Finite Group Databasebeta

← #4319993

Subgroups — #4319993

Order 512.

2650 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2611C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic
3#16675C212yesmaximal
4#16668C212yesmaximal
5#16608C212yesmaximal, complemented
6#16565C212yesmaximal, complemented
7#15884C212yesmaximal, complemented
8#15860C214yesmaximal, complemented
9#15856C212yesmaximal, complemented
10#15119C212yesmaximal, complemented
11#15114C212yesmaximal
12#14901C212yesmaximal, complemented
13#14874C212yesmaximal
14#14383C214yesmaximal
15#13957C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2611—28no—
21C2×#400C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#398C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#395C2212yes—
26C2×#395C2212yes—
27C2×#395C2214yes—
28#2310—28no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#382C2211yescharacteristic
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#376C2211yescharacteristic
43C2×#376C2211yescharacteristic
44#1446C2214yes—
45#1446C2212yes—
46#1446C2212yes—
47#1445C2214yes—
48#1445C2214yes—
49#1444C2212yes—
50#1444C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2212yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1437C2212yes—
68#1436C2212yes—
69#1436C2214yes—
70C4⋊#95C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8—24no—
78C23×Q8—28no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8—44no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C2×#125—24no—
87C2×#125—42no—
88C2×#125—42no—
89C2×#117—24no—
90C2×#117—42no—
91C2×#117—42no—
92C2×#117—44no—
93C2×#117D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#400—24no—
119#399—24no—
120#398—24no—
121#397—24no—
122#395—44no—
123#395—44no—
124#395—48no—
125#391—28no—
126#391—28no—
127#391—28no—
128#391—28no—
129#389—28no—
130#388—28no—
131#387—28no—
132#387—48no—
133#385—28no—
134#385—216no—
135#382—42no—
136#381—28no—
137#381—28no—
138#376—24no—
139#376—42no—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C25C2411yescharacteristic
172C22×Q8—232no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—28no—
176C22×Q8—28no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—28no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—44no—
183C22×Q8—432no—
184C22×Q8—432no—
185C22×Q8—416no—
186C22×Q8—416no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4C2×D414yes—
198C23×C4#4118yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213#125—416no—
214#117—416no—
215#117—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4C2×D414yes—
268C2×C22⋊C4C2×D414yes—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42C2×D414yes—
281C2×C42C2×D414yes—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24C22×D412yes—
320C24C2×#4114yes—
321C24C2511yescharacteristic
322C2×Q8—216no—
323C2×Q8—216no—
324C2×Q8—416no—
325C2×Q8—416no—
326C2×Q8—416no—
327C2×Q8—416no—
328C2×Q8—4128no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—48no—
333C22×C4—216no—
334C22×C4—216no—
335C22×C4—28no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—216no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
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—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C2×#4118yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C42−1+418yes—
422C22×C42−1+414yes—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C4⋊C4—84no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C42—28no—
443C42—28no—
444C42—48no—
445C42—48no—
446C42—48no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—28no—
453C23—24no—
454C23—28no—
455C23—42no—
456C23—44no—
457C23—42no—
458C23—44no—
459C23—41no—
460C23—42no—
461C23—41no—
462C23#52514yes—
463C23C23×D411yescharacteristic
464C23C22×#4112yes—
465C23C2×H(5,𝔽2)11yescharacteristic
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×H(5,𝔽2)12yes—
468C23C2×2−1+412yes—
469C23C2×2−1+414yes—
470C23#58512yes—
471Q8—432no—
472Q8—432no—
473C2×C4—232no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—432no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4#53414yes—
552C2×C4#53514yes—
553C22—24no—
554C22—28no—
555C22—42no—
556C22—42no—
557C22—48no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22C2×#52511yescharacteristic
562C22C2×#53411yescharacteristic
563C22C2×#53511yescharacteristic
564C22#348912yes—
565C22#349012yes—
566C22#349012yes—
567C22#349514yes—
568C22#349912yes—
569C22#350914yes—
570C22#353812yes—
571C22#353814yes—
572C22#354712yes—
573C22#354714yes—
574C22#355011yescharacteristic
575C22#355012yes—
576C22#355111yescharacteristic
577C4—24no—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—48no—
584C4—48no—
585C4—44no—
586C4—82no—
587C4—84no—
588C4—82no—
589C4—84no—
590C4—82no—
591C2—44no—
592C2#3505011yescharacteristic
593C2#3608612yes—
594C2#3737611yescharacteristic
595C2#3738611yescharacteristic
596C2#4299412yes—
597C2#4703412yes—
598C2#4756712yes—
599C2#5147214yes—
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
5. #16608C2semidirect
6. #16565C2semidirect
7. #15884C2semidirect
8. #15860C2semidirect
9. #15856C2semidirect
10. #15119C2semidirect
12. #14901C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26112
C2×#23102
#166752
#166682
#166082
#165652
#158842
#158602
#158562
#151192
#151142
#149012
#148742
#143832
#139572