Finite Group Databasebeta

← #391703

Subgroups — #391703

Order 512.

2570 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C2×#1779C211yesmaximal, characteristic
3#14680C211yesmaximal, characteristic
4#11200C212yesmaximal, complemented
5#11199C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#10970C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#103C2214yescomplemented
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18#2848—24no—
19#2848C418yescomplemented
20C2×#385—21no—
21C2×#385—21no—
22C2×#385—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#379—21no—
30C2×#379—21no—
31#1779—24no—
32C2×#340C2214yescomplemented
33C2×#340C2214yescomplemented
34C2×#340C2214yescomplemented
35C22⋊#95—21no—
36C22⋊#95—21no—
37C22⋊#95—21no—
38C26D411yescharacteristic
39C23×Q8C2311yescharacteristic
40C24×C4—44no—
41C24×C4—44no—
42C24×C4—44no—
43C24×C4C2×C414yes—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D414yes—
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2314yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—44no—
56C2×C22⋊Q8—44no—
57C2×C22⋊Q8—44no—
58C2×C22⋊Q8C2×C414yes—
59C2×C22⋊Q8C2×C414yes—
60C2×C22⋊Q8C2×C414yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C2×#103—44no—
78C2×#103—44no—
79C2×#103—44no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—44no—
84C2×C22⋊C8—44no—
85C2×C22⋊C8—44no—
86#385—28no—
87#385—28no—
88#385—28no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—44no—
93#380—44no—
94#380—44no—
95#380C2×C414yes—
96#380C2×C414yes—
97#380C2×C414yes—
98#379—28no—
99#379—44no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106#340—44no—
107#340—44no—
108#340—44no—
109C25—24no—
110C25—22no—
111C25—22no—
112C25—24no—
113C25—42no—
114C25—42no—
115C25—42no—
116C25—42no—
117C25C22⋊C414yes—
118C25C2×D411yescharacteristic
119C25C2×D411yescharacteristic
120C25C2×D411yescharacteristic
121C22×Q8—21no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8C22×C412yes—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—48no—
141C23×C4—48no—
142C23×C4—48no—
143C23×C4—48no—
144C23×C4—48no—
145C23×C4—48no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—416no—
156C23×C4—416no—
157C23×C4—416no—
158C23×C4—416no—
159C23×C4C22⋊C414yes—
160C23×C4C22⋊C414yes—
161C23×C4C22⋊C414yes—
162C23×C4C22×C412yes—
163C23×C4C22×C412yes—
164C23×C4C22×C412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2411yescharacteristic
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—24no—
175C22×C8—24no—
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—24no—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182C22⋊Q8—48no—
183C22⋊Q8—48no—
184C22⋊Q8—48no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
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—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228#103—416no—
229C22⋊C8—48no—
230C22⋊C8—48no—
231C22⋊C8—48no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—44no—
236#95—44no—
237#95—44no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—28no—
259C24—28no—
260C24—28no—
261C24—44no—
262C24—44no—
263C24—44no—
264C24—44no—
265C24—48no—
266C24—416no—
267C24—48no—
268C24—48no—
269C24—48no—
270C24—416no—
271C24—48no—
272C24—416no—
273C24—48no—
274C24—416no—
275C24C2×C22⋊C412yes—
276C24C2×C22⋊C412yes—
277C24C2×C22⋊C412yes—
278C24#13514yes—
279C24AGL(1,ℤ/8)12yes—
280C24#13712yes—
281C24#13714yes—
282C24C22×D411yescharacteristic
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
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—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—416no—
334C22×C4—416no—
335C22×C4—416no—
336C22×C4—464no—
337C22×C4—464no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4H(5,𝔽2)14yes—
377C22×C4H(5,𝔽2)14yes—
378C22×C4H(5,𝔽2)14yes—
379C2×C8—44no—
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—44no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—28no—
429C23—24no—
430C23—28no—
431C23—24no—
432C23—28no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—28no—
439C23—24no—
440C23—416no—
441C23—416no—
442C23—416no—
443C23—416no—
444C23—416no—
445C23—416no—
446C23—416no—
447C23—464no—
448C23—464no—
449C23—42no—
450C23—44no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23#41614yes—
466C23#41714yes—
467C23#41914yes—
468C23C22×C22⋊C411yescharacteristic
469C23#53412yes—
470C23#53412yes—
471C23#53412yes—
472C23C2×#13512yes—
473C23C2×AGL(1,ℤ/8)11yescharacteristic
474C23C2×#13712yes—
475C23C2×#13711yescharacteristic
476C23#57512yes—
477Q8—48no—
478Q8—84no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—416no—
489C2×C4—416no—
490C2×C4—416no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—48no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—84no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4#51812yes—
526C2×C4#51812yes—
527C2×C4#51812yes—
528C2×C4#53412yes—
529C2×C4#53412yes—
530C2×C4#53412yes—
531C2×C4#56014yes—
532C2×C4#56014yes—
533C2×C4#56014yes—
534C2×C4H(3,𝔽4)14yes—
535C8—84no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—464no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—48no—
559C22—48no—
560C22—48no—
561C22—48no—
562C22—48no—
563C22—48no—
564C22#230811yescharacteristic
565C22#230811yescharacteristic
566C22#230811yescharacteristic
567C22#281214yes—
568C22C2×#41611yescharacteristic
569C22C2×#41911yescharacteristic
570C22C2×#41711yescharacteristic
571C22#289611yescharacteristic
572C22#318812yes—
573C22#319014yes—
574C22#319012yes—
575C22#319114yes—
576C22#319312yes—
577C22#319412yes—
578C22#319512yes—
579C22#319614yes—
580C22#319612yes—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—164no—
589C4#237514yes—
590C4#281214yes—
591C2—44no—
592C2—44no—
593C2—44no—
594C2#1465711yescharacteristic
595C2#2176011yescharacteristic
596C2#2176211yescharacteristic
597C2#2176311yescharacteristic
598C2#2176511yescharacteristic
599C2#2176811yescharacteristic
600C2#2176911yescharacteristic
601C2#3270914yes—
602C2#3271114yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11200C2semidirect
5. #11199C2semidirect
6. #11199C2semidirect
7. #10970C2semidirect
8. #10970C2semidirect
9. #10970C2semidirect
14. C22×#103C22semidirect
19. #2848C4semidirect
32. C2×#340C22semidirect
33. C2×#340C22semidirect
34. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#17792
#146802
#112002
#111992
#109702