Finite Group Databasebeta

← #394867

Subgroups — #394867

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#2439C211yesmaximal, characteristic
3C2×#2057C212yesmaximal, complemented
4C2×#1852C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#11327C212yesmaximal, complemented
7#11325C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#428—28no—
17C2×#428—24no—
18C22×#107C2212yes—
19C22×#107C2212yes—
20C22×#106C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22#2439—24no—
23#2439—24no—
24C2×#396—21no—
25C2×#393—21no—
26C2×#391—21no—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#387—21no—
31C2×#385—21no—
32C2×#382C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#2057—28no—
37#1852—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1442—21no—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C23×Q8—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—24no—
55C22×M16—24no—
56C22×M16Q814yescomplemented
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—42no—
63C2×#117—22no—
64C2×#117—24no—
65C2×#117—42no—
66C2×#117D414yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—24no—
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⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
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
90#428—232no—
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107—24no—
95C2×#107—42no—
96C2×#107D418yes—
97C2×#106—24no—
98C2×#106—24no—
99C2×#106—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8Q814yescomplemented
103#396—28no—
104#393—28no—
105#391—28no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—24no—
111#389—42no—
112#387—28no—
113#385—44no—
114#382—22no—
115#382—22no—
116#380—22no—
117#380—22no—
118#380D414yes—
119#380Q814yescomplemented
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95D411yescharacteristic
132C2×#95D411yescharacteristic
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—42no—
138#336—24no—
139#336—24no—
140C25C2×D411yescharacteristic
141C22×Q8—24no—
142C22×Q8—44no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—82no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—82no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×Q812yes—
169C23×C4C2×Q812yes—
170C23×C4C2×Q812yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C2×M16—216no—
174C2×M16—28no—
175C2×M16—28no—
176C2×M16—28no—
177C2×M16—216no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C2×D414yes—
189C22×C8C2×Q814yes—
190C22×C8C2×Q814yes—
191C22×C8#4114yes—
192C4⋊Q8—48no—
193C4⋊Q8—44no—
194C4⋊Q8—44no—
195#117—28no—
196#117—44no—
197#117—48no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×Q814yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—82no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—44no—
264#107—216no—
265#107—48no—
266#106—216no—
267C22⋊C8—28no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—22no—
274#95—22no—
275#95—24no—
276#95—44no—
277#95—44no—
278#95—82no—
279C24—22no—
280C24—22no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24#12012yes—
288C24C22⋊Q812yes—
289C24C4⋊Q812yes—
290C24C22×D411yescharacteristic
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—88no—
294C2×Q8—84no—
295C2×Q8—88no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
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—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—48no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4#12012yes—
370C22×C4C22⋊Q814yes—
371C22×C4C22⋊Q814yes—
372C22×C4C22⋊Q812yes—
373C22×C4C4⋊Q814yes—
374C22×C4C4⋊Q814yes—
375C22×C4C4⋊Q812yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4#13712yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×Q811yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4H(5,𝔽2)12yes—
387C22×C42−1+412yes—
388M16—232no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8C22⋊Q818yes—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—88no—
426C4⋊C4—84no—
427C4⋊C4#12014yes—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—84no—
433C42—48no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—42no—
439C42—42no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23#39514yes—
463C23C2×#12011yescharacteristic
464C23C2×C22⋊Q811yescharacteristic
465C23C2×C4⋊Q811yescharacteristic
466C23D4212yes—
467C23#54712yes—
468C23C2×AGL(1,ℤ/8)11yescharacteristic
469C23C2×#13711yescharacteristic
470C23#57511yescharacteristic
471C23#57511yescharacteristic
472C23#57612yes—
473C23#57812yes—
474Q8—816no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—816no—
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—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4#39514yes—
541C2×C4C2×#12011yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×C22⋊Q812yes—
544C2×C4C2×C22⋊Q811yescharacteristic
545C2×C4#52511yescharacteristic
546C2×C4#52512yes—
547C2×C4C2×C4⋊Q811yescharacteristic
548C2×C4C2×C4⋊Q811yescharacteristic
549C2×C4#53411yescharacteristic
550C2×C4#53611yescharacteristic
551C2×C4#53912yes—
552C2×C4#54412yes—
553C2×C4D4212yes—
554C2×C4#54712yes—
555C2×C4D4×Q812yes—
556C2×C4D4×Q812yes—
557C2×C4#55412yes—
558C2×C4#55412yes—
559C8—28no—
560C8—44no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22C2×#39511yescharacteristic
573C22#240311yescharacteristic
574C22#240611yescharacteristic
575C22#244012yes—
576C22#244912yes—
577C22#300611yescharacteristic
578C22#300711yescharacteristic
579C22#300811yescharacteristic
580C22#301111yescharacteristic
581C22#307711yescharacteristic
582C22#307811yescharacteristic
583C22#308312yes—
584C22#316412yes—
585C22#316511yescharacteristic
586C22#316611yescharacteristic
587C22#326412yes—
588C22#327012yes—
589C22#327312yes—
590C22#327712yes—
591C22#334712yes—
592C22#334912yes—
593C22#335312yes—
594C22#335712yes—
595C4—24no—
596C4—42no—
597C4—44no—
598C4—42no—
599C4—81no—
600C4—82no—
601C4—81no—
602C4—84no—
603C4—82no—
604C4—82no—
605C4—84no—
606C4#243411yescharacteristic
607C4#244012yes—
608C4#244912yes—
609C4#258311yescharacteristic
610C4#262711yescharacteristic
611C4#284811yescharacteristic
612C2—44no—
613C2#1472011yescharacteristic
614C2#2355411yescharacteristic
615C2#2355511yescharacteristic
616C2#2386111yescharacteristic
617C2#2386711yescharacteristic
618C2#2410511yescharacteristic
619C2#2410711yescharacteristic
620C2#2607412yes—
621C2#2607912yes—
622C2#2635212yes—
623C2#2635812yes—
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
3. C2×#2057C2semidirect
6. #11327C2semidirect
7. #11325C2semidirect
8. #11219C2semidirect
56. C22×M16Q8semidirect
102. C2×C22⋊C8Q8semidirect
119. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#24392
C2×#20572
C2×#18522
#141302
#113272
#113252
#112192
#107312
#107222
#107152