Finite Group Databasebeta

← #387946

Subgroups — #387946

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3#13906C211yesmaximal, characteristic
4#11207C212yesmaximal, complemented
5#11199C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#10885C212yesmaximal, complemented
8#10730C212yesmaximal, complemented
9#10722C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15#2487—24no—
16#2487C418yescomplemented
17C2×#400—21no—
18C2×#399—21no—
19C2×#390—21no—
20C2×#389C2212yes—
21C2×#386—21no—
22C2×#385—21no—
23C2×#384C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1870—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1439—21no—
40#1439—21no—
41C4⋊#95—21no—
42C23×D4D411yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—24no—
48C22×M16—41no—
49C22×M16—41no—
50C22×M16—42no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C2×C4×D4C2×C414yes—
58C2×C4×D4D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4—41no—
66C22×C4⋊C4—41no—
67C22×C4⋊C4—41no—
68C22×C4⋊C4C2×C414yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C2×C22⋊C8—24no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—41no—
87C2×C22⋊C8—41no—
88#400—28no—
89#399—28no—
90#390—44no—
91#389—24no—
92#389C2×C418yes—
93#386—28no—
94#385—28no—
95#384—41no—
96#384—41no—
97#380—24no—
98#380—22no—
99#380—24no—
100#380—24no—
101#380—42no—
102#380—42no—
103#380C2×C418yes—
104#380C2×C414yes—
105C4×C4⋊C4—41no—
106C4×C4⋊C4—41no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113#340—24no—
114#340—24no—
115#340—42no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#336—24no—
122C25C2×D411yescharacteristic
123C25C2×D411yescharacteristic
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—41no—
129C22×D4—41no—
130C22×D4C22⋊C414yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2411yescharacteristic
165C2×M16—44no—
166C2×M16—48no—
167C2×M16—48no—
168C2×M16—44no—
169C2×M16—44no—
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×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8#4114yes—
184C4×D4—44no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—44no—
188C2×C4⋊C4—22no—
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—24no—
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—28no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4#4114yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4C22⋊C414yes—
247C2×C22⋊C4C22⋊C414yes—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42C22×C412yes—
256C2×C42C2×D412yes—
257C22⋊C8—48no—
258C22⋊C8—44no—
259#95—28no—
260#95—24no—
261#95—28no—
262#95—44no—
263#95—44no—
264#95—42no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24C2×C22⋊C412yes—
284C24C2×C22⋊C412yes—
285C24AGL(1,ℤ/8)12yes—
286C24#13712yes—
287C24C22×D411yescharacteristic
288C2×D4—28no—
289C2×D4—28no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—48no—
351C22×C4—42no—
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—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4#13712yes—
385C22×C4#13712yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C4H(5,𝔽2)12yes—
395C22×C42−1+412yes—
396M16—416no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—88no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—84no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—22no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23#41714yes—
489C23C22×C22⋊C411yescharacteristic
490C23#53411yescharacteristic
491C23#53411yescharacteristic
492C23#54612yes—
493C23#54712yes—
494C23C2×AGL(1,ℤ/8)11yescharacteristic
495C23C2×#13711yescharacteristic
496C23C2×#13711yescharacteristic
497C23C2×#13711yescharacteristic
498C23#57612yes—
499C23#57712yes—
500D4—416no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—816no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4#41914yes—
564C2×C4C4×#4112yes—
565C2×C4#51811yescharacteristic
566C2×C4#51811yescharacteristic
567C2×C4#52012yes—
568C2×C4#53411yescharacteristic
569C2×C4#53411yescharacteristic
570C2×C4#53812yes—
571C2×C4#53912yes—
572C2×C4#54012yes—
573C2×C4#54212yes—
574C2×C4#54612yes—
575C2×C4#54712yes—
576C2×C4#55111yescharacteristic
577C2×C4#55312yes—
578C2×C4#55911yescharacteristic
579C2×C4#56011yescharacteristic
580C2×C4#56212yes—
581C2×C4#56311yescharacteristic
582C8—44no—
583C8—82no—
584C22—24no—
585C22—28no—
586C22—42no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—82no—
601C22—81no—
602C22—81no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22#230811yescharacteristic
607C22#233912yes—
608C22#270812yes—
609C22#270911yescharacteristic
610C22#271711yescharacteristic
611C22C2×#41911yescharacteristic
612C22C2×#41711yescharacteristic
613C22#289712yes—
614C22#319311yescharacteristic
615C22#319311yescharacteristic
616C22#319511yescharacteristic
617C22#319511yescharacteristic
618C22#319812yes—
619C22#319911yescharacteristic
620C22#320011yescharacteristic
621C22#331812yes—
622C22#332012yes—
623C22#333112yes—
624C22#333312yes—
625C22#334512yes—
626C22#334612yes—
627C22#335612yes—
628C22#335712yes—
629C4—24no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—84no—
634C4—84no—
635C4—84no—
636C4—162no—
637C4#233512yes—
638C4#237411yescharacteristic
639C4#237611yescharacteristic
640C4#269711yescharacteristic
641C4#270112yes—
642C4#272411yescharacteristic
643C2—44no—
644C2—82no—
645C2#1450511yescharacteristic
646C2#2176511yescharacteristic
647C2#2177011yescharacteristic
648C2#2249312yes—
649C2#2250212yes—
650C2#3201012yes—
651C2#3201412yes—
652C2#3203611yescharacteristic
653C2#3203711yescharacteristic
654C2#3226811yescharacteristic
655C2#3226911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11207C2semidirect
5. #11199C2semidirect
6. #11199C2semidirect
7. #10885C2semidirect
8. #10730C2semidirect
9. #10722C2semidirect
16. #2487C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#18702
#139062
#112072
#111992
#108852
#107302
#107222