Finite Group Databasebeta

← #4866488

Subgroups — #4866488

Order 512.

2072 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2C2×#2621C211yesmaximal, characteristic
3#17189C214yesmaximal
4#17048C212yesmaximal, complemented
5#16858C214yesmaximal
6#16688C212yesmaximal, complemented
7#16663C212yesmaximal, complemented
8#16629C212yesmaximal, complemented
9#16603C212yesmaximal, complemented
10#16254C212yesmaximal
11#15856C212yesmaximal, complemented
12#15790C212yesmaximal, complemented
13#15001C212yesmaximal
14#14872C212yesmaximal, complemented
15#13938C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2621—28no—
21#2499—28no—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#384C2211yescharacteristic
38C2×#383C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43#1446C2212yes—
44#1446C2212yes—
45#1445C2214yes—
46#1445C2214yes—
47#1444C2212yes—
48#1444C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1437C2212yes—
71#1436C2214yes—
72#1436C2212yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8—42no—
87C2×C22⋊Q8—42no—
88C2×#117—42no—
89C2×#117—42no—
90C2×#117D418yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116#398—28no—
117#398—28no—
118#397—28no—
119#397—28no—
120#397—28no—
121#397—28no—
122#395—44no—
123#395—44no—
124#395—48no—
125#391—24no—
126#391—24no—
127#391—28no—
128#387—28no—
129#387—28no—
130#387—48no—
131#384—24no—
132#383—42no—
133#382—24no—
134#382—24no—
135C4×C4⋊C4—42no—
136C4×C22⋊C4—24no—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C22×Q8—24no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—44no—
177C22×Q8—42no—
178C22×Q8—44no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—42no—
183C23×C4C2×D414yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C4⋊Q8—416no—
199C22⋊Q8—416no—
200C22⋊Q8—432no—
201#117—416no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4C2×D414yes—
259C2×C22⋊C4C2×D414yes—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—44no—
270C2×C42C2×D414yes—
271C2×C42C2×D414yes—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—28no—
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—42no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—22no—
309C24—22no—
310C24C22×D412yes—
311C24C2511yescharacteristic
312C2×Q8—44no—
313C2×Q8—48no—
314C2×Q8—44no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—44no—
318C2×Q8—84no—
319C2×Q8—84no—
320C2×Q8—88no—
321C2×Q8—88no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
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—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—44no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C4H(5,𝔽2)14yes—
403C22×C42−1+414yes—
404C22×C42−1+418yes—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—416no—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—416no—
417C4⋊C4—88no—
418C4⋊C4—88no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—416no—
424C22⋊C4—88no—
425C42—28no—
426C42—28no—
427C42—44no—
428C42—44no—
429C42—84no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—42no—
440C23—41no—
441C23—42no—
442C23—44no—
443C23—41no—
444C23—44no—
445C23—44no—
446C23C23×D411yescharacteristic
447C23C2×H(5,𝔽2)11yescharacteristic
448C23C2×H(5,𝔽2)11yescharacteristic
449C23C2×H(5,𝔽2)12yes—
450C23C2×2−1+412yes—
451C23C2×2−1+414yes—
452C23#58512yes—
453C23#58512yes—
454Q8—816no—
455Q8—88no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—416no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4#53414yes—
534C2×C4#53514yes—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—84no—
541C22—82no—
542C22—82no—
543C22C2×#53411yescharacteristic
544C22C2×#53511yescharacteristic
545C22#344911yescharacteristic
546C22#348912yes—
547C22#349912yes—
548C22#353412yes—
549C22#353412yes—
550C22#353812yes—
551C22#354114yes—
552C22#354712yes—
553C22#355011yescharacteristic
554C22#355012yes—
555C22#355014yes—
556C22#355111yescharacteristic
557C22#356314yes—
558C22#356414yes—
559C4—24no—
560C4—24no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—48no—
567C4—82no—
568C4—84no—
569C4—82no—
570C4—84no—
571C4—82no—
572C4—84no—
573C2—82no—
574C2#3737611yescharacteristic
575C2#3738611yescharacteristic
576C2#3744011yescharacteristic
577C2#4230612yes—
578C2#4291112yes—
579C2#5641912yes—
580C2#5840614yes—
581C2#5894112yes—
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
4. #17048C2semidirect
6. #16688C2semidirect
7. #16663C2semidirect
8. #16629C2semidirect
9. #16603C2semidirect
11. #15856C2semidirect
12. #15790C2semidirect
14. #14872C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
C2×#26212
#171892
#170482
#168582
#166882
#166632
#166292
#166032
#162542
#158562
#157902
#150012
#148722
#139382