Finite Group Databasebeta

← #627065

Subgroups — #627065

Order 512.

4370 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic
2C2×#2627C212yesmaximal, complemented
3C22×#388C211yesmaximal, characteristic
4#15672C218yesmaximal, complemented
5#15279C218yesmaximal, complemented
6#14934C212yesmaximal, complemented
7#14914C212yesmaximal, complemented
8#14702C212yesmaximal, complemented
9#14338C211yesmaximal, characteristic
10#14127C212yesmaximal
11#13869C212yesmaximal
12C2×#543—216no—
13C2×#543—28no—
14C22×C22⋊Q8C2214yes—
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2214yes—
17C22×C22⋊Q8C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2627—216no—
25C2×#400C2214yescomplemented
26C2×#398C2218yescomplemented
27C2×#395C2214yescomplemented
28C2×#391C2214yescomplemented
29C2×#388—28no—
30C2×#388C2218yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#381C2214yes—
34C2×#380C2212yes—
35C2×#379C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38#1446C2214yescomplemented
39#1446C2214yescomplemented
40#1444C2214yescomplemented
41#1442C2218yescomplemented
42#1442C2214yescomplemented
43#1441C2218yescomplemented
44#1440C2218yes—
45#1439C2218yes—
46#1439C2218yes—
47#1436C2218yescomplemented
48C4⋊#95C2214yescomplemented
49C22⋊#95C2212yes—
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2214yes—
53C22⋊#95C2214yes—
54C22⋊#95C2214yes—
55C26C2311yescharacteristic
56C23×Q8C2312yes—
57C23×Q8C2312yes—
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C24×C4C2312yes—
62C24×C4C2312yes—
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66#543—2128no—
67C2×C22⋊Q8—216no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—216no—
72C2×C22⋊Q8—216no—
73C2×C22⋊Q8—216no—
74C2×C22⋊Q8—216no—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8D4116yes—
78C22×C4⋊C4—28no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2318yes—
86C22×C4⋊C4C2314yes—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—216no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4D418yes—
95C22×C22⋊C4Q818yescomplemented
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42—24no—
105C22×C42—44no—
106C22×C42C2314yes—
107#400—216no—
108#398—232no—
109#395—216no—
110#391—48no—
111#388—232no—
112#385—216no—
113#385—48no—
114#381—216no—
115#380—28no—
116#379—28no—
117C2×#95—24no—
118C2×#95—24no—
119C2×#95C2318yes—
120C2×#95C2314yes—
121C2×#95C2318yes—
122C2×#95C2318yes—
123C2×#95C2318yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C25—24no—
132C25—24no—
133C25—42no—
134C25—42no—
135C25—42no—
136C25—42no—
137C25C2×D414yes—
138C25C2×Q814yes—
139C25#4114yes—
140C25C2412yes—
141C25C2411yescharacteristic
142C22×Q8—28no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—44no—
147C22×Q8—44no—
148C22×Q8—44no—
149C22×Q8—44no—
150C22×Q8C2×D418yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—28no—
163C23×C4—28no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4C2×D414yes—
181C23×C4C2×D418yes—
182C23×C4C2×D414yes—
183C23×C4C2×Q814yes—
184C23×C4C2×Q814yes—
185C23×C4C2418yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C22⋊Q8—2128no—
195C22⋊Q8—2128no—
196C22⋊Q8—264no—
197C22⋊Q8—264no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—216no—
212C2×C4⋊C4—416no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C2×D4116yes—
218C2×C4⋊C4C2×D418yes—
219C2×C22⋊C4—264no—
220C2×C22⋊C4—232no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—232no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—416no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4C2×D4116yes—
245C2×C22⋊C4C2×D4116yes—
246C2×C22⋊C4C2×Q8116yes—
247C2×C42—28no—
248C2×C42—28no—
249C2×C42—44no—
250C2×C42—416no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—216no—
255#95—216no—
256#95—216no—
257#95—216no—
258#95—48no—
259#95—48no—
260#95—44no—
261#95—44no—
262#95—48no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—28no—
274C24—28no—
275C24—216no—
276C24—216no—
277C24—44no—
278C24—44no—
279C24—44no—
280C24—44no—
281C24—48no—
282C24—48no—
283C24—48no—
284C24—48no—
285C24—88no—
286C24—88no—
287C24—88no—
288C24—88no—
289C24C22⋊Q8116yes—
290C24C22×D414yes—
291C24C22×D412yes—
292C24C22×D414yes—
293C24C22×Q814yes—
294C24C22×Q812yes—
295C24C2×#4114yes—
296C24C2×#4112yes—
297C242−1+414yes—
298C24C2511yescharacteristic
299C2×Q8—216no—
300C2×Q8—216no—
301C2×Q8—416no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—88no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—816no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—216no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—28no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—216no—
338C22×C4—216no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—416no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—416no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—816no—
370C22×C4—816no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4#120116yes—
378C22×C4C22×D418yes—
379C22×C4C22×D414yes—
380C22×C4C22×D418yes—
381C22×C4C22×D414yes—
382C22×C4C22×D414yes—
383C22×C4C22×D414yes—
384C22×C4C22×Q814yes—
385C22×C4C22×Q814yes—
386C22×C4H(5,𝔽2)14yes—
387C4⋊C4—216no—
388C4⋊C4—232no—
389C4⋊C4—232no—
390C4⋊C4—216no—
391C4⋊C4—416no—
392C4⋊C4—88no—
393C4⋊C4—816no—
394C22⋊C4—232no—
395C22⋊C4—232no—
396C22⋊C4—232no—
397C22⋊C4—232no—
398C22⋊C4—416no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—88no—
402C42—416no—
403C23—24no—
404C23—24no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—28no—
411C23—28no—
412C23—24no—
413C23—28no—
414C23—28no—
415C23—24no—
416C23—24no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—28no—
426C23—28no—
427C23—232no—
428C23—416no—
429C23—416no—
430C23—48no—
431C23—48no—
432C23—48no—
433C23—48no—
434C23—88no—
435C23—88no—
436C23—88no—
437C23—88no—
438C23—832no—
439C23—832no—
440C23C2×#12014yes—
441C23C2×C22⋊Q818yes—
442C23C2×C22⋊Q814yes—
443C23#53614yes—
444C23#54818yes—
445C23D4×Q818yes—
446C23C23×D411yescharacteristic
447C23C23×D412yes—
448C23C23×Q811yescharacteristic
449C23C22×#4111yescharacteristic
450C23C2×H(5,𝔽2)12yes—
451C23C2×2−1+412yes—
452C23#58512yes—
453C23#58512yes—
454C23#58512yes—
455Q8—816no—
456Q8—816no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—216no—
471C2×C4—216no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—416no—
475C2×C4—416no—
476C2×C4—416no—
477C2×C4—42no—
478C2×C4—416no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—88no—
506C2×C4—816no—
507C2×C4—816no—
508C2×C4—88no—
509C2×C4C2×#12018yes—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—216no—
517C22—216no—
518C22—28no—
519C22—28no—
520C22—28no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—416no—
526C22—416no—
527C22—832no—
528C22#240918yes—
529C22C22×#12011yescharacteristic
530C22C22×C22⋊Q811yescharacteristic
531C22C2×#53611yescharacteristic
532C22#345812yes—
533C22#345912yes—
534C22C2×#54812yes—
535C22C2×D4×Q812yes—
536C22#348411yescharacteristic
537C22#348511yescharacteristic
538C22#348511yescharacteristic
539C22#348511yescharacteristic
540C22#349112yes—
541C22#349614yes—
542C22#350014yes—
543C22#350112yes—
544C22#353514yes—
545C22#356414yes—
546C4—28no—
547C4—48no—
548C4—44no—
549C4—48no—
550C4—44no—
551C4—44no—
552C4—44no—
553C4—82no—
554C4—88no—
555C2—28no—
556C2—48no—
557C2C2×#240911yescharacteristic
558C2#3498511yescharacteristic
559C2#3509011yescharacteristic
560C2#3534711yescharacteristic
561C2#3561611yescharacteristic
562C2#3672011yescharacteristic
563C2#3697011yescharacteristic
564C2#3936414yes—
565C2#3942214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2627C2semidirect
4. #15672C2semidirect
5. #15279C2semidirect
6. #14934C2semidirect
7. #14914C2semidirect
8. #14702C2semidirect
25. C2×#400C22semidirect
26. C2×#398C22semidirect
27. C2×#395C22semidirect
28. C2×#391C22semidirect
38. #1446C22semidirect
39. #1446C22semidirect
40. #1444C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect
47. #1436C22semidirect
48. C4⋊#95C22semidirect
95. C22×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#26272
C22×#3882
#156722
#152792
#149342
#149142
#147022
#143382
#141272
#138692