Finite Group Databasebeta

← #4868579

Subgroups — #4868579

Order 512.

2294 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2564C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3#17227C212yesmaximal, complemented
4#17143C212yesmaximal, complemented
5#16986C212yesmaximal, complemented
6#16678C212yesmaximal, complemented
7#16668C212yesmaximal, complemented
8#16666C212yesmaximal, complemented
9#16567C212yesmaximal, complemented
10#16396C214yesmaximal, complemented
11#16366C212yesmaximal, complemented
12#15982C214yesmaximal
13#14936C212yesmaximal, complemented
14#14843C212yesmaximal, complemented
15#14345C211yesmaximal, characteristic
16C22×#123C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2564—28no—
21#2486—28no—
22C2×#400C2211yescharacteristic
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#399C2211yescharacteristic
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#395C2211yescharacteristic
29C2×#393C2211yescharacteristic
30C2×#389C2212yes—
31C2×#387C2212yes—
32C2×#386C2212yes—
33C2×#385C2212yes—
34C2×#385C2214yes—
35C2×#383C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#381C2212yes—
38C2×#381C2214yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C22×#95C2211yescharacteristic
43#1446C2214yes—
44#1445C2212yes—
45#1445C2214yes—
46#1445C2214yes—
47#1445C2212yes—
48#1445C2212yes—
49#1444C2212yes—
50#1444C2212yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#123—28no—
84C2×#123—44no—
85C2×#123—44no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—42no—
88C2×C22⋊Q8—42no—
89C2×C4×D4—42no—
90C2×C4×D4—42no—
91C2×C4×D4D418yes—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#400—24no—
118#400—44no—
119#400—44no—
120#399—24no—
121#398—28no—
122#397—28no—
123#395—24no—
124#393—24no—
125#389—28no—
126#387—28no—
127#386—28no—
128#385—28no—
129#385—48no—
130#383—42no—
131#382—42no—
132#381—28no—
133#381—48no—
134#380—28no—
135#380—24no—
136#380—28no—
137C2×#95—24no—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×Q8—22no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—44no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4C2×D414yes—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213#123—432no—
214C22⋊Q8—416no—
215C4×D4—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C2×D414yes—
275C2×C22⋊C4C2×D414yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42C2×D414yes—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
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—44no—
301#95—42no—
302#95—42no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—24no—
322C24C22×D412yes—
323C24C22×D412yes—
324C24H(5,𝔽2)14yes—
325C242−1+414yes—
326C24C2511yescharacteristic
327C2×Q8—44no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—44no—
334C2×D4—48no—
335C2×D4—48no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
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—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—416no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—88no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+414yes—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—416no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—84no—
437C4⋊C4—88no—
438C4⋊C4—84no—
439C4⋊C4—88no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—416no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—416no—
447C22⋊C4—84no—
448C22⋊C4—88no—
449C42—28no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—44no—
479C23—44no—
480C23#53514yes—
481C23#53614yes—
482C23C23×D411yescharacteristic
483C23C2×H(5,𝔽2)11yescharacteristic
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×H(5,𝔽2)12yes—
486C23C2×2−1+412yes—
487C23C2×2−1+412yes—
488C23#58514yes—
489C23#58512yes—
490Q8—88no—
491D4—416no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—416no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
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—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—88no—
565C2×C4—816no—
566C2×C4—816no—
567C2×C4#53414yes—
568C22—24no—
569C22—24no—
570C22—28no—
571C22—28no—
572C22—42no—
573C22—42no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—48no—
583C22—48no—
584C22C2×#53411yescharacteristic
585C22C2×#53511yescharacteristic
586C22C2×#53611yescharacteristic
587C22#348512yes—
588C22#348912yes—
589C22#353312yes—
590C22#353314yes—
591C22#353312yes—
592C22#353514yes—
593C22#353612yes—
594C22#353612yes—
595C22#354712yes—
596C22#355011yescharacteristic
597C22#355111yescharacteristic
598C22#356314yes—
599C22#356414yes—
600C4—24no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—44no—
606C4—82no—
607C4—84no—
608C4—84no—
609C4—82no—
610C4—84no—
611C4—82no—
612C4—84no—
613C2—44no—
614C2—44no—
615C2#3696511yescharacteristic
616C2#3737611yescharacteristic
617C2#3745111yescharacteristic
618C2#4140912yes—
619C2#4141112yes—
620C2#5616412yes—
621C2#5832314yes—
622C2#5839012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17227C2semidirect
4. #17143C2semidirect
5. #16986C2semidirect
6. #16678C2semidirect
7. #16668C2semidirect
8. #16666C2semidirect
9. #16567C2semidirect
10. #16396C2semidirect
11. #16366C2semidirect
13. #14936C2semidirect
14. #14843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25642
C2×#24862
#172272
#171432
#169862
#166782
#166682
#166662
#165672
#163962
#163662
#159822
#149362
#148432
#143452