Finite Group Databasebeta

← #5026755

Subgroups — #5026755

Order 512.

3996 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#562C211yesmaximal, characteristic, complemented
2C2×#2610C212yesmaximal, complemented
3C2×#2629C211yesmaximal, characteristic, complemented
4#15397C214yesmaximal, complemented
5#15368C214yesmaximal, complemented
6#15319C214yesmaximal, complemented
7#15315C214yesmaximal, complemented
8#15280C214yesmaximal, complemented
9#15279C214yesmaximal, complemented
10#14273C212yesmaximal, complemented
11#14170C211yesmaximal, characteristic, complemented
12C2×#562—28no—
13C2×#562—216no—
14C22×#125C2211yescharacteristic
15C22×#124C2212yes—
16C22×#123C2214yes—
17C22×C22⋊Q8C2214yes—
18C22×#121C2214yes—
19C22×#117C2211yescharacteristic
20#2629—28no—
21#2610—216no—
22C2×#398C2214yes—
23C2×#396C2214yes—
24C2×#395C2212yes—
25C2×#394C2212yes—
26C2×#394C2214yes—
27C2×#393C2212yes—
28C2×#392C2212yes—
29C2×#392C2214yes—
30C2×#391C2214yes—
31C2×#390C2214yes—
32C2×#390C2214yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#385C2214yes—
39C2×#384C2211yescharacteristic
40C2×#380C2212yes—
41C2×#376C2212yes—
42#1445C2214yes—
43#1444C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2214yes—
61C23×Q8C2312yes—
62C23×D4C2314yes—
63C23×D4C2314yes—
64C23×D4C2312yes—
65C24×C4C2311yescharacteristic
66C24×C4C2314yes—
67#562—2128no—
68C2×#125—28no—
69C2×#125D418yescomplemented
70C2×#124—28no—
71C2×#124—28no—
72C2×#124D4116yescomplemented
73C2×#123—216no—
74C2×#123—216no—
75C2×#123—216no—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—216no—
79C2×#121—216no—
80C2×#121—216no—
81C2×#121—216no—
82C2×#117—28no—
83C2×#117—24no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2314yes—
107C22×C42C2311yescharacteristic
108#398—216no—
109#396—216no—
110#395—28no—
111#394—28no—
112#394—48no—
113#393—28no—
114#392—28no—
115#392—48no—
116#391—48no—
117#390—216no—
118#390—48no—
119#389—28no—
120#388—216no—
121#388—216no—
122#386—216no—
123#386—48no—
124#385—48no—
125#384—24no—
126#380—28no—
127#376—28no—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C25C2414yes—
144C25C2411yescharacteristic
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8—44no—
148C22×Q8—44no—
149C22×Q8C2×D418yes—
150C22×D4—28no—
151C22×D4—28no—
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—28no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4C2×D418yes—
169C23×C4—22no—
170C23×C4—28no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—24no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190#125—232no—
191#124—264no—
192#123—2128no—
193C22⋊Q8—2128no—
194#121—2128no—
195#117—416no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C2×D4116yes—
219C2×C4⋊C4C2×D418yes—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
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—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
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—44no—
240C2×C22⋊C4—44no—
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×C22⋊C4C2×D4116yes—
258C2×C22⋊C4C2×D4116yes—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—28no—
263C2×C42—44no—
264C2×C42C2×D414yes—
265C2×C42C2×D418yes—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281C24—21no—
282C24—22no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—21no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—22no—
300C24—24no—
301C24C22×D418yes—
302C242−1+412yes—
303C24C2511yescharacteristic
304C2×Q8—216no—
305C2×Q8—416no—
306C2×Q8—88no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×D4—232no—
310C2×D4—216no—
311C2×D4—416no—
312C2×D4—416no—
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—88no—
316C2×D4—84no—
317C2×D4—84no—
318C2×D4—88no—
319C2×D4—88no—
320C2×D4—88no—
321C2×D4—88no—
322C2×D4—88no—
323C2×D4—88no—
324C2×D4—88no—
325C2×D4—88no—
326C2×D4—88no—
327C2×D4—88no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—232no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4C22×D418yes—
387C22×C4C22×D418yes—
388C22×C4C22×D414yes—
389C22×C4C22×D414yes—
390C22×C4C22×D414yes—
391C22×C4C22×D412yes—
392C22×C4C22×D414yes—
393C22×C4H(5,𝔽2)18yes—
394C22×C42−1+412yes—
395C4⋊C4—232no—
396C4⋊C4—232no—
397C4⋊C4—216no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C4⋊C4—88no—
403C22⋊C4—232no—
404C22⋊C4—232no—
405C22⋊C4—232no—
406C22⋊C4—416no—
407C22⋊C4—416no—
408C22⋊C4—416no—
409C22⋊C4—48no—
410C22⋊C4—88no—
411C22⋊C4—88no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C42—416no—
417C42—44no—
418C42#120116yes—
419C23—28no—
420C23—22no—
421C23—28no—
422C23—28no—
423C23—28no—
424C23—24no—
425C23—28no—
426C23—28no—
427C23—28no—
428C23—28no—
429C23—41no—
430C23—44no—
431C23—41no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—42no—
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—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23C23×D411yescharacteristic
468C23C23×D412yes—
469C23C2×H(5,𝔽2)14yes—
470C23C2×2−1+411yescharacteristic
471C23C2×2−1+411yescharacteristic
472C23#58512yes—
473C23#58514yes—
474Q8—816no—
475D4—816no—
476D4—832no—
477D4—832no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—416no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—832no—
533C2×C4C2×#12014yes—
534C2×C4C2×#12018yes—
535C22—44no—
536C22—42no—
537C22—48no—
538C22—48no—
539C22—48no—
540C22—48no—
541C22—48no—
542C22—48no—
543C22—48no—
544C22—48no—
545C22—84no—
546C22—84no—
547C22—81no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—82no—
555C22—81no—
556C22—84no—
557C22C22×#12011yescharacteristic
558C22#348611yescharacteristic
559C22#348711yescharacteristic
560C22#348814yes—
561C22#348912yes—
562C22#348912yes—
563C22#349614yes—
564C22#349812yes—
565C22#350012yes—
566C22#353514yes—
567C22#354214yes—
568C22#354714yes—
569C22#356614yes—
570C4—24no—
571C4—28no—
572C4—42no—
573C4—48no—
574C4—48no—
575C4—48no—
576C4—84no—
577C4—82no—
578C4—84no—
579C2—82no—
580C2—88no—
581C2#3551312yes—
582C2#3565911yescharacteristic
583C2#5158514yes—
584C2#5159014yes—
585C2#5175112yes—
586C2#5175411yescharacteristic
587C2#5190511yescharacteristic
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. C22×#562C2semidirect
2. C2×#2610C2semidirect
3. C2×#2629C2semidirect
4. #15397C2semidirect
5. #15368C2semidirect
6. #15319C2semidirect
7. #15315C2semidirect
8. #15280C2semidirect
9. #15279C2semidirect
10. #14273C2semidirect
11. #14170C2semidirect
69. C2×#125D4semidirect
72. C2×#124D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5622
C2×#26102
C2×#26292
#153972
#153682
#153192
#153152
#152802
#152792
#142732
#141702