Finite Group Databasebeta

← #192636

Subgroups — #192636

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic, complemented
3#20800C211yesmaximal, characteristic, complemented
4#20737C211yesmaximal, characteristic
5C2×#1489C211yesmaximal, characteristic
6#8888C212yesmaximal, complemented
7#8886C212yesmaximal, complemented
8#8884C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8043C211yesmaximal, characteristic, complemented
11#7896C211yesmaximal, characteristic, complemented
12C2×#520—21no—
13C2×C4×#41C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#428—22no—
16C2×#423C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18Q8×C4⋊C4—21no—
19Q8×C4⋊C4C2212yes—
20D4×C4⋊C4—22no—
21D4×C4⋊C4—21no—
22D4×C4⋊C4—22no—
23D4×C4⋊C4C2212yes—
24#2317—22no—
25#2317—21no—
26#2285—21no—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C2×#384—21no—
30C2×#384C2211yescharacteristic
31#1948—21no—
32#1948C2212yes—
33#1947—21no—
34#1947C2212yes—
35#1847—22no—
36#1845C2212yes—
37#1844C2212yes—
38#1825—21no—
39#1809—22no—
40#1803C2211yescharacteristic
41#1803C2211yescharacteristic
42#1802C2211yescharacteristic
43#1802C2211yescharacteristic
44#1788—21no—
45C4×#103—21no—
46C4×#102—21no—
47#1489C418yescomplemented
48C2×Q8⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×#41D411yescharacteristic
55C22×M16—21no—
56C22×M16—21no—
57#520—44no—
58#520—44no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41C2×C414yes—
62C4×#41Q814yes—
63C2×C4×Q8—22no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—24no—
69C2×C4×D4—22no—
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—22no—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117D411yescharacteristic
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4D412yes—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#431—22no—
86#431—22no—
87#428—48no—
88C2×#107—24no—
89C2×#107—22no—
90C2×#106—24no—
91C2×#106—22no—
92#423—22no—
93#423C2×C414yescomplemented
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2312yes—
98C2×#103—21no—
99C2×#103—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×C22⋊C8—21no—
103C4×M16—22no—
104C4×M16C2×C414yescomplemented
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2312yes—
108#389—24no—
109#389—22no—
110#389—24no—
111#389—22no—
112#385—24no—
113#385—24no—
114#385—24no—
115#384—22no—
116#384—22no—
117#384—22no—
118#384—21no—
119#384—21no—
120#384D412yes—
121#384D411yescharacteristic
122#384D411yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2312yes—
126#382—22no—
127#382—21no—
128#382—21no—
129#380—22no—
130#380—21no—
131#380—22no—
132#380—21no—
133#380—21no—
134#380D412yes—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2312yes—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—22no—
146#340—22no—
147Q8⋊C8C2×C418yescomplemented
148D4⋊C8C2×C418yescomplemented
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41C4⋊C414yes—
155C22×Q8C2×D411yescharacteristic
156C22×D4—21no—
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—24no—
167C2×M16—24no—
168C2×M16—24no—
169C2×M16—24no—
170C2×M16—81no—
171C2×M16—81no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C4×Q8—22no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—44no—
180C4×Q8—44no—
181C4×Q8C4⋊C414yes—
182C4×Q8C22×C412yes—
183C4×Q8C2×Q812yes—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—22no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—416no—
192C4×D4—44no—
193C4×D4—44no—
194C4×D4C4⋊C414yes—
195C4×D4C22×C412yes—
196C4×D4C2×Q812yes—
197#117—22no—
198#117—24no—
199#117—24no—
200#117—48no—
201#117—42no—
202#117—42no—
203#117—44no—
204#117C4⋊C414yes—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—81no—
232C2×C4⋊C4—81no—
233C2×C4⋊C4—84no—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—41no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—41no—
253C2×C42C22×C412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×Q812yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42C2411yescharacteristic
261#107—48no—
262#107—82no—
263#106—48no—
264#106—82no—
265C4⋊C8—24no—
266C4⋊C8—22no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8C22×C414yes—
272#103—42no—
273#103—42no—
274#102—42no—
275#102—42no—
276C22⋊C8—42no—
277#97—24no—
278C4×C8—22no—
279C4×C8—42no—
280C4×C8C22×C414yes—
281#95—24no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—41no—
286#95—41no—
287C24—21no—
288C24#12011yescharacteristic
289#41—28no—
290#41—28no—
291#41—28no—
292#41—28no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8C2×C4⋊C412yes—
297C2×Q8AGL(1,ℤ/8)12yes—
298C2×Q8#13712yes—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4C2×C4⋊C412yes—
306C2×D4AGL(1,ℤ/8)12yes—
307C2×D4#13712yes—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—48no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—84no—
343C22×C4—84no—
344C22×C4C2×C4⋊C412yes—
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4C22⋊Q812yes—
351C22×C4C22⋊Q812yes—
352C22×C4#12312yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356M16—216no—
357M16—84no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4C2×C4⋊C412yes—
405C4⋊C4C2×C4⋊C412yes—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—24no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—82no—
413C42—22no—
414C42—22no—
415C42—24no—
416C42—24no—
417C42—48no—
418C42—42no—
419C42—42no—
420C42—82no—
421C42C2×C4⋊C412yes—
422C42C4×D412yes—
423C42C4×D412yes—
424C42C22⋊Q812yes—
425C42C22⋊Q812yes—
426C42#12312yes—
427C42C23×C411yescharacteristic
428C42C22×Q811yescharacteristic
429C42C2×#4111yescharacteristic
430C42C2×#4111yescharacteristic
431C23—21no—
432C23—22no—
433C23—21no—
434C23—22no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23#38512yes—
445C23C2×#12011yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—22no—
449Q8#42814yescomplemented
450D4—216no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4#42814yescomplemented
455C2×C4—216no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—164no—
502C2×C4—168no—
503C2×C4#38512yes—
504C2×C4#38512yes—
505C2×C4#38512yes—
506C2×C4#45312yes—
507C2×C4#45612yes—
508C2×C4C22×C4⋊C411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C22⋊Q811yescharacteristic
512C2×C4C2×C22⋊Q811yescharacteristic
513C2×C4C2×#12311yescharacteristic
514C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
515C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
516C2×C4C2×#13711yescharacteristic
517C2×C4C2×#13711yescharacteristic
518C2×C4#57611yescharacteristic
519C2×C4#57711yescharacteristic
520C2×C4#57711yescharacteristic
521C2×C4#57811yescharacteristic
522C8—28no—
523C8—44no—
524C8—81no—
525C8—81no—
526C22—22no—
527C22—24no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22C2×#38511yescharacteristic
541C22#300311yescharacteristic
542C22#300411yescharacteristic
543C22#301211yescharacteristic
544C22#301311yescharacteristic
545C22C2×#45311yescharacteristic
546C22C2×#45611yescharacteristic
547C4—22no—
548C4—24no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4—161no—
558C4—161no—
559C4#185914yes—
560C4C2×#42811yescharacteristic
561C4C2×#42811yescharacteristic
562C4#294611yescharacteristic
563C4#294711yescharacteristic
564C4#308311yescharacteristic
565C4#308411yescharacteristic
566C4#309211yescharacteristic
567C4#309211yescharacteristic
568C2—42no—
569C2—44no—
570C2#1856611yescharacteristic
571C2#1856711yescharacteristic
572C2#1859011yescharacteristic
573C2#1859111yescharacteristic
574C2#2084411yescharacteristic
575C2C2×#185911yescharacteristic
576C2#2087811yescharacteristic
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. #21964C2semidirect
3. #20800C2semidirect
6. #8888C2semidirect
7. #8886C2semidirect
8. #8884C2semidirect
9. #8882C2semidirect
10. #8043C2semidirect
11. #7896C2semidirect
47. #1489C4semidirect
93. #423C2×C4semidirect
104. C4×M16C2×C4semidirect
147. Q8⋊C8C2×C4semidirect
148. D4⋊C8C2×C4semidirect
449. Q8#428semidirect
454. D4#428semidirect

Maximal subgroups

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

subgroupindex
#341832
#219642
#208002
#207372
C2×#14892
#88882
#88862
#88842
#88822
#80432
#78962