Finite Group Databasebeta

← #394833

Subgroups — #394833

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#2059C212yesmaximal, complemented
4C2×#1853C211yesmaximal, characteristic
5C2×#1852C211yesmaximal, characteristic
6#11328C212yesmaximal, complemented
7#11327C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10#10731C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2629—24no—
21#2629—24no—
22C2×#398—21no—
23C2×#398—21no—
24C2×#396—21no—
25C2×#396—21no—
26#2317—28no—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#385—21no—
32C2×#385—21no—
33C2×#384—21no—
34C2×#384C2211yescharacteristic
35C2×#382—21no—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38#2059—28no—
39#1853—24no—
40#1852—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—24no—
56C2×#125—42no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—24no—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117—42no—
64C2×#117Q814yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×#107—24no—
95C2×#107—24no—
96C2×#107—42no—
97C2×#106—24no—
98C2×#106—24no—
99C2×#106—42no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8Q814yes—
105C2×C22⋊C8Q814yes—
106#398—28no—
107#398—28no—
108#396—28no—
109#396—28no—
110#389—24no—
111#389—24no—
112#389—22no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—24no—
117#389D418yes—
118#385—28no—
119#385—28no—
120#384—28no—
121#384—22no—
122#384—22no—
123#382—28no—
124#380—22no—
125#380—22no—
126#380D414yes—
127#380D414yes—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#336—24no—
141#336—42no—
142C25C2×D411yescharacteristic
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—82no—
157C23×C4—82no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×Q812yes—
166C23×C4C2×Q812yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C2×Q814yes—
183C22×C8C2×Q814yes—
184#125—48no—
185#125—48no—
186#117—216no—
187#117—216no—
188#117—48no—
189#117—48no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×Q814yes—
248C2×C4⋊C4#4114yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—82no—
261C2×C22⋊C4—82no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42C2×D414yes—
273#107—48no—
274#106—48no—
275C22⋊C8—28no—
276C22⋊C8—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—28no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24#12012yes—
292C24C22⋊Q812yes—
293C24C4⋊Q812yes—
294C24C22×D411yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—48no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—88no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—88no—
368C22×C4#12012yes—
369C22×C4#12012yes—
370C22×C4#12012yes—
371C22×C4C22⋊Q814yes—
372C22×C4C22⋊Q814yes—
373C22×C4C22⋊Q812yes—
374C22×C4C4⋊Q814yes—
375C22×C4C4⋊Q814yes—
376C22×C4C4⋊Q812yes—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×Q811yescharacteristic
381C22×C4C22×Q811yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C42−1+412yes—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—88no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—88no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4C22⋊Q818yes—
428C22⋊C4—216no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C42—216no—
434C42—24no—
435C42—24no—
436C42—42no—
437C42—42no—
438C23—24no—
439C23—24no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23#39514yes—
461C23C2×#12011yescharacteristic
462C23C2×#12011yescharacteristic
463C23C2×C22⋊Q811yescharacteristic
464C23#52511yescharacteristic
465C23C2×C4⋊Q811yescharacteristic
466C23#53411yescharacteristic
467C23#53611yescharacteristic
468C23#57612yes—
469C23#57712yes—
470C23#57712yes—
471C23#57812yes—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—816no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4#39514yes—
538C2×C4C2×C22⋊Q812yes—
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×C22⋊Q811yescharacteristic
541C2×C4#52512yes—
542C2×C4C2×C4⋊Q811yescharacteristic
543C2×C4C2×C4⋊Q811yescharacteristic
544C2×C4#53912yes—
545C2×C4#54412yes—
546C2×C4D4212yes—
547C2×C4D4212yes—
548C2×C4#54712yes—
549C2×C4#54712yes—
550C2×C4D4×Q812yes—
551C2×C4D4×Q812yes—
552C2×C4#55412yes—
553C2×C4#55412yes—
554C8—44no—
555C8—44no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22C2×#39511yescharacteristic
568C22#240311yescharacteristic
569C22#240611yescharacteristic
570C22#243411yescharacteristic
571C22#258311yescharacteristic
572C22#262711yescharacteristic
573C22#284811yescharacteristic
574C22#300811yescharacteristic
575C22#300911yescharacteristic
576C22#301011yescharacteristic
577C22#301111yescharacteristic
578C22#301212yes—
579C22#301312yes—
580C22#308312yes—
581C22#308412yes—
582C22#311912yes—
583C22#312012yes—
584C22#316312yes—
585C22#316412yes—
586C22#319812yes—
587C22#320112yes—
588C22#324112yes—
589C22#324212yes—
590C4—24no—
591C4—42no—
592C4—42no—
593C4—48no—
594C4—82no—
595C4—82no—
596C4—81no—
597C4—82no—
598C4—84no—
599C4—81no—
600C4—82no—
601C4#244012yes—
602C4#244012yes—
603C4#244912yes—
604C4#244912yes—
605C2—44no—
606C2#1472011yescharacteristic
607C2#2355511yescharacteristic
608C2#2355611yescharacteristic
609C2#2392411yescharacteristic
610C2#2392711yescharacteristic
611C2#2406911yescharacteristic
612C2#2407111yescharacteristic
613C2#2582212yes—
614C2#2930912yes—
615C2#2997912yes—
616C2#3296812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2059C2semidirect
6. #11328C2semidirect
7. #11327C2semidirect
8. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#23172
C2×#20592
C2×#18532
C2×#18522
#113282
#113272
#112072
#107312
#107132