Finite Group Databasebeta

← #205237

Subgroups — #205237

Order 512.

1112 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3184C211yesmaximal, characteristic
2#21888C211yesmaximal, characteristic, complemented
3#21807C211yesmaximal, characteristic, complemented
4#20482C211yesmaximal, characteristic, complemented
5C2×#1565C211yesmaximal, characteristic, complemented
6C2×#1543C211yesmaximal, characteristic, complemented
7C2×#1540C211yesmaximal, characteristic, complemented
8#9747C211yesmaximal, characteristic, complemented
9#9746C211yesmaximal, characteristic, complemented
10#9743C211yesmaximal, characteristic, complemented
11#9741C211yesmaximal, characteristic, complemented
12#9738C211yesmaximal, characteristic, complemented
13#9735C211yesmaximal, characteristic, complemented
14#9734C211yesmaximal, characteristic, complemented
15#9729C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3184—24no—
19#3184C418yescomplemented
20C2×#498C2211yescharacteristic
21C2×#493C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#469—21no—
25C2×#469—21no—
26C2×#442—21no—
27C2×#442—21no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2283C2211yescharacteristic
33#1987C2211yescharacteristic
34#1987C2211yescharacteristic
35#1982C2211yescharacteristic
36#1982C2211yescharacteristic
37#1982C2211yescharacteristic
38#1982C2211yescharacteristic
39#1981C2211yescharacteristic
40#1981C2211yescharacteristic
41#1958—22no—
42#1893—22no—
43#1804—22no—
44#1788—22no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1753—21no—
50#1753—21no—
51#1746—21no—
52#1726—21no—
53#1565—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1543—24no—
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61#1540—24no—
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#331C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×#41—21no—
70C22×D8—21no—
71C22×D8—21no—
72C22×M16D411yescharacteristic
73C22×M16D411yescharacteristic
74C23×C8—21no—
75#532—22no—
76#532—22no—
77#532—22no—
78#532—22no—
79#532—22no—
80#532—22no—
81#532C2×C414yes—
82#532C2×C414yes—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#124—21no—
88C2×#124—21no—
89C2×#121—21no—
90C2×#121—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C22×C42C2311yescharacteristic
98#498—22no—
99#498C2×C414yes—
100#493—22no—
101#493C2×C414yes—
102#486—22no—
103#486—22no—
104#486C2×C414yes—
105#486C2×C414yes—
106#469—28no—
107#469—28no—
108#442—28no—
109#442—28no—
110#431—22no—
111#431—22no—
112#431—22no—
113#431D414yes—
114C2×#106—21no—
115C2×#106—21no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×#103—22no—
126C2×#103—22no—
127C2×#102—21no—
128C2×#102—21no—
129C2×#102—21no—
130C2×#102—21no—
131C2×#102—22no—
132C2×#102—22no—
133C2×#102—22no—
134C2×#102—22no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C4×M16C2×C414yes—
138C4×M16D414yes—
139C2×#97—21no—
140C2×#97—21no—
141C2×#97D411yescharacteristic
142C2×#97D411yescharacteristic
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8—21no—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#383—21no—
150#382D412yes—
151#381—21no—
152#380D412yes—
153C4×C4⋊C4C2311yescharacteristic
154C4×C4⋊C4C2311yescharacteristic
155C4×C22⋊C4—21no—
156#376—21no—
157#340—22no—
158#340—22no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#335—22no—
164#335—22no—
165#334—22no—
166#334—22no—
167#333—22no—
168#333—22no—
169#332—22no—
170#332—22no—
171#331—22no—
172#331—22no—
173#331—22no—
174#331—22no—
175C2×#41—24no—
176C2×#41—24no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41—82no—
180C2×#41—82no—
181C2×#41—84no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—41no—
191C22×D4—41no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4#4111yescharacteristic
197C2×D8—24no—
198C2×D8—24no—
199C2×D8—42no—
200C2×D8—42no—
201C2×D8—42no—
202C2×D8—42no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—41no—
206C2×M16—41no—
207C2×M16C22⋊C414yes—
208C2×M16C22⋊C414yes—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—42no—
216C22×C8—82no—
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8C22×C412yes—
228C4⋊Q8C22×C412yes—
229#127—21no—
230#127—21no—
231#127—21no—
232#127—21no—
233#127—21no—
234#127—21no—
235#127C22×C412yes—
236#127C22×C412yes—
237#124—24no—
238#124—24no—
239#124—42no—
240#124—42no—
241#121—24no—
242#121—24no—
243#121—24no—
244#121—24no—
245#121—42no—
246#121—42no—
247#121—42no—
248#121—42no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42C22×C412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D412yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42#4111yescharacteristic
276C2×C42#4112yes—
277C2×C42C2411yescharacteristic
278#106—24no—
279#106—24no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—42no—
291C4⋊C8C2×D412yes—
292C4⋊C8C2×D412yes—
293#103—82no—
294#103—82no—
295#102—24no—
296#102—24no—
297#102—24no—
298#102—24no—
299#102—82no—
300#102—82no—
301#102—82no—
302#102—82no—
303C22⋊C8—24no—
304C22⋊C8—42no—
305#97—42no—
306#97—42no—
307#97C22⋊C414yes—
308#97C22⋊C414yes—
309C4×C8—24no—
310C4×C8—42no—
311C4×C8—42no—
312C4×C8—42no—
313C4×C8C22×C412yes—
314C4×C8C22×C412yes—
315C4×C8C2×D412yes—
316C4×C8C2×D412yes—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—21no—
321#95—21no—
322C24—41no—
323C24—41no—
324C24#12111yescharacteristic
325#41—88no—
326#41—84no—
327#41—84no—
328#41—88no—
329#41—88no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—82no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—84no—
366C2×D4—84no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4C2×C22⋊C412yes—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4#12112yes—
386C22×C4#12111yescharacteristic
387C22×C4#12111yescharacteristic
388C22×C4#12111yescharacteristic
389C22×C4#12412yes—
390C22×C4#12712yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394D8—48no—
395D8—48no—
396M16—28no—
397M16—44no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—84no—
419C2×C8C2×C22⋊C412yes—
420C2×C8C2×C22⋊C412yes—
421C2×C8C2×C22⋊C412yes—
422C2×C8C2×C22⋊C412yes—
423C2×C8#13512yes—
424C2×C8#13512yes—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—82no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42C2×C22⋊C412yes—
447C42C4×D412yes—
448C42C4×D412yes—
449C42#12112yes—
450C42#12412yes—
451C42#12712yes—
452C42C23×C411yescharacteristic
453C42C22×D411yescharacteristic
454C42C22×D411yescharacteristic
455C42C2×#4111yescharacteristic
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23—42no—
461C23—42no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23#39012yes—
471C23C2×#12111yescharacteristic
472Q8—44no—
473Q8—82no—
474Q8—82no—
475Q8—81no—
476Q8—81no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—44no—
482D4—44no—
483D4—82no—
484D4—82no—
485D4—82no—
486D4—82no—
487D4—88no—
488D4—88no—
489D4—81no—
490D4—81no—
491D4—81no—
492D4—81no—
493D4—81no—
494D4—81no—
495D4—81no—
496D4—81no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—88no—
520C2×C4—88no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#39012yes—
524C2×C4#39012yes—
525C2×C4#39012yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12411yescharacteristic
531C2×C4C2×#12711yescharacteristic
532C2×C4C2×#13511yescharacteristic
533C2×C4C2×#13511yescharacteristic
534C2×C4#57511yescharacteristic
535C2×C4#57511yescharacteristic
536C2×C4#57612yes—
537C2×C4#57712yes—
538C8—22no—
539C8—22no—
540C8—22no—
541C8—42no—
542C8—42no—
543C8—81no—
544C8—81no—
545C8—81no—
546C8—81no—
547C8#41614yes—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—162no—
560C22—162no—
561C22—162no—
562C22—162no—
563C22C2×#39011yescharacteristic
564C22#303511yescharacteristic
565C22#303511yescharacteristic
566C22#306212yes—
567C22#306812yes—
568C4—22no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—22no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4C2×#41611yescharacteristic
578C4#289611yescharacteristic
579C4#294612yes—
580C4#303511yescharacteristic
581C4#303511yescharacteristic
582C4#315411yescharacteristic
583C4#315511yescharacteristic
584C2—42no—
585C2—162no—
586C2—162no—
587C2#1924812yes—
588C2#1925012yes—
589C2#2123511yescharacteristic
590C2#2124911yescharacteristic
591C2#2125711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3184C2
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. #21888C2semidirect
3. #21807C2semidirect
4. #20482C2semidirect
5. C2×#1565C2semidirect
6. C2×#1543C2semidirect
7. C2×#1540C2semidirect
8. #9747C2semidirect
9. #9746C2semidirect
10. #9743C2semidirect
11. #9741C2semidirect
12. #9738C2semidirect
13. #9735C2semidirect
14. #9734C2semidirect
15. #9729C2semidirect
19. #3184C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31842
#218882
#218072
#204822
C2×#15652
C2×#15432
C2×#15402
#97472
#97462
#97432
#97412
#97382
#97352
#97342
#97292