Finite Group Databasebeta

← #205251

Subgroups — #205251

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3182C211yesmaximal, characteristic
2#21389C211yesmaximal, characteristic, complemented
3#21257C211yesmaximal, characteristic
4#20480C211yesmaximal, characteristic, complemented
5C2×#1565C211yesmaximal, characteristic, complemented
6C2×#1553C211yesmaximal, characteristic
7C2×#1552C211yesmaximal, characteristic, complemented
8#9748C211yesmaximal, characteristic, complemented
9#9746C211yesmaximal, characteristic, complemented
10#9744C211yesmaximal, characteristic, complemented
11#9740C211yesmaximal, characteristic, complemented
12#9738C211yesmaximal, characteristic, complemented
13#9736C211yesmaximal, characteristic, complemented
14#9732C211yesmaximal, characteristic, complemented
15#9731C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3182—24no—
19#3182C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#468—21no—
25C2×#468—21no—
26C2×#440—21no—
27C2×#440—21no—
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×#416—22no—
32C2×C4×M16C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1988C2211yescharacteristic
35#1988C2211yescharacteristic
36#1983—22no—
37#1982C2211yescharacteristic
38#1982C2211yescharacteristic
39#1981C2211yescharacteristic
40#1981C2211yescharacteristic
41#1981C2211yescharacteristic
42#1981C2211yescharacteristic
43#1873—22no—
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C4×#103—22no—
49#1753—21no—
50#1753—21no—
51#1745—21no—
52C4×#97—21no—
53#1565—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1553—24no—
59#1552—24no—
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×SD16—21no—
70C22×SD16—21no—
71C22×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73C23×C8—21no—
74#533—22no—
75#533—22no—
76#533—22no—
77#533C2×C414yes—
78#532—22no—
79#532—22no—
80#532—22no—
81#532C2×C414yes—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#125—21no—
87C2×#124—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×#121—21no—
91C2×#121—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×#117—21no—
97C2×#117D412yes—
98C22×C42C2311yescharacteristic
99#500—22no—
100#500C2×C414yes—
101#492—22no—
102#492C2×C414yes—
103#487—22no—
104#487C2×C414yes—
105#486—22no—
106#486C2×C414yes—
107#468—28no—
108#468—28no—
109#440—28no—
110#440—28no—
111#432—22no—
112#432D414yes—
113#431—22no—
114#431—22no—
115C2×#107—21no—
116C2×#107—21no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126#416—48no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#103—22no—
130C2×#103—22no—
131C2×#103—22no—
132C2×#103—22no—
133C2×#102—21no—
134C2×#102—21no—
135C2×#102—22no—
136C2×#102—22no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C4×M16C2×C414yes—
140C4×M16D414yes—
141C2×#97—21no—
142C2×#97—21no—
143C2×#97—21no—
144C2×#97D411yescharacteristic
145C2×#97D411yescharacteristic
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C4×C4⋊C4D412yes—
152C4×C4⋊C4C2311yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156#376—21no—
157C43—21no—
158#340—22no—
159#340—22no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#335—22no—
165#335—22no—
166#334—22no—
167#334—22no—
168#332—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#332—22no—
174#331—22no—
175#331—22no—
176C2×#41—24no—
177C2×#41—44no—
178C2×#41—44no—
179C2×#41—42no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—41no—
187C23×C4—41no—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4#4111yescharacteristic
191C2×SD16—24no—
192C2×SD16—24no—
193C2×SD16—42no—
194C2×SD16—42no—
195C2×SD16—42no—
196C2×SD16—42no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—41no—
200C2×M16—41no—
201C2×M16C22⋊C414yes—
202C2×M16C22⋊C414yes—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—44no—
210C22×C8—42no—
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8C22×C412yes—
225C4⋊Q8C22×C412yes—
226C4⋊Q8C22×C412yes—
227#127—21no—
228#127—21no—
229#127—21no—
230#127C22×C412yes—
231#125—24no—
232#125—42no—
233#124—24no—
234#124—42no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—42no—
238C22⋊Q8—42no—
239#121—24no—
240#121—24no—
241#121—42no—
242#121—42no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—42no—
250C4×D4—42no—
251#117—24no—
252#117—24no—
253#117—42no—
254#117—42no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—41no—
274C2×C42C22×C412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D412yes—
279C2×C42#4111yescharacteristic
280C2×C42#4112yes—
281C2×C42C2411yescharacteristic
282#107—24no—
283#107—24no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—42no—
295C4⋊C8C2×D412yes—
296C4⋊C8C2×D412yes—
297#103—24no—
298#103—24no—
299#103—44no—
300#103—44no—
301#103—44no—
302#103—44no—
303#102—24no—
304#102—24no—
305#102—44no—
306#102—44no—
307C22⋊C8—24no—
308C22⋊C8—42no—
309#97—24no—
310#97—42no—
311#97—42no—
312#97C22⋊C414yes—
313#97C22⋊C414yes—
314C4×C8—42no—
315C4×C8—42no—
316C4×C8—42no—
317C4×C8C22×C412yes—
318C4×C8C22×C412yes—
319C4×C8C2×D412yes—
320C4×C8C2×D412yes—
321#95—22no—
322#95—21no—
323#95—21no—
324C24—41no—
325C24#12111yescharacteristic
326#41—416no—
327#41—84no—
328#41—84no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—44no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—42no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—82no—
357C2×D4—84no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—44no—
371C22×C4—84no—
372C22×C4C2×C22⋊C412yes—
373C22×C4C4×D412yes—
374C22×C4C4×D412yes—
375C22×C4#12112yes—
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12412yes—
380C22×C4#12712yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384SD16—48no—
385SD16—48no—
386M16—28no—
387M16—44no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—84no—
409C2×C8C2×C22⋊C412yes—
410C2×C8C2×C22⋊C412yes—
411C2×C8C2×C22⋊C412yes—
412C2×C8C2×C22⋊C412yes—
413C2×C8#13512yes—
414C2×C8#13512yes—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4AGL(1,ℤ/8)14yes—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C42—44no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—44no—
443C42—82no—
444C42C2×C22⋊C412yes—
445C42C4×D412yes—
446C42C4×D412yes—
447C42#12112yes—
448C42#12412yes—
449C42#12712yes—
450C42C23×C411yescharacteristic
451C42C22×D411yescharacteristic
452C42C22×D411yescharacteristic
453C42C2×#4111yescharacteristic
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—42no—
459C23—81no—
460C23—81no—
461C23—82no—
462C23—82no—
463C23#39012yes—
464C23C2×#12111yescharacteristic
465Q8—42no—
466Q8—42no—
467Q8—44no—
468Q8—44no—
469Q8—81no—
470Q8—81no—
471Q8—81no—
472Q8—81no—
473Q8—81no—
474Q8—81no—
475D4—42no—
476D4—42no—
477D4—44no—
478D4—82no—
479D4—82no—
480D4—88no—
481D4—81no—
482D4—81no—
483D4—81no—
484D4—81no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—88no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4#39012yes—
514C2×C4#39012yes—
515C2×C4#39012yes—
516C2×C4C22×C22⋊C411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×C4×D411yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12411yescharacteristic
521C2×C4C2×#12711yescharacteristic
522C2×C4C2×#13511yescharacteristic
523C2×C4C2×#13511yescharacteristic
524C2×C4C2×AGL(1,ℤ/8)12yes—
525C2×C4#57511yescharacteristic
526C2×C4#57511yescharacteristic
527C2×C4#57512yes—
528C8—22no—
529C8—22no—
530C8—22no—
531C8—42no—
532C8—42no—
533C8—81no—
534C8—81no—
535C8—81no—
536C8—81no—
537C8#41614yescomplemented
538C22—22no—
539C22—22no—
540C22—22no—
541C22—42no—
542C22—42no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—162no—
547C22—162no—
548C22C2×#39011yescharacteristic
549C22#303511yescharacteristic
550C22#303511yescharacteristic
551C22#305512yes—
552C22#306912yes—
553C4—21no—
554C4—21no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—44no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4C2×#41611yescharacteristic
564C4#289611yescharacteristic
565C4C4×AGL(1,ℤ/8)12yes—
566C4#303511yescharacteristic
567C4#303511yescharacteristic
568C4#315111yescharacteristic
569C4#315311yescharacteristic
570C2—42no—
571C2—162no—
572C2#1917812yes—
573C2#1918212yes—
574C2#2123611yescharacteristic
575C2#2125011yescharacteristic
576C2#2125711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21389C2semidirect
4. #20480C2semidirect
5. C2×#1565C2semidirect
7. C2×#1552C2semidirect
8. #9748C2semidirect
9. #9746C2semidirect
10. #9744C2semidirect
11. #9740C2semidirect
12. #9738C2semidirect
13. #9736C2semidirect
14. #9732C2semidirect
15. #9731C2semidirect
19. #3182C4semidirect
537. C8#416semidirect

Maximal subgroups

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

subgroupindex
C2×#31822
#213892
#212572
#204802
C2×#15652
C2×#15532
C2×#15522
#97482
#97462
#97442
#97402
#97382
#97362
#97322
#97312