Finite Group Databasebeta

← #192743

Subgroups — #192743

Order 512.

1118 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×#1477C211yesmaximal, characteristic
6#8897C212yesmaximal, complemented
7#8895C212yesmaximal, complemented
8#8889C212yesmaximal, complemented
9#8887C212yesmaximal, complemented
10#8045C211yesmaximal, characteristic, complemented
11#7901C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×#107—21no—
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×#417—21no—
18#2357—21no—
19#2356—21no—
20#2355C2211yescharacteristic
21#2355C2211yescharacteristic
22#2354C2211yescharacteristic
23#2354C2211yescharacteristic
24#2353—22no—
25#2352—21no—
26#2352—21no—
27#2298—22no—
28#2297—21no—
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#2283—21no—
32C4×#117—21no—
33C4×#117—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1948—22no—
36#1947—21no—
37#1947—21no—
38#1947C2212yes—
39#1947C2212yes—
40#1853—22no—
41#1844C2212yes—
42#1834C2212yes—
43#1803—21no—
44#1803C2212yes—
45#1802—21no—
46#1802C2212yes—
47#1736—21no—
48C2×#340—21no—
49#1477C418yescomplemented
50C2×Q8⋊C8C2212yes—
51C2×D4⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×#41D411yescharacteristic
57C22×M16—21no—
58C23×C8—21no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41C2×C414yes—
62C4×#41Q814yes—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C2×#117—21no—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82#431—22no—
83#431C2×C414yes—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—44no—
87C2×#106—24no—
88#423—22no—
89#423—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C414yes—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97#417—44no—
98C2×#103—22no—
99C2×#102—22no—
100C2×C22⋊C8—21no—
101C2×C4×C8C2312yes—
102#391—21no—
103#391—21no—
104#390—21no—
105#390—21no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#386—22no—
115#386—22no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384C2312yes—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#383—22no—
123#382—21no—
124#382—21no—
125#382—21no—
126#381—22no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380D412yes—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4D411yescharacteristic
142C4×C4⋊C4D412yes—
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150#376—22no—
151#376—22no—
152C2×#95—21no—
153C43—22no—
154#340—22no—
155#340—22no—
156#340—24no—
157#340—22no—
158#340—22no—
159Q8⋊C8C2×C418yescomplemented
160D4⋊C8C2×C418yescomplemented
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41C4⋊C414yes—
167C22×Q8C2×D411yescharacteristic
168C22×D4—21no—
169C22×D4C2×D411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—41no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—42no—
182C22×C8—24no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—42no—
187C22×C8—42no—
188C4×Q8—22no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8C4⋊C414yes—
192C4×Q8C22×C412yes—
193C4×Q8C2×Q812yes—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—22no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4C4⋊C414yes—
202C4×D4C22×C412yes—
203C4×D4C2×Q812yes—
204#117—22no—
205#117—24no—
206#117—24no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211#117—42no—
212#117—42no—
213#117C4⋊C414yes—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C22×C412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×Q812yes—
274C2×C42#4112yes—
275C2×C42#4112yes—
276C2×C42C2411yescharacteristic
277#107—44no—
278#107—44no—
279#106—84no—
280C4⋊C8—24no—
281C4⋊C8—24no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—42no—
286C4⋊C8C22×C414yes—
287#103—44no—
288#102—44no—
289C22⋊C8—24no—
290C4×C8—22no—
291C4×C8C22×C414yes—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—21no—
301#95—21no—
302#95—22no—
303#95—22no—
304#95—21no—
305#95—21no—
306C24—21no—
307C24#12011yescharacteristic
308#41—28no—
309#41—28no—
310#41—44no—
311#41—44no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8C2×C4⋊C412yes—
318C2×D4—24no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4C2×C4⋊C412yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4C2×C4⋊C412yes—
365C22×C4C4×D412yes—
366C22×C4C4×D412yes—
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4C22⋊Q812yes—
371C22×C4C22⋊Q812yes—
372C22×C4#12312yes—
373C22×C4#13512yes—
374C22×C4#13512yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4#13712yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380M16—84no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—48no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—24no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4C2×C4⋊C412yes—
423C4⋊C4C2×C4⋊C412yes—
424C4⋊C4AGL(1,ℤ/8)12yes—
425C4⋊C4#13712yes—
426C22⋊C4—24no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—24no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C42—22no—
436C42—22no—
437C42—24no—
438C42—24no—
439C42—44no—
440C42—44no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—42no—
446C42—42no—
447C42—44no—
448C42—82no—
449C42—82no—
450C42C2×C4⋊C412yes—
451C42C4×D412yes—
452C42C4×D412yes—
453C42C22⋊Q812yes—
454C42C22⋊Q812yes—
455C42#12312yes—
456C42C23×C411yescharacteristic
457C42C22×Q811yescharacteristic
458C42C2×#4111yescharacteristic
459C42C2×#4111yescharacteristic
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23#38512yes—
472C23#44912yes—
473C23#45212yes—
474C23C2×#12011yescharacteristic
475Q8—22no—
476Q8—22no—
477Q8—41no—
478Q8—41no—
479D4—22no—
480D4—22no—
481D4—48no—
482D4—41no—
483D4—41no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—164no—
535C2×C4#38512yes—
536C2×C4#38512yes—
537C2×C4#38512yes—
538C2×C4#42714yes—
539C2×C4#42814yes—
540C2×C4#44912yes—
541C2×C4#45212yes—
542C2×C4#48214yes—
543C2×C4#48514yes—
544C2×C4C22×C4⋊C411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C2×C4C2×C22⋊Q811yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C2×C4C2×#13511yescharacteristic
551C2×C4C2×#13511yescharacteristic
552C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
553C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
554C2×C4C2×#13711yescharacteristic
555C2×C4C2×#13711yescharacteristic
556C2×C4#57511yescharacteristic
557C2×C4#57511yescharacteristic
558C8—44no—
559C8—44no—
560C8—82no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22#185414yes—
575C22C2×#38511yescharacteristic
576C22C2×#44911yescharacteristic
577C22C2×#45211yescharacteristic
578C22#300611yescharacteristic
579C22#300711yescharacteristic
580C22#302111yescharacteristic
581C22#302211yescharacteristic
582C4—21no—
583C4—22no—
584C4—21no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—82no—
592C4—81no—
593C4—82no—
594C4—81no—
595C4—162no—
596C4#185414yes—
597C4C2×#42711yescharacteristic
598C4C2×#42811yescharacteristic
599C4C4×AGL(1,ℤ/8)11yescharacteristic
600C4C4×#13711yescharacteristic
601C4#307911yescharacteristic
602C4#308011yescharacteristic
603C4C2×#48211yescharacteristic
604C4C2×#48511yescharacteristic
605C2—24no—
606C2—82no—
607C2#1879111yescharacteristic
608C2#1879211yescharacteristic
609C2#1880311yescharacteristic
610C2#1880411yescharacteristic
611C2C2×#185411yescharacteristic
612C2#2086511yescharacteristic
613C2#2087511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21981C2semidirect
3. C2×#1947C2semidirect
6. #8897C2semidirect
7. #8895C2semidirect
8. #8889C2semidirect
9. #8887C2semidirect
10. #8045C2semidirect
11. #7901C2semidirect
49. #1477C4semidirect
159. Q8⋊C8C2×C4semidirect
160. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341212
#219812
C2×#19472
#206482
C2×#14772
#88972
#88952
#88892
#88872
#80452
#79012