Finite Group Databasebeta

← #6489034

Subgroups — #6489034

Order 512.

1732 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34621C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#32967C212yesmaximal, complemented
4#32912C212yesmaximal
5#30139C214yesmaximal, complemented
6#29135C212yesmaximal, complemented
7#28392C212yesmaximal, complemented
8#28287C212yesmaximal, complemented
9#28252C212yesmaximal, complemented
10#27679C212yesmaximal, complemented
11#24518C214yesmaximal, complemented
12#24209C212yesmaximal, complemented
13#22152C212yesmaximal, complemented
14#22146C212yesmaximal, complemented
15#20812C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#519C2212yes—
18C2×#479C2214yes—
19C2×#460C2214yes—
20C2×#461C2212yes—
21C2×#461C2212yes—
22C2×#448C2214yes—
23C2×#433C2212yes—
24#2651—22no—
25#2648—22no—
26#2647C2211yescharacteristic
27#2644C2212yes—
28#2641C2211yescharacteristic
29#2629C2211yescharacteristic
30#2624C2212yes—
31#2619C2211yescharacteristic
32#2395—22no—
33#2392—22no—
34#2365—22no—
35#2365—22no—
36#2344—22no—
37#2344—22no—
38#2327C2212yes—
39#2327C2212yes—
40#2310C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×#127C2211yescharacteristic
44C4×#125C2211yescharacteristic
45C4×#121C2212yes—
46#2293C2212yes—
47#2285C2211yescharacteristic
48C4×#117C2211yescharacteristic
49#2101C2214yes—
50#2092C2214yes—
51#2068C2214yes—
52#2066C2214yes—
53#2055C2214yes—
54#2055C2214yes—
55#2043C2214yes—
56#2013C2214yes—
57#1981C2212yes—
58#1981C2212yes—
59#1973C2214yes—
60#1968C2214yes—
61#1948C2212yes—
62#1947C2212yes—
63#1935C2214yes—
64#1933C2214yes—
65#1880C2214yes—
66#1880C2214yes—
67#1875C2214yes—
68#1865C2214yes—
69#1863C2214yes—
70#1854C2214yes—
71#1842C2212yes—
72#1832C2212yes—
73#1805C2214yes—
74#1803C2214yes—
75#1802C2212yes—
76#1802C2212yes—
77#1798C2212yes—
78#1797C2212yes—
79#1787C2214yes—
80#1768C2212yes—
81#1767C2211yescharacteristic
82C4×C4⋊C8C2211yescharacteristic
83C23×Q8C2312yes—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92C2×#125C2311yescharacteristic
93C2×#121C2312yes—
94#519—48no—
95C2×C4×Q8—24no—
96C2×C4×Q8C2314yes—
97C2×C4×Q8C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4C2312yes—
101C2×#117—24no—
102C2×#117—22no—
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C42C2311yescharacteristic
109#479—48no—
110#461—28no—
111#461—28no—
112#460—48no—
113#448—48no—
114#433—44no—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2312yes—
131C2×C4×C8C2312yes—
132#400—22no—
133#399—22no—
134#398—22no—
135#398—22no—
136#397—22no—
137#397—22no—
138#397—22no—
139#397—22no—
140#396—22no—
141#396—22no—
142#394—22no—
143#394—22no—
144#394—22no—
145#394—22no—
146#394—22no—
147#392—22no—
148#391—24no—
149#391C2314yes—
150#391C2314yes—
151#390—22no—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—24no—
169#387—24no—
170#387—24no—
171#387—24no—
172#386—22no—
173#384C2311yescharacteristic
174#384C2312yes—
175#382C2311yescharacteristic
176#382C2312yes—
177#381—24no—
178#380—21no—
179#380—21no—
180#380C2312yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4D412yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4D412yes—
194C4×C22⋊C4D412yes—
195#376C2311yescharacteristic
196#376C2311yescharacteristic
197C43C2311yescharacteristic
198C43C2311yescharacteristic
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2312yes—
205#336C2312yes—
206C22×Q8—28no—
207C22×Q8—22no—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—44no—
211C22×Q8C2414yes—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C2412yes—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2411yescharacteristic
220C2×SD16—28no—
221C2×SD16—28no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C2×SD16—44no—
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—28no—
229C4⋊Q8—28no—
230C4⋊Q8—28no—
231C4⋊Q8—24no—
232C4⋊Q8#4118yes—
233#127—24no—
234#125—24no—
235#121—24no—
236#121—24no—
237C4×Q8—28no—
238C4×Q8—28no—
239C4×Q8—44no—
240C4×Q8—44no—
241C4×Q8—416no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—84no—
245C4×D4—84no—
246#117—22no—
247#117—22no—
248#117—24no—
249#117—416no—
250#117—44no—
251#117—84no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
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—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305#107—44no—
306#106—44no—
307C4⋊C8—24no—
308C4⋊C8—24no—
309C4⋊C8—28no—
310C4⋊C8—44no—
311#103—28no—
312#103—28no—
313#103—44no—
314#103—44no—
315#103—44no—
316#103—44no—
317#102—44no—
318#102—44no—
319C22⋊C8—28no—
320#97—42no—
321C4×C8—42no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D412yes—
338#95C2×D412yes—
339C24—22no—
340C24C22×D411yescharacteristic
341C2×Q8—28no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—416no—
348C2×Q8—416no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8C2×#4118yes—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—82no—
363C2×D4—82no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—28no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392C22×C4C2511yescharacteristic
393SD16—88no—
394SD16—88no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C4⋊C4—28no—
402C4⋊C4—28no—
403C4⋊C4—28no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C42—22no—
440C42—22no—
441C42—24no—
442C42—24no—
443C42—22no—
444C42—21no—
445C42—21no—
446C42—22no—
447C42—21no—
448C42—21no—
449C42—42no—
450C42—42no—
451C42—44no—
452C42—48no—
453C42—48no—
454C42—44no—
455C42C2×#4114yes—
456C42H(5,𝔽2)14yes—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23#53712yes—
469C23C23×D411yescharacteristic
470Q8—28no—
471Q8—28no—
472Q8—416no—
473Q8—44no—
474Q8—44no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—88no—
480D4—88no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—44no—
503C2×C4—416no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—88no—
520C2×C4#52512yes—
521C2×C4#52512yes—
522C2×C4#53514yes—
523C2×C4#53712yes—
524C2×C4#57714yes—
525C2×C4C22×#4112yes—
526C2×C4C2×H(5,𝔽2)11yescharacteristic
527C2×C4C2×H(5,𝔽2)12yes—
528C2×C4C2×2−1+411yescharacteristic
529C8—44no—
530C8—44no—
531C22—22no—
532C22—24no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22C2×#52511yescharacteristic
544C22C2×#53511yescharacteristic
545C22C2×#53711yescharacteristic
546C22C2×#57712yes—
547C22#358612yes—
548C4—22no—
549C4—22no—
550C4—24no—
551C4—41no—
552C4—41no—
553C4—48no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—82no—
558C4—82no—
559C4—84no—
560C4—82no—
561C4#312614yes—
562C4#351912yes—
563C4#352012yes—
564C2—42no—
565C2—162no—
566C2#3523611yescharacteristic
567C2C2×#312611yescharacteristic
568C2#6056611yescharacteristic
569C2#6083612yes—
570C2#6092312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34209C2semidirect
3. #32967C2semidirect
5. #30139C2semidirect
6. #29135C2semidirect
7. #28392C2semidirect
8. #28287C2semidirect
9. #28252C2semidirect
10. #27679C2semidirect
11. #24518C2semidirect
12. #24209C2semidirect
13. #22152C2semidirect
14. #22146C2semidirect
15. #20812C2semidirect

Maximal subgroups

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

subgroupindex
#346212
#342092
#329672
#329122
#301392
#291352
#283922
#282872
#282522
#276792
#245182
#242092
#221522
#221462
#208122