Finite Group Databasebeta

← #5026740

Subgroups — #5026740

Order 512.

3632 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#563C211yesmaximal, characteristic, complemented
2C2×#2611C212yesmaximal, complemented
3C2×#2620C211yesmaximal, characteristic
4#16093C212yesmaximal, complemented
5#15983C212yesmaximal, complemented
6#15950C212yesmaximal
7#15940C214yesmaximal, complemented
8#15939C214yesmaximal, complemented
9#15614C214yesmaximal, complemented
10#15613C214yesmaximal, complemented
11#15265C212yesmaximal, complemented
12#14283C212yesmaximal, complemented
13#14169C211yesmaximal, characteristic, complemented
14C2×#563—216no—
15C2×#563—28no—
16C22×C4⋊Q8C2212yes—
17C22×#126C2214yes—
18C22×#125C2212yes—
19C22×#124C2211yescharacteristic
20C22×#123C2212yes—
21C22×C22⋊Q8C2214yes—
22C22×C22⋊Q8C2211yescharacteristic
23C22×#121C2211yescharacteristic
24C22×#117C2211yescharacteristic
25#2620—28no—
26#2611—216no—
27C2×#400C2212yes—
28C2×#399C2212yes—
29C2×#398C2212yes—
30C2×#397C2212yes—
31C2×#396C2214yes—
32C2×#395C2212yes—
33C2×#394C2214yes—
34C2×#393C2212yes—
35C2×#391C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#389C2214yes—
38C2×#388C2214yes—
39C2×#387C2214yes—
40C2×#386C2212yes—
41C2×#385C2214yes—
42C2×#385C2212yes—
43C2×#384C2211yescharacteristic
44C2×#376C2212yes—
45#1446C2212yes—
46#1446C2214yes—
47#1444C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2212yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2212yes—
63#1437C2212yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C22⋊#95C2214yes—
71C23×Q8C2312yes—
72C23×Q8C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77#563—2128no—
78C2×C4⋊Q8—28no—
79C2×C4⋊Q8—216no—
80C2×#126—216no—
81C2×#126—216no—
82C2×#126—216no—
83C2×#125—28no—
84C2×#125—28no—
85C2×#125D4116yescomplemented
86C2×#124—28no—
87C2×#124D418yes—
88C2×#123—28no—
89C2×#123—216no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—216no—
94C2×C22⋊Q8—42no—
95C2×C22⋊Q8—42no—
96C2×#121—24no—
97C2×#121—42no—
98C2×#121—42no—
99C2×#117—28no—
100C2×#117—24no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129#400—28no—
130#399—28no—
131#398—28no—
132#397—28no—
133#396—216no—
134#395—44no—
135#394—216no—
136#393—44no—
137#391—24no—
138#390—24no—
139#389—216no—
140#388—216no—
141#387—216no—
142#386—44no—
143#385—216no—
144#385—44no—
145#384—24no—
146#376—28no—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C22×Q8—28no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—42no—
174C22×Q8—44no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8C2×D414yes—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4C2×D414yes—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—28no—
193C23×C4—24no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C4⋊Q8—264no—
210#126—2128no—
211#125—264no—
212#124—232no—
213#123—264no—
214C22⋊Q8—2128no—
215C22⋊Q8—416no—
216#121—416no—
217#117—416no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D4116yes—
252C2×C4⋊C4C2×D4116yes—
253C2×C4⋊C4C2×D418yes—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
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—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4C2×D4116yes—
287C2×C42—24no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—44no—
294C2×C42C2×D418yes—
295C2×C42C2×D414yes—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—44no—
315C24—22no—
316C24—24no—
317C24—21no—
318C24—22no—
319C24—21no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24C22×D414yes—
330C24H(5,𝔽2)12yes—
331C24C2511yescharacteristic
332C2×Q8—216no—
333C2×Q8—28no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×Q8—88no—
341C2×Q8—88no—
342C2×D4—28no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—88no—
350C2×D4—84no—
351C2×D4—88no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—216no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
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—24no—
406C22×C4—24no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4C22×D418yes—
416C22×C4C22×D414yes—
417C22×C4C22×D418yes—
418C22×C4C22×D414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D414yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C42−1+418yes—
425C4⋊C4—232no—
426C4⋊C4—216no—
427C4⋊C4—232no—
428C4⋊C4—232no—
429C4⋊C4—216no—
430C4⋊C4—416no—
431C4⋊C4—416no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—88no—
437C22⋊C4—232no—
438C22⋊C4—216no—
439C22⋊C4—232no—
440C22⋊C4—416no—
441C22⋊C4—48no—
442C22⋊C4—416no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—84no—
449C42—216no—
450C42—416no—
451C42—44no—
452C42#120116yes—
453C23—24no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23C23×D411yescharacteristic
490C23C23×D412yes—
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×2−1+414yes—
494C23#58512yes—
495C23#58514yes—
496Q8—88no—
497Q8—816no—
498D4—88no—
499D4—816no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—44no—
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—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
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—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—816no—
563C2×C4C2×#12014yes—
564C2×C4C2×#12018yes—
565C22—42no—
566C22—44no—
567C22—48no—
568C22—42no—
569C22—44no—
570C22—48no—
571C22—48no—
572C22—42no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22C22×#12011yescharacteristic
582C22#348411yescharacteristic
583C22#348511yescharacteristic
584C22#348812yes—
585C22#348812yes—
586C22#348914yes—
587C22#349714yes—
588C22#349912yes—
589C22#350012yes—
590C22#353312yes—
591C22#353812yes—
592C22#354412yes—
593C22#355112yes—
594C22#356414yes—
595C22#356514yes—
596C4—28no—
597C4—24no—
598C4—48no—
599C4—48no—
600C4—42no—
601C4—48no—
602C4—44no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—84no—
607C2—82no—
608C2—84no—
609C2#3551211yescharacteristic
610C2#3566012yes—
611C2#5154211yescharacteristic
612C2#5154512yes—
613C2#5158714yes—
614C2#5159114yes—
615C2#5197811yescharacteristic
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. C22×#563C2semidirect
2. C2×#2611C2semidirect
4. #16093C2semidirect
5. #15983C2semidirect
7. #15940C2semidirect
8. #15939C2semidirect
9. #15614C2semidirect
10. #15613C2semidirect
11. #15265C2semidirect
12. #14283C2semidirect
13. #14169C2semidirect
85. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5632
C2×#26112
C2×#26202
#160932
#159832
#159502
#159402
#159392
#156142
#156132
#152652
#142832
#141692