Finite Group Databasebeta

← #625848

Subgroups — #625848

Order 512.

5058 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×#2405C212yesmaximal
2C2×#2315C211yesmaximal, characteristic, complemented
3C22×#386C211yesmaximal, characteristic
4#15383C218yesmaximal, complemented
5#15269C218yesmaximal, complemented
6#15014C212yesmaximal, complemented
7#14820C212yesmaximal, complemented
8#14680C212yesmaximal, complemented
9#14338C211yesmaximal, characteristic, complemented
10#14275C212yesmaximal, complemented
11#13869C212yesmaximal, complemented
12C24×D4C2211yescharacteristic
13C25×C4C2211yescharacteristic
14C22×#123C2214yes—
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2405—216no—
20C2×#397C2214yescomplemented
21C2×#394C2218yes—
22#2315—28no—
23C2×#386—28no—
24C2×#386C2214yes—
25C2×#386C2218yes—
26C2×#385C2214yes—
27C2×#385C2214yescomplemented
28C2×#385C2214yescomplemented
29C2×#381C2214yescomplemented
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2214yescomplemented
34C2×#380C2214yescomplemented
35C2×#379C2212yes—
36C2×#379C2214yes—
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39#1445C2214yescomplemented
40#1444C2214yescomplemented
41#1444C2214yescomplemented
42#1442C2214yescomplemented
43#1442C2218yescomplemented
44#1441C2218yescomplemented
45#1440C2218yescomplemented
46#1440C2218yescomplemented
47#1439C2218yescomplemented
48#1439C2214yescomplemented
49C22⋊#95C2212yes—
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2218yescomplemented
53C22⋊#95C2214yescomplemented
54C26C2311yescharacteristic
55C26C2311yescharacteristic
56C23×D4—24no—
57C23×D4—24no—
58C23×D4—44no—
59C23×D4—44no—
60C23×D4D418yes—
61C23×D4C2314yes—
62C24×C4—24no—
63C24×C4—28no—
64C24×C4—44no—
65C24×C4—44no—
66C24×C4C2311yescharacteristic
67C24×C4C2314yes—
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C2×#123—216no—
75C2×#123—48no—
76C2×#123—48no—
77C22×C4⋊C4C2318yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—28no—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—44no—
90C22×C22⋊C4—44no—
91C22×C22⋊C4D418yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2318yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2318yes—
101C22×C22⋊C4C2314yes—
102#397—48no—
103#394—232no—
104#386—216no—
105#386—232no—
106#385—216no—
107#385—216no—
108#385—216no—
109#381—48no—
110#380—28no—
111#380—28no—
112#380—28no—
113#380—48no—
114#380—48no—
115#379—216no—
116#379—28no—
117C2×#95—24no—
118C2×#95—24no—
119C2×#95C2318yes—
120C2×#95C2318yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2318yes—
127C2×#95C2314yes—
128C2×#95C2318yes—
129C2×#95C2314yes—
130C2×#95C2318yes—
131C25—22no—
132C25—22no—
133C25—24no—
134C25—24no—
135C25—24no—
136C25—24no—
137C25—24no—
138C25—24no—
139C25—42no—
140C25—42no—
141C25—42no—
142C25—42no—
143C25—42no—
144C25—42no—
145C25C2×D414yes—
146C25C2×D414yes—
147C25C2×D414yes—
148C25#4114yes—
149C25C2412yes—
150C25C2412yes—
151C25C2411yescharacteristic
152C22×D4—232no—
153C22×D4—28no—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—432no—
160C22×D4—416no—
161C22×D4—416no—
162C22×D4—432no—
163C22×D4C2×D4116yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—28no—
168C23×C4—24no—
169C23×C4—28no—
170C23×C4—24no—
171C23×C4—28no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—416no—
199C23×C4—464no—
200C23×C4—416no—
201C23×C4—416no—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4#4118yes—
206C23×C4C2418yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214#123—464no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C22⋊C4—232no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—216no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—416no—
256C2×C22⋊C4—416no—
257C2×C22⋊C4—432no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4C2×D4116yes—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—216no—
268#95—28no—
269#95—216no—
270#95—44no—
271#95—48no—
272#95—48no—
273#95—44no—
274#95—48no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—28no—
280C24—28no—
281C24—216no—
282C24—28no—
283C24—216no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—22no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—28no—
299C24—28no—
300C24—216no—
301C24—216no—
302C24—216no—
303C24—48no—
304C24—44no—
305C24—48no—
306C24—44no—
307C24—44no—
308C24—48no—
309C24—44no—
310C24—48no—
311C24—48no—
312C24—416no—
313C24—416no—
314C24—416no—
315C24—416no—
316C24—44no—
317C24—48no—
318C24—44no—
319C24—44no—
320C24—44no—
321C24—48no—
322C24—88no—
323C24—88no—
324C24—88no—
325C24—88no—
326C24#121116yes—
327C24C22×D414yes—
328C24C22×D414yes—
329C24C22×D414yes—
330C24C22×D412yes—
331C24C22×D412yes—
332C24C2×#4114yes—
333C24C2×#4112yes—
334C24C2×#4114yes—
335C24H(5,𝔽2)14yes—
336C24H(5,𝔽2)14yes—
337C242−1+414yes—
338C24C2511yescharacteristic
339C2×D4—232no—
340C2×D4—232no—
341C2×D4—416no—
342C2×D4—416no—
343C2×D4—416no—
344C2×D4—416no—
345C2×D4—416no—
346C2×D4—4128no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—232no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—416no—
388C22×C4—48no—
389C22×C4—416no—
390C22×C4—416no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—416no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—4128no—
402C22×C4—44no—
403C22×C4—416no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—88no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4#123116yescomplemented
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4114yes—
425C22×C4H(5,𝔽2)14yes—
426C4⋊C4—432no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—88no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C22⋊C4—232no—
433C22⋊C4—416no—
434C22⋊C4—416no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—432no—
439C22⋊C4—416no—
440C22⋊C4—88no—
441C22⋊C4—816no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—28no—
449C23—24no—
450C23—28no—
451C23—24no—
452C23—28no—
453C23—28no—
454C23—232no—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—232no—
480C23—416no—
481C23—416no—
482C23—464no—
483C23—416no—
484C23—464no—
485C23—416no—
486C23—416no—
487C23—416no—
488C23—416no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—48no—
497C23—44no—
498C23—48no—
499C23—416no—
500C23—44no—
501C23—48no—
502C23—44no—
503C23—48no—
504C23—88no—
505C23—832no—
506C23—88no—
507C23—88no—
508C23—88no—
509C23—832no—
510C23C2×#12118yes—
511C23C2×#12114yes—
512C23C2×#12314yes—
513C23#53418yes—
514C23#53614yes—
515C23#54718yes—
516C23#55218yes—
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C22×#4111yescharacteristic
520C23C22×#4112yes—
521C23C2×H(5,𝔽2)12yes—
522C23C2×H(5,𝔽2)12yes—
523C23C2×H(5,𝔽2)12yes—
524C23C2×2−1+412yes—
525C23#58512yes—
526D4—464no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—416no—
539C2×C4—464no—
540C2×C4—416no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—88no—
571C2×C4—88no—
572C2×C4—816no—
573C2×C4—816no—
574C2×C4C2×#12318yes—
575C22—24no—
576C22—216no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—216no—
583C22—28no—
584C22—28no—
585C22—28no—
586C22—28no—
587C22—216no—
588C22—464no—
589C22—44no—
590C22—44no—
591C22—48no—
592C22—48no—
593C22—416no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—48no—
603C22—416no—
604C22—832no—
605C22#248618yes—
606C22C22×#12111yescharacteristic
607C22C22×#12311yescharacteristic
608C22C2×#53412yes—
609C22C2×#53611yescharacteristic
610C22#345912yes—
611C22C2×#54712yes—
612C22C2×#55212yes—
613C22#348411yescharacteristic
614C22#348411yescharacteristic
615C22#348411yescharacteristic
616C22#348511yescharacteristic
617C22#349014yes—
618C22#349012yes—
619C22#349414yes—
620C22#349612yes—
621C22#353514yes—
622C22#353614yes—
623C4—28no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—82no—
628C4—84no—
629C4—84no—
630C4—88no—
631C2—28no—
632C2—48no—
633C2—48no—
634C2C2×#248611yescharacteristic
635C2#3498111yescharacteristic
636C2#3498411yescharacteristic
637C2#3535011yescharacteristic
638C2#3554011yescharacteristic
639C2#3671811yescharacteristic
640C2#3672011yescharacteristic
641C2#3786014yes—
642C2#3793214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2315C2semidirect
4. #15383C2semidirect
5. #15269C2semidirect
6. #15014C2semidirect
7. #14820C2semidirect
8. #14680C2semidirect
9. #14338C2semidirect
10. #14275C2semidirect
11. #13869C2semidirect
20. C2×#397C22semidirect
27. C2×#385C22semidirect
28. C2×#385C22semidirect
29. C2×#381C22semidirect
33. C2×#380C22semidirect
34. C2×#380C22semidirect
39. #1445C22semidirect
40. #1444C22semidirect
41. #1444C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
52. C22⋊#95C22semidirect
53. C22⋊#95C22semidirect
419. C22×C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24052
C2×#23152
C22×#3862
#153832
#152692
#150142
#148202
#146802
#143382
#142752
#138692