Finite Group Databasebeta

← #395921

Subgroups — #395921

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#2068C212yesmaximal, complemented
3C2×#1894C211yesmaximal, characteristic
4C2×#1873C211yesmaximal, characteristic
5C2×#1851C211yesmaximal, characteristic
6#14139C211yesmaximal, characteristic
7#13951C211yesmaximal, characteristic
8#11231C212yesmaximal, complemented
9#11219C212yesmaximal, complemented
10#11122C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#428—28no—
17C2×#428—24no—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#2068—28no—
30#1894—24no—
31#1873—24no—
32#1851—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1441—21no—
40#1441—21no—
41#1439—22no—
42C4⋊#95—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45#1434—21no—
46C22⋊#95—22no—
47C23×Q8C2311yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—24no—
54C22×M16D414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8Q814yes—
64C2×#117—22no—
65C2×#117—24no—
66C2×#117—42no—
67C2×#117D414yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D412yes—
84C22×C42—21no—
85C22×C42C2312yes—
86#428—232no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#107D418yes—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—42no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106#391—24no—
107#391—42no—
108#389—24no—
109#389—42no—
110#380—22no—
111#380—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124#340—24no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—42no—
129#336—24no—
130C25C2×D411yescharacteristic
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8C2×Q812yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—44no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—81no—
155C23×C4—81no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×Q812yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4#4112yes—
165C23×C4C2411yescharacteristic
166C2×M16—216no—
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—28no—
170C2×M16—216no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8C2×D414yes—
186C22×C8C2×D414yes—
187C22×C8#4114yes—
188C22⋊Q8—28no—
189C22⋊Q8—28no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192#117—28no—
193#117—44no—
194#117—48no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×Q812yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—81no—
242C2×C22⋊C4—81no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—44no—
251#107—216no—
252#106—216no—
253C4⋊C8—48no—
254#103—44no—
255#103—48no—
256#103—44no—
257C22⋊C8—44no—
258#95—22no—
259#95—22no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—42no—
264#95—44no—
265#95—82no—
266#95—82no—
267#95—82no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24#12112yes—
277C24#12112yes—
278C24C22⋊Q812yes—
279C24C22×D411yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
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—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—48no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
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—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
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—88no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4#12114yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4C22⋊Q814yes—
372C22×C4C22⋊Q814yes—
373C22×C4C22⋊Q812yes—
374C22×C4#12614yes—
375C22×C4#13714yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×Q811yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4H(5,𝔽2)12yes—
385C22×C42−1+412yes—
386M16—232no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8#12118yes—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—48no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—88no—
424C4⋊C4#12014yes—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—84no—
429C42—48no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23#39614yes—
457C23C2×#12111yescharacteristic
458C23C2×#12111yescharacteristic
459C23C2×C22⋊Q811yescharacteristic
460C23D4212yes—
461C23#54712yes—
462C23C2×#13712yes—
463C23#57512yes—
464C23#57611yescharacteristic
465C23#57711yescharacteristic
466C23#57711yescharacteristic
467C23#57811yescharacteristic
468Q8—42no—
469Q8—42no—
470Q8—44no—
471Q8—81no—
472Q8—81no—
473Q8—82no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—816no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#39614yes—
536C2×C4C2×#12011yescharacteristic
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×#12112yes—
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×C22⋊Q811yescharacteristic
541C2×C4#52511yescharacteristic
542C2×C4#52512yes—
543C2×C4C2×#12611yescharacteristic
544C2×C4#53411yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53712yes—
547C2×C4#53912yes—
548C2×C4#54612yes—
549C2×C4#54812yes—
550C2×C4D4×Q812yes—
551C2×C4#55212yes—
552C2×C4#55312yes—
553C2×C4#55412yes—
554C8—28no—
555C8—81no—
556C8—81no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22C2×#39611yescharacteristic
569C22#241011yescharacteristic
570C22#241311yescharacteristic
571C22#241612yes—
572C22#244912yes—
573C22#303911yescharacteristic
574C22#304011yescharacteristic
575C22#304511yescharacteristic
576C22#304611yescharacteristic
577C22#305612yes—
578C22#306211yescharacteristic
579C22#306311yescharacteristic
580C22#307812yes—
581C22#308311yescharacteristic
582C22#308411yescharacteristic
583C22#326712yes—
584C22#327012yes—
585C22#327312yes—
586C22#327812yes—
587C22#334412yes—
588C22#334912yes—
589C22#335112yes—
590C22#335712yes—
591C4—24no—
592C4—44no—
593C4—44no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—162no—
600C4#240911yescharacteristic
601C4#241712yes—
602C4#248112yes—
603C4#257911yescharacteristic
604C4#261811yescharacteristic
605C4#284711yescharacteristic
606C2—44no—
607C2#1472911yescharacteristic
608C2#2361611yescharacteristic
609C2#2361911yescharacteristic
610C2#2437311yescharacteristic
611C2#2437411yescharacteristic
612C2#2459311yescharacteristic
613C2#2459811yescharacteristic
614C2#2479112yes—
615C2#2479512yes—
616C2#2635812yes—
617C2#2636012yes—
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
2. C2×#2068C2semidirect
8. #11231C2semidirect
9. #11219C2semidirect
10. #11122C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#20682
C2×#18942
C2×#18732
C2×#18512
#141392
#139512
#112312
#112192
#111222
#107302