5,𝔽2)12yes—
500C23C2×2−1+412yes—
501C23#58512yes—
502Q8—464no—
503Q8—88no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—416no—
538C2×C4—416no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—88no—
582C2×C4C2×C22⋊Q814yes—
583C2×C4C2×C22⋊Q814yes—
584C2×C4C2×C4⋊Q818yes—
585C22—24no—
586C22—216no—
587C22—216no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22#256718yes—
594C22C22×C22⋊Q811yescharacteristic
595C22C22×C4⋊Q811yescharacteristic
596C22C2×#53511yescharacteristic
597C22C2×#55412yes—
598C22C2×Q8212yes—
599C22#348411yescharacteristic
600C22#348411yescharacteristic
601C22#348511yescharacteristic
602C22#348611yescharacteristic
603C22#350112yes—
604C22#350114yes—
605C22#350314yes—
606C22#350312yes—
607C22#350512yes—
608C22#351112yes—
609C22#353214yes—
610C22#353514yes—
611C4—24no—
612C4—24no—
613C4—28no—
614C4—44no—
615C4—48no—
616C4—48no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—82no—
623C4—82no—
624C4—82no—
625C2—28no—
626C2C2×#256711yescharacteristic
627C2#3534911yescharacteristic
628C2#3539111yescharacteristic
629C2#3557411yescharacteristic
630C2#3557611yescharacteristic
631C2#3670311yescharacteristic
632C2#3711811yescharacteristic
633C2#4355114yes—
634C2#4355314yes—
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. #15627C2semidirect
4. #15625C2semidirect
5. #15592C2semidirect
6. #15510C2semidirect
92. C2×C22⋊Q8Q8semidirect
139. #380Q8semidirect
189. C22×Q8C2×Q8semidirect
275. C2×C4⋊C4C2×Q8semidirect
276. C2×C4⋊C4C2×Q8semidirect
277. C2×C4⋊C4C2×Q8semidirect
326. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#3952
C2×#23292
#156272
#156252
#155922
#155102
#150922
#148742
#141592
#141492
#141392
#141302
#139652
#138802
#137782