Finite Group Databasebeta

← #205441

Subgroups — #205441

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3207C211yesmaximal, characteristic
2#21884C211yesmaximal, characteristic, complemented
3#21809C211yesmaximal, characteristic, complemented
4#20538C211yesmaximal, characteristic, complemented
5C2×#1573C211yesmaximal, characteristic, complemented
6C2×#1553C211yesmaximal, characteristic, complemented
7C2×#1540C211yesmaximal, characteristic, complemented
8#9748C211yesmaximal, characteristic, complemented
9#9746C211yesmaximal, characteristic, complemented
10#9744C211yesmaximal, characteristic, complemented
11#9743C211yesmaximal, characteristic, complemented
12#9738C211yesmaximal, characteristic, complemented
13#9736C211yesmaximal, characteristic, complemented
14#9734C211yesmaximal, characteristic, complemented
15#9732C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3207—24no—
19#3207C418yescomplemented
20C2×#493C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#463—21no—
25C2×#463—21no—
26C2×#449—21no—
27C2×#449—21no—
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32#2285C2211yescharacteristic
33#1988C2211yescharacteristic
34#1988C2211yescharacteristic
35#1982C2211yescharacteristic
36#1982C2211yescharacteristic
37#1982C2211yescharacteristic
38#1982C2211yescharacteristic
39#1981C2211yescharacteristic
40#1981C2211yescharacteristic
41#1959—22no—
42#1893—22no—
43#1802—22no—
44#1788—22no—
45#1770—21no—
46#1770C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4—21no—
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1750—21no—
52#1750—21no—
53#1573—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1553—24no—
59#1540—24no—
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#331C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×#41—21no—
70C22×SD16—21no—
71C22×SD16—21no—
72C22×D8—21no—
73C22×D8—21no—
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—
97C2×#117—21no—
98C22×C42C2311yescharacteristic
99#493—22no—
100#493C2×C414yes—
101#492—22no—
102#492C2×C414yes—
103#487—22no—
104#487C2×C414yes—
105#486—22no—
106#486C2×C414yes—
107#463—28no—
108#463—28no—
109#449—28no—
110#449—28no—
111#432—22no—
112#432—22no—
113#431—22no—
114#431C2×C414yes—
115#431D414yes—
116#431D414yes—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8D411yescharacteristic
121C2×C4⋊C8D411yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×#103—21no—
129C2×#103—21no—
130C2×#103—22no—
131C2×#103—22no—
132C2×#102—21no—
133C2×#102—21no—
134C2×#102—22no—
135C2×#102—22no—
136C2×#102—22no—
137C2×#102—22no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×#97—21no—
143C2×#97—21no—
144C2×#97—21no—
145C2×C4×C8—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150#384D412yes—
151#382—21no—
152#380—21no—
153#380D412yes—
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2311yescharacteristic
156C4×C4⋊C4C2311yescharacteristic
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#332—22no—
168#332—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—81no—
180C2×#41—81no—
181C2×#41—82no—
182C2×#41—82no—
183C2×#41—81no—
184C2×#41—81no—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—41no—
194C22×D4—41no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4#4111yescharacteristic
200C2×SD16—24no—
201C2×SD16—24no—
202C2×SD16—81no—
203C2×SD16—81no—
204C2×SD16—81no—
205C2×SD16—81no—
206C2×D8—24no—
207C2×D8—24no—
208C2×D8—81no—
209C2×D8—81no—
210C2×D8—81no—
211C2×D8—81no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—42no—
219C22×C8—82no—
220C22×C8C2×D411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8C22×C412yes—
231C4⋊Q8C22×C412yes—
232#127—21no—
233#127—21no—
234#127—21no—
235#127—21no—
236#127—21no—
237#127—21no—
238#127C22×C412yes—
239#127C22×C412yes—
240#124—24no—
241#124—24no—
242#124—42no—
243#124—42no—
244#121—24no—
245#121—24no—
246#121—24no—
247#121—24no—
248#121—42no—
249#121—42no—
250#121—42no—
251#121—42no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260#117—81no—
261#117—81no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42C22×C412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D412yes—
282C2×C42#4111yescharacteristic
283C2×C42#4112yes—
284C2×C42C2411yescharacteristic
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—21no—
295C4⋊C8—42no—
296C4⋊C8C22⋊C414yes—
297C4⋊C8C22⋊C414yes—
298C4⋊C8C2×D412yes—
299C4⋊C8C2×D412yes—
300#103—24no—
301#103—24no—
302#103—82no—
303#103—82no—
304#102—24no—
305#102—24no—
306#102—82no—
307#102—82no—
308#102—82no—
309#102—82no—
310C22⋊C8—24no—
311C22⋊C8—42no—
312C22⋊C8C22⋊C414yes—
313C22⋊C8C22⋊C414yes—
314#97—42no—
315#97—81no—
316#97—81no—
317C4×C8—24no—
318C4×C8—81no—
319C4×C8—81no—
320C4×C8C22×C412yes—
321C4×C8C22×C412yes—
322C4×C8C2×D412yes—
323C4×C8C2×D412yes—
324#95—22no—
325#95—21no—
326#95—21no—
327C24—41no—
328C24—41no—
329C24#12111yescharacteristic
330#41—84no—
331#41—84no—
332#41—88no—
333#41—84no—
334#41—84no—
335#41—84no—
336#41—84no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—42no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—84no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—84no—
394C22×C4C2×C22⋊C412yes—
395C22×C4C4×D412yes—
396C22×C4C4×D412yes—
397C22×C4#12112yes—
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12412yes—
402C22×C4#12712yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4111yescharacteristic
406SD16—84no—
407SD16—84no—
408D8—84no—
409D8—84no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—84no—
433C2×C8C2×C22⋊C412yes—
434C2×C8C2×C22⋊C412yes—
435C2×C8C2×C22⋊C412yes—
436C2×C8C2×C22⋊C412yes—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—44no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—82no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—82no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42C2×C22⋊C412yes—
461C42C4×D412yes—
462C42C4×D412yes—
463C42#12112yes—
464C42#12412yes—
465C42#12712yes—
466C42C23×C411yescharacteristic
467C42C22×D411yescharacteristic
468C42C22×D411yescharacteristic
469C42C2×#4111yescharacteristic
470C23—21no—
471C23—21no—
472C23—22no—
473C23—22no—
474C23—42no—
475C23—42no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23#39012yes—
485C23C2×#12111yescharacteristic
486Q8—42no—
487Q8—42no—
488Q8—44no—
489Q8—81no—
490Q8—81no—
491Q8—81no—
492Q8—81no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—44no—
498D4—44no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—82no—
503D4—88no—
504D4—88no—
505D4—81no—
506D4—81no—
507D4—81no—
508D4—81no—
509D4—81no—
510D4—81no—
511D4—81no—
512D4—81no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—88no—
536C2×C4—88no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4#39012yes—
541C2×C4#39012yes—
542C2×C4#39012yes—
543C2×C4C22×C22⋊C411yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×#12411yescharacteristic
548C2×C4C2×#12711yescharacteristic
549C2×C4#57611yescharacteristic
550C2×C4#57611yescharacteristic
551C2×C4#57612yes—
552C2×C4#57711yescharacteristic
553C2×C4#57711yescharacteristic
554C2×C4#57712yes—
555C8—22no—
556C8—22no—
557C8—22no—
558C8—22no—
559C8—42no—
560C8—42no—
561C8—81no—
562C8—81no—
563C8—81no—
564C8—81no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—162no—
577C22—162no—
578C22—162no—
579C22—162no—
580C22C2×#39011yescharacteristic
581C22#304111yescharacteristic
582C22#304311yescharacteristic
583C22#306212yes—
584C22#306712yes—
585C4—21no—
586C4—21no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4#289711yescharacteristic
595C4#289711yescharacteristic
596C4#294612yes—
597C4#304511yescharacteristic
598C4#304511yescharacteristic
599C4#315411yescharacteristic
600C4#315511yescharacteristic
601C2—42no—
602C2—162no—
603C2—162no—
604C2#1955712yes—
605C2#1956912yes—
606C2#2123511yescharacteristic
607C2#2124811yescharacteristic
608C2#2126311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3207C2
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. #21884C2semidirect
3. #21809C2semidirect
4. #20538C2semidirect
5. C2×#1573C2semidirect
6. C2×#1553C2semidirect
7. C2×#1540C2semidirect
8. #9748C2semidirect
9. #9746C2semidirect
10. #9744C2semidirect
11. #9743C2semidirect
12. #9738C2semidirect
13. #9736C2semidirect
14. #9734C2semidirect
15. #9732C2semidirect
19. #3207C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32072
#218842
#218092
#205382
C2×#15732
C2×#15532
C2×#15402
#97482
#97462
#97442
#97432
#97382
#97362
#97342
#97322