Finite Group Databasebeta

← #192374

Subgroups — #192374

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2325C211yesmaximal, characteristic
2#21942C211yesmaximal, characteristic, complemented
3#21764C211yesmaximal, characteristic
4C2×#1507C211yesmaximal, characteristic
5#13965C211yesmaximal, characteristic, complemented
6#8873C214yesmaximal, complemented
7#8866C214yesmaximal, complemented
8#8019C211yesmaximal, characteristic, complemented
9#7909C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C22×C4×D4—21no—
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14#2325—22no—
15#2325—22no—
16#2325—22no—
17#2325C2214yes—
18C2×#388—21no—
19C2×#387—21no—
20C2×#386—21no—
21C2×#386—21no—
22C2×#385—21no—
23C2×#385—21no—
24C2×#385—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C22×#95C2211yescharacteristic
29#1895—22no—
30#1874C2212yes—
31#1874C2212yes—
32#1853—21no—
33#1853C2212yescomplemented
34#1852—21no—
35#1852C2212yescomplemented
36#1824—21no—
37#1809—22no—
38#1786—21no—
39#1507C418yescomplemented
40C2×#327C2214yes—
41#1444C2212yescomplemented
42#1444C2212yescomplemented
43C4⋊#95—22no—
44#1294C2214yescomplemented
45#1280C2214yescomplemented
46C23×D4—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—21no—
53C22×M16—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121C2×C414yes—
59C2×#121D412yes—
60C2×#121Q814yes—
61C2×#121C2312yes—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—48no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C42—21no—
81C22×C42—21no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#102—22no—
85C2×#102—22no—
86#407—22no—
87#407—22no—
88#407—22no—
89#407C2×C414yescomplemented
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2312yes—
94#388—24no—
95#388—24no—
96#387—24no—
97#387—24no—
98#386—24no—
99#386—24no—
100#386—24no—
101#386—24no—
102#385—24no—
103#385—24no—
104#385—44no—
105#385—44no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380D412yes—
113#380C2312yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4C2×C414yescomplemented
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D412yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#327C2×C4116yescomplemented
134C25—21no—
135C25C2×D411yescharacteristic
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4—81no—
145C22×D4—81no—
146C22×D4C4⋊C414yescomplemented
147C22×D4C2×D412yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—42no—
161C23×C4—44no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—81no—
165C23×C4—81no—
166C23×C4—81no—
167C23×C4—81no—
168C23×C4—82no—
169C23×C4—82no—
170C23×C4C22×C412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×Q812yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182C2×M16—24no—
183C2×M16—42no—
184C2×M16—81no—
185C2×M16—81no—
186C2×M16—81no—
187C2×M16—81no—
188C22×C8—22no—
189C22×C8—22no—
190#121—24no—
191#121—24no—
192#121—24no—
193#121—24no—
194#121—24no—
195#121—22no—
196#121—22no—
197#121C4⋊C418yescomplemented
198#121C22×C414yes—
199#121C2×Q814yes—
200C4×D4—416no—
201C4×D4—48no—
202C4×D4—48no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
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—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—81no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C4⋊C414yescomplemented
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4—82no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—44no—
250C2×C42—81no—
251C2×C42—81no—
252#107—82no—
253#106—82no—
254#102—82no—
255#102—82no—
256C22⋊C8—24no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—42no—
261C22⋊C8C22×C414yescomplemented
262#95—24no—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—44no—
272#95—44no—
273#95—41no—
274#95—41no—
275#95C22×C414yescomplemented
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—24no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24C4×D412yes—
291C24#12011yescharacteristic
292C24#12011yescharacteristic
293C24C22⋊Q812yes—
294C24C22×D411yescharacteristic
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—28no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—416no—
304C2×D4—416no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4C2×C4⋊C414yes—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—48no—
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—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—88no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—88no—
384C22×C4—816no—
385C22×C4C2×C4⋊C412yes—
386C22×C4C2×C4⋊C412yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4C4×D412yes—
390C22×C4#12011yescharacteristic
391C22×C4#12011yescharacteristic
392C22×C4C22⋊Q812yes—
393C22×C4C22⋊Q812yes—
394C22×C4C22⋊Q812yes—
395C22×C4#12312yes—
396C22×C4#12312yes—
397C22×C4AGL(1,ℤ/8)14yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×Q811yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404M16—84no—
405M16—84no—
406C2×C8—24no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—48no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4C2×C4⋊C414yes—
426C22⋊C4—216no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—28no—
430C22⋊C4—48no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C42—48no—
438C42—84no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—24no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—48no—
465C23—48no—
466C23—48no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23#38512yes—
478C23#38512yes—
479C23#45312yes—
480C23#45612yes—
481C23C2×C4×D411yescharacteristic
482C23C2×#12011yescharacteristic
483C23C2×C22⋊Q811yescharacteristic
484C23C2×AGL(1,ℤ/8)12yes—
485C23#57512yes—
486D4—44no—
487D4—416no—
488D4—48no—
489D4—48no—
490D4—82no—
491D4—84no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
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—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—816no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4#38512yes—
559C2×C4#38512yes—
560C2×C4C22×C4⋊C411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×C22⋊Q811yescharacteristic
563C2×C4C2×#12311yescharacteristic
564C8—44no—
565C8—82no—
566C22—21no—
567C22—21no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—21no—
578C22—22no—
579C22—21no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—48no—
584C22—48no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—816no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22#185914yes—
598C22C2×#38511yescharacteristic
599C22C4×AGL(1,ℤ/8)12yes—
600C22#300311yescharacteristic
601C22#300511yescharacteristic
602C22C2×#45311yescharacteristic
603C22C2×#45611yescharacteristic
604C22#302611yescharacteristic
605C22#302611yescharacteristic
606C22#307712yes—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—22no—
612C4—22no—
613C4—48no—
614C4—44no—
615C4—44no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—161no—
620C4—161no—
621C2—22no—
622C2—24no—
623C2—22no—
624C2—42no—
625C2—84no—
626C2#1831611yescharacteristic
627C2#1832111yescharacteristic
628C2#1832511yescharacteristic
629C2#1832811yescharacteristic
630C2C2×#185911yescharacteristic
631C2#2087311yescharacteristic
632C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21942C2semidirect
5. #13965C2semidirect
6. #8873C2semidirect
7. #8866C2semidirect
8. #8019C2semidirect
9. #7909C2semidirect
33. #1853C22semidirect
35. #1852C22semidirect
39. #1507C4semidirect
41. #1444C22semidirect
42. #1444C22semidirect
44. #1294C22semidirect
45. #1280C22semidirect
89. #407C2×C4semidirect
115. C4×C22⋊C4C2×C4semidirect
133. #327C2×C4semidirect
146. C22×D4C4⋊C4semidirect
197. #121C4⋊C4semidirect
223. C2×C4⋊C4C4⋊C4semidirect
261. C22⋊C8C22×C4semidirect
275. #95C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23252
#219422
#217642
C2×#15072
#139652
#88732
#88662
#80192
#79092