Finite Group Databasebeta

← #581624

Subgroups — #581624

Order 512.

2734 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2312C211yesmaximal, characteristic
2#14673C211yesmaximal, characteristic
3#14509C214yesmaximal, complemented
4#14393C214yesmaximal, complemented
5#14372C214yesmaximal, complemented
6#14127C211yesmaximal, characteristic
7#14038C214yesmaximal, complemented
8#13986C214yesmaximal, complemented
9#13917C214yesmaximal, complemented
10#13869C211yesmaximal, characteristic
11#13869C211yesmaximal, characteristic
12#13775C211yesmaximal, characteristic
13#13775C211yesmaximal, characteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17#2312C4116yescomplemented
18C2×#388C2214yes—
19C2×#388C2214yes—
20C2×#387C2214yescomplemented
21C2×#385C2212yes—
22C2×#385C2212yes—
23C2×#382C2214yescomplemented
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#380C2212yes—
27C2×#379C2211yescharacteristic
28C2×#379C2211yescharacteristic
29C2×#379C2211yescharacteristic
30C2×#379C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2212yes—
34#1445C2214yescomplemented
35#1443C2214yescomplemented
36#1443C2214yescomplemented
37#1443C2214yescomplemented
38#1442C2214yescomplemented
39#1441C2214yescomplemented
40#1440C2214yes—
41#1440C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1437C2214yes—
47#1437C2214yes—
48#1437C2214yes—
49#1437C2214yes—
50#1436C2214yescomplemented
51#1436C2214yescomplemented
52#1436C2214yescomplemented
53#1436C2214yescomplemented
54#1434C2214yescomplemented
55#1434C2214yescomplemented
56#1434C2214yescomplemented
57C22⋊#95C2212yes—
58C22⋊#95C2212yes—
59C22⋊#95C2212yes—
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C4×#95C2214yescomplemented
68C26C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—44no—
83C22×C22⋊C4—44no—
84C22×C22⋊C4C2×C418yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C42C2314yes—
104C22×C42C2314yes—
105C22×C42C2314yes—
106#388C2×C4132yescomplemented
107#388C2×C4132yescomplemented
108#387—48no—
109#385—44no—
110#385—44no—
111#382—48no—
112#381—44no—
113#381—44no—
114#380C2×C4116yescomplemented
115#379—24no—
116#379—24no—
117#379C2×C418yes—
118#379C2×C418yes—
119C4×C22⋊C4—44no—
120C4×C22⋊C4—44no—
121C4×C22⋊C4C2×C4116yescomplemented
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C25—42no—
145C25—42no—
146C25—42no—
147C25—42no—
148C25—42no—
149C25—42no—
150C25C22×C414yes—
151C25#4114yes—
152C25#4114yes—
153C25C2411yescharacteristic
154C25C2411yescharacteristic
155C25C2411yescharacteristic
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C22×C414yes—
181C23×C4C22×C414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C22×C4116yes—
201C2×C22⋊C4—232no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—416no—
241C2×C22⋊C4—416no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4C22×C418yes—
251C2×C22⋊C4C22×C418yes—
252C2×C22⋊C4C22×C418yes—
253C2×C22⋊C4C22×C418yes—
254C2×C22⋊C4C22×C418yes—
255C2×C22⋊C4C22×C418yes—
256C2×C22⋊C4C22×C418yes—
257C2×C22⋊C4C22×C418yes—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42C22×C4116yes—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95C22×C4116yes—
288#95C22×C4116yes—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—44no—
308C24—44no—
309C24—48no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24—48no—
318C24—48no—
319C24—88no—
320C24—88no—
321C24—88no—
322C24—88no—
323C24#117116yes—
324C24C23×C412yes—
325C24C23×C412yes—
326C24C23×C412yes—
327C24C2×#4112yes—
328C24C2×#4112yes—
329C24C2×#4112yes—
330C24C2×#4112yes—
331C24C2×#4112yes—
332C24C2×#4112yes—
333C24C2511yescharacteristic
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—88no—
395C22×C4—88no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4C23×C414yes—
413C22×C4C23×C418yes—
414C22×C4C23×C418yes—
415C22×C4C23×C414yes—
416C22×C4H(5,𝔽2)14yes—
417C22×C4H(5,𝔽2)14yes—
418C4⋊C4—416no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C22⋊C4—216no—
422C22⋊C4—216no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C42—416no—
440C42—88no—
441C42—88no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—416no—
479C23—416no—
480C23—416no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—88no—
494C23—88no—
495C23—88no—
496C23—832no—
497C23—832no—
498C23—88no—
499C23C2×#11714yes—
500C23C2×#11714yes—
501C23C2×#11714yes—
502C23C24×C411yescharacteristic
503C23C22×#4111yescharacteristic
504C23C22×#4111yescharacteristic
505C23C2×H(5,𝔽2)12yes—
506C23C2×H(5,𝔽2)12yes—
507C23#58512yes—
508C23#58512yes—
509C23#58512yes—
510C23#58512yes—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—88no—
570C2×C4—88no—
571C2×C4—88no—
572C2×C4—88no—
573C2×C4—88no—
574C2×C4—88no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—48no—
597C22—48no—
598C22—48no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—832no—
603C22C22×#11711yescharacteristic
604C22#342812yes—
605C22C4×H(5,𝔽2)12yes—
606C22C4×H(5,𝔽2)12yes—
607C22#349011yescharacteristic
608C22#349011yescharacteristic
609C22#349111yescharacteristic
610C22#349111yescharacteristic
611C22#349111yescharacteristic
612C22#349111yescharacteristic
613C22#349112yes—
614C22#350512yes—
615C22#350512yes—
616C22#355814yes—
617C22#356014yes—
618C22#356214yes—
619C22#356614yes—
620C4—44no—
621C4—44no—
622C4—48no—
623C4—48no—
624C4—44no—
625C4—44no—
626C4—82no—
627C4—82no—
628C4—84no—
629C4—84no—
630C2—44no—
631C2—44no—
632C2—44no—
633C2#3417711yescharacteristic
634C2#3425111yescharacteristic
635C2#3425111yescharacteristic
636C2#3445414yes—
637C2#4638511yescharacteristic
638C2#4638811yescharacteristic
639C2#4639011yescharacteristic
640C2#4640014yes—
641C2#4683811yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14509C2semidirect
4. #14393C2semidirect
5. #14372C2semidirect
7. #14038C2semidirect
8. #13986C2semidirect
9. #13917C2semidirect
17. #2312C4semidirect
20. C2×#387C22semidirect
23. C2×#382C22semidirect
34. #1445C22semidirect
35. #1443C22semidirect
36. #1443C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1441C22semidirect
50. #1436C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect
53. #1436C22semidirect
54. #1434C22semidirect
55. #1434C22semidirect
56. #1434C22semidirect
67. C4×#95C22semidirect
106. #388C2×C4semidirect
107. #388C2×C4semidirect
114. #380C2×C4semidirect
121. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23122
#146732
#145092
#143932
#143722
#141272
#140382
#139862
#139172
#138692
#137752