Finite Group Databasebeta

← #405401

Subgroups — #405401

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1850C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#14139C211yesmaximal, characteristic
6#13924C211yesmaximal, characteristic
7#11256C212yesmaximal, complemented
8#11232C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#11077C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#484—24no—
17C2×#484—24no—
18C2×#484—24no—
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×C4⋊C8C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C2×#391C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1892—24no—
30#1850—24no—
31C2×#340—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1445—22no—
39#1441—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×Q8C2311yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8D414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4D414yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42C2312yes—
88#484—232no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—42no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—42no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8D414yes—
110#391—24no—
111#391—42no—
112#380—22no—
113#380—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—42no—
131#336—24no—
132C25C2×D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8C2×D412yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C22×C8—28no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—48no—
189C22×C8—48no—
190C22×C8—44no—
191C22×C8—44no—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8#4114yes—
195C22⋊Q8—28no—
196C22⋊Q8—28no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4#4114yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4Q1614yes—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262#107—28no—
263#107—28no—
264C4⋊C8—48no—
265#103—28no—
266#103—28no—
267#103—48no—
268C22⋊C8—28no—
269#95—24no—
270#95—44no—
271#95—44no—
272#95—42no—
273#95—82no—
274#95—82no—
275#95—82no—
276#95—82no—
277#95—82no—
278#95—82no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24#12112yes—
284C24#12112yes—
285C24#12112yes—
286C24C2×Q1612yes—
287C24#13512yes—
288C24#13712yes—
289C24#13712yes—
290C24C22×D411yescharacteristic
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—48no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
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—42no—
365C22×C4—42no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—84no—
379C22×C4—88no—
380C22×C4—88no—
381C22×C4—84no—
382C22×C4#12014yes—
383C22×C4#12112yes—
384C22×C4#12112yes—
385C22×C4#12112yes—
386C22×C4#12314yes—
387C22×C4#12314yes—
388C22×C4#12414yes—
389C22×C4C2×Q1614yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C42−1+412yes—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—416no—
408C2×C8—42no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—48no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C22⋊C4—28no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—84no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23#39414yes—
466C23#46214yes—
467C23#46214yes—
468C23#46414yes—
469C23#46414yes—
470C23#46714yes—
471C23#47014yes—
472C23C2×#12111yescharacteristic
473C23C2×#12111yescharacteristic
474C23C2×#12111yescharacteristic
475C23#54612yes—
476C23#54712yes—
477C23C22×Q1611yescharacteristic
478C23C2×#13511yescharacteristic
479C23C2×#13711yescharacteristic
480C23C2×#13711yescharacteristic
481C23#57612yes—
482C23#57712yes—
483Q8—42no—
484Q8—42no—
485Q8—44no—
486Q8—81no—
487Q8—81no—
488Q8—82no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
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—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—816no—
527C2×C4—816no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4#39414yes—
550C2×C4C2×#12011yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4#52511yescharacteristic
555C2×C4C2×#12411yescharacteristic
556C2×C4#52912yes—
557C2×C4#52912yes—
558C2×C4#53511yescharacteristic
559C2×C4#53611yescharacteristic
560C2×C4#53912yes—
561C2×C4#53912yes—
562C2×C4#54612yes—
563C2×C4#54612yes—
564C2×C4#54712yes—
565C2×C4#54712yes—
566C2×C4#55112yes—
567C2×C4#55212yes—
568C8—42no—
569C8—44no—
570C8—42no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—24no—
578C22—28no—
579C22—24no—
580C22—24no—
581C22#205618yes—
582C22C2×#39411yescharacteristic
583C22#241111yescharacteristic
584C22#241311yescharacteristic
585C22#242512yes—
586C22#242512yes—
587C22C2×#46211yescharacteristic
588C22C2×#46211yescharacteristic
589C22C2×#46411yescharacteristic
590C22C2×#46411yescharacteristic
591C22#304111yescharacteristic
592C22#304211yescharacteristic
593C22#304512yes—
594C22C2×#46711yescharacteristic
595C22C2×#47011yescharacteristic
596C22#306212yes—
597C22#329712yes—
598C22#329912yes—
599C22#331412yes—
600C22#332012yes—
601C22#333712yes—
602C22#333812yes—
603C22#334412yes—
604C22#334612yes—
605C4—24no—
606C4—44no—
607C4—82no—
608C4—84no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—162no—
614C4#240411yescharacteristic
615C4#240611yescharacteristic
616C4#241912yes—
617C4#248812yes—
618C4#262311yescharacteristic
619C4#285311yescharacteristic
620C2—28no—
621C2#1491211yescharacteristic
622C2C2×#205611yescharacteristic
623C2#2354411yescharacteristic
624C2#2447911yescharacteristic
625C2#2448211yescharacteristic
626C2#2457911yescharacteristic
627C2#2458211yescharacteristic
628C2#2520612yes—
629C2#2521012yes—
630C2#2523412yes—
631C2#2524812yes—
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
7. #11256C2semidirect
8. #11232C2semidirect
9. #11190C2semidirect
10. #11077C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#18922
C2×#18502
C22×#3402
#141392
#139242
#112562
#112322
#111902
#110772
#107302