Finite Group Databasebeta

← #4465880

Subgroups — #4465880

Order 512.

2660 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2620C211yesmaximal, characteristic
3#17124C214yesmaximal, complemented
4#16989C212yesmaximal, complemented
5#16549C214yesmaximal, complemented
6#16251C214yesmaximal, complemented
7#15784C212yesmaximal, complemented
8#15671C212yesmaximal, complemented
9#15625C212yesmaximal, complemented
10#15276C214yesmaximal, complemented
11#15260C212yesmaximal
12#14893C212yesmaximal, complemented
13#14503C211yesmaximal, characteristic, complemented
14C22×#124C2211yescharacteristic
15C22×#123C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2683—28no—
21#2620—28no—
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#397C2214yes—
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#395C2212yes—
28C2×#395C2214yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#393C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#390C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#384C2211yescharacteristic
40#1446C2214yes—
41#1445C2212yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2212yes—
62#1437C2212yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C2×#124—24no—
76C2×#124—44no—
77C2×#123—44no—
78C2×#123—48no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—42no—
82C2×C22⋊Q8—48no—
83C2×C22⋊Q8—48no—
84C2×C22⋊Q8—42no—
85C2×#121—24no—
86C2×#121—42no—
87C2×#121—42no—
88C2×#117—24no—
89C2×#117—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#400—44no—
118#399—44no—
119#397—48no—
120#396—216no—
121#396—216no—
122#395—28no—
123#395—48no—
124#394—216no—
125#394—48no—
126#393—44no—
127#391—24no—
128#391—44no—
129#390D418yes—
130#389—28no—
131#389—48no—
132#386—28no—
133#385—44no—
134#384—24no—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2412yes—
156C25C2411yescharacteristic
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—28no—
160C22×Q8—24no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—44no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—42no—
173C22×D4—44no—
174C22×D4—42no—
175C22×D4—44no—
176C22×D4C2×D414yes—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198#124—416no—
199#123—432no—
200C22⋊Q8—464no—
201C22⋊Q8—416no—
202#121—416no—
203#117—416no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D4116yes—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42C2×D414yes—
278#95—28no—
279#95—28no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—44no—
287#95—44no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95—44no—
297#95—42no—
298C24—21no—
299C24—22no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—22no—
304C24—24no—
305C24—21no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24C22×D414yes—
313C24H(5,𝔽2)12yes—
314C24C2511yescharacteristic
315C2×Q8—416no—
316C2×Q8—48no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×Q8—88no—
324C2×Q8—84no—
325C2×D4—28no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—84no—
331C2×D4—84no—
332C2×D4—88no—
333C2×D4—84no—
334C2×D4—88no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—44no—
395C22×C4—416no—
396C22×C4—44no—
397C22×C4—88no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4C22×D414yes—
401C22×C4C22×D414yes—
402C22×C4C22×D412yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)14yes—
405C22×C4H(5,𝔽2)18yes—
406C22×C42−1+418yes—
407C22×C42−1+414yes—
408C4⋊C4—216no—
409C4⋊C4—44no—
410C4⋊C4—416no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—416no—
414C4⋊C4—44no—
415C4⋊C4—416no—
416C4⋊C4—416no—
417C4⋊C4—84no—
418C4⋊C4—88no—
419C4⋊C4—88no—
420C22⋊C4—416no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—416no—
425C22⋊C4—48no—
426C22⋊C4—416no—
427C22⋊C4—416no—
428C22⋊C4—48no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C22⋊C4—84no—
432C42—28no—
433C42—44no—
434C42—84no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—28no—
439C23—24no—
440C23—28no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23C23×D411yescharacteristic
472C23C2×H(5,𝔽2)11yescharacteristic
473C23C2×H(5,𝔽2)14yes—
474C23C2×H(5,𝔽2)12yes—
475C23C2×H(5,𝔽2)11yescharacteristic
476C23C2×2−1+412yes—
477C23C2×2−1+414yes—
478Q8—88no—
479Q8—816no—
480D4—88no—
481D4—816no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—416no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
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—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
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#53614yes—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—48no—
552C22—42no—
553C22—44no—
554C22—48no—
555C22—48no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22—82no—
561C22—84no—
562C22—84no—
563C22C2×#53611yescharacteristic
564C22#348411yescharacteristic
565C22#348412yes—
566C22#348411yescharacteristic
567C22#349912yes—
568C22#353212yes—
569C22#353614yes—
570C22#353612yes—
571C22#353612yes—
572C22#353712yes—
573C22#353812yes—
574C22#354412yes—
575C22#354814yes—
576C22#355014yes—
577C22#355214yes—
578C4—24no—
579C4—48no—
580C4—44no—
581C4—44no—
582C4—44no—
583C4—42no—
584C4—48no—
585C4—82no—
586C4—84no—
587C4—82no—
588C4—84no—
589C2—84no—
590C2—82no—
591C2#3671811yescharacteristic
592C2#4242712yes—
593C2#4642312yes—
594C2#5154211yescharacteristic
595C2#5200411yescharacteristic
596C2#5494114yes—
597C2#5500014yes—
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×#2683C2semidirect
3. #17124C2semidirect
4. #16989C2semidirect
5. #16549C2semidirect
6. #16251C2semidirect
7. #15784C2semidirect
8. #15671C2semidirect
9. #15625C2semidirect
10. #15276C2semidirect
12. #14893C2semidirect
13. #14503C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#26202
#171242
#169892
#165492
#162512
#157842
#156712
#156252
#152762
#152602
#148932
#145032