Finite Group Databasebeta

← #4868759

Subgroups — #4868759

Order 512.

2192 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2564C211yesmaximal, characteristic
3#17198C212yesmaximal, complemented
4#17106C212yesmaximal, complemented
5#16689C212yesmaximal, complemented
6#16686C212yesmaximal, complemented
7#16507C212yesmaximal, complemented
8#16399C212yesmaximal, complemented
9#16393C212yesmaximal, complemented
10#16365C212yesmaximal, complemented
11#16299C214yesmaximal, complemented
12#15285C214yesmaximal
13#15088C212yesmaximal, complemented
14#15019C212yesmaximal, complemented
15#14344C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#2564—28no—
20#2563—28no—
21C2×#400C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2214yes—
26C2×#395C2211yescharacteristic
27C2×#395C2214yes—
28C2×#393C2211yescharacteristic
29C2×#393C2212yes—
30C2×#393C2212yes—
31C2×#390C2212yes—
32C2×#387C2212yes—
33C2×#386C2212yes—
34C2×#385C2212yes—
35C2×#382C2212yes—
36C2×#381C2211yescharacteristic
37C2×#381C2214yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1446C2214yes—
45#1446C2214yes—
46#1445C2214yes—
47#1445C2212yes—
48#1445C2212yes—
49#1444C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1440C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2212yes—
72#1434C2212yes—
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8—42no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—42no—
88C2×C22⋊Q8—42no—
89C2×C4×D4—42no—
90C2×C4×D4—42no—
91C2×C4×D4D418yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#400—24no—
116#399—24no—
117#398—44no—
118#398—44no—
119#397—48no—
120#395—24no—
121#395—48no—
122#393—28no—
123#393—28no—
124#393—24no—
125#390—28no—
126#387—28no—
127#386—28no—
128#385—28no—
129#382—28no—
130#381—48no—
131#381—42no—
132#380—28no—
133#380—28no—
134#380—28no—
135#380—42no—
136C4×C4⋊C4—24no—
137C2×#95—24no—
138C2×#95C2312yes—
139C2×#95C2311yescharacteristic
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4C2×D414yes—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—24no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C4⋊Q8—416no—
211C22⋊Q8—416no—
212C4×D4—416no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D414yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C2×D414yes—
269C2×C22⋊C4C2×D414yes—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42C2×D414yes—
276#95—24no—
277#95—22no—
278#95—22no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24C22×D412yes—
320C24C22×D412yes—
321C242−1+414yes—
322C24C2511yescharacteristic
323C2×Q8—44no—
324C2×Q8—48no—
325C2×Q8—84no—
326C2×Q8—88no—
327C2×Q8—84no—
328C2×Q8—84no—
329C2×Q8—88no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
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—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—416no—
406C22×C4—48no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—88no—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4H(5,𝔽2)14yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+414yes—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—84no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C42—28no—
444C42—48no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—28no—
450C23—24no—
451C23—24no—
452C23—28no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—41no—
463C23—41no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—41no—
473C23—41no—
474C23—44no—
475C23—44no—
476C23#53614yes—
477C23C23×D411yescharacteristic
478C23C2×H(5,𝔽2)11yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×H(5,𝔽2)12yes—
481C23C2×2−1+412yes—
482C23C2×2−1+412yes—
483C23#58514yes—
484C23#58512yes—
485Q8—816no—
486Q8—88no—
487D4—416no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—88no—
559C2×C4—88no—
560C2×C4—816no—
561C2×C4#53414yes—
562C2×C4#53614yes—
563C22—24no—
564C22—28no—
565C22—44no—
566C22—42no—
567C22—48no—
568C22—44no—
569C22—44no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—44no—
574C22—44no—
575C22—42no—
576C22—82no—
577C22—84no—
578C22—82no—
579C22C2×#53411yescharacteristic
580C22C2×#53611yescharacteristic
581C22C2×#53611yescharacteristic
582C22#348512yes—
583C22#348812yes—
584C22#353514yes—
585C22#353812yes—
586C22#353812yes—
587C22#354512yes—
588C22#354512yes—
589C22#354612yes—
590C22#354714yes—
591C22#354911yescharacteristic
592C22#355111yescharacteristic
593C22#356414yes—
594C22#356514yes—
595C4—24no—
596C4—24no—
597C4—42no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—82no—
602C4—84no—
603C4—84no—
604C4—84no—
605C4—82no—
606C4—84no—
607C4—82no—
608C2—44no—
609C2—82no—
610C2#3711911yescharacteristic
611C2#3727311yescharacteristic
612C2#3745311yescharacteristic
613C2#4141212yes—
614C2#4268112yes—
615C2#5398012yes—
616C2#5561612yes—
617C2#5751414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17198C2semidirect
4. #17106C2semidirect
5. #16689C2semidirect
6. #16686C2semidirect
7. #16507C2semidirect
8. #16399C2semidirect
9. #16393C2semidirect
10. #16365C2semidirect
11. #16299C2semidirect
13. #15088C2semidirect
14. #15019C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25642
#171982
#171062
#166892
#166862
#165072
#163992
#163932
#163652
#162992
#152852
#150882
#150192
#143442