Finite Group Databasebeta

← #4431143

Subgroups — #4431143

Order 512.

2616 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic, complemented
2C2×#2624C211yesmaximal, characteristic
3C2×#2618C211yesmaximal, characteristic
4C2×#2346C211yesmaximal, characteristic, complemented
5#16550C214yesmaximal, complemented
6#16519C212yesmaximal, complemented
7#16517C212yesmaximal, complemented
8#15921C214yesmaximal, complemented
9#15612C214yesmaximal, complemented
10#14903C212yesmaximal, complemented
11#14898C212yesmaximal, complemented
12#14853C214yesmaximal, complemented
13#14675C211yesmaximal, characteristic, complemented
14#14290C211yesmaximal, characteristic
15#13923C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2624—28no—
21#2618—28no—
22#2484—28no—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#397C2212yes—
29C2×#396C2214yes—
30C2×#396C2214yes—
31C2×#394C2214yes—
32C2×#394C2212yes—
33#2346—28no—
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#382C2212yes—
43C2×#381C2212yes—
44C2×#381C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×#376C2212yes—
50#1445C2211yescharacteristic
51#1445C2211yescharacteristic
52#1444C2212yes—
53#1444C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2214yes—
69#1438C2211yescharacteristic
70#1438C2211yescharacteristic
71#1437C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76#1434C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—28no—
85C2×#123—48no—
86C2×#121—24no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117D418yes—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123#400—28no—
124#399—28no—
125#398—28no—
126#398—28no—
127#398—48no—
128#397—28no—
129#396—216no—
130#396—48no—
131#394—216no—
132#394—28no—
133#390—28no—
134#389—28no—
135#389—28no—
136#388—216no—
137#387—28no—
138#387—28no—
139#386—28no—
140#385—28no—
141#382—28no—
142#381—24no—
143#381—28no—
144#380—24no—
145#380D418yes—
146C4×C4⋊C4—28no—
147C4×C22⋊C4—24no—
148#376—28no—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4#4114yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212#123—432no—
213#121—416no—
214#117—232no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
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—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
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×C4⋊C4C2×D418yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4C2×D418yes—
286C2×C22⋊C4C2×D418yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D418yes—
295#95—22no—
296#95—28no—
297#95—22no—
298#95—28no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C2×D418yes—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24C22×D412yes—
336C24C22×D412yes—
337C24C2×#4112yes—
338C24C2511yescharacteristic
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—44no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—88no—
347C2×D4—88no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
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—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4#121116yes—
421C22×C4C22×D414yes—
422C22×C4C22×D418yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C42−1+414yes—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—416no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—44no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C22⋊C4—216no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C22⋊C4—84no—
456C22⋊C4—88no—
457C42—216no—
458C42—416no—
459C42—48no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23C2×#12114yes—
497C23C23×D411yescharacteristic
498C23C23×D411yescharacteristic
499C23C22×#4111yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×2−1+412yes—
502C23#58514yes—
503C23#58512yes—
504C23#58512yes—
505D4—88no—
506D4—816no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—416no—
528C2×C4—416no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—88no—
579C2×C4C2×#12114yes—
580C2×C4C2×#12114yes—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—44no—
585C22—44no—
586C22—48no—
587C22—44no—
588C22—44no—
589C22—82no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—82no—
596C22—82no—
597C22—84no—
598C22C22×#12111yescharacteristic
599C22#344912yes—
600C22#346112yes—
601C22#348411yescharacteristic
602C22#348512yes—
603C22#348711yescharacteristic
604C22#348812yes—
605C22#348912yes—
606C22#349112yes—
607C22#349412yes—
608C22#349911yescharacteristic
609C22#350011yescharacteristic
610C22#354714yes—
611C22#355114yes—
612C22#356414yes—
613C22#356714yes—
614C4—24no—
615C4—24no—
616C4—44no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—48no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—84no—
625C4—82no—
626C4—82no—
627C2—44no—
628C2—162no—
629C2#3508911yescharacteristic
630C2#3530811yescharacteristic
631C2#3556311yescharacteristic
632C2#4162014yes—
633C2#4215512yes—
634C2#4231412yes—
635C2#4878312yes—
636C2#5016712yes—
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×#2484C2semidirect
4. C2×#2346C2semidirect
5. #16550C2semidirect
6. #16519C2semidirect
7. #16517C2semidirect
8. #15921C2semidirect
9. #15612C2semidirect
10. #14903C2semidirect
11. #14898C2semidirect
12. #14853C2semidirect
13. #14675C2semidirect
15. #13923C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#26242
C2×#26182
C2×#23462
#165502
#165192
#165172
#159212
#156122
#149032
#148982
#148532
#146752
#142902
#139232