Finite Group Databasebeta

← #396490

Subgroups — #396490

Order 512.

2260 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#391C211yesmaximal, characteristic
2C2×#2097C212yesmaximal, complemented
3C2×#1853C211yesmaximal, characteristic
4C2×#1852C211yesmaximal, characteristic
5C2×#1787C211yesmaximal, characteristic
6C2×#1787C211yesmaximal, characteristic
7#14682C211yesmaximal, characteristic
8#11124C212yesmaximal, complemented
9#11122C212yesmaximal, complemented
10#11052C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#391—22no—
24C2×#391—28no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C22×#95—22no—
32#2097—28no—
33#1853—24no—
34#1852—24no—
35#1787—24no—
36#1787—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434—22no—
44C22⋊#95—22no—
45C23×Q8—24no—
46C23×Q8—24no—
47C23×Q8—21no—
48C23×Q8—22no—
49C23×Q8—22no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—42no—
54C23×Q8—42no—
55C23×Q8Q814yes—
56C23×Q8C2312yes—
57C24×C4—21no—
58C24×C4—21no—
59C24×C4D412yes—
60C24×C4C2311yescharacteristic
61C23×C8C2312yes—
62C23×C8C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42—24no—
78C22×C42—42no—
79C22×C42D412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—42no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8D414yes—
105C2×C22⋊C8D414yes—
106#391—216no—
107#391—24no—
108#391—24no—
109#391—24no—
110#391D418yes—
111#382—24no—
112#382—42no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117C2×#95—216no—
118C2×#95—24no—
119C2×#95—24no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—21no—
126#340—24no—
127#340—24no—
128#336—24no—
129#336—42no—
130C25C2×D411yescharacteristic
131C22×Q8—24no—
132C22×Q8—216no—
133C22×Q8—24no—
134C22×Q8—216no—
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—28no—
140C22×Q8—28no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—48no—
150C22×Q8—48no—
151C22×Q8—48no—
152C22×Q8—48no—
153C22×Q8—48no—
154C22×Q8—48no—
155C22×Q8—48no—
156C22×Q8—48no—
157C22×Q8—48no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—48no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—42no—
165C22×Q8C2×D414yes—
166C22×Q8C2×Q814yes—
167C22×Q8#4114yes—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×Q812yes—
203C23×C4#4112yes—
204C23×C4#4112yes—
205C23×C4#4112yes—
206C23×C4C2411yescharacteristic
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C2×D414yes—
220C22×C8C2×D414yes—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C2×D414yes—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—48no—
262C2×C42—48no—
263C2×C42—42no—
264C2×C42C2×D414yes—
265#107—48no—
266#106—48no—
267#103—48no—
268#103—48no—
269#103—48no—
270#103—48no—
271C22⋊C8—28no—
272C22⋊C8—28no—
273#95—216no—
274#95—216no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—44no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24C22⋊Q812yes—
288C24#12312yes—
289C24#12712yes—
290C24#13514yes—
291C24#13714yes—
292C24C22×D411yescharacteristic
293C2×Q8—216no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—432no—
310C2×Q8—432no—
311C2×Q8—48no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—432no—
317C2×Q8—432no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—48no—
331C2×Q8C22⋊Q818yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—416no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—84no—
409C22×C4—88no—
410C22×C4—84no—
411C22×C4#12014yes—
412C22×C4C22⋊Q814yes—
413C22×C4C22⋊Q814yes—
414C22×C4C22⋊Q812yes—
415C22×C4#12314yes—
416C22×C4#12314yes—
417C22×C4#12312yes—
418C22×C4#12712yes—
419C22×C4#13714yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×Q811yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4112yes—
427C22×C4C2×#4111yescharacteristic
428C22×C4H(5,𝔽2)12yes—
429C22×C4H(5,𝔽2)12yes—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—48no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C42—24no—
460C42—24no—
461C42—48no—
462C42—42no—
463C42—42no—
464C23—24no—
465C23—24no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23#39914yes—
483C23#47918yes—
484C23#48518yes—
485C23#49514yes—
486C23#49714yes—
487C23C2×#12011yescharacteristic
488C23C2×C22⋊Q811yescharacteristic
489C23C2×#12311yescharacteristic
490C23#52511yescharacteristic
491C23C2×#12711yescharacteristic
492C23#53411yescharacteristic
493C23#53511yescharacteristic
494C23C2×#13512yes—
495C23C2×#13712yes—
496C23C2×#13712yes—
497C23#57512yes—
498Q8—28no—
499Q8—28no—
500Q8—432no—
501Q8—44no—
502Q8—44no—
503Q8—44no—
504Q8—44no—
505Q8—44no—
506Q8—44no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—432no—
525C2×C4—432no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—816no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—84no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4#39914yes—
567C2×C4C2×C22⋊Q812yes—
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×#12311yescharacteristic
571C2×C4C2×#12311yescharacteristic
572C2×C4#52512yes—
573C2×C4#53812yes—
574C2×C4#54312yes—
575C2×C4D4212yes—
576C2×C4D4212yes—
577C2×C4#54612yes—
578C2×C4#54612yes—
579C2×C4#54612yes—
580C2×C4#54612yes—
581C2×C4#55112yes—
582C2×C4#55112yes—
583C8—44no—
584C8—44no—
585C22—28no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—28no—
593C22—28no—
594C22—28no—
595C22#209718yescomplemented
596C22C2×#39911yescharacteristic
597C22#240511yescharacteristic
598C22#243311yescharacteristic
599C22#248211yescharacteristic
600C22#253911yescharacteristic
601C22#262611yescharacteristic
602C22#284811yescharacteristic
603C22#300212yes—
604C22#300412yes—
605C22C2×#47912yes—
606C22#307812yes—
607C22C2×#48512yes—
608C22#309512yes—
609C22#310812yes—
610C22#311712yes—
611C22C2×#49511yescharacteristic
612C22C2×#49711yescharacteristic
613C22#315211yescharacteristic
614C22#315311yescharacteristic
615C22#319012yes—
616C22#319612yes—
617C22#321112yes—
618C22#321812yes—
619C4—24no—
620C4—48no—
621C4—44no—
622C4—44no—
623C4—84no—
624C4—161no—
625C4—161no—
626C4—161no—
627C4—161no—
628C4#241812yes—
629C4#241812yes—
630C4#243712yes—
631C4#243712yes—
632C2—28no—
633C2#1473411yescharacteristic
634C2C2×#209711yescharacteristic
635C2#2373211yescharacteristic
636C2#2401012yes—
637C2#2572411yescharacteristic
638C2#2572611yescharacteristic
639C2#2758912yes—
640C2#2830611yescharacteristic
641C2#2831511yescharacteristic
642C2#2995712yes—
643C2#3296012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2097C2semidirect
8. #11124C2semidirect
9. #11122C2semidirect
10. #11052C2semidirect
595. C22#2097semidirect

Maximal subgroups

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

subgroupindex
C22×#3912
C2×#20972
C2×#18532
C2×#18522
C2×#17872
#146822
#111242
#111222
#110522
#107122