Finite Group Databasebeta

← #877915

Subgroups — #877915

Order 512.

2366 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2763C212yesmaximal
2#16987C212yesmaximal
3#16823C212yesmaximal
4#15944C214yesmaximal, complemented
5#15691C214yesmaximal, complemented
6#15146C212yesmaximal, complemented
7#15128C212yesmaximal, complemented
8#15125C211yesmaximal, characteristic
9#15125C211yesmaximal, characteristic
10#14697C212yesmaximal
11#14514C212yesmaximal
12#14053C212yesmaximal, complemented
13#13911C212yesmaximal, complemented
14#13876C212yesmaximal
15C4×#385C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2763—216no—
20C2×#400C2214yes—
21C2×#395C2214yes—
22C2×#393C2212yes—
23C2×#393C2212yes—
24C2×#391C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#385C2214yes—
29C2×#385C2211yescharacteristic
30C2×#385C2211yescharacteristic
31C2×#384C2212yes—
32C2×#382C2214yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39#1446C2212yes—
40#1446C2212yes—
41#1446C2214yes—
42#1446C2214yes—
43#1444C2214yes—
44#1442C2212yes—
45#1442C2212yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1436C2214yes—
58#1436C2212yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2214yes—
74C22⋊#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C23×Q8C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8—42no—
84C2×C4⋊Q8—44no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C42—44no—
104C22×C42—42no—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2314yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111#400—216no—
112#395—216no—
113#393—28no—
114#393—28no—
115#391—216no—
116#389—44no—
117#389—44no—
118#388—216no—
119#385—216no—
120#385D418yes—
121#385Q818yescomplemented
122#384—44no—
123#382—216no—
124#380—28no—
125#380—44no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—44no—
128C4×C22⋊C4—42no—
129C4×C22⋊C4Q818yescomplemented
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C25C2411yescharacteristic
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—28no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—44no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—41no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—41no—
179C23×C4—41no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—41no—
183C23×C4C2×D414yes—
184C23×C4C2×Q814yes—
185C23×C4C2×Q814yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C4⋊Q8—416no—
201C4⋊Q8—416no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4C2×Q818yes—
235C2×C4⋊C4#4118yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×D414yes—
249C2×C22⋊C4C2×D414yes—
250C2×C22⋊C4C2×Q814yes—
251C2×C22⋊C4C2×Q814yes—
252C2×C22⋊C4C2×Q814yes—
253C2×C22⋊C4C2×Q814yes—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—24no—
258C2×C42—28no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—44no—
269C2×C42—416no—
270C2×C42C2×Q818yes—
271#95—24no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—28no—
280#95—28no—
281#95—42no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—42no—
286#95—42no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—41no—
292#95—41no—
293#95—42no—
294#95—42no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95C2×D418yes—
299#95C2×Q818yes—
300#95C2×Q818yes—
301C24—22no—
302C24—21no—
303C24—22no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24C22×D412yes—
310C24C22×Q812yes—
311C24C22×Q812yes—
312C24C2511yescharacteristic
313C2×Q8—48no—
314C2×Q8—416no—
315C2×Q8—48no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8—88no—
319C2×Q8—88no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—88no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—216no—
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—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4C22×Q812yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q814yes—
414C22×C4C22×Q814yes—
415C22×C4C22×Q812yes—
416C22×C4C22×Q814yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q812yes—
419C22×C4C2×#4118yes—
420C22×C4C2×#4114yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+412yes—
424C4⋊C4—216no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—88no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—416no—
440C22⋊C4—88no—
441C22⋊C4C4⋊Q8116yescomplemented
442C42—216no—
443C42—416no—
444C42—48no—
445C42—44no—
446C42—48no—
447C42—44no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—24no—
460C23—42no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23C2×C4⋊Q814yes—
471C23C23×D411yescharacteristic
472C23C23×Q811yescharacteristic
473C23C23×Q811yescharacteristic
474C23C22×#4112yes—
475C23C2×H(5,𝔽2)12yes—
476C23C2×H(5,𝔽2)11yescharacteristic
477C23C2×2−1+411yescharacteristic
478C23#58511yescharacteristic
479C23#58511yescharacteristic
480C23#58512yes—
481C23#58512yes—
482Q8—816no—
483Q8—816no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—416no—
513C2×C4—416no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—416no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
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—44no—
567C2×C4C2×C4⋊Q818yes—
568C2×C4#54218yes—
569C2×C4#54414yes—
570C2×C4#55114yes—
571C2×C4#55418yes—
572C2×C4H(3,𝔽4)14yes—
573C22—24no—
574C22—28no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22C22×C4⋊Q811yescharacteristic
585C22C2×#54212yes—
586C22C2×#54411yescharacteristic
587C22#345811yescharacteristic
588C22D4×#4112yes—
589C22C2×#55111yescharacteristic
590C22C2×#55412yes—
591C22Q8×#4112yes—
592C22#349611yescharacteristic
593C22#349612yes—
594C22#349711yescharacteristic
595C22#350111yescharacteristic
596C22#350112yes—
597C22#350211yescharacteristic
598C22#350412yes—
599C22#352112yes—
600C22#352412yes—
601C22C2×H(3,𝔽4)11yescharacteristic
602C22#354011yescharacteristic
603C22#354011yescharacteristic
604C22#354012yes—
605C22#354012yes—
606C22#355712yes—
607C4—28no—
608C4—44no—
609C4—42no—
610C4—42no—
611C4—44no—
612C4—44no—
613C4—42no—
614C4—48no—
615C4—42no—
616C4—42no—
617C4—84no—
618C4—84no—
619C4—84no—
620C4#276318yescomplemented
621C2—44no—
622C2#3516711yescharacteristic
623C2#3626112yes—
624C2#3742811yescharacteristic
625C2#3810612yes—
626C2#3941112yes—
627C2#3966512yes—
628C2#4547911yescharacteristic
629C2#4549411yescharacteristic
630C2C2×#276311yescharacteristic
631C2#5937511yescharacteristic
632C2#5937611yescharacteristic
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
4. #15944C2semidirect
5. #15691C2semidirect
6. #15146C2semidirect
7. #15128C2semidirect
12. #14053C2semidirect
13. #13911C2semidirect
121. #385Q8semidirect
129. C4×C22⋊C4Q8semidirect
441. C22⋊C4C4⋊Q8semidirect
620. C4#2763semidirect

Maximal subgroups

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

subgroupindex
C2×#27632
#169872
#168232
#159442
#156912
#151462
#151282
#151252
#146972
#145142
#140532
#139112
#138762
C4×#3852