Finite Group Databasebeta

← #6526685

Subgroups — #6526685

Order 512.

2120 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2C2×#3103C211yesmaximal, characteristic
3C2×#2567C211yesmaximal, characteristic, complemented
4#34206C211yesmaximal, characteristic
5#28115C214yesmaximal, complemented
6#27645C214yesmaximal, complemented
7#25453C214yesmaximal, complemented
8#25322C214yesmaximal, complemented
9#24815C214yesmaximal, complemented
10#24680C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21834C211yesmaximal, characteristic, complemented
13#20734C211yesmaximal, characteristic, complemented
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×#520—21no—
17C22×C4×Q8C2211yescharacteristic
18#3227—28no—
19C2×#498C2212yes—
20C2×#486C2212yes—
21#3103—28no—
22C2×#481C2212yes—
23C2×#481C2214yes—
24C2×#475C2212yes—
25C2×#476C2214yes—
26C2×#460C2212yes—
27C2×#448C2212yes—
28C2×C4×SD16C2214yes—
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2567—22no—
32#2567—22no—
33#2567C2214yes—
34#2567C2214yes—
35C2×#395—22no—
36#2367—22no—
37#2366C2212yes—
38#2365C2212yes—
39#2354C2212yes—
40#2352—22no—
41D4×C4⋊C4C2212yes—
42#2325—22no—
43#2311—21no—
44C4×#121C2214yes—
45C2×#391C2212yes—
46C2×#389—21no—
47C2×#389—21no—
48#2285C2211yescharacteristic
49C2×#384C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2084C2218yes—
52#2071C2218yes—
53#2066C2218yes—
54#2055C2218yes—
55#1990C2212yes—
56#1990C2212yes—
57#1987C2214yes—
58#1948C2218yes—
59#1939C2214yes—
60#1939C2214yes—
61#1927C2218yes—
62#1877C2218yes—
63#1824C2214yes—
64#1824C2212yes—
65#1803C2212yes—
66#1803C2212yes—
67#1802C2218yes—
68#1797C2214yes—
69#1787C2212yes—
70C4×#103C2214yes—
71C22×#41—21no—
72C23×Q8—21no—
73C23×Q8C2311yescharacteristic
74C22×SD16C2314yes—
75#532—24no—
76#532D418yes—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2311yescharacteristic
84C2×#127C2311yescharacteristic
85C2×#124—21no—
86C2×#121—21no—
87C2×#121C2312yes—
88#520—48no—
89C2×C4×Q8—22no—
90C2×C4×Q8—21no—
91C2×C4×Q8—21no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×C4×D4—24no—
97C2×C4×D4—22no—
98C2×C4×D4—21no—
99C2×C4×D4C2312yes—
100C2×C4×D4C2314yes—
101C2×#117—21no—
102C2×#117—22no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C42—21no—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#498—28no—
116#486—28no—
117#481—28no—
118#481—216no—
119#476—216no—
120#475—28no—
121#460—28no—
122#448—28no—
123C4×SD16—216no—
124#431—24no—
125#431—24no—
126C2×#106C2314yes—
127C2×#106C2318yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2318yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2318yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#395—28no—
141#395—28no—
142#391—22no—
143#391—24no—
144#391C2318yes—
145#390—22no—
146#390C2314yes—
147#389—24no—
148#389—24no—
149#389—24no—
150#389—24no—
151#389C2314yes—
152#388—24no—
153#388—24no—
154#387—24no—
155#386—24no—
156#386—24no—
157#385—24no—
158#384—24no—
159#384C2318yes—
160#384C2311yescharacteristic
161#384C2311yescharacteristic
162#382—21no—
163#380—21no—
164#380—22no—
165#380—22no—
166#380C2312yes—
167C4×C4⋊C4—21no—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C22⋊C4—24no—
175C2×#95—22no—
176C2×#95—22no—
177C2×#95D412yes—
178#340C2318yes—
179#340C2314yes—
180#340C2318yes—
181#340C2314yes—
182#336C2318yes—
183C2×#41—28no—
184C2×#41—44no—
185C2×#41—44no—
186C22×Q8—24no—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—42no—
194C22×Q8—44no—
195C22×Q8—82no—
196C22×Q8—82no—
197C22×Q8C2412yes—
198C22×D4—22no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4C2412yes—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—21no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D412yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2411yescharacteristic
212C2×SD16—28no—
213C2×SD16—28no—
214C2×SD16—44no—
215C22×C8C2414yes—
216C22×C8C2414yes—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219C4⋊Q8—22no—
220C4⋊Q8—24no—
221C4⋊Q8—24no—
222C4⋊Q8—44no—
223C4⋊Q8—44no—
224C4⋊Q8C2×D414yes—
225#127—22no—
226#127C2×D414yes—
227#124—28no—
228#121—28no—
229#121—24no—
230#121C2×D418yes—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—48no—
236C4×Q8—44no—
237C4×Q8—44no—
238C4×D4—28no—
239C4×D4—24no—
240C4×D4—48no—
241C4×D4—416no—
242C4×D4—48no—
243C4×D4C2×D418yes—
244#117—44no—
245#117—48no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—84no—
269C2×C4⋊C4—84no—
270C2×C4⋊C4SD1618yescomplemented
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C2×D414yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D414yes—
296C2×C42#4114yes—
297C2×C42C2414yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300#106—28no—
301#106—216no—
302C4⋊C8—24no—
303C4⋊C8—24no—
304C4⋊C8—48no—
305#103—28no—
306#103—28no—
307#103—44no—
308#102—28no—
309#102—216no—
310C22⋊C8—24no—
311C22⋊C8—24no—
312C4×C8—24no—
313C4×C8—24no—
314#95—28no—
315#95—28no—
316#95—22no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95C2×D414yes—
321#95C2×D414yes—
322C24—22no—
323C24C22×D411yescharacteristic
324#41—432no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—82no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8—82no—
344C2×Q8—816no—
345C2×Q82−1+414yes—
346C2×D4—28no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—81no—
357C2×D4—81no—
358C2×D4C22×D414yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4#12114yes—
390C22×C4#12114yes—
391C22×C4C2×SD1618yes—
392C22×C4C2×SD1614yes—
393C22×C4#13714yes—
394C22×C4C22×D414yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C22×D414yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4112yes—
403C22×C4C2511yescharacteristic
404SD16—416no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—48no—
429C4⋊C4—42no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—88no—
433C4⋊C4—88no—
434C4⋊C4C2×SD1618yes—
435C4⋊C4C2×SD1618yes—
436C4⋊C4C22×D414yes—
437C22⋊C4—28no—
438C22⋊C4—28no—
439C22⋊C4—44no—
440C22⋊C4—48no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C42—28no—
444C42—44no—
445C42—44no—
446C42—44no—
447C42—48no—
448C42—44no—
449C42#12114yes—
450C42#12114yes—
451C42C22×D412yes—
452C42C22×D412yes—
453C42C2×#4112yes—
454C42C2×#4112yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23C2×#12112yes—
467C23C22×SD1612yes—
468C23C2×#13712yes—
469C23C23×D411yescharacteristic
470Q8—24no—
471Q8—24no—
472Q8—48no—
473Q8—42no—
474Q8—42no—
475Q8—48no—
476Q8—88no—
477Q8—88no—
478D4—28no—
479D4—416no—
480D4—44no—
481D4—48no—
482D4—84no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—416no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—168no—
522C2×C4#461116yes—
523C2×C4C2×#12112yes—
524C2×C4C2×#12112yes—
525C2×C4C2×#12112yes—
526C2×C4C2×#12112yes—
527C2×C4C2×#12112yes—
528C2×C4C22×SD1618yes—
529C2×C4C22×SD1612yes—
530C2×C4C22×SD1612yes—
531C2×C4C2×#13712yes—
532C2×C4C2×#13712yes—
533C2×C4C23×D411yescharacteristic
534C2×C4C22×#4111yescharacteristic
535C2×C4C2×2−1+412yes—
536C2×C4#58512yes—
537C8—44no—
538C8—44no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22C2×#46114yes—
553C22C22×#12111yescharacteristic
554C22#348511yescharacteristic
555C22#348611yescharacteristic
556C22C23×SD1611yescharacteristic
557C22C22×#13711yescharacteristic
558C22#358511yescharacteristic
559C22#358511yescharacteristic
560C4—28no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—44no—
565C4—44no—
566C4—44no—
567C4—82no—
568C4—82no—
569C4—84no—
570C4—82no—
571C4C2×#46114yes—
572C4C2×#46114yes—
573C4#348912yes—
574C4#349512yes—
575C2—24no—
576C2—84no—
577C2#3563311yescharacteristic
578C2C22×#46111yescharacteristic
579C2#6042611yescharacteristic
580C2#6169711yescharacteristic
581C2#6171911yescharacteristic
582C2#6182011yescharacteristic
583C2#6183011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2567C2semidirect
5. #28115C2semidirect
6. #27645C2semidirect
7. #25453C2semidirect
8. #25322C2semidirect
9. #24815C2semidirect
10. #24680C2semidirect
11. #21951C2semidirect
12. #21834C2semidirect
13. #20734C2semidirect
270. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
C2×#31032
C2×#25672
#342062
#281152
#276452
#254532
#253222
#248152
#246802
#219512
#218342
#207342