Finite Group Databasebeta

← #192765

Subgroups — #192765

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic, complemented
3#21981C211yesmaximal, characteristic, complemented
4#20743C211yesmaximal, characteristic
5C2×#1488C211yesmaximal, characteristic
6#8892C212yesmaximal, complemented
7#8890C212yesmaximal, complemented
8#8888C212yesmaximal, complemented
9#8886C212yesmaximal, complemented
10#8040C211yesmaximal, characteristic, complemented
11#7900C211yesmaximal, characteristic, complemented
12C2×#520—21no—
13C2×C4×#41C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17Q8×C4⋊C4—21no—
18Q8×C4⋊C4C2212yes—
19D4×C4⋊C4—22no—
20D4×C4⋊C4—21no—
21D4×C4⋊C4—22no—
22D4×C4⋊C4C2212yes—
23#2317—22no—
24#2317—21no—
25#2285—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384—21no—
29C2×#384C2211yescharacteristic
30#1948—22no—
31#1948C2212yes—
32#1947C2212yes—
33#1854—22no—
34#1853—22no—
35#1847—21no—
36#1844C2212yes—
37#1836—21no—
38#1834C2212yes—
39#1825—21no—
40#1803C2211yescharacteristic
41#1803C2211yescharacteristic
42#1802C2211yescharacteristic
43#1802C2211yescharacteristic
44#1788—21no—
45#1763—21no—
46#1762—21no—
47#1488C418yescomplemented
48C2×Q8⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×#41D411yescharacteristic
55#520—44no—
56#520—44no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41C2×C414yes—
60C4×#41Q814yes—
61C2×C4×Q8—22no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—24no—
67C2×C4×D4—22no—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—22no—
71C2×#117—21no—
72C2×#117—21no—
73C2×#117—21no—
74C2×#117D411yescharacteristic
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4D412yes—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#432—22no—
84#432—22no—
85#432—22no—
86#432C2×C414yescomplemented
87#431—22no—
88#431C2×C414yescomplemented
89C2×#107—22no—
90C2×#106—24no—
91C2×#106—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2312yes—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8C2312yes—
106#389—24no—
107#389—22no—
108#389—24no—
109#389—22no—
110#385—24no—
111#385—24no—
112#385—24no—
113#384—22no—
114#384—22no—
115#384—22no—
116#384—21no—
117#384—21no—
118#384D412yes—
119#384D411yescharacteristic
120#384D411yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2312yes—
124#382—22no—
125#382—21no—
126#382—21no—
127#380—22no—
128#380—21no—
129#380—22no—
130#380—21no—
131#380—21no—
132#380D412yes—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2312yes—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—24no—
144#340—22no—
145#340—22no—
146Q8⋊C8C2×C418yescomplemented
147D4⋊C8C2×C418yescomplemented
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41C4⋊C414yes—
154C22×Q8C2×D411yescharacteristic
155C22×D4—21no—
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C4×Q8—22no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—44no—
173C4×Q8—44no—
174C4×Q8C4⋊C414yes—
175C4×Q8C22×C412yes—
176C4×Q8C2×Q812yes—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—22no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—416no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4C4⋊C414yes—
188C4×D4C22×C412yes—
189C4×D4C2×Q812yes—
190#117—22no—
191#117—24no—
192#117—24no—
193#117—48no—
194#117—42no—
195#117—42no—
196#117—82no—
197#117C4⋊C414yes—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—84no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—84no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D414yes—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—41no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—41no—
244C2×C42C22×C412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×Q812yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42C2411yescharacteristic
252#107—82no—
253#106—82no—
254#106—84no—
255C4⋊C8—24no—
256C4⋊C8—22no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8C22×C414yes—
262#103—42no—
263#103—42no—
264#102—42no—
265#102—42no—
266C22⋊C8—24no—
267C22⋊C8—24no—
268C22⋊C8—42no—
269#97—24no—
270#97—42no—
271C4×C8—22no—
272C4×C8C22×C414yes—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—41no—
278#95—41no—
279C24—21no—
280C24#12011yescharacteristic
281#41—28no—
282#41—28no—
283#41—28no—
284#41—28no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8C2×C4⋊C412yes—
289C2×Q8AGL(1,ℤ/8)12yes—
290C2×Q8#13712yes—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4C2×C4⋊C412yes—
298C2×D4AGL(1,ℤ/8)12yes—
299C2×D4#13712yes—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—48no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4C2×C4⋊C412yes—
333C22×C4C4×D412yes—
334C22×C4C4×D412yes—
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4#12011yescharacteristic
338C22×C4C22⋊Q812yes—
339C22×C4C22⋊Q812yes—
340C22×C4#12312yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—48no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4C2×C4⋊C412yes—
387C4⋊C4C2×C4⋊C412yes—
388C22⋊C4—24no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—24no—
392C22⋊C4—44no—
393C22⋊C4—48no—
394C22⋊C4—82no—
395C42—22no—
396C42—22no—
397C42—24no—
398C42—24no—
399C42—48no—
400C42—42no—
401C42—42no—
402C42—82no—
403C42C2×C4⋊C412yes—
404C42C4×D412yes—
405C42C4×D412yes—
406C42C22⋊Q812yes—
407C42C22⋊Q812yes—
408C42#12312yes—
409C42C23×C411yescharacteristic
410C42C22×Q811yescharacteristic
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C23—21no—
414C23—22no—
415C23—21no—
416C23—22no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23#38512yes—
427C23C2×#12011yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—22no—
431Q8#42814yes—
432D4—216no—
433D4—22no—
434D4—22no—
435D4—22no—
436D4#42814yes—
437C2×C4—216no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—48no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—164no—
482C2×C4—168no—
483C2×C4#38512yes—
484C2×C4#38512yes—
485C2×C4#38512yes—
486C2×C4C22×C4⋊C411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×C22⋊Q811yescharacteristic
490C2×C4C2×C22⋊Q811yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
493C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
494C2×C4C2×#13711yescharacteristic
495C2×C4C2×#13711yescharacteristic
496C2×C4#57712yes—
497C2×C4#57712yes—
498C8—44no—
499C8—44no—
500C8—81no—
501C8—81no—
502C22—22no—
503C22—24no—
504C22—22no—
505C22—22no—
506C22—41no—
507C22—42no—
508C22—41no—
509C22—42no—
510C22—42no—
511C22—41no—
512C22—42no—
513C22—41no—
514C22—42no—
515C22—42no—
516C22C2×#38511yescharacteristic
517C22#300311yescharacteristic
518C22#300411yescharacteristic
519C22#300911yescharacteristic
520C22#301011yescharacteristic
521C22#301911yescharacteristic
522C22#302011yescharacteristic
523C4—22no—
524C4—24no—
525C4—21no—
526C4—22no—
527C4—21no—
528C4—22no—
529C4—42no—
530C4—42no—
531C4—82no—
532C4—84no—
533C4—82no—
534C4—161no—
535C4—161no—
536C4C2×#42811yescharacteristic
537C4C2×#42811yescharacteristic
538C4#295212yes—
539C4#308412yes—
540C4#309211yescharacteristic
541C4#309211yescharacteristic
542C2—42no—
543C2—44no—
544C2#1881511yescharacteristic
545C2#1881611yescharacteristic
546C2#1882611yescharacteristic
547C2#1882911yescharacteristic
548C2#2084411yescharacteristic
549C2#2086411yescharacteristic
550C2#2088211yescharacteristic
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. #21988C2semidirect
3. #21981C2semidirect
6. #8892C2semidirect
7. #8890C2semidirect
8. #8888C2semidirect
9. #8886C2semidirect
10. #8040C2semidirect
11. #7900C2semidirect
47. #1488C4semidirect
86. #432C2×C4semidirect
88. #431C2×C4semidirect
146. Q8⋊C8C2×C4semidirect
147. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341832
#219882
#219812
#207432
C2×#14882
#88922
#88902
#88882
#88862
#80402
#79002