Finite Group Databasebeta

← #6519606

Subgroups — #6519606

Order 512.

2078 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic, complemented
2C2×#2989C211yesmaximal, characteristic
3#35279C211yesmaximal, characteristic
4#34952C211yesmaximal, characteristic
5#28108C214yesmaximal, complemented
6#25394C214yesmaximal, complemented
7#25302C214yesmaximal, complemented
8#24760C214yesmaximal, complemented
9#24660C214yesmaximal, complemented
10#24273C214yesmaximal, complemented
11#21959C211yesmaximal, characteristic, complemented
12#21804C211yesmaximal, characteristic, complemented
13#20734C211yesmaximal, characteristic, complemented
14C23×Q16C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#470C2218yes—
19C2×#462—28no—
20C2×#462C2214yes—
21C2×#462C2214yes—
22#2989—28no—
23C2×#446C2212yes—
24C2×#436C2212yes—
25C2×#435C2214yes—
26C2×#432C2212yes—
27C2×#431C2211yescharacteristic
28C2×#423C2212yes—
29C22×C4⋊C8C2211yescharacteristic
30C22×#103C2212yes—
31#2857C2214yes—
32#2628—22no—
33#2582C2212yes—
34#2581—21no—
35#2580—21no—
36#2579—22no—
37#2568C2212yes—
38#2567—21no—
39#2563—21no—
40#2542C2212yes—
41#2538C2212yes—
42#2487—22no—
43#2487C2212yes—
44#2409C2212yes—
45#2409C2214yes—
46#2396—22no—
47#2317—21no—
48#2311—21no—
49C4×C22⋊Q8C2214yes—
50#2297—21no—
51#2296—21no—
52#2285C2211yescharacteristic
53#2285C2211yescharacteristic
54#2066C2218yes—
55#2056C2218yes—
56#2035C2218yes—
57#2001C2218yes—
58#1981C2212yes—
59#1981C2212yes—
60#1971C2218yes—
61#1956C2214yes—
62#1948C2214yes—
63#1948C2214yes—
64#1935C2218yes—
65#1894C2218yes—
66#1824C2212yes—
67#1807C2214yes—
68#1803C2212yes—
69#1803C2212yes—
70#1787C2214yes—
71#1787C2212yes—
72C4×#103C2214yes—
73C22×#41D411yescharacteristic
74C23×Q8C2311yescharacteristic
75C23×Q8C2312yes—
76C22×Q16—28no—
77C22×Q16—48no—
78C22×Q16C2318yes—
79C22×M16C2314yes—
80C23×C8C2312yes—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C22⋊Q8C2314yes—
88C2×C22⋊Q8C2318yes—
89C4×#41—24no—
90C4×#41—24no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8D411yescharacteristic
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4D412yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2311yescharacteristic
103C2×#117—21no—
104C2×#117—21no—
105C2×#117D411yescharacteristic
106C2×#117D412yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C42—21no—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#470—416no—
115#462—216no—
116#462—216no—
117#446—28no—
118#436—28no—
119#435—216no—
120#432—28no—
121#431—24no—
122C2×#106C2318yes—
123#423—28no—
124C2×C4⋊C8D418yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×#103—28no—
131C2×#103C2314yes—
132C2×#103C2318yes—
133C2×#103C2318yes—
134C2×#103C2314yes—
135C2×#103C2318yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140#400—24no—
141#400—22no—
142#399—22no—
143#399—24no—
144#398—24no—
145#396—24no—
146#395—24no—
147#395—22no—
148#393—24no—
149#393—22no—
150#391—22no—
151#391—22no—
152#391C2314yes—
153#391C2318yes—
154#390—22no—
155#390—22no—
156#389—22no—
157#389—22no—
158#389—24no—
159#389—24no—
160#389C2314yes—
161#387—24no—
162#386—24no—
163#386—22no—
164#386—22no—
165#385—24no—
166#385—24no—
167#385—22no—
168#385—22no—
169#384—22no—
170#384C2312yes—
171#384C2312yes—
172#382—22no—
173#382—21no—
174#382—21no—
175#380—24no—
176#380—24no—
177#380—21no—
178#380—21no—
179#380C2312yes—
180#380C2312yes—
181#380C2314yes—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—24no—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—24no—
188#340C2318yes—
189#340C2314yes—
190#340C2314yes—
191C2×#41—24no—
192C2×#41—24no—
193C2×#41—24no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—42no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8C2×D411yescharacteristic
205C22×Q8C2414yes—
206C22×Q8C2412yes—
207C22×D4C2×D412yes—
208C22×D4C2×D411yescharacteristic
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—21no—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C2×Q16—216no—
219C2×Q16—216no—
220C2×Q16—48no—
221C2×Q16—432no—
222C2×M16—28no—
223C2×M16—48no—
224C22×C8—44no—
225C22×C8C2×D418yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228C22×C8C2414yes—
229C4⋊Q8—24no—
230C4⋊Q8—24no—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—44no—
234C4⋊Q8—44no—
235C22⋊Q8—28no—
236C22⋊Q8—44no—
237C22⋊Q8—48no—
238C22⋊Q8—48no—
239C4×Q8—24no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—44no—
247C4×Q8—44no—
248C4×D4—28no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—22no—
253C4×D4—22no—
254#117—24no—
255#117—28no—
256#117—44no—
257#117—44no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D411yescharacteristic
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2414yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—44no—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D414yes—
304C2×C42#4114yes—
305C2×C42C2412yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#106—48no—
309C4⋊C8—28no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8C2×D418yes—
313C4⋊C8C2×D418yes—
314#103—28no—
315#103—216no—
316#103—48no—
317#103—44no—
318#103—48no—
319C22⋊C8—24no—
320C22⋊C8—28no—
321#97—28no—
322C4×C8—24no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—28no—
332#95—28no—
333C24C22×D412yes—
334C24C22×D411yescharacteristic
335#41—416no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—28no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—28no—
347C2×Q8—22no—
348C2×Q8—28no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—84no—
357C2×Q8—88no—
358C2×Q8H(5,𝔽2)14yes—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—82no—
387C22×C4—84no—
388C22×C4#12114yes—
389C22×C4#12114yes—
390C22×C4#13714yes—
391C22×C4#13714yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4C2511yescharacteristic
403Q16—432no—
404M16—88no—
405C2×C8—28no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—48no—
411C2×C8C22×D418yes—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—82no—
434C4⋊C4—84no—
435C4⋊C4—88no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—48no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—84no—
443C22⋊C4—88no—
444C42—24no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—44no—
449C42—44no—
450C42—44no—
451C42—44no—
452C42#12114yes—
453C42#12114yes—
454C42C22×D412yes—
455C42C22×D412yes—
456C42C2×#4112yes—
457C42C2×#4112yes—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23C2×#12112yes—
466C23C2×#13712yes—
467C23C2×#13712yes—
468C23C23×D411yescharacteristic
469Q8—24no—
470Q8—24no—
471Q8—48no—
472Q8—42no—
473Q8—42no—
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—48no—
478Q8—88no—
479D4—48no—
480D4—44no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—88no—
517C2×C4#470116yescomplemented
518C2×C4C2×#12112yes—
519C2×C4C2×#12112yes—
520C2×C4C2×#12112yes—
521C2×C4C2×#12112yes—
522C2×C4C2×#12112yes—
523C2×C4C2×#13712yes—
524C2×C4C2×#13712yes—
525C2×C4C2×#13712yes—
526C2×C4C2×#13712yes—
527C2×C4C23×D411yescharacteristic
528C2×C4C22×#4111yescharacteristic
529C2×C4C2×H(5,𝔽2)12yes—
530C2×C4#58512yes—
531C8—28no—
532C8—42no—
533C8—42no—
534C22—24no—
535C22—24no—
536C22—22no—
537C22—22no—
538C22—44no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22C2×#47014yes—
544C22C22×#12111yescharacteristic
545C22#348812yes—
546C22C22×#13711yescharacteristic
547C22C22×#13711yescharacteristic
548C22#358612yes—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—44no—
556C4—84no—
557C4—84no—
558C4—82no—
559C4—84no—
560C4C2×#47014yes—
561C4C2×#47014yes—
562C4#348411yescharacteristic
563C4#348711yescharacteristic
564C4#349412yes—
565C2—24no—
566C2—82no—
567C2#3556311yescharacteristic
568C2C22×#47011yescharacteristic
569C2#6045511yescharacteristic
570C2#6196412yes—
571C2#6196612yes—
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×#462C2semidirect
5. #28108C2semidirect
6. #25394C2semidirect
7. #25302C2semidirect
8. #24760C2semidirect
9. #24660C2semidirect
10. #24273C2semidirect
11. #21959C2semidirect
12. #21804C2semidirect
13. #20734C2semidirect
517. C2×C4#470semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C2×#29892
#352792
#349522
#281082
#253942
#253022
#247602
#246602
#242732
#219592
#218042
#207342