Finite Group Databasebeta

← #5025122

Subgroups — #5025122

Order 512.

3396 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic, complemented
2C2×#2598C212yesmaximal, complemented
3C2×#2629C211yesmaximal, characteristic, complemented
4C2×#2633C212yesmaximal, complemented
5#15696C214yesmaximal, complemented
6#15513C214yesmaximal, complemented
7#15505C214yesmaximal, complemented
8#15493C214yesmaximal, complemented
9#15340C214yesmaximal, complemented
10#15315C214yesmaximal, complemented
11#13830C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#121C2214yes—
14C22×C4×D4C2212yes—
15C22×#117C2211yescharacteristic
16#2633—216no—
17#2629—28no—
18#2598—216no—
19#2537—28no—
20C2×#400C2212yes—
21C2×#399C2214yes—
22C2×#399C2212yes—
23C2×#399C2214yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#397C2214yes—
27C2×#396C2214yes—
28C2×#394C2214yes—
29C2×#394C2212yes—
30C2×#394C2214yes—
31C2×#392C2212yes—
32C2×#392C2214yes—
33C2×#390C2214yes—
34C2×#390C2214yes—
35C2×#389C2212yes—
36C2×#388C2214yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#384C2211yescharacteristic
43C2×#383C2212yes—
44C2×#380C2212yes—
45C2×#380C2214yes—
46#1445C2214yes—
47#1444C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2214yes—
61C23×D4C2312yes—
62C23×D4C2314yes—
63C23×D4C2314yes—
64C24×C4C2311yescharacteristic
65C24×C4C2314yes—
66C2×#125—24no—
67C2×#125—28no—
68C2×#121—216no—
69C2×#121—48no—
70C2×#121—48no—
71C2×C4×D4—28no—
72C2×C4×D4—28no—
73C2×C4×D4—28no—
74C2×#117—28no—
75C2×#117—24no—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2311yescharacteristic
100C22×C42C2314yes—
101C22×C42C2311yescharacteristic
102#400—28no—
103#399—28no—
104#399—216no—
105#399—48no—
106#398—216no—
107#398—216no—
108#397—216no—
109#396—216no—
110#394—216no—
111#394—28no—
112#394—216no—
113#392—28no—
114#392—48no—
115#390—48no—
116#390—48no—
117#389—28no—
118#388—216no—
119#386—216no—
120#386—216no—
121#385—216no—
122#385—216no—
123#385—216no—
124#384D418yescomplemented
125#383D4116yescomplemented
126#380—28no—
127#380—216no—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C25C2414yes—
143C25C2411yescharacteristic
144C22×D4—28no—
145C22×D4—28no—
146C22×D4—28no—
147C22×D4—28no—
148C22×D4—28no—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—44no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184#125—416no—
185#121—464no—
186C4×D4—432no—
187#117—416no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C2×D4116yes—
217C2×C4⋊C4C2×D418yes—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
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—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42C2×D418yes—
263C2×C42C2×D414yes—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95C2×D418yes—
275#95C2×D418yes—
276#95C2×D4116yes—
277#95C2×D4116yes—
278C24—21no—
279C24—24no—
280C24—24no—
281C24—22no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—21no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—22no—
297C24—24no—
298C24H(5,𝔽2)18yes—
299C242−1+412yes—
300C24C2511yescharacteristic
301C2×D4—216no—
302C2×D4—416no—
303C2×D4—416no—
304C2×D4—48no—
305C2×D4—416no—
306C2×D4—416no—
307C2×D4—416no—
308C2×D4—416no—
309C2×D4—416no—
310C2×D4—416no—
311C2×D4—48no—
312C2×D4—88no—
313C2×D4—88no—
314C2×D4—88no—
315C2×D4—88no—
316C2×D4—88no—
317C2×D4—88no—
318C2×D4—88no—
319C2×D4—88no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—28no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—416no—
371C22×C4—416no—
372C22×C4—44no—
373C22×C4—416no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C22×D418yes—
379C22×C4C22×D418yes—
380C22×C4C22×D418yes—
381C22×C4C22×D414yes—
382C22×C4C22×D414yes—
383C22×C4C22×D414yes—
384C22×C4C22×D412yes—
385C22×C4C22×D414yes—
386C22×C42−1+412yes—
387C4⋊C4—48no—
388C4⋊C4—416no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—416no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C22⋊C4—416no—
398C22⋊C4—416no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—416no—
402C22⋊C4—416no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C42—44no—
411C42—88no—
412C42#127116yes—
413C23—28no—
414C23—22no—
415C23—28no—
416C23—28no—
417C23—24no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—28no—
422C23—28no—
423C23—41no—
424C23—44no—
425C23—42no—
426C23—44no—
427C23—41no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23C23×D411yescharacteristic
462C23C23×D412yes—
463C23C2×H(5,𝔽2)14yes—
464C23C2×2−1+411yescharacteristic
465C23C2×2−1+411yescharacteristic
466C23#58512yes—
467C23#58514yes—
468D4—432no—
469D4—832no—
470D4—832no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—22no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—416no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—832no—
526C2×C4C2×#12714yes—
527C2×C4C2×#12718yes—
528C22—48no—
529C22—44no—
530C22—48no—
531C22—42no—
532C22—48no—
533C22—48no—
534C22—48no—
535C22—48no—
536C22—48no—
537C22—48no—
538C22—84no—
539C22—81no—
540C22—84no—
541C22—81no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—82no—
550C22C22×#12711yescharacteristic
551C22#348412yes—
552C22#348511yescharacteristic
553C22#348611yescharacteristic
554C22#348712yes—
555C22#348814yes—
556C22#348814yes—
557C22#348912yes—
558C22#348912yes—
559C22#353414yes—
560C22#353514yes—
561C22#355914yes—
562C22#356614yes—
563C4—24no—
564C4—28no—
565C4—48no—
566C4—42no—
567C4—48no—
568C4—48no—
569C4—82no—
570C4—84no—
571C4—84no—
572C2—82no—
573C2—88no—
574C2#3556512yes—
575C2#3563411yescharacteristic
576C2#5063814yes—
577C2#5085914yes—
578C2#5200612yes—
579C2#5200811yescharacteristic
580C2#5208411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2537C2semidirect
2. C2×#2598C2semidirect
3. C2×#2629C2semidirect
4. C2×#2633C2semidirect
5. #15696C2semidirect
6. #15513C2semidirect
7. #15505C2semidirect
8. #15493C2semidirect
9. #15340C2semidirect
10. #15315C2semidirect
11. #13830C2semidirect
124. #384D4semidirect
125. #383D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#25982
C2×#26292
C2×#26332
#156962
#155132
#155052
#154932
#153402
#153152
#138302