Finite Group Databasebeta

← #642017

Subgroups — #642017

Order 512.

2538 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3#16993C211yesmaximal, characteristic, complemented
4#16992C211yesmaximal, characteristic, complemented
5#16991C212yesmaximal, complemented
6#16990C211yesmaximal, characteristic, complemented
7#16989C211yesmaximal, characteristic, complemented
8#16988C212yesmaximal, complemented
9#15099C214yesmaximal, complemented
10#15098C212yesmaximal, complemented
11#15097C212yesmaximal, complemented
12#14766C212yesmaximal
13#14758C212yesmaximal
14#14275C214yesmaximal
15#13878C212yesmaximal
16#13878C212yesmaximal
17C2×#1434C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2855—28no—
23C2×#398C2212yes—
24C2×#395C2212yes—
25#2296—28no—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2214yes—
42C2×#380C2214yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C22×#95C2212yes—
46C22×#95C2212yes—
47C22×#95C2212yes—
48#1446C2212yes—
49#1445C2212yes—
50#1444C2212yes—
51#1444C2212yes—
52#1444C2212yes—
53#1444C2212yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2214yes—
58#1444C2214yes—
59#1442C2212yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2212yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1434C2212yes—
79#1434C2211yescharacteristic
80#1434C2211yescharacteristic
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2314yes—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—44no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—28no—
95C2×#117—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42—44no—
119C22×C42D418yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#398—44no—
127#395—44no—
128#391—24no—
129#391—24no—
130#389—24no—
131#389—24no—
132#389—24no—
133#389—24no—
134#389—24no—
135#389—24no—
136#387—28no—
137#387—28no—
138#385—28no—
139#385—28no—
140#385—28no—
141#381—28no—
142#380—24no—
143#380—216no—
144#380—48no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—28no—
147C2×#95—44no—
148C2×#95—44no—
149C2×#95Q8116yescomplemented
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—42no—
187C22×Q8—42no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4C2×Q818yes—
207C23×C4C2×Q8116yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C22⋊Q8—416no—
225#117—232no—
226#117—416no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
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⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4#4118yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42—416no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42C2×D414yes—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42C2×D414yes—
294#95—24no—
295#95—24no—
296#95—24no—
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—28no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—44no—
310#95—42no—
311#95—44no—
312#95—42no—
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#95C2×Q8116yes—
324#95C2×Q818yes—
325#95C2×Q818yes—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24C22×D412yes—
332C24C22×Q814yes—
333C24C2×#4114yes—
334C24H(5,𝔽2)14yes—
335C24C2511yescharacteristic
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—84no—
340C2×Q8—84no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—232no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
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—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—232no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4C4⋊Q8116yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×Q818yes—
436C22×C4C22×Q818yes—
437C22×C4C22×Q814yes—
438C22×C4C22×Q814yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C22⋊C4—216no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C42—216no—
466C42—44no—
467C42—44no—
468C42—44no—
469C42—44no—
470C42—48no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—24no—
483C23—24no—
484C23#52514yes—
485C23C2×C4⋊Q814yes—
486C23#53414yes—
487C23#554116yes—
488C23C23×D411yescharacteristic
489C23C23×Q812yes—
490C23C22×#4112yes—
491C23C2×H(5,𝔽2)12yes—
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×2−1+411yescharacteristic
494C23#58512yes—
495C23#58512yes—
496C23#58512yes—
497Q8—88no—
498C2×C4—232no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—416no—
530C2×C4—416no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
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—42no—
567C2×C4—42no—
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—832no—
581C2×C4C2×C4⋊Q814yes—
582C2×C4C2×C4⋊Q814yes—
583C2×C4#53714yes—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—216no—
591C22—216no—
592C22#258318yes—
593C22C2×#52511yescharacteristic
594C22C22×C4⋊Q811yescharacteristic
595C22C2×#53411yescharacteristic
596C22C2×#53711yescharacteristic
597C22#344911yescharacteristic
598C22#344911yescharacteristic
599C22#344911yescharacteristic
600C22#345814yes—
601C22C2×#55414yes—
602C22#350112yes—
603C22#350212yes—
604C22#351412yes—
605C22#351512yes—
606C22#352114yes—
607C22#353314yes—
608C22#353412yes—
609C22#353512yes—
610C4—24no—
611C4—24no—
612C4—24no—
613C4—44no—
614C4—44no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—48no—
619C4—48no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C2—28no—
627C2#3464211yescharacteristic
628C2C2×#258311yescharacteristic
629C2#3513011yescharacteristic
630C2#3514411yescharacteristic
631C2#3517311yescharacteristic
632C2#3745211yescharacteristic
633C2#3745411yescharacteristic
634C2#3810214yes—
635C2#3810314yes—
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. #16993C2semidirect
4. #16992C2semidirect
5. #16991C2semidirect
6. #16990C2semidirect
7. #16989C2semidirect
8. #16988C2semidirect
9. #15099C2semidirect
10. #15098C2semidirect
11. #15097C2semidirect
149. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22962
#169932
#169922
#169912
#169902
#169892
#169882
#150992
#150982
#150972
#147662
#147582
#142752
#138782
C2×#14342