Finite Group Databasebeta

← #626124

Subgroups — #626124

Order 512.

4526 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2405C212yesmaximal
2C2×#2315C211yesmaximal, characteristic, complemented
3C22×#390C211yesmaximal, characteristic
4#15300C218yesmaximal, complemented
5#15271C218yesmaximal, complemented
6#14821C212yesmaximal, complemented
7#14820C212yesmaximal, complemented
8#14697C212yesmaximal, complemented
9#14338C212yesmaximal, complemented
10#14338C211yesmaximal, characteristic, complemented
11#13869C212yesmaximal, complemented
12C24×D4C2211yescharacteristic
13C22×#123C2214yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2405—216no—
22C2×#398C2214yescomplemented
23C2×#397C2214yescomplemented
24C2×#394C2218yes—
25#2315—28no—
26C2×#390—28no—
27C2×#390C2218yes—
28C2×#389C2214yescomplemented
29C2×#386C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yescomplemented
32C2×#385C2214yescomplemented
33C2×#381C2214yescomplemented
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#379C2212yes—
38C2×#379C2214yes—
39#1441C2218yescomplemented
40#1441C2214yescomplemented
41#1440C2218yescomplemented
42#1440C2218yescomplemented
43#1439C2214yescomplemented
44#1439C2218yescomplemented
45#1439C2218yescomplemented
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1436C2218yescomplemented
49C4⋊#95C2214yescomplemented
50C22⋊#95C2214yes—
51C22⋊#95C2212yes—
52C22⋊#95C2214yes—
53C22⋊#95C2214yescomplemented
54C26C2311yescharacteristic
55C26C2311yescharacteristic
56C23×D4—24no—
57C23×D4—24no—
58C23×D4—44no—
59C23×D4—44no—
60C23×D4D418yes—
61C23×D4C2314yes—
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C2×#123—216no—
70C2×#123—48no—
71C2×#123—48no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—44no—
74C22×C4⋊C4C2318yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—28no—
82C22×C22⋊C4—28no—
83C22×C22⋊C4—28no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—44no—
90C22×C22⋊C4—44no—
91C22×C22⋊C4D418yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2318yes—
101C22×C22⋊C4C2318yes—
102C22×C22⋊C4C2314yes—
103C22×C42—24no—
104C22×C42—44no—
105C22×C42C2314yes—
106#398—48no—
107#397—48no—
108#394—232no—
109#390—232no—
110#389—48no—
111#386—216no—
112#385—216no—
113#385—216no—
114#385—216no—
115#381—48no—
116#380—28no—
117#380—28no—
118#380—28no—
119#379—216no—
120#379—28no—
121C2×#95C2318yes—
122C2×#95C2318yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2318yes—
128C2×#95C2314yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C25—24no—
132C25—24no—
133C25—22no—
134C25—22no—
135C25—24no—
136C25—24no—
137C25—42no—
138C25—42no—
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—432no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—416no—
161C22×D4—416no—
162C22×D4—432no—
163C22×D4C2×D4116yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4C2×D414yes—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4#4118yes—
195C23×C4C2418yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203#123—464no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—216no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—216no—
209C2×C4⋊C4—216no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—416no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C22⋊C4—232no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—216no—
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—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—216no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—416no—
252C2×C22⋊C4—416no—
253C2×C22⋊C4—432no—
254C2×C22⋊C4—416no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4C2×D4116yes—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—44no—
263C2×C42—416no—
264#95—28no—
265#95—28no—
266#95—216no—
267#95—28no—
268#95—216no—
269#95—48no—
270#95—48no—
271#95—44no—
272#95—44no—
273#95—48no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—28no—
285C24—28no—
286C24—216no—
287C24—216no—
288C24—28no—
289C24—22no—
290C24—24no—
291C24—22no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—216no—
298C24—44no—
299C24—48no—
300C24—48no—
301C24—44no—
302C24—44no—
303C24—48no—
304C24—44no—
305C24—48no—
306C24—48no—
307C24—48no—
308C24—44no—
309C24—44no—
310C24—44no—
311C24—48no—
312C24—48no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—48no—
317C24—88no—
318C24—88no—
319C24—88no—
320C24—88no—
321C24—88no—
322C24—88no—
323C24—88no—
324C24—88no—
325C24#121116yes—
326C24C22×D414yes—
327C24C22×D414yes—
328C24C22×D414yes—
329C24C22×D412yes—
330C24C22×D412yes—
331C24C2×#4114yes—
332C24C2×#4112yes—
333C24C2×#4114yes—
334C242−1+414yes—
335C24C2511yescharacteristic
336C2×D4—232no—
337C2×D4—232no—
338C2×D4—416no—
339C2×D4—416no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4—416no—
343C2×D4—4128no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—216no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—216no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—28no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—416no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—416no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—416no—
396C22×C4—416no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—88no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4#123116yescomplemented
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4C22×D414yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4114yes—
414C22×C4H(5,𝔽2)14yes—
415C22×C4H(5,𝔽2)14yes—
416C22×C4H(5,𝔽2)14yes—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—432no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C22⋊C4—232no—
425C22⋊C4—416no—
426C22⋊C4—416no—
427C22⋊C4—416no—
428C22⋊C4—432no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C22⋊C4—416no—
433C22⋊C4—88no—
434C22⋊C4—816no—
435C42—416no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—28no—
445C23—28no—
446C23—28no—
447C23—28no—
448C23—24no—
449C23—28no—
450C23—24no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—232no—
461C23—24no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—416no—
466C23—416no—
467C23—416no—
468C23—416no—
469C23—48no—
470C23—48no—
471C23—48no—
472C23—48no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—48no—
483C23—48no—
484C23—416no—
485C23—48no—
486C23—48no—
487C23—88no—
488C23—832no—
489C23—832no—
490C23—88no—
491C23—88no—
492C23—832no—
493C23—88no—
494C23—88no—
495C23—88no—
496C23—88no—
497C23—832no—
498C23—88no—
499C23C2×#12118yes—
500C23C2×#12114yes—
501C23C2×#12314yes—
502C23#53614yes—
503C23#54718yes—
504C23#55218yes—
505C23C23×D411yescharacteristic
506C23C23×D411yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4112yes—
509C23C2×H(5,𝔽2)12yes—
510C23C2×H(5,𝔽2)12yes—
511C23C2×H(5,𝔽2)12yes—
512C23C2×2−1+412yes—
513C23#58512yes—
514D4—464no—
515C2×C4—232no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—416no—
532C2×C4—416no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—88no—
559C2×C4—816no—
560C2×C4—816no—
561C2×C4—88no—
562C2×C4C2×#12318yes—
563C2×C4#53418yes—
564C22—24no—
565C22—216no—
566C22—24no—
567C22—24no—
568C22—216no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—28no—
573C22—28no—
574C22—28no—
575C22—44no—
576C22—416no—
577C22—44no—
578C22—44no—
579C22—416no—
580C22—44no—
581C22—416no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—832no—
586C22—832no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—88no—
591C22—82no—
592C22#248618yes—
593C22C22×#12111yescharacteristic
594C22C22×#12311yescharacteristic
595C22C2×#53412yes—
596C22C2×#53611yescharacteristic
597C22#345912yes—
598C22C2×#54712yes—
599C22C2×#55212yes—
600C22#348411yescharacteristic
601C22#348411yescharacteristic
602C22#348411yescharacteristic
603C22#348511yescharacteristic
604C22#349012yes—
605C22#349612yes—
606C22#350514yes—
607C22#350814yes—
608C22#353614yes—
609C22#356114yes—
610C4—28no—
611C4—28no—
612C4—44no—
613C4—44no—
614C4—88no—
615C4—84no—
616C4—84no—
617C4—82no—
618C2—28no—
619C2—48no—
620C2—84no—
621C2C2×#248611yescharacteristic
622C2#3498111yescharacteristic
623C2#3498411yescharacteristic
624C2#3535111yescharacteristic
625C2#3555411yescharacteristic
626C2#3671811yescharacteristic
627C2#3696611yescharacteristic
628C2#3948614yes—
629C2#3954014yes—
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. #15300C2semidirect
5. #15271C2semidirect
6. #14821C2semidirect
7. #14820C2semidirect
8. #14697C2semidirect
9. #14338C2semidirect
10. #14338C2semidirect
11. #13869C2semidirect
22. C2×#398C22semidirect
23. C2×#397C22semidirect
28. C2×#389C22semidirect
31. C2×#385C22semidirect
32. C2×#385C22semidirect
33. C2×#381C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
41. #1440C22semidirect
42. #1440C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1436C22semidirect
49. C4⋊#95C22semidirect
53. C22⋊#95C22semidirect
408. 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×#3902
#153002
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