Finite Group Databasebeta

← #204611

Subgroups — #204611

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3C2×#1513C214yesmaximal, complemented
4#14682C211yesmaximal, characteristic, complemented
5#9684C212yesmaximal, complemented
6#9683C212yesmaximal, complemented
7#9681C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525C2212yes—
10C23×C42C2211yescharacteristic
11C2×#407C2212yes—
12C2×C4×M16—21no—
13#2857C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2856C411yescharacteristic, complemented
16#2856C2211yescharacteristic
17#2854—21no—
18#2854—21no—
19#2850C412yescomplemented
20#2850C2212yes—
21#2847—21no—
22#2847—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121—21no—
27#2297C412yescomplemented
28#2285C411yescharacteristic, complemented
29C2×#382—21no—
30C2×#380C2212yes—
31#1749—21no—
32#1749C2212yes—
33C4×C22⋊C8—21no—
34C4×C22⋊C8C2212yes—
35#1728—21no—
36#1513—216no—
37C2×#328C2218yes—
38C2×#327C2218yes—
39C4⋊#95—21no—
40#1434C2212yes—
41#1434C2212yes—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C22×#41C2×C412yes—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49#532—44no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#124C2×C412yes—
53#525—28no—
54#525—28no—
55C2×#123—22no—
56C2×C22⋊Q8C2×C412yes—
57C2×C22⋊Q8C2312yes—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121C2×C412yes—
65C2×#121C2312yes—
66C2×#120—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×#117C2×C412yes—
72C22×C4⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—44no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#407—24no—
88#407D418yescomplemented
89C2×C22⋊C8—22no—
90C2×C22⋊C8C2314yes—
91C4×M16—44no—
92C2×#97—22no—
93C2×C4×C8—22no—
94#391C2×C412yes—
95#391C2×C412yes—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390C2×C412yes—
101#390C2×C412yes—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#382—44no—
119#382C2×C412yes—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—28no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95C2314yes—
136#336—22no—
137#336—22no—
138#328—216no—
139#327—216no—
140C25C2×D411yescharacteristic
141C2×#41—28no—
142C2×#41—44no—
143C2×#41—44no—
144C22×Q8C22×C412yes—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4C22×C412yes—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D414yes—
179C23×C4#4114yes—
180C23×C4C2411yescharacteristic
181C2×M16—28no—
182C2×M16—44no—
183C2×M16—44no—
184C22×C8—24no—
185C4⋊Q8—42no—
186#127—42no—
187#124—44no—
188#123—48no—
189C22⋊Q8—28no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192#121—28no—
193#121—24no—
194#121—24no—
195#121—44no—
196#121—44no—
197#121—44no—
198#121—44no—
199#121—44no—
200#121—44no—
201#120—44no—
202#120—44no—
203C4×D4—44no—
204C4×D4—48no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208#117—28no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C22⋊C8—28no—
266C22⋊C8—24no—
267C22⋊C8C2×D418yes—
268#97—44no—
269C4×C8—44no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—28no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C411yescharacteristic
290C24#12412yes—
291C24#12712yes—
292C24C22×D411yescharacteristic
293#41—816no—
294C2×Q8—24no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×D4—24no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—42no—
315C2×D4—42no—
316C22×C4—22no—
317C22×C4—22no—
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—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—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—41no—
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—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—416no—
370C22×C4—416no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C4×D414yes—
392C22×C4C4×D414yes—
393C22×C4#12114yes—
394C22×C4#12114yes—
395C22×C4#12412yes—
396C22×C4#12712yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C2×#4112yes—
401M16—416no—
402C2×C8—28no—
403C2×C8—44no—
404C2×C8—44no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—88no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—88no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—44no—
445C42#9914yes—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—24no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23#39012yes—
480C23#39012yes—
481C23C22×C22⋊C411yescharacteristic
482C23C2×#12411yescharacteristic
483C23C2×#12711yescharacteristic
484C23#57512yes—
485C23#57512yes—
486Q8—84no—
487D4—84no—
488D4—84no—
489D4—84no—
490D4—88no—
491D4—88no—
492D4—84no—
493D4—84no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—88no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4#39012yes—
558C2×C4#39012yes—
559C2×C4C2×#9912yes—
560C2×C4C2×#9912yes—
561C2×C4C2×#9912yes—
562C2×C4#41012yes—
563C2×C4#42112yes—
564C2×C4#42112yes—
565C2×C4C2×C4×D412yes—
566C2×C4C2×#12112yes—
567C8—48no—
568C22—21no—
569C22—21no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—48no—
583C22—48no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22C2×#39011yescharacteristic
600C22C22×#9911yescharacteristic
601C22C2×#41011yescharacteristic
602C22#289611yescharacteristic
603C22#289611yescharacteristic
604C22C2×#42111yescharacteristic
605C22C2×#42111yescharacteristic
606C22#314212yes—
607C22#315312yes—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—164no—
623C4#196414yes—
624C2—22no—
625C2—22no—
626C2—42no—
627C2—42no—
628C2—162no—
629C2#1755811yescharacteristic
630C2#1755911yescharacteristic
631C2#1758211yescharacteristic
632C2#1759411yescharacteristic
633C2#2120111yescharacteristic
634C2C2×#196411yescharacteristic
635C2#2125211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
3. C2×#1513C2semidirect
4. #14682C2semidirect
5. #9684C2semidirect
6. #9683C2semidirect
7. #9681C2semidirect
8. C2×#532C4semidirect
13. #2857C4semidirect
15. #2856C4semidirect
19. #2850C4semidirect
27. #2297C4semidirect
28. #2285C4semidirect
88. #407D4semidirect

Maximal subgroups

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

subgroupindex
#366242
C4×#4072
C2×#15132
#146822
#96842
#96832
#96812