Finite Group Databasebeta

← #764111

Subgroups — #764111

Order 512.

4490 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic
2C2×#2583C212yesmaximal, complemented
3C22×#396C211yesmaximal, characteristic
4#15614C218yesmaximal, complemented
5#15276C218yesmaximal, complemented
6#14914C212yesmaximal, complemented
7#14720C212yesmaximal, complemented
8#14139C212yesmaximal, complemented
9#14127C212yesmaximal
10#14127C212yesmaximal
11#13775C211yesmaximal, characteristic
12C2×#543—216no—
13C2×#543—28no—
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2214yes—
16C22×C22⋊Q8C2214yes—
17C22×C22⋊Q8C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25#2583—216no—
26C2×#398C2214yescomplemented
27C2×#396—28no—
28C2×#396C2218yes—
29C2×#395C2214yescomplemented
30C2×#395C2214yescomplemented
31C2×#391C2214yescomplemented
32C2×#389C2218yescomplemented
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#379C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C22×#95C2211yescharacteristic
40#1446C2214yescomplemented
41#1444C2214yescomplemented
42#1444C2218yescomplemented
43#1441C2218yescomplemented
44#1441C2218yescomplemented
45#1441C2214yescomplemented
46#1439C2218yes—
47#1439C2218yes—
48#1437C2218yes—
49C4⋊#95C2214yescomplemented
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2214yes—
53C22⋊#95C2214yes—
54C22⋊#95C2212yes—
55C26C2311yescharacteristic
56C23×Q8C2312yes—
57C23×Q8C2312yes—
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2312yes—
63C24×C4C2312yes—
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⋊C4—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—44no—
83C22×C4⋊C4—44no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2318yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2318yes—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—216no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4D418yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2314yes—
110#398—216no—
111#396—232no—
112#395—216no—
113#395—48no—
114#391—48no—
115#389—232no—
116#385—216no—
117#385—216no—
118#385—48no—
119#385—48no—
120#379Q8116yescomplemented
121C4×C22⋊C4—28no—
122C2×#95—24no—
123C2×#95C2314yes—
124C2×#95C2318yes—
125C2×#95C2314yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2314yes—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C2×#95C2314yes—
133C25—24no—
134C25—24no—
135C25—42no—
136C25—42no—
137C25—42no—
138C25—42no—
139C25C2×D414yes—
140C25C2×Q818yes—
141C25C2412yes—
142C25C2411yescharacteristic
143C22×Q8—28no—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8—44no—
148C22×Q8—44no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×Q8C2×D418yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—28no—
160C23×C4—28no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
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—42no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4C2×D414yes—
182C23×C4C2×D418yes—
183C23×C4C2×D414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2418yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C22⋊Q8—2128no—
194C22⋊Q8—2128no—
195C22⋊Q8—264no—
196C22⋊Q8—264no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—216no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—416no—
220C2×C4⋊C4—416no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4C2×D4116yes—
225C2×C4⋊C4C2×D418yes—
226C2×C22⋊C4—264no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—232no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—416no—
248C2×C22⋊C4—416no—
249C2×C22⋊C4C2×D4116yes—
250C2×C22⋊C4C2×D4116yes—
251C2×C22⋊C4C2×Q8116yes—
252C2×C22⋊C4C2×Q8116yes—
253C2×C22⋊C4C2×Q8116yes—
254C2×C42—28no—
255C2×C42—44no—
256C2×C42—44no—
257#95—28no—
258#95—28no—
259#95—216no—
260#95—216no—
261#95—216no—
262#95—44no—
263#95—48no—
264#95—48no—
265#95—44no—
266#95—48no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—28no—
278C24—28no—
279C24—232no—
280C24—44no—
281C24—44no—
282C24—44no—
283C24—44no—
284C24—416no—
285C24—416no—
286C24—88no—
287C24—88no—
288C24—88no—
289C24—88no—
290C24C4⋊Q8116yes—
291C24C22×D414yes—
292C24C22×D412yes—
293C24C22×D414yes—
294C24C22×Q814yes—
295C24C22×Q814yes—
296C24C22×Q814yes—
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—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—24no—
325C22×C4—28no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—216no—
338C22×C4—216no—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—416no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—416no—
354C22×C4—416no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—816no—
366C22×C4—816no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4#120116yes—
375C22×C4C22×D418yes—
376C22×C4C22×D418yes—
377C22×C4C22×D414yes—
378C22×C4C22×D414yes—
379C22×C4C22×D414yes—
380C22×C4C22×D414yes—
381C22×C4C22×Q8116yes—
382C22×C4H(5,𝔽2)14yes—
383C22×C4H(5,𝔽2)14yes—
384C4⋊C4—216no—
385C4⋊C4—232no—
386C4⋊C4—232no—
387C4⋊C4—216no—
388C4⋊C4—432no—
389C4⋊C4—416no—
390C4⋊C4—416no—
391C4⋊C4—416no—
392C4⋊C4—816no—
393C22⋊C4—232no—
394C22⋊C4—232no—
395C22⋊C4—232no—
396C22⋊C4—232no—
397C22⋊C4—416no—
398C22⋊C4—416no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C42—88no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—28no—
407C23—28no—
408C23—28no—
409C23—28no—
410C23—28no—
411C23—28no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—28no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—28no—
421C23—28no—
422C23—28no—
423C23—28no—
424C23—232no—
425C23—416no—
426C23—416no—
427C23—48no—
428C23—48no—
429C23—48no—
430C23—48no—
431C23—864no—
432C23—88no—
433C23—88no—
434C23—88no—
435C23—88no—
436C23C2×#12014yes—
437C23C2×C4⋊Q818yes—
438C23C2×C4⋊Q814yes—
439C23#53614yes—
440C23D4×Q8116yes—
441C23C23×D411yescharacteristic
442C23C23×D412yes—
443C23C23×Q812yes—
444C23C2×H(5,𝔽2)12yes—
445C23C2×H(5,𝔽2)12yes—
446C23C2×2−1+412yes—
447C23#58512yes—
448C23#58512yes—
449Q8—816no—
450Q8—816no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—216no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—216no—
465C2×C4—216no—
466C2×C4—216no—
467C2×C4—216no—
468C2×C4—28no—
469C2×C4—416no—
470C2×C4—416no—
471C2×C4—416no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—88no—
498C2×C4—816no—
499C2×C4—816no—
500C2×C4—88no—
501C2×C4—88no—
502C2×C4C2×#12018yes—
503C22—232no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—28no—
511C22—28no—
512C22—28no—
513C22—416no—
514C22—416no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—832no—
520C22#243418yes—
521C22C22×#12011yescharacteristic
522C22C22×C4⋊Q811yescharacteristic
523C22C2×#53611yescharacteristic
524C22C2×D4×Q814yes—
525C22#348411yescharacteristic
526C22#348411yescharacteristic
527C22#348511yescharacteristic
528C22#348511yescharacteristic
529C22#348814yes—
530C22#349614yes—
531C22#350114yes—
532C22#350114yes—
533C22#353614yes—
534C22#356414yes—
535C4—28no—
536C4—416no—
537C4—48no—
538C4—44no—
539C4—44no—
540C4—44no—
541C4—48no—
542C4—82no—
543C4—82no—
544C2—28no—
545C2—48no—
546C2C2×#243411yescharacteristic
547C2#3534711yescharacteristic
548C2#3539111yescharacteristic
549C2#3539111yescharacteristic
550C2#3551211yescharacteristic
551C2#3671811yescharacteristic
552C2#3697011yescharacteristic
553C2#4302518yes—
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×#2583C2semidirect
4. #15614C2semidirect
5. #15276C2semidirect
6. #14914C2semidirect
7. #14720C2semidirect
8. #14139C2semidirect
26. C2×#398C22semidirect
29. C2×#395C22semidirect
30. C2×#395C22semidirect
31. C2×#391C22semidirect
32. C2×#389C22semidirect
40. #1446C22semidirect
41. #1444C22semidirect
42. #1444C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
49. C4⋊#95C22semidirect
120. #379Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#25832
C22×#3962
#156142
#152762
#149142
#147202
#141392
#141272
#137752