Finite Group Databasebeta

← #192700

Subgroups — #192700

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34243C211yesmaximal, characteristic
2#21867C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic, complemented
4#20800C211yesmaximal, characteristic, complemented
5C2×#1483C211yesmaximal, characteristic
6#8889C212yesmaximal, complemented
7#8887C212yesmaximal, complemented
8#8884C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#7904C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#428—22no—
15C22×#107—21no—
16C22×C4⋊C8C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2370—21no—
19#2367—21no—
20#2359—22no—
21#2355C2211yescharacteristic
22#2354C2211yescharacteristic
23#2352—21no—
24Q8×C4⋊C4C2211yescharacteristic
25D4×C4⋊C4—21no—
26D4×C4⋊C4C2211yescharacteristic
27#2334—22no—
28#2314—21no—
29C4×C22⋊Q8—21no—
30C4×#121—21no—
31#2285C2211yescharacteristic
32C2×#384C2211yescharacteristic
33#2274—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1992—21no—
36#1959—21no—
37#1947—21no—
38#1947—21no—
39#1947C2212yes—
40#1947C2212yes—
41#1874—21no—
42#1872—21no—
43#1847—22no—
44#1845C2212yes—
45#1844C2212yes—
46#1803C2212yes—
47#1802C2212yes—
48C2×#340—21no—
49#1483C418yescomplemented
50C2×Q8⋊C8C2212yes—
51C2×D4⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×M16—21no—
57C23×C8—21no—
58#529—22no—
59#529—22no—
60#529C2×C414yes—
61#529Q814yes—
62C2×#126—21no—
63C2×#125D411yescharacteristic
64C2×#124D411yescharacteristic
65C2×#123—21no—
66C2×C22⋊Q8D411yescharacteristic
67C2×#121D411yescharacteristic
68C2×C4×Q8—21no—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C2×#117—21no—
76C2×#117D412yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#432—22no—
86#432—22no—
87#428—48no—
88C2×#107—24no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—44no—
92C2×#106—24no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8C2×C414yescomplemented
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×#103—22no—
99C2×#102—22no—
100C2×C22⋊C8—21no—
101C4×M16—22no—
102C4×M16C2×C414yescomplemented
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8C2312yes—
106#391—21no—
107#390—21no—
108#389—22no—
109#389—22no—
110#389—21no—
111#389—21no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#386—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#384—21no—
125#384—21no—
126#384—21no—
127#384—21no—
128#384C2312yes—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#382—21no—
132#382D412yes—
133#380—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4D411yescharacteristic
148C4×C4⋊C4D411yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C2×#95—21no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159Q8⋊C8C2×C418yescomplemented
160D4⋊C8C2×C418yescomplemented
161C22×Q8C2×D411yescharacteristic
162C22×D4—21no—
163C22×D4C2×D411yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—41no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C2×M16—24no—
173C2×M16—24no—
174C2×M16—24no—
175C22×C8—22no—
176C22×C8—24no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—42no—
180C22×C8—42no—
181#126—24no—
182#126—24no—
183#125—22no—
184#125C4⋊C414yes—
185#124—22no—
186#124C4⋊C414yes—
187#123—24no—
188#123—24no—
189C22⋊Q8—22no—
190C22⋊Q8C4⋊C414yes—
191#121—22no—
192#121C4⋊C414yes—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—82no—
196C4×Q8C22×C412yes—
197C4×Q8C2×Q812yes—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—82no—
203C4×D4—84no—
204C4×D4C22×C412yes—
205C4×D4C2×Q812yes—
206#117—24no—
207#117—24no—
208#117—42no—
209#117—42no—
210#117—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
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—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—82no—
267C2×C42C22×C412yes—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×Q812yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274C2×C42C2411yescharacteristic
275#107—44no—
276#107—48no—
277#107—44no—
278#106—48no—
279C4⋊C8—24no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8C22×C414yes—
284#103—82no—
285#102—82no—
286C22⋊C8—42no—
287#97—24no—
288#97—42no—
289C4×C8—22no—
290C4×C8C22×C414yes—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—21no—
298#95—21no—
299#95—22no—
300#95—22no—
301#95—21no—
302#95—21no—
303C24—21no—
304C24#12011yescharacteristic
305C2×Q8—21no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8C2×C4⋊C412yes—
309C2×D4—24no—
310C2×D4—21no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4C2×C4⋊C412yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4C2×C4⋊C412yes—
353C22×C4C4×D412yes—
354C22×C4C4×D412yes—
355C22×C4#12011yescharacteristic
356C22×C4#12011yescharacteristic
357C22×C4#12011yescharacteristic
358C22×C4C22⋊Q812yes—
359C22×C4C22⋊Q812yes—
360C22×C4#12312yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364M16—216no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—48no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4C2×C4⋊C412yes—
410C4⋊C4C2×C4⋊C412yes—
411C4⋊C4AGL(1,ℤ/8)12yes—
412C4⋊C4#13712yes—
413C22⋊C4—24no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—82no—
422C22⋊C4—82no—
423C42—22no—
424C42—22no—
425C42—44no—
426C42—42no—
427C42—42no—
428C42—81no—
429C42—81no—
430C42—82no—
431C42—82no—
432C42—82no—
433C42C2×C4⋊C412yes—
434C42C4×D412yes—
435C42C4×D412yes—
436C42C22⋊Q812yes—
437C42C22⋊Q812yes—
438C42#12312yes—
439C42C23×C411yescharacteristic
440C42C22×Q811yescharacteristic
441C42C2×#4111yescharacteristic
442C42C2×#4111yescharacteristic
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23#38512yes—
455C23#45312yes—
456C23#45612yes—
457C23C2×#12011yescharacteristic
458Q8—24no—
459Q8—42no—
460D4—24no—
461D4—48no—
462D4—42no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—164no—
516C2×C4—164no—
517C2×C4#38512yes—
518C2×C4#38512yes—
519C2×C4#38512yes—
520C2×C4C22×C4⋊C411yescharacteristic
521C2×C4C2×C4×D411yescharacteristic
522C2×C4C2×C4×D411yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×#12311yescharacteristic
526C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
527C2×C4C2×#13711yescharacteristic
528C2×C4#57511yescharacteristic
529C2×C4#57511yescharacteristic
530C2×C4#57612yes—
531C2×C4#57812yes—
532C8—28no—
533C8—44no—
534C8—82no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22#185914yes—
549C22C2×#38511yescharacteristic
550C22#300611yescharacteristic
551C22#300711yescharacteristic
552C22#300811yescharacteristic
553C22#301111yescharacteristic
554C22C2×#45311yescharacteristic
555C22C2×#45611yescharacteristic
556C4—22no—
557C4—21no—
558C4—21no—
559C4—22no—
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—81no—
567C4—82no—
568C4—82no—
569C4—81no—
570C4—162no—
571C4#291312yes—
572C4C4×AGL(1,ℤ/8)11yescharacteristic
573C4C4×#13711yescharacteristic
574C4#307911yescharacteristic
575C4#308011yescharacteristic
576C4#309812yes—
577C2—24no—
578C2—82no—
579C2#1856211yescharacteristic
580C2#1856511yescharacteristic
581C2#1858711yescharacteristic
582C2#1858811yescharacteristic
583C2#2084711yescharacteristic
584C2C2×#185911yescharacteristic
585C2#2087511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
3. C2×#1947C2semidirect
4. #20800C2semidirect
6. #8889C2semidirect
7. #8887C2semidirect
8. #8884C2semidirect
9. #8882C2semidirect
10. #8044C2semidirect
11. #7904C2semidirect
49. #1483C4semidirect
95. C2×C4⋊C8C2×C4semidirect
102. C4×M16C2×C4semidirect
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
#342432
#218672
C2×#19472
#208002
C2×#14832
#88892
#88872
#88842
#88822
#80442
#79042