Finite Group Databasebeta

← #4933906

Subgroups — #4933906

Order 512.

2470 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
1C2×#2804C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3#16823C214yesmaximal, complemented
4#16694C212yesmaximal, complemented
5#15946C212yesmaximal, complemented
6#15783C212yesmaximal, complemented
7#15475C214yesmaximal, complemented
8#15378C214yesmaximal, complemented
9#15097C212yesmaximal, complemented
10#14728C214yesmaximal, complemented
11#14513C212yesmaximal
12#14015C212yesmaximal
13#13832C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2212yes—
17#2804—28no—
18#2563—28no—
19C2×#400C2212yes—
20C2×#398C2212yes—
21C2×#398C2212yes—
22C2×#395C2212yes—
23C2×#393C2214yes—
24C2×#393C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#385C2214yes—
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#380C2214yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1446C2214yes—
39#1446C2214yes—
40#1446C2212yes—
41#1446C2212yes—
42#1445C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1437C2214yes—
58#1436C2212yes—
59#1436C2214yes—
60#1436C2212yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63#1434C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—42no—
77C2×C4⋊Q8—42no—
78C2×C4×D4—24no—
79C2×C4×D4—42no—
80C2×C4×D4—42no—
81C22×C4⋊C4—48no—
82C22×C4⋊C4—44no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105#400—28no—
106#398—44no—
107#398—44no—
108#395—44no—
109#393—216no—
110#393—216no—
111#391—28no—
112#391—44no—
113#390—24no—
114#390D418yes—
115#386—28no—
116#386Q8116yescomplemented
117#385—216no—
118#385—44no—
119#385—48no—
120#382—24no—
121#382—24no—
122#380—216no—
123C4×C4⋊C4—24no—
124C2×#95C2312yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25C2412yes—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—42no—
154C22×Q8—44no—
155C22×Q8—44no—
156C22×Q8—44no—
157C22×Q8—44no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×D4—41no—
161C22×D4—41no—
162C22×D4—42no—
163C22×D4C2×D414yes—
164C22×D4C2×Q818yes—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—28no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4C2×Q818yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C4⋊Q8—416no—
193C4×D4—416no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—432no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4C2×D4116yes—
247C2×C22⋊C4C2×Q818yes—
248C2×C22⋊C4C2×Q8116yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42C2×D414yes—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—28no—
266#95—28no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95—44no—
282#95C2×Q8116yes—
283C24—22no—
284C24—24no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—24no—
290C24—22no—
291C24C22×D414yes—
292C24C22×Q814yes—
293C24C22×Q814yes—
294C24C2511yescharacteristic
295C2×Q8—48no—
296C2×Q8—48no—
297C2×Q8—48no—
298C2×Q8—84no—
299C2×Q8—88no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—88no—
303C2×Q8—88no—
304C2×Q8—88no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4C4⋊Q8116yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—416no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—416no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—416no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—84no—
378C22×C4—88no—
379C22×C4—84no—
380C22×C4—88no—
381C22×C4—84no—
382C22×C4—88no—
383C22×C4C22×D414yes—
384C22×C4C22×D414yes—
385C22×C4C22×D412yes—
386C22×C4C22×Q818yes—
387C22×C4C22×Q818yes—
388C22×C4C22×Q814yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)14yes—
391C22×C4H(5,𝔽2)12yes—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—416no—
398C4⋊C4—416no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—88no—
402C4⋊C4—88no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C22⋊C4—88no—
408C22⋊C4—84no—
409C22⋊C4—88no—
410C42—28no—
411C42—44no—
412C42—44no—
413C42—84no—
414C23—24no—
415C23—28no—
416C23—24no—
417C23—28no—
418C23—24no—
419C23—28no—
420C23—24no—
421C23—28no—
422C23—24no—
423C23—24no—
424C23—28no—
425C23—28no—
426C23—42no—
427C23—44no—
428C23—42no—
429C23—44no—
430C23—42no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23C2×C4⋊Q818yes—
437C23C23×D411yescharacteristic
438C23C23×Q812yes—
439C23C2×H(5,𝔽2)11yescharacteristic
440C23C2×H(5,𝔽2)11yescharacteristic
441C23C2×H(5,𝔽2)12yes—
442C23#58512yes—
443C23#58514yes—
444C23#58512yes—
445Q8—816no—
446Q8—816no—
447D4—416no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—432no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—816no—
515C2×C4—816no—
516C2×C4C2×C4⋊Q814yes—
517C2×C4H(3,𝔽4)14yes—
518C22—28no—
519C22—216no—
520C22—44no—
521C22—48no—
522C22—48no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—48no—
529C22C22×C4⋊Q811yescharacteristic
530C22#348411yescharacteristic
531C22#348511yescharacteristic
532C22#348512yes—
533C22#348812yes—
534C22#350112yes—
535C22#350112yes—
536C22#350114yes—
537C22#350414yes—
538C22C2×H(3,𝔽4)11yescharacteristic
539C22#353312yes—
540C22#354012yes—
541C22#354014yes—
542C22#354511yescharacteristic
543C22#355812yes—
544C22#356614yes—
545C4—24no—
546C4—24no—
547C4—44no—
548C4—42no—
549C4—48no—
550C4—48no—
551C4—44no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—82no—
556C4—84no—
557C2—48no—
558C2#3539111yescharacteristic
559C2#3595012yes—
560C2#4549412yes—
561C2#4597711yescharacteristic
562C2#4797814yes—
563C2#5017112yes—
564C2#5090212yes—
565C2#5205411yescharacteristic
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. #16823C2semidirect
4. #16694C2semidirect
5. #15946C2semidirect
6. #15783C2semidirect
7. #15475C2semidirect
8. #15378C2semidirect
9. #15097C2semidirect
10. #14728C2semidirect
116. #386Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#25632
#168232
#166942
#159462
#157832
#154752
#153782
#150972
#147282
#145132
#140152
#138322