Finite Group Databasebeta

← #467005

Subgroups — #467005

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1863C211yesmaximal, characteristic, complemented
4#15181C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11199C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic
11#10789C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21#2538—24no—
22#2538—24no—
23C2×#396—22no—
24C2×#390—21no—
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31#2099—24no—
32#1863—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1442—21no—
42#1439—22no—
43#1438—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4C2312yes—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—42no—
56C2×C22⋊Q8—42no—
57C2×C4×Q8—22no—
58C2×C4×Q8—42no—
59C2×C4×Q8—42no—
60C2×C4×Q8D414yes—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—42no—
64C2×C4×D4—42no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C2×#107—22no—
84C2×#107—42no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99#396—216no—
100#390—28no—
101#389—28no—
102#389—22no—
103#389—22no—
104#384—22no—
105#384D414yes—
106#380—24no—
107#380—24no—
108#380—24no—
109#380—42no—
110#380—42no—
111#380—42no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—42no—
122#340—42no—
123#340—42no—
124#336—22no—
125#336—22no—
126C25C2×D412yes—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8C2×D412yes—
136C22×Q8#4114yes—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—42no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8#4112yes—
183C22⋊Q8—48no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186C22⋊Q8—48no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—48no—
190C4×D4—44no—
191C4×D4—44no—
192C4×D4—48no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
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—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4#4112yes—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
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—44no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4—82no—
247C2×C22⋊C4—82no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—44no—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254#107—44no—
255#103—28no—
256#103—44no—
257#103—48no—
258C22⋊C8—48no—
259C22⋊C8—48no—
260#95—28no—
261#95—44no—
262#95—44no—
263#95—42no—
264#95—42no—
265#95—81no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24#13714yes—
281C24C22×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8#12414yes—
300C2×D4—28no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—44no—
304C2×D4—44no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
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—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
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—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4#12012yes—
381C22×C4#12012yes—
382C22×C4#13512yes—
383C22×C4#13712yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C42−1+411yescharacteristic
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
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—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—416no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—48no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C22⋊C4—48no—
436C22⋊C4—416no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C42—22no—
441C42—22no—
442C42—24no—
443C42—48no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23C2×#12011yescharacteristic
480C23#53412yes—
481C23#54611yescharacteristic
482C23#54612yes—
483C23#54711yescharacteristic
484C23C2×#13511yescharacteristic
485C23C2×#13711yescharacteristic
486C23C2×#13712yes—
487C23#57612yes—
488C23#57711yescharacteristic
489C23#57811yescharacteristic
490Q8—24no—
491Q8—24no—
492Q8—42no—
493Q8—44no—
494Q8—44no—
495Q8—42no—
496D4—416no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4#45214yes—
556C2×C4#52511yescharacteristic
557C2×C4C2×#12411yescharacteristic
558C2×C4#53411yescharacteristic
559C2×C4#53411yescharacteristic
560C2×C4#53511yescharacteristic
561C2×C4#53512yes—
562C2×C4#53711yescharacteristic
563C2×C4#53812yes—
564C2×C4#53812yes—
565C2×C4#53912yes—
566C2×C4#54012yes—
567C2×C4#54612yes—
568C2×C4#54811yescharacteristic
569C2×C4#55011yescharacteristic
570C2×C4#55312yes—
571C2×C4#55312yes—
572C2×C4#56012yes—
573C2×C4#56212yes—
574C8—42no—
575C8—82no—
576C8—81no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22#240711yescharacteristic
593C22#243511yescharacteristic
594C22#261311yescharacteristic
595C22#268812yes—
596C22#270812yes—
597C22C2×#45211yescharacteristic
598C22#300411yescharacteristic
599C22#300811yescharacteristic
600C22#301311yescharacteristic
601C22#319012yes—
602C22#319512yes—
603C22#319812yes—
604C22#320312yes—
605C22#330411yescharacteristic
606C22#330512yes—
607C22#331212yes—
608C22#331412yes—
609C22#332012yes—
610C22#332811yescharacteristic
611C22#333312yes—
612C22#334211yescharacteristic
613C22#334612yes—
614C22#335611yescharacteristic
615C4—24no—
616C4—42no—
617C4—44no—
618C4—84no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—162no—
623C4—161no—
624C4#259111yescharacteristic
625C4#263511yescharacteristic
626C4#263911yescharacteristic
627C4#271112yes—
628C4#271412yes—
629C4#281311yescharacteristic
630C2—84no—
631C2#1655711yescharacteristic
632C2#2412311yescharacteristic
633C2#2413011yescharacteristic
634C2#2588311yescharacteristic
635C2#2589411yescharacteristic
636C2#2959911yescharacteristic
637C2#2960711yescharacteristic
638C2#3139512yes—
639C2#3139812yes—
640C2#3200912yes—
641C2#3201412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×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
2. C2×#2099C2semidirect
3. C2×#1863C2semidirect
5. #11357C2semidirect
6. #11199C2semidirect
7. #11096C2semidirect
8. #10970C2semidirect
9. #10970C2semidirect
11. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#20992
C2×#18632
#151812
#113572
#111992
#110962
#109702
#109642
#107892