Finite Group Databasebeta

← #405532

Subgroups — #405532

Order 512.

1520 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4#14130C211yesmaximal, characteristic
5#11385C212yesmaximal, complemented
6#11328C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18#2441—24no—
19#2441—24no—
20C2×#399—21no—
21C2×#396—21no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#387—21no—
27C2×#385—21no—
28C2×#384C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1852—24no—
33#1736—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—21no—
44#1442—21no—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—42no—
52C22×M16—42no—
53C22×M16Q814yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—24no—
59C2×#125—42no—
60C2×#117—22no—
61C2×#117—24no—
62C2×#117—42no—
63C2×#117D414yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
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—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—42no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8Q814yes—
96#399—28no—
97#396—28no—
98#389—28no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—24no—
104#389—42no—
105#387—28no—
106#385—44no—
107#384—22no—
108#384—22no—
109#380—22no—
110#380—22no—
111#380D414yes—
112#380Q814yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130#340—42no—
131#336—42no—
132#336—42no—
133C25C2×D411yescharacteristic
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—82no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×Q812yes—
157C23×C4C2×Q812yes—
158C23×C4C2×Q812yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C2×M16—28no—
162C2×M16—48no—
163C2×M16—44no—
164C2×M16—44no—
165C2×M16—48no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
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×C8C2×Q814yes—
179C22×C8C2×Q814yes—
180#125—48no—
181#125—44no—
182#125—44no—
183#117—28no—
184#117—44no—
185#117—48no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
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—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×Q814yes—
234C2×C4⋊C4#4114yes—
235C2×C4⋊C4#4114yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—82no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—44no—
259#107—48no—
260C22⋊C8—28no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—24no—
267#95—44no—
268#95—44no—
269#95—82no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—22no—
278C24#12012yes—
279C24C22⋊Q812yes—
280C24C4⋊Q812yes—
281C24C22×D411yescharacteristic
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4#12012yes—
356C22×C4C22⋊Q814yes—
357C22×C4C22⋊Q814yes—
358C22×C4C22⋊Q812yes—
359C22×C4C4⋊Q814yes—
360C22×C4C4⋊Q814yes—
361C22×C4C4⋊Q812yes—
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4#13712yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q811yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374M16—416no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—88no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4#12314yes—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—84no—
422C42—48no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—42no—
428C42—42no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23#39514yes—
452C23C2×#12011yescharacteristic
453C23C2×C22⋊Q811yescharacteristic
454C23C2×C4⋊Q811yescharacteristic
455C23#54612yes—
456C23#54612yes—
457C23C2×AGL(1,ℤ/8)11yescharacteristic
458C23C2×#13711yescharacteristic
459C23#57511yescharacteristic
460C23#57511yescharacteristic
461C23#57612yes—
462C23#57812yes—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—816no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#39514yes—
529C2×C4C2×C22⋊Q811yescharacteristic
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4#52511yescharacteristic
533C2×C4#52912yes—
534C2×C4C2×C4⋊Q811yescharacteristic
535C2×C4C2×C4⋊Q811yescharacteristic
536C2×C4#53312yes—
537C2×C4#53411yescharacteristic
538C2×C4#53511yescharacteristic
539C2×C4#53812yes—
540C2×C4#54312yes—
541C2×C4#54612yes—
542C2×C4#54612yes—
543C2×C4#55412yes—
544C2×C4#55412yes—
545C2×C4#55412yes—
546C2×C4#55412yes—
547C8—44no—
548C8—44no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22C2×#39511yescharacteristic
561C22#240411yescharacteristic
562C22#240511yescharacteristic
563C22#245012yes—
564C22#254912yes—
565C22#300611yescharacteristic
566C22#300711yescharacteristic
567C22#300811yescharacteristic
568C22#301111yescharacteristic
569C22#307711yescharacteristic
570C22#307811yescharacteristic
571C22#308312yes—
572C22#316412yes—
573C22#316511yescharacteristic
574C22#316611yescharacteristic
575C22#330912yes—
576C22#331212yes—
577C22#331512yes—
578C22#331812yes—
579C22#332212yes—
580C22#332712yes—
581C22#333012yes—
582C22#333312yes—
583C4—24no—
584C4—42no—
585C4—42no—
586C4—44no—
587C4—84no—
588C4—82no—
589C4—84no—
590C4—81no—
591C4—82no—
592C4—82no—
593C4—81no—
594C4#245012yes—
595C4#250211yescharacteristic
596C4#254912yes—
597C4#258311yescharacteristic
598C4#262611yescharacteristic
599C4#285511yescharacteristic
600C2—44no—
601C2#1491411yescharacteristic
602C2#2355411yescharacteristic
603C2#2355511yescharacteristic
604C2#2397311yescharacteristic
605C2#2398311yescharacteristic
606C2#2404611yescharacteristic
607C2#2405411yescharacteristic
608C2#2643212yes—
609C2#2643812yes—
610C2#2878112yes—
611C2#2878612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11385C2semidirect
6. #11328C2semidirect
7. #11219C2semidirect
8. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#18522
C2×#17362
#141302
#113852
#113282
#112192
#112072
#107312
#107222
#107152