Finite Group Databasebeta

← #575607

Subgroups — #575607

Order 512.

3710 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#563C211yesmaximal, characteristic
2C2×#2372C211yesmaximal, characteristic
3C2×#2326C211yesmaximal, characteristic
4#14505C212yesmaximal
5#14475C214yesmaximal, complemented
6#14450C212yesmaximal, complemented
7#14364C214yesmaximal, complemented
8#14283C212yesmaximal, complemented
9#14170C212yesmaximal, complemented
10#14102C214yesmaximal, complemented
11#14052C214yesmaximal, complemented
12#14024C212yesmaximal, complemented
13#13968C212yesmaximal
14C2×#563—28no—
15C2×#563—28no—
16C2×#563C4116yescomplemented
17C22×C4⋊Q8C2212yes—
18C22×#126C2214yes—
19C22×#125C2212yes—
20C22×#124C2211yescharacteristic
21C22×#123C2212yes—
22C22×C22⋊Q8C2214yes—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25#2372—28no—
26#2326—28no—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#386C2212yes—
35C2×#385C2214yes—
36C2×#384C2211yescharacteristic
37C2×#381C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×#376C2211yescharacteristic
40C2×#376C2212yes—
41C22×#95C2212yes—
42#1446C2214yes—
43#1445C2212yes—
44#1445C2212yes—
45#1445C2212yes—
46#1445C2212yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65#1434C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C4×#95C2212yes—
72C23×Q8C2312yes—
73C23×Q8C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78#563—2128no—
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8C2×C4116yescomplemented
82C2×#126—216no—
83C2×#126—216no—
84C2×#126C2×C4132yescomplemented
85C2×#125—28no—
86C2×#125—28no—
87C2×#125C2×C4116yescomplemented
88C2×#124C2×C418yes—
89C2×#124D418yes—
90C2×#124D418yes—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123C2×C4116yes—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8C2×C4132yescomplemented
97C2×C4×Q8—24no—
98C2×C4×Q8—42no—
99C2×C4×Q8—42no—
100C2×C4×D4—24no—
101C2×C4×D4—42no—
102C2×C4×D4—42no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#391—24no—
128#391—28no—
129#391—44no—
130#391—44no—
131#390—24no—
132#388—216no—
133#388—216no—
134#386—28no—
135#385—216no—
136#384—24no—
137#381—44no—
138C4×C22⋊C4—216no—
139#376—24no—
140#376—28no—
141C2×#95—44no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C25C2412yes—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8C22×C418yes—
177C22×Q8C22×C414yes—
178C22×Q8C2×D414yes—
179C22×Q8C2×D414yes—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4C22×C414yes—
185C22×D4C2×D414yes—
186C22×D4C2×D414yes—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4C22×C418yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C4⋊Q8—264no—
217#126—2128no—
218#125—264no—
219#124C22⋊C4164yes—
220#123—264no—
221C22⋊Q8—2128no—
222C4×Q8—416no—
223C4×D4—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C4116yes—
251C2×C4⋊C4C22×C4116yes—
252C2×C4⋊C4C22×C4116yes—
253C2×C4⋊C4C22×C418yes—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
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—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4C22×C4116yes—
271C2×C22⋊C4C22×C418yes—
272C2×C22⋊C4C22×C4116yes—
273C2×C22⋊C4C2×D4116yes—
274C2×C22⋊C4C2×D4116yes—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—42no—
287C2×C42—44no—
288C2×C42—41no—
289C2×C42—41no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42C22×C418yes—
293C2×C42C22×C414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24C23×C414yes—
328C24C22×D414yes—
329C24C22×D414yes—
330C24C2511yescharacteristic
331C2×Q8—216no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—84no—
341C2×Q8—84no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8C2×C22⋊C4116yes—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4C2×C22⋊C4116yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
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—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—24no—
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—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—88no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4C23×C418yes—
426C22×C4C23×C418yes—
427C22×C4C23×C414yes—
428C22×C4C23×C414yes—
429C22×C4C23×C412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C42−1+418yes—
439C4⋊C4—216no—
440C4⋊C4—232no—
441C4⋊C4—232no—
442C4⋊C4—232no—
443C4⋊C4—216no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—84no—
448C4⋊C4—88no—
449C22⋊C4—232no—
450C22⋊C4—216no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—84no—
454C22⋊C4—88no—
455C22⋊C4C2×C22⋊C4164yes—
456C42—28no—
457C42—216no—
458C42—416no—
459C42—48no—
460C42—44no—
461C42—84no—
462C42C2×C22⋊C4116yes—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—28no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23C22×C22⋊C418yes—
486C23C24×C411yescharacteristic
487C23C23×D411yescharacteristic
488C23C23×D411yescharacteristic
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×2−1+414yes—
492C23#58511yescharacteristic
493C23#58511yescharacteristic
494C23#58514yes—
495Q8—416no—
496Q8—816no—
497D4—416no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
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—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—816no—
566C2×C4—816no—
567C2×C4—816no—
568C2×C4C22×C22⋊C418yes—
569C2×C4C22×C22⋊C418yes—
570C2×C4C22×C22⋊C414yes—
571C22—28no—
572C22—216no—
573C22—44no—
574C22—44no—
575C22—48no—
576C22—48no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—48no—
582C22C23×C22⋊C411yescharacteristic
583C22C4×H(5,𝔽2)11yescharacteristic
584C22C4×H(5,𝔽2)11yescharacteristic
585C22C4×2−1+414yes—
586C22#348411yescharacteristic
587C22#348511yescharacteristic
588C22#348914yes—
589C22#349611yescharacteristic
590C22#349611yescharacteristic
591C22#349912yes—
592C22#350012yes—
593C22#353312yes—
594C22#353812yes—
595C22#354412yes—
596C22#355112yes—
597C22#356414yes—
598C22#356514yes—
599C4—24no—
600C4—28no—
601C4—28no—
602C4—42no—
603C4—42no—
604C4—48no—
605C4—48no—
606C4—48no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—84no—
611C2—48no—
612C2#3425611yescharacteristic
613C2#3425912yes—
614C2#3451412yes—
615C2#3452112yes—
616C2#5154211yescharacteristic
617C2#5154512yes—
618C2#5158612yes—
619C2#5158912yes—
620C2#5197811yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #14475C2semidirect
6. #14450C2semidirect
7. #14364C2semidirect
8. #14283C2semidirect
9. #14170C2semidirect
10. #14102C2semidirect
11. #14052C2semidirect
12. #14024C2semidirect
16. C2×#563C4semidirect
81. C2×C4⋊Q8C2×C4semidirect
84. C2×#126C2×C4semidirect
87. C2×#125C2×C4semidirect
96. C2×C22⋊Q8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5632
C2×#23722
C2×#23262
#145052
#144752
#144502
#143642
#142832
#141702
#141022
#140522
#140242
#139682