Finite Group Databasebeta

← #4389613

Subgroups — #4389613

Order 512.

2670 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2660C211yesmaximal, characteristic, complemented
3C2×#2633C211yesmaximal, characteristic
4C2×#2336C211yesmaximal, characteristic, complemented
5#17061C212yesmaximal, complemented
6#16800C212yesmaximal
7#15765C214yesmaximal, complemented
8#15739C214yesmaximal, complemented
9#15614C214yesmaximal, complemented
10#15354C214yesmaximal, complemented
11#15163C212yesmaximal
12#15151C212yesmaximal, complemented
13#14663C211yesmaximal, characteristic, complemented
14#13964C211yesmaximal, characteristic, complemented
15#13959C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2214yes—
19C22×C4×D4C2211yescharacteristic
20#2660—28no—
21#2633—28no—
22#2542—28no—
23C2×#399C2212yes—
24C2×#399C2214yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#397C2212yes—
30C2×#396C2214yes—
31C2×#395C2214yes—
32C2×#394C2212yes—
33#2336—28no—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2214yes—
38C2×#388C2214yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#385C2214yes—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#383C2211yescharacteristic
48C2×#383C2211yescharacteristic
49C2×#382C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×#376C2211yescharacteristic
53#1446C2212yes—
54#1446C2212yes—
55#1445C2212yes—
56#1445C2214yes—
57#1445C2212yes—
58#1445C2212yes—
59#1445C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2212yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2214yes—
76#1434C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×#125—24no—
86C2×#125—24no—
87C2×#125D418yes—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—48no—
90C2×C22⋊Q8—48no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#399—28no—
125#399—216no—
126#398—28no—
127#398—28no—
128#398—28no—
129#398—48no—
130#397—28no—
131#396—48no—
132#395—216no—
133#394—28no—
134#390—24no—
135#390—24no—
136#389—28no—
137#389—48no—
138#388—48no—
139#387—28no—
140#387—28no—
141#387—28no—
142#387—28no—
143#385—216no—
144#384—28no—
145#384—24no—
146#384—24no—
147#383—24no—
148#383—24no—
149#382—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4D418yes—
152#376—24no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25C2412yes—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—42no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C4⋊Q8—416no—
210#125—232no—
211C22⋊Q8—464no—
212C4×D4—416no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D418yes—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42#4114yes—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—22no—
306#95—24no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C2×D418yes—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—22no—
334C242−1+414yes—
335C24C2511yescharacteristic
336C2×Q8—416no—
337C2×Q8—48no—
338C2×Q8—88no—
339C2×Q8—84no—
340C2×Q8—84no—
341C2×D4—28no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—44no—
345C2×D4—44no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
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—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—416no—
406C22×C4—48no—
407C22×C4—88no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4C22×D414yes—
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4H(5,𝔽2)14yes—
424C22×C4H(5,𝔽2)14yes—
425C22×C42−1+418yes—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—416no—
432C4⋊C4—416no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—416no—
439C4⋊C4—84no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—216no—
451C42—44no—
452C42—44no—
453C42—48no—
454C42—84no—
455C42#121116yes—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—28no—
460C23—28no—
461C23—42no—
462C23—44no—
463C23—42no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23C23×D411yescharacteristic
480C23C23×D411yescharacteristic
481C23C22×#4111yescharacteristic
482C23C2×H(5,𝔽2)12yes—
483C23C2×H(5,𝔽2)12yes—
484C23C2×2−1+412yes—
485C23C2×2−1+414yes—
486C23#58512yes—
487Q8—816no—
488D4—416no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—816no—
561C2×C4C2×#12114yes—
562C2×C4C2×#12114yes—
563C2×C4C2×#12114yes—
564C2×C4#53518yes—
565C22—44no—
566C22—48no—
567C22—44no—
568C22—48no—
569C22—44no—
570C22—82no—
571C22—84no—
572C22—82no—
573C22—84no—
574C22—82no—
575C22—82no—
576C22C22×#12111yescharacteristic
577C22C2×#53512yes—
578C22#348511yescharacteristic
579C22#348611yescharacteristic
580C22#348612yes—
581C22#348912yes—
582C22#349512yes—
583C22#349812yes—
584C22#349912yes—
585C22#350811yescharacteristic
586C22#350811yescharacteristic
587C22#350912yes—
588C22#353514yes—
589C22#353814yes—
590C22#355214yes—
591C22#355914yes—
592C4—24no—
593C4—24no—
594C4—28no—
595C4—24no—
596C4—44no—
597C4—48no—
598C4—44no—
599C4—44no—
600C4—82no—
601C4—84no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C2—84no—
607C2#3527211yescharacteristic
608C2#3563311yescharacteristic
609C2#3566711yescharacteristic
610C2#4137512yes—
611C2#4279212yes—
612C2#4769214yes—
613C2#5091412yes—
614C2#5122612yes—
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
2. C2×#2660C2semidirect
4. C2×#2336C2semidirect
5. #17061C2semidirect
7. #15765C2semidirect
8. #15739C2semidirect
9. #15614C2semidirect
10. #15354C2semidirect
12. #15151C2semidirect
13. #14663C2semidirect
14. #13964C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#26602
C2×#26332
C2×#23362
#170612
#168002
#157652
#157392
#156142
#153542
#151632
#151512
#146632
#139642
#139592