Finite Group Databasebeta

← #920473

Subgroups — #920473

Order 512.

2246 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C22×#400C211yesmaximal, characteristic
3#17225C214yesmaximal, complemented
4#16911C214yesmaximal, complemented
5#16689C212yesmaximal, complemented
6#16653C212yesmaximal, complemented
7#15946C212yesmaximal, complemented
8#15884C212yesmaximal, complemented
9#14919C212yesmaximal
10#14919C212yesmaximal
11#14914C212yesmaximal
12#14263C212yesmaximal
13#14040C212yesmaximal
14#13962C212yesmaximal
15#13870C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2503—28no—
21C2×#400—28no—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C22×#95C2212yes—
41C22×#95C2211yescharacteristic
42C22×#95C2212yes—
43#1446C2214yes—
44#1445C2214yes—
45#1445C2214yes—
46#1445C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1436C2212yes—
66#1436C2214yes—
67#1436C2212yes—
68C4⋊#95C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—42no—
85C2×C22⋊Q8—42no—
86C2×#117—28no—
87C2×#117—42no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#400—28no—
114#400—28no—
115#400—28no—
116#400—28no—
117#400—28no—
118#400—28no—
119#391—24no—
120#391—24no—
121#385—28no—
122#385—44no—
123#385—44no—
124#382—24no—
125#382—24no—
126#381—28no—
127#381—28no—
128#381—28no—
129#381—42no—
130C4×C22⋊C4—24no—
131C2×#95—28no—
132C2×#95—28no—
133C2×#95—24no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
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—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C22×Q8—22no—
170C22×Q8—24no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—42no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2×D414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C22⋊Q8—416no—
214#117—416no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4#4118yes—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—28no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270#95—24no—
271#95—24no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—42no—
301#95—42no—
302#95—44no—
303#95—44no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24C22×D412yes—
308C24C2×#4114yes—
309C24H(5,𝔽2)12yes—
310C24H(5,𝔽2)12yes—
311C242−1+414yes—
312C24C2511yescharacteristic
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—84no—
317C2×Q8—84no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—216no—
325C22×C4—216no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C2×#4118yes—
420C22×C42−1+414yes—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C22⋊C4—216no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C42—48no—
438C42—44no—
439C42—44no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—28no—
444C23—28no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—28no—
450C23—28no—
451C23#53414yes—
452C23#54218yes—
453C23#54718yes—
454C23#55114yes—
455C23#56314yes—
456C23C23×D411yescharacteristic
457C23C22×#4112yes—
458C23C2×H(5,𝔽2)11yescharacteristic
459C23C2×H(5,𝔽2)11yescharacteristic
460C23C2×2−1+412yes—
461C23C2×2−1+412yes—
462C23#58512yes—
463C23#58514yes—
464Q8—88no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—416no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—416no—
493C2×C4—416no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
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—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—816no—
546C2×C4—816no—
547C2×C4#53614yes—
548C22—28no—
549C22—216no—
550C22—24no—
551C22—24no—
552C22—28no—
553C22—28no—
554C22—28no—
555C22#271718yes—
556C22C2×#53411yescharacteristic
557C22C2×#53611yescharacteristic
558C22#344911yescharacteristic
559C22C2×#54212yes—
560C22C2×#54712yes—
561C22C2×#55111yescharacteristic
562C22#348712yes—
563C22#351514yes—
564C22#352114yes—
565C22C2×#56311yescharacteristic
566C22#353512yes—
567C22#353512yes—
568C22#354512yes—
569C22#354512yes—
570C22#356314yes—
571C22#356414yes—
572C4—24no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—48no—
579C4—82no—
580C4—84no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4—82no—
585C4—84no—
586C2—28no—
587C2#3718711yescharacteristic
588C2#3744111yescharacteristic
589C2#3804712yes—
590C2#3891912yes—
591C2#4153514yes—
592C2#4372712yes—
593C2C2×#271711yescharacteristic
594C2#5281712yes—
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. #17225C2semidirect
4. #16911C2semidirect
5. #16689C2semidirect
6. #16653C2semidirect
7. #15946C2semidirect
8. #15884C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C22×#4002
#172252
#169112
#166892
#166532
#159462
#158842
#149192
#149142
#142632
#140402
#139622
#138702