Finite Group Databasebeta

← #4308997

Subgroups — #4308997

Order 512.

2302 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic
3#16649C214yesmaximal, complemented
4#16641C214yesmaximal, complemented
5#16625C214yesmaximal, complemented
6#16617C214yesmaximal, complemented
7#15073C212yesmaximal
8#15019C211yesmaximal, characteristic
9#15014C211yesmaximal, characteristic
10#14934C211yesmaximal, characteristic
11#14919C212yesmaximal
12#14914C211yesmaximal, characteristic
13#14705C212yesmaximal
14#14342C211yesmaximal, characteristic
15#14265C212yesmaximal
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2627—28no—
21C2×#400C2212yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#395C2212yes—
25C2×#387—28no—
26C2×#387C2214yes—
27C2×#387C2214yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#381C2212yes—
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×#381C2212yes—
35C2×#380C2211yescharacteristic
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1446C2212yes—
41#1446C2212yes—
42#1445C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1445C2214yes—
46#1445C2212yes—
47#1445C2212yes—
48#1444C2212yes—
49#1444C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—44no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4D418yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C42—24no—
108C22×C42—44no—
109C22×C42C2314yes—
110#400—28no—
111#398—28no—
112#398—28no—
113#395—28no—
114#387—216no—
115#387—216no—
116#385—28no—
117#385—28no—
118#385—44no—
119#381—28no—
120#381—28no—
121#381—44no—
122#381—44no—
123#380—24no—
124C2×#95—24no—
125C2×#95—24no—
126C2×#95—24no—
127C2×#95—24no—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25C2411yescharacteristic
164C22×Q8—22no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—42no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4C2×D414yes—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4#4114yes—
198C23×C4#4114yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C22⋊Q8—416no—
214C2×C4⋊C4—232no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—416no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D418yes—
243C2×C4⋊C4C2×D418yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C42—28no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42—416no—
260#95—24no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—42no—
285#95—42no—
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—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24C22×D412yes—
300C24C2×#4112yes—
301C24C2×#4112yes—
302C24H(5,𝔽2)14yes—
303C242−1+414yes—
304C24C2511yescharacteristic
305C2×Q8—44no—
306C2×Q8—82no—
307C2×Q8—82no—
308C2×Q8—88no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—216no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
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—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—416no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—416no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4C22×D414yes—
408C22×C4C22×D412yes—
409C22×C4C22×D414yes—
410C22×C4C22×D412yes—
411C22×C4C2×#4114yes—
412C22×C4C2×#4114yes—
413C22×C42−1+414yes—
414C4⋊C4—232no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C42—416no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—28no—
439C23—28no—
440C23—28no—
441C23—28no—
442C23#52514yes—
443C23#53414yes—
444C23#53614yes—
445C23#53918yes—
446C23#54218yes—
447C23C23×D411yescharacteristic
448C23C22×#4111yescharacteristic
449C23C22×#4111yescharacteristic
450C23C2×H(5,𝔽2)12yes—
451C23C2×2−1+412yes—
452C23C2×2−1+412yes—
453C23#58512yes—
454C23#58512yes—
455C23#58512yes—
456Q8—88no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—416no—
476C2×C4—416no—
477C2×C4—416no—
478C2×C4—416no—
479C2×C4—416no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
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—88no—
537C2×C4—88no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—216no—
545C22—216no—
546C22#263818yes—
547C22C2×#52511yescharacteristic
548C22C2×#53411yescharacteristic
549C22C2×#53611yescharacteristic
550C22C2×#53912yes—
551C22C2×#54212yes—
552C22#345512yes—
553C22#346112yes—
554C22#348511yescharacteristic
555C22#348511yescharacteristic
556C22#348611yescharacteristic
557C22#348711yescharacteristic
558C22#349112yes—
559C22#350712yes—
560C22#353314yes—
561C22#353514yes—
562C22#356314yes—
563C22#356414yes—
564C4—44no—
565C4—44no—
566C4—44no—
567C4—44no—
568C4—44no—
569C4—42no—
570C4—42no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—84no—
577C4—84no—
578C2—28no—
579C2C2×#263811yescharacteristic
580C2#3509211yescharacteristic
581C2#3509411yescharacteristic
582C2#3695911yescharacteristic
583C2#3696211yescharacteristic
584C2#3696511yescharacteristic
585C2#3699914yes—
586C2#3718611yescharacteristic
587C2#3750714yes—
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
3. #16649C2semidirect
4. #16641C2semidirect
5. #16625C2semidirect
6. #16617C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3872
#166492
#166412
#166252
#166172
#150732
#150192
#150142
#149342
#149192
#149142
#147052
#143422
#142652