Finite Group Databasebeta

← #6526610

Subgroups — #6526610

Order 512.

1990 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C2×#2989C211yesmaximal, characteristic
3#35316C211yesmaximal, characteristic
4#34952C211yesmaximal, characteristic
5#28105C214yesmaximal, complemented
6#27663C214yesmaximal, complemented
7#25479C214yesmaximal, complemented
8#25348C214yesmaximal, complemented
9#24845C214yesmaximal, complemented
10#24710C214yesmaximal, complemented
11#21961C211yesmaximal, characteristic, complemented
12#21833C211yesmaximal, characteristic, complemented
13#20734C211yesmaximal, characteristic, complemented
14C2×#536—21no—
15C22×C4⋊Q8C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#475—28no—
19C2×#475C2214yes—
20C2×#475C2214yes—
21#2989—28no—
22C2×#446C2212yes—
23C2×#436C2212yes—
24C2×#435C2214yes—
25C2×#432C2212yes—
26C2×#431C2211yescharacteristic
27C22×#106C2212yes—
28C2×#423C2212yes—
29C22×C4⋊C8C2211yescharacteristic
30C22×#103C2212yes—
31#2857C2214yes—
32#2621—22no—
33#2582C2212yes—
34#2581—21no—
35#2580—21no—
36#2579—22no—
37#2547—21no—
38#2547C2212yes—
39#2546—21no—
40#2546C2212yes—
41#2538C2212yes—
42#2487C2212yes—
43#2486—22no—
44#2486C2214yes—
45#2409C2212yes—
46#2396—22no—
47#2310—21no—
48C4×C22⋊Q8C2214yes—
49#2297—21no—
50#2296—21no—
51#2285C2211yescharacteristic
52#2285C2211yescharacteristic
53#2088C2218yes—
54#2077C2218yes—
55#2068C2218yes—
56#2036C2218yes—
57#1990C2214yes—
58#1988C2212yes—
59#1988C2212yes—
60#1947C2214yes—
61#1947C2214yes—
62#1943C2218yes—
63#1935C2218yes—
64#1894C2218yes—
65#1824C2212yes—
66#1823C2214yes—
67#1809C2218yes—
68#1808C2214yes—
69#1803C2212yes—
70#1803C2212yes—
71#1787C2212yes—
72C4×#103C2214yes—
73C22×#41D411yescharacteristic
74C23×Q8—21no—
75C23×Q8C2311yescharacteristic
76C22×M16C2314yes—
77C23×C8C2312yes—
78#536—48no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#124—22no—
87C2×#123—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8C2314yes—
90C4×#41—24no—
91C4×#41—24no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8D411yescharacteristic
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2311yescharacteristic
100C2×C4×D4D412yes—
101C2×C4×D4D411yescharacteristic
102C2×C4×D4C2312yes—
103C2×C4×D4C2311yescharacteristic
104C2×#117—21no—
105C2×#117—21no—
106C2×#117D411yescharacteristic
107C2×#117D412yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C42—21no—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#475—216no—
116#475—216no—
117#446—28no—
118#436—28no—
119#435—216no—
120#432—28no—
121#431—24no—
122C2×#107C2318yes—
123C2×#106—28no—
124C2×#106C2318yes—
125#423—28no—
126C2×C4⋊C8D418yes—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#103—28no—
133C2×#103C2318yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2312yes—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140#400—22no—
141#399—22no—
142#398—24no—
143#398—24no—
144#398—24no—
145#397—24no—
146#397—24no—
147#396—24no—
148#395—22no—
149#393—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—24no—
154#391C2314yes—
155#390—22no—
156#389—22no—
157#389—22no—
158#389—24no—
159#389C2318yes—
160#389C2314yes—
161#387—24no—
162#386—22no—
163#386—22no—
164#385—24no—
165#385—22no—
166#385—22no—
167#384C2312yes—
168#384C2312yes—
169#382—21no—
170#382—21no—
171#381—24no—
172#380—21no—
173#380—24no—
174#380—21no—
175#380C2312yes—
176#380C2312yes—
177#380C2314yes—
178#380C2318yes—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—24no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C4×C22⋊C4—24no—
185#340C2314yes—
186#340C2314yes—
187#340C2318yes—
188#340C2318yes—
189#340C2318yes—
190C2×#41—24no—
191C2×#41—24no—
192C2×#41—24no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—42no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8C2×D411yescharacteristic
204C22×Q8C2412yes—
205C22×D4C2×D412yes—
206C22×D4C2×D411yescharacteristic
207C23×C4—22no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C2×M16—28no—
217C2×M16—48no—
218C22×C8—44no—
219C22×C8C2×D418yes—
220C22×C8C2412yes—
221C22×C8C2412yes—
222C22×C8C2414yes—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—24no—
226C4⋊Q8—24no—
227C4⋊Q8—48no—
228C4⋊Q8—44no—
229C4⋊Q8—44no—
230#124—48no—
231#123—416no—
232C22⋊Q8—28no—
233C22⋊Q8—44no—
234C22⋊Q8—416no—
235C4×Q8—24no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—44no—
243C4×Q8—44no—
244C4×D4—28no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—22no—
249C4×D4—22no—
250#117—24no—
251#117—28no—
252#117—44no—
253#117—44no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D411yescharacteristic
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2414yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—44no—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D414yes—
301C2×C42#4114yes—
302C2×C42C2412yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305#107—48no—
306#106—216no—
307C4⋊C8—28no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8C2×D418yes—
311C4⋊C8C2×D418yes—
312#103—28no—
313#103—216no—
314#103—44no—
315C22⋊C8—24no—
316C22⋊C8—28no—
317#97—28no—
318C4×C8—24no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—28no—
328#95—28no—
329C24C22×D412yes—
330C24C22×D411yescharacteristic
331#41—416no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—28no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—44no—
342C2×Q8—48no—
343C2×Q8—44no—
344C2×Q8—42no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—816no—
352C2×Q82−1+414yes—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—82no—
383C22×C4#12114yes—
384C22×C4#12114yes—
385C22×C4AGL(1,ℤ/8)14yes—
386C22×C4#13714yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2511yescharacteristic
398M16—88no—
399C2×C8—28no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—48no—
405C2×C8C22×D418yes—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—416no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—48no—
435C22⋊C4—416no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—84no—
439C42—24no—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—44no—
444C42—44no—
445C42—44no—
446C42—44no—
447C42#12114yes—
448C42#12114yes—
449C42C22×D412yes—
450C42C22×D412yes—
451C42C2×#4112yes—
452C42C2×#4112yes—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23C2×#12112yes—
461C23C2×AGL(1,ℤ/8)12yes—
462C23C2×#13712yes—
463C23C23×D411yescharacteristic
464Q8—24no—
465Q8—24no—
466Q8—48no—
467Q8—42no—
468Q8—42no—
469Q8—44no—
470Q8—88no—
471Q8—88no—
472D4—48no—
473D4—44no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—44no—
489C2×C4—48no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—88no—
511C2×C4#468116yes—
512C2×C4C2×#12112yes—
513C2×C4C2×#12112yes—
514C2×C4C2×#12112yes—
515C2×C4C2×#12112yes—
516C2×C4C2×#12112yes—
517C2×C4C2×AGL(1,ℤ/8)12yes—
518C2×C4C2×AGL(1,ℤ/8)12yes—
519C2×C4C2×#13712yes—
520C2×C4C2×#13712yes—
521C2×C4C23×D411yescharacteristic
522C2×C4C22×#4111yescharacteristic
523C2×C4C2×2−1+412yes—
524C2×C4#58512yes—
525C8—28no—
526C8—42no—
527C8—42no—
528C22—22no—
529C22—22no—
530C22—24no—
531C22—24no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22C2×#46814yes—
538C22C22×#12111yescharacteristic
539C22#348912yes—
540C22C22×AGL(1,ℤ/8)11yescharacteristic
541C22C22×#13711yescharacteristic
542C22#358612yes—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—44no—
550C4—84no—
551C4—82no—
552C4—84no—
553C4—84no—
554C4C2×#46814yes—
555C4C2×#46814yes—
556C4#348511yescharacteristic
557C4#348611yescharacteristic
558C4#349512yes—
559C2—24no—
560C2—82no—
561C2#3563311yescharacteristic
562C2C22×#46811yescharacteristic
563C2#6045511yescharacteristic
564C2#6211112yes—
565C2#6211312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#475C2semidirect
5. #28105C2semidirect
6. #27663C2semidirect
7. #25479C2semidirect
8. #25348C2semidirect
9. #24845C2semidirect
10. #24710C2semidirect
11. #21961C2semidirect
12. #21833C2semidirect
13. #20734C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C2×#29892
#353162
#349522
#281052
#276632
#254792
#253482
#248452
#247102
#219612
#218332
#207342