Finite Group Databasebeta

← #4465879

Subgroups — #4465879

Order 512.

2276 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2620C211yesmaximal, characteristic
3#17122C214yesmaximal, complemented
4#16891C212yesmaximal, complemented
5#16837C214yesmaximal, complemented
6#16267C214yesmaximal, complemented
7#15984C212yesmaximal
8#15798C212yesmaximal, complemented
9#15784C212yesmaximal, complemented
10#15670C212yesmaximal, complemented
11#15385C214yesmaximal, complemented
12#14887C212yesmaximal, complemented
13#14503C211yesmaximal, characteristic, complemented
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18#2804—28no—
19#2620—28no—
20C2×#400C2212yes—
21C2×#397C2212yes—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#395C2212yes—
25C2×#394C2214yes—
26C2×#393C2214yes—
27C2×#393C2212yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×#384C2211yescharacteristic
35C2×#382C2212yes—
36C2×#381C2212yes—
37#1446C2214yes—
38#1446C2214yes—
39#1445C2214yes—
40#1445C2212yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2212yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2212yes—
54#1440C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2212yes—
60#1438C2212yes—
61#1437C2214yes—
62#1437C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C2×#124—24no—
76C2×#124—44no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—42no—
79C2×C22⋊Q8—42no—
80C2×#121—24no—
81C2×#121—42no—
82C2×#121—42no—
83C2×#117—24no—
84C2×#117—44no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108#400—28no—
109#397—48no—
110#397—44no—
111#396—216no—
112#395—44no—
113#394—216no—
114#393—216no—
115#393—44no—
116#391—24no—
117#391—28no—
118#390D418yes—
119#386—28no—
120#386—44no—
121#385—216no—
122#384—24no—
123#382—44no—
124#381—44no—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25C2412yes—
150C25C2411yescharacteristic
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—42no—
157C22×Q8—44no—
158C22×Q8—44no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4C2×D414yes—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—44no—
180C23×C4—42no—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192#124—416no—
193C22⋊Q8—416no—
194#121—416no—
195#117—416no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×D4116yes—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42C2×D414yes—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—24no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—44no—
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—42no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—44no—
280#95—44no—
281#95—44no—
282C24—21no—
283C24—22no—
284C24—24no—
285C24—22no—
286C24—21no—
287C24—22no—
288C24—22no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24C22×D414yes—
297C24H(5,𝔽2)12yes—
298C24C2511yescharacteristic
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—48no—
302C2×Q8—44no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—88no—
308C2×Q8—88no—
309C2×D4—28no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—88no—
317C2×D4—84no—
318C2×D4—88no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—28no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
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—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—48no—
378C22×C4—48no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—48no—
382C22×C4—84no—
383C22×C4—88no—
384C22×C4—84no—
385C22×C4—88no—
386C22×C4C22×D414yes—
387C22×C4C22×D414yes—
388C22×C4C22×D412yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)14yes—
391C22×C42−1+414yes—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—84no—
397C4⋊C4—88no—
398C4⋊C4—84no—
399C4⋊C4—88no—
400C22⋊C4—416no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C22⋊C4—84no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C22⋊C4—88no—
411C22⋊C4—88no—
412C42—28no—
413C42—44no—
414C42—48no—
415C23—24no—
416C23—24no—
417C23—22no—
418C23—24no—
419C23—28no—
420C23—28no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23C23×D411yescharacteristic
452C23C2×H(5,𝔽2)11yescharacteristic
453C23C2×H(5,𝔽2)11yescharacteristic
454C23C2×H(5,𝔽2)12yes—
455C23C2×2−1+412yes—
456C23#58514yes—
457C23#58514yes—
458Q8—88no—
459Q8—816no—
460D4—88no—
461D4—816no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—416no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
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—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—816no—
526C2×C4#53614yes—
527C22—44no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—48no—
532C22—42no—
533C22—44no—
534C22—48no—
535C22—48no—
536C22—82no—
537C22—82no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22C2×#53611yescharacteristic
544C22#348411yescharacteristic
545C22#348411yescharacteristic
546C22#348512yes—
547C22#350012yes—
548C22#353312yes—
549C22#353612yes—
550C22#353612yes—
551C22#354012yes—
552C22#354012yes—
553C22#354512yes—
554C22#355114yes—
555C22#355714yes—
556C22#356014yes—
557C22#356514yes—
558C4—24no—
559C4—44no—
560C4—44no—
561C4—44no—
562C4—42no—
563C4—84no—
564C4—82no—
565C4—84no—
566C4—82no—
567C4—84no—
568C4—84no—
569C2—84no—
570C2—82no—
571C2#3671811yescharacteristic
572C2#4264712yes—
573C2#4642712yes—
574C2#5174911yescharacteristic
575C2#5205411yescharacteristic
576C2#5523114yes—
577C2#5655414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2804C2semidirect
3. #17122C2semidirect
4. #16891C2semidirect
5. #16837C2semidirect
6. #16267C2semidirect
8. #15798C2semidirect
9. #15784C2semidirect
10. #15670C2semidirect
11. #15385C2semidirect
12. #14887C2semidirect
13. #14503C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#26202
#171222
#168912
#168372
#162672
#159842
#157982
#157842
#156702
#153852
#148872
#145032