Finite Group Databasebeta

← #4386437

Subgroups — #4386437

Order 512.

3272 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2816C211yesmaximal, characteristic
2C2×#2436C211yesmaximal, characteristic
3C2×#2434C211yesmaximal, characteristic, complemented
4C2×#2310C211yesmaximal, characteristic, complemented
5#16661C214yesmaximal, complemented
6#15500C214yesmaximal, complemented
7#15499C214yesmaximal, complemented
8#15322C214yesmaximal, complemented
9#15321C214yesmaximal, complemented
10#14890C214yesmaximal, complemented
11#14292C211yesmaximal, characteristic, complemented
12#14290C211yesmaximal, characteristic
13#14165C211yesmaximal, characteristic, complemented
14C24×Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×#124C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#117C2212yes—
20#2816—28no—
21#2436—28no—
22#2434—28no—
23C2×#399C2212yes—
24C2×#398C2212yes—
25C2×#396C2214yes—
26C2×#395C2214yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#393C2212yes—
30C2×#393C2214yes—
31#2310—28no—
32C2×#391C2212yes—
33C2×#391C2214yes—
34C2×#391C2212yes—
35C2×#391C2214yes—
36C2×#388C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#381C2212yes—
42C2×#381C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#1446C2212yes—
47#1445C2214yes—
48#1445C2212yes—
49#1445C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1438C2212yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2214yes—
71C23×Q8—24no—
72C23×Q8—24no—
73C23×Q8—44no—
74C23×Q8—44no—
75C23×Q8D418yes—
76C23×Q8C2312yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—44no—
86C2×C4⋊Q8—44no—
87C2×#124—28no—
88C2×#124—44no—
89C2×#124—44no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#117—28no—
97C2×#117—44no—
98C2×#117—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122#399—28no—
123#398—28no—
124#396—48no—
125#395—216no—
126#394—216no—
127#394—28no—
128#393—216no—
129#393—28no—
130#391—28no—
131#391—28no—
132#391—216no—
133#391—216no—
134#388—48no—
135#386—28no—
136#386—28no—
137#385—28no—
138#385—28no—
139#381—24no—
140#381—28no—
141#381D418yes—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—24no—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C22×Q8—232no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—432no—
187C22×Q8—432no—
188C22×Q8—416no—
189C22×Q8—416no—
190C22×Q8C2×D418yes—
191C22×Q8C2×D418yes—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C4⋊Q8—432no—
229#124—432no—
230C22⋊Q8—416no—
231#121—416no—
232#117—432no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
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×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
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—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C2×D418yes—
291C2×C42—28no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—44no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
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—
321#95C2×D418yes—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
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×Q8—216no—
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—4128no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—84no—
357C2×Q8—82no—
358C2×Q8—82no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—84no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4—88no—
368C2×D4—88no—
369C22×C4—216no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—216no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
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—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4#121116yes—
439C22×C4C22×D414yes—
440C22×C4C22×D418yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C42−1+418yes—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—48no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—416no—
472C42—48no—
473C42—48no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23C2×#12114yes—
510C23C23×D411yescharacteristic
511C23C23×D411yescharacteristic
512C23C22×#4111yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×H(5,𝔽2)12yes—
515C23C2×2−1+414yes—
516C23#58514yes—
517Q8—432no—
518Q8—432no—
519Q8—88no—
520D4—88no—
521D4—816no—
522C2×C4—232no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—416no—
547C2×C4—416no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4C2×#12114yes—
590C2×C4C2×#12114yes—
591C2×C4#53418yes—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—48no—
596C22—44no—
597C22—48no—
598C22—44no—
599C22—84no—
600C22—84no—
601C22—82no—
602C22—82no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—84no—
608C22C22×#12111yescharacteristic
609C22C2×#53412yes—
610C22#348412yes—
611C22#348612yes—
612C22#348712yes—
613C22#349011yescharacteristic
614C22#349011yescharacteristic
615C22#349912yes—
616C22#350012yes—
617C22#350912yes—
618C22#351012yes—
619C22#354114yes—
620C22#354714yes—
621C22#355414yes—
622C22#356114yes—
623C4—28no—
624C4—24no—
625C4—24no—
626C4—48no—
627C4—48no—
628C4—44no—
629C4—84no—
630C4—82no—
631C4—82no—
632C4—84no—
633C4—82no—
634C2—44no—
635C2—162no—
636C2#3498111yescharacteristic
637C2#3566711yescharacteristic
638C2#3567711yescharacteristic
639C2#4280614yes—
640C2#4645512yes—
641C2#4703212yes—
642C2#5132714yes—
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
3. C2×#2434C2semidirect
4. C2×#2310C2semidirect
5. #16661C2semidirect
6. #15500C2semidirect
7. #15499C2semidirect
8. #15322C2semidirect
9. #15321C2semidirect
10. #14890C2semidirect
11. #14292C2semidirect
13. #14165C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28162
C2×#24362
C2×#24342
C2×#23102
#166612
#155002
#154992
#153222
#153212
#148902
#142922
#142902
#141652