Finite Group Databasebeta

← #4866008

Subgroups — #4866008

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2611C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic
3#17168C212yesmaximal
4#17160C212yesmaximal
5#17048C212yesmaximal, complemented
6#17032C212yesmaximal, complemented
7#17004C212yesmaximal, complemented
8#16634C212yesmaximal, complemented
9#16560C212yesmaximal, complemented
10#16447C214yesmaximal
11#16400C212yesmaximal
12#15684C214yesmaximal, complemented
13#15121C212yesmaximal, complemented
14#14841C212yesmaximal
15#13939C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2628—28no—
20#2611—28no—
21C2×#400C2212yes—
22C2×#400C2211yescharacteristic
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#399C2211yescharacteristic
27C2×#398C2211yescharacteristic
28C2×#397C2211yescharacteristic
29C2×#395C2212yes—
30C2×#393C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2212yes—
34C2×#387C2212yes—
35C2×#385C2212yes—
36C2×#382C2211yescharacteristic
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×#376C2211yescharacteristic
41#1446C2214yes—
42#1446C2214yes—
43#1445C2214yes—
44#1445C2212yes—
45#1445C2212yes—
46#1444C2212yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1440C2212yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2212yes—
68#1439C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2212yes—
75#1434C2214yes—
76#1434C2212yes—
77C23×Q8C2312yes—
78C24×C4C2311yescharacteristic
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×#125—24no—
83C2×#125—42no—
84C2×#125—42no—
85C2×#117—42no—
86C2×#117—42no—
87C2×#117D418yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113#400—24no—
114#400—28no—
115#400—44no—
116#400—44no—
117#399—24no—
118#399—28no—
119#398—24no—
120#397—24no—
121#395—28no—
122#393—28no—
123#389—28no—
124#389—28no—
125#388—28no—
126#387—28no—
127#385—28no—
128#382—24no—
129#380—28no—
130#380—42no—
131C4×C22⋊C4—42no—
132#376—24no—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2311yescharacteristic
137C2×#95C2312yes—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C25C2411yescharacteristic
166C22×Q8—24no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—44no—
170C22×Q8—44no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—42no—
175C23×C4C2×D414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C4⋊Q8—416no—
191#125—416no—
192#117—416no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4C2×D414yes—
250C2×C22⋊C4C2×D414yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C2×D414yes—
262C2×C42C2×D414yes—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—24no—
275#95—24no—
276#95—28no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—24no—
301C24—22no—
302C24C22×D412yes—
303C24C2511yescharacteristic
304C2×Q8—48no—
305C2×Q8—88no—
306C2×Q8—88no—
307C2×Q8—84no—
308C2×Q8—84no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
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—22no—
378C22×C4—24no—
379C22×C4—44no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C42−1+414yes—
391C22×C42−1+414yes—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—84no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—88no—
412C22⋊C4—84no—
413C42—28no—
414C42—28no—
415C42—48no—
416C42—84no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23C23×D411yescharacteristic
434C23C2×H(5,𝔽2)11yescharacteristic
435C23C2×H(5,𝔽2)11yescharacteristic
436C23C2×2−1+412yes—
437C23C2×2−1+412yes—
438C23#58512yes—
439C23#58512yes—
440C23#58514yes—
441Q8—816no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—416no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4#53414yes—
520C2×C4#53614yes—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—42no—
525C22—44no—
526C22—82no—
527C22—82no—
528C22—84no—
529C22C2×#53411yescharacteristic
530C22C2×#53611yescharacteristic
531C22#344911yescharacteristic
532C22#348912yes—
533C22#350012yes—
534C22#353512yes—
535C22#353512yes—
536C22#354012yes—
537C22#354712yes—
538C22#355011yescharacteristic
539C22#355111yescharacteristic
540C22#355112yes—
541C22#356314yes—
542C22#356514yes—
543C22#356514yes—
544C22#356614yes—
545C4—24no—
546C4—24no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—44no—
552C4—82no—
553C4—84no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4—84no—
558C4—84no—
559C2—82no—
560C2#3737611yescharacteristic
561C2#3741911yescharacteristic
562C2#3744711yescharacteristic
563C2#4230912yes—
564C2#4297812yes—
565C2#5683412yes—
566C2#5696012yes—
567C2#5849814yes—
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
5. #17048C2semidirect
6. #17032C2semidirect
7. #17004C2semidirect
8. #16634C2semidirect
9. #16560C2semidirect
12. #15684C2semidirect
13. #15121C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26112
C2×#26282
#171682
#171602
#170482
#170322
#170042
#166342
#165602
#164472
#164002
#156842
#151212
#148412
#139392