Finite Group Databasebeta

← #4378525

Subgroups — #4378525

Order 512.

2428 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#2326C211yesmaximal, characteristic
3#16892C212yesmaximal, complemented
4#16582C214yesmaximal, complemented
5#16393C212yesmaximal, complemented
6#15983C212yesmaximal
7#15856C212yesmaximal, complemented
8#15790C212yesmaximal, complemented
9#15262C212yesmaximal
10#15122C212yesmaximal, complemented
11#14880C212yesmaximal, complemented
12#14840C212yesmaximal, complemented
13#14786C212yesmaximal, complemented
14#14384C214yesmaximal, complemented
15#13940C211yesmaximal, characteristic
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2486—28no—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#395C2214yes—
27#2326—28no—
28C2×#391C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#387C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#384C2211yescharacteristic
38C2×#381C2212yes—
39C2×#381C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44#1445C2212yes—
45#1445C2214yes—
46#1445C2212yes—
47#1445C2214yes—
48#1444C2212yes—
49#1444C2214yes—
50#1443C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1440C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2212yes—
73C4⋊#95C2214yes—
74#1434C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×#124—24no—
83C2×#124—28no—
84C2×#123—28no—
85C2×#123—48no—
86C2×C4×Q8—24no—
87C2×C4×Q8—42no—
88C2×C4×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⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123#398—28no—
124#398—44no—
125#398—44no—
126#397—28no—
127#395—48no—
128#391—24no—
129#390—24no—
130#389—28no—
131#388—28no—
132#388—28no—
133#387—48no—
134#385—28no—
135#385—28no—
136#385—216no—
137#384—24no—
138#381—28no—
139#381—42no—
140#380—28no—
141#380—24no—
142C4×C22⋊C4—42no—
143#376—24no—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4C2×D414yes—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—44no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208#124—232no—
209#123—432no—
210C4×Q8—416no—
211C4×D4—416no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4C2×D414yes—
272C2×C22⋊C4C2×D414yes—
273C2×C22⋊C4#4118yes—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—44no—
285C2×C42C2×D414yes—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—28no—
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—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—24no—
316C24—24no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—24no—
322C24—24no—
323C24C22×D412yes—
324C24C22×D412yes—
325C24C2×#4114yes—
326C24C2511yescharacteristic
327C2×Q8—28no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
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—28no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
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—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4114yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C4H(5,𝔽2)14yes—
420C22×C42−1+418yes—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—416no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—416no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C22⋊C4—216no—
434C22⋊C4—216no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—48no—
438C22⋊C4—416no—
439C22⋊C4—48no—
440C22⋊C4—416no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—28no—
445C42—28no—
446C42—48no—
447C42—44no—
448C42—84no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—28no—
453C23—24no—
454C23—28no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—42no—
477C23—44no—
478C23#52514yes—
479C23C23×D411yescharacteristic
480C23C22×#4112yes—
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×2−1+414yes—
485C23#58514yes—
486Q8—416no—
487D4—416no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—416no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
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—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—88no—
559C2×C4—816no—
560C2×C4#53414yes—
561C2×C4#53514yes—
562C22—24no—
563C22—28no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—48no—
568C22—48no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—84no—
575C22—84no—
576C22C2×#52511yescharacteristic
577C22C2×#53411yescharacteristic
578C22C2×#53511yescharacteristic
579C22#348512yes—
580C22#349514yes—
581C22#349912yes—
582C22#351514yes—
583C22#351912yes—
584C22#351912yes—
585C22#353312yes—
586C22#353312yes—
587C22#353812yes—
588C22#354114yes—
589C22#355012yes—
590C22#355414yes—
591C4—24no—
592C4—24no—
593C4—44no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—44no—
598C4—48no—
599C4—84no—
600C4—82no—
601C4—82no—
602C4—84no—
603C4—84no—
604C2—44no—
605C2—82no—
606C2#3507611yescharacteristic
607C2#3602112yes—
608C2#3695511yescharacteristic
609C2#3738611yescharacteristic
610C2#4264812yes—
611C2#4707514yes—
612C2#5136514yes—
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. #16892C2semidirect
4. #16582C2semidirect
5. #16393C2semidirect
7. #15856C2semidirect
8. #15790C2semidirect
10. #15122C2semidirect
11. #14880C2semidirect
12. #14840C2semidirect
13. #14786C2semidirect
14. #14384C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23262
#168922
#165822
#163932
#159832
#158562
#157902
#152622
#151222
#148802
#148402
#147862
#143842
#139402