Finite Group Databasebeta

← #6481755

Subgroups — #6481755

Order 512.

1940 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic, complemented
2C2×#3103C211yesmaximal, characteristic
3#35180C211yesmaximal, characteristic
4C2×#2355C211yesmaximal, characteristic, complemented
5#33110C214yesmaximal, complemented
6#29719C214yesmaximal, complemented
7#24683C214yesmaximal, complemented
8#24679C214yesmaximal, complemented
9#24633C214yesmaximal, complemented
10#24629C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21860C211yesmaximal, characteristic, complemented
13#21859C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17#3104—28no—
18#3103—28no—
19C2×#484C2212yes—
20C2×#481C2212yes—
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×#462C2212yes—
24C2×#460C2212yes—
25C2×#451C2212yes—
26C2×#448C2212yes—
27C2×C4×Q16C2214yes—
28C2×C4×SD16C2214yes—
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2850C2214yes—
32#2639—22no—
33#2592—22no—
34#2568C2212yes—
35#2563C2212yes—
36#2562—22no—
37#2548C2212yes—
38#2547C2212yes—
39#2355—22no—
40#2355—22no—
41#2355C2214yes—
42#2355C2214yes—
43#2311—21no—
44#2311—21no—
45C2×#391C2212yes—
46C2×#387—22no—
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2074C2218yes—
53#2056C2218yes—
54#2055C2218yes—
55#2015C2218yes—
56#1990C2212yes—
57#1990C2212yes—
58#1983C2218yes—
59#1957C2212yes—
60#1956C2212yes—
61#1949C2218yes—
62#1939C2214yes—
63#1939C2214yes—
64#1927C2218yes—
65#1925C2218yes—
66#1874C2214yes—
67#1872C2214yes—
68#1803C2214yes—
69#1802C2214yes—
70#1797C2214yes—
71C22×#41—21no—
72C23×Q8C2311yescharacteristic
73C22×Q16C2314yes—
74C22×SD16C2314yes—
75#533—24no—
76#533—24no—
77#532—24no—
78#532—24no—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#127C2311yescharacteristic
83C2×#125—21no—
84C2×#124—21no—
85C2×#124C2314yes—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8C2312yes—
88C2×#121—21no—
89C2×#121C2312yes—
90C2×C4×Q8—21no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—24no—
94C2×C4×Q8D418yes—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×D4—21no—
99C2×C4×D4C2312yes—
100C2×#117—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C42—21no—
109C22×C42—22no—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112#484—28no—
113#481—28no—
114#477—28no—
115#475—28no—
116#462—28no—
117#460—28no—
118#451—28no—
119#448—28no—
120C4×Q16—216no—
121C4×SD16—216no—
122#431—24no—
123#431—24no—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2318yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2318yes—
133C2×#102C2314yes—
134C2×#102C2318yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#400—24no—
140#397—24no—
141#394—24no—
142#393—24no—
143#391—22no—
144#391—22no—
145#391—24no—
146#391C2318yes—
147#391C2314yes—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#388—24no—
152#388—24no—
153#387—28no—
154#387—28no—
155#386—24no—
156#386—24no—
157#384—21no—
158#384—21no—
159#384C2312yes—
160#384C2312yes—
161#382—24no—
162#382—24no—
163#382C2314yes—
164#380—22no—
165#380—22no—
166#380—22no—
167#380—22no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4C2318yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C2×#95—22no—
177C2×#95—22no—
178C2×#95—22no—
179C2×#95D412yes—
180#340C2318yes—
181#340C2318yes—
182#336C2318yes—
183C2×#41—44no—
184C2×#41—44no—
185C2×#41—44no—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×Q8—22no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8C2×D414yes—
193C22×Q8C2412yes—
194C22×Q8C2412yes—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4C2412yes—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2411yescharacteristic
208C2×Q16—28no—
209C2×Q16—44no—
210C2×Q16—44no—
211C2×SD16—28no—
212C2×SD16—44no—
213C2×SD16—44no—
214C22×C8C2414yes—
215C22×C8C2414yes—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222#127—22no—
223#127—22no—
224#125—44no—
225#124—48no—
226#124—44no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—44no—
230#121—24no—
231#121—24no—
232#121—44no—
233C4×Q8—28no—
234C4×Q8—28no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—24no—
239C4×Q8—44no—
240C4×Q8C2×D418yes—
241C4×Q8C2×D418yes—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—44no—
245#117—44no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—82no—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C42—28no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—28no—
288C2×C42—44no—
289C2×C42—84no—
290C2×C42C2×D414yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D414yes—
294C2×C42#4114yes—
295C2×C42C2414yes—
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298#107—28no—
299#106—28no—
300C4⋊C8—24no—
301C4⋊C8—24no—
302C4⋊C8—48no—
303#103—28no—
304#103—28no—
305#103—28no—
306#103—48no—
307#102—28no—
308#102—48no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C4×C8—24no—
312C4×C8—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95C2×D414yes—
321#95C2×D414yes—
322C24—22no—
323C24C22×D411yescharacteristic
324#41—816no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—48no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—48no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8C22×D414yes—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—81no—
358C2×D4—81no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4#12114yes—
388C22×C4#12114yes—
389C22×C4#13514yes—
390C22×C4#13714yes—
391C22×C4C22×D414yes—
392C22×C4C22×D414yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2511yescharacteristic
401Q16—416no—
402SD16—416no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—28no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4C22×D414yes—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C42—28no—
442C42—28no—
443C42—28no—
444C42—44no—
445C42—44no—
446C42—88no—
447C42#12114yes—
448C42#12114yes—
449C42C22×D412yes—
450C42C22×D412yes—
451C42C2×#4112yes—
452C42C2×#4112yes—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×#12112yes—
465C23C2×#13512yes—
466C23C2×#13712yes—
467C23C23×D411yescharacteristic
468Q8—24no—
469Q8—24no—
470Q8—48no—
471Q8—42no—
472Q8—42no—
473Q8—44no—
474Q8—44no—
475Q8—84no—
476Q8—84no—
477D4—44no—
478D4—44no—
479D4—84no—
480D4—88no—
481D4—84no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
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—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—88no—
520C2×C4—88no—
521C2×C4#464116yes—
522C2×C4C2×#12112yes—
523C2×C4C2×#12112yes—
524C2×C4C2×#12112yes—
525C2×C4C2×#12112yes—
526C2×C4C2×#12112yes—
527C2×C4C2×#13512yes—
528C2×C4C2×#13512yes—
529C2×C4C2×#13712yes—
530C2×C4C2×#13712yes—
531C2×C4C23×D411yescharacteristic
532C2×C4C22×#4111yescharacteristic
533C2×C4#58512yes—
534C2×C4#58512yes—
535C8—44no—
536C8—44no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22C2×#46414yes—
551C22C22×#12111yescharacteristic
552C22#348511yescharacteristic
553C22#348711yescharacteristic
554C22C22×#13511yescharacteristic
555C22C22×#13711yescharacteristic
556C22#358511yescharacteristic
557C22#358611yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—44no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—82no—
566C4—82no—
567C4—84no—
568C4—82no—
569C4C2×#46414yes—
570C4C2×#46414yes—
571C4#344912yes—
572C4#349312yes—
573C2—24no—
574C2—162no—
575C2#3515011yescharacteristic
576C2C22×#46411yescharacteristic
577C2#6042511yescharacteristic
578C2#6170111yescharacteristic
579C2#6171911yescharacteristic
580C2#6188511yescharacteristic
581C2#6189311yescharacteristic
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
1. C2×#3104C2semidirect
4. C2×#2355C2semidirect
5. #33110C2semidirect
6. #29719C2semidirect
7. #24683C2semidirect
8. #24679C2semidirect
9. #24633C2semidirect
10. #24629C2semidirect
11. #21951C2semidirect
12. #21860C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C2×#31032
#351802
C2×#23552
#331102
#297192
#246832
#246792
#246332
#246292
#219512
#218602
#218592