Finite Group Databasebeta

← #4335007

Subgroups — #4335007

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2599C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#16965C212yesmaximal, complemented
4#16677C212yesmaximal, complemented
5#16462C212yesmaximal
6#16318C212yesmaximal
7#16113C214yesmaximal, complemented
8#15790C212yesmaximal, complemented
9#15625C212yesmaximal, complemented
10#14892C212yesmaximal
11#14880C212yesmaximal, complemented
12#14818C212yesmaximal, complemented
13#14786C212yesmaximal
14#14526C214yesmaximal
15#13938C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2599—28no—
21C2×#398C2211yescharacteristic
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#397C2214yes—
25C2×#397C2211yescharacteristic
26C2×#395C2211yescharacteristic
27C2×#395C2214yes—
28C2×#393C2211yescharacteristic
29#2317—28no—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2214yes—
33C2×#388C2212yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#383C2211yescharacteristic
42C2×#383C2211yescharacteristic
43C2×#382C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45#1445C2212yes—
46#1445C2214yes—
47#1445C2214yes—
48#1445C2212yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2212yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2212yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4×Q8—24no—
81C2×C4×Q8—42no—
82C2×C4×Q8—42no—
83C2×#117—24no—
84C2×#117—28no—
85C2×#117—42no—
86C2×#117—42no—
87C2×#117D418yes—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117#398—24no—
118#398—44no—
119#398—44no—
120#397—24no—
121#397—216no—
122#395—24no—
123#395—48no—
124#393—24no—
125#389—28no—
126#389—28no—
127#389—48no—
128#388—28no—
129#387—28no—
130#387—28no—
131#385—28no—
132#385—28no—
133#385—28no—
134#385—48no—
135#384—28no—
136#383—24no—
137#383—42no—
138#382—28no—
139C4×C4⋊C4—42no—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C25C2411yescharacteristic
171C22×Q8—22no—
172C22×Q8—24no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—42no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4C2×D414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C4×Q8—416no—
201#117—232no—
202#117—416no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4C2×D414yes—
261C2×C22⋊C4C2×D414yes—
262C2×C22⋊C4#4118yes—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42C2×D414yes—
275C2×C42C2×D414yes—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—24no—
310C24—22no—
311C24—22no—
312C24C22×D412yes—
313C24C2×#4114yes—
314C24C2511yescharacteristic
315C2×Q8—48no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—48no—
319C2×Q8—44no—
320C22×C4—216no—
321C22×C4—216no—
322C22×C4—28no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
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—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—48no—
392C22×C4—88no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4—88no—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C2×#4118yes—
403C22×C4H(5,𝔽2)14yes—
404C22×C4H(5,𝔽2)12yes—
405C22×C4H(5,𝔽2)12yes—
406C22×C42−1+418yes—
407C22×C42−1+414yes—
408C4⋊C4—216no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—416no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C22⋊C4—216no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—416no—
428C22⋊C4—48no—
429C42—216no—
430C42—28no—
431C42—28no—
432C42—48no—
433C42—84no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—28no—
441C23—28no—
442C23—41no—
443C23—42no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—44no—
449C23#52514yes—
450C23C23×D411yescharacteristic
451C23C22×#4112yes—
452C23C2×H(5,𝔽2)11yescharacteristic
453C23C2×H(5,𝔽2)11yescharacteristic
454C23C2×H(5,𝔽2)12yes—
455C23C2×2−1+412yes—
456C23C2×2−1+414yes—
457C23#58512yes—
458Q8—416no—
459C2×C4—232no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
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—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
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—816no—
537C2×C4#53414yes—
538C2×C4#53614yes—
539C22—28no—
540C22—24no—
541C22—42no—
542C22—44no—
543C22—48no—
544C22—42no—
545C22—44no—
546C22—44no—
547C22C2×#52511yescharacteristic
548C22C2×#53411yescharacteristic
549C22C2×#53611yescharacteristic
550C22#348812yes—
551C22#349212yes—
552C22#349212yes—
553C22#349912yes—
554C22#351514yes—
555C22#352014yes—
556C22#353711yescharacteristic
557C22#353812yes—
558C22#354011yescharacteristic
559C22#354212yes—
560C22#355012yes—
561C22#355214yes—
562C22#355514yes—
563C4—24no—
564C4—24no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—44no—
570C4—48no—
571C4—48no—
572C4—84no—
573C4—84no—
574C4—82no—
575C4—82no—
576C4—82no—
577C2—44no—
578C2#3505711yescharacteristic
579C2#3600112yes—
580C2#3726911yescharacteristic
581C2#3738611yescharacteristic
582C2#4291812yes—
583C2#4675812yes—
584C2#4714012yes—
585C2#4970614yes—
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. #16965C2semidirect
4. #16677C2semidirect
7. #16113C2semidirect
8. #15790C2semidirect
9. #15625C2semidirect
11. #14880C2semidirect
12. #14818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25992
C2×#23172
#169652
#166772
#164622
#163182
#161132
#157902
#156252
#148922
#148802
#148182
#147862
#145262
#139382