Finite Group Databasebeta

← #205320

Subgroups — #205320

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3182C211yesmaximal, characteristic
2#21893C211yesmaximal, characteristic
3#21839C211yesmaximal, characteristic, complemented
4#20441C211yesmaximal, characteristic, complemented
5C2×#1569C211yesmaximal, characteristic, complemented
6C2×#1551C211yesmaximal, characteristic
7C2×#1550C211yesmaximal, characteristic, complemented
8#9747C211yesmaximal, characteristic, complemented
9#9745C211yesmaximal, characteristic, complemented
10#9744C211yesmaximal, characteristic, complemented
11#9739C211yesmaximal, characteristic, complemented
12#9737C211yesmaximal, characteristic, complemented
13#9735C211yesmaximal, characteristic, complemented
14#9732C211yesmaximal, characteristic, complemented
15#9730C211yesmaximal, 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×#423C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×#405—21no—
32C2×C4×M16C2211yescharacteristic
33#2285C2211yescharacteristic
34#1992—22no—
35#1988C2211yescharacteristic
36#1988C2211yescharacteristic
37#1987C2211yescharacteristic
38#1987C2211yescharacteristic
39#1987C2211yescharacteristic
40#1987C2211yescharacteristic
41#1982C2211yescharacteristic
42#1982C2211yescharacteristic
43#1895—22no—
44#1825—22no—
45#1802—22no—
46#1771—21no—
47#1771—21no—
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1729—21no—
53#1569—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1551—24no—
59#1550—24no—
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#333C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×SD16—21no—
70C22×SD16—21no—
71C22×M16—21no—
72C22×M16D411yescharacteristic
73C22×M16D411yescharacteristic
74C23×C8—21no—
75#533—22no—
76#533—22no—
77#533—22no—
78#533C2×C414yes—
79#532—22no—
80#532—22no—
81#532—22no—
82#532C2×C414yes—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#125—21no—
88C2×#124—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×#117—21no—
98C2×#117—21no—
99C22×C42C2311yescharacteristic
100#500—22no—
101#500C2×C414yes—
102#492—22no—
103#492C2×C414yes—
104#487—22no—
105#487C2×C414yes—
106#486—22no—
107#486C2×C414yes—
108#468—28no—
109#468—28no—
110#440—28no—
111#440—28no—
112#432—22no—
113#432—22no—
114C2×#107—21no—
115C2×#107—21no—
116#423—22no—
117#423—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8D414yes—
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×#102—21no—
131C2×#102—21no—
132C2×#102—22no—
133C2×#102—22no—
134C2×C22⋊C8—21no—
135#405—44no—
136C4×M16C2×C414yes—
137C4×M16D414yes—
138C2×#97—21no—
139C2×#97—21no—
140C2×#97—21no—
141C2×#97D411yescharacteristic
142C2×#97D411yescharacteristic
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#384D412yes—
148#382—21no—
149#380—21no—
150#380D412yes—
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2311yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
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#333—22no—
169#333—22no—
170#333—22no—
171#333—22no—
172#332—22no—
173#332—22no—
174#331—22no—
175#331—22no—
176C2×#41—24no—
177C2×#41—42no—
178C2×#41—84no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—41no—
186C23×C4—41no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4111yescharacteristic
190C2×SD16—24no—
191C2×SD16—24no—
192C2×SD16—42no—
193C2×SD16—42no—
194C2×SD16—42no—
195C2×SD16—42no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—41no—
199C2×M16—41no—
200C2×M16—42no—
201C2×M16—82no—
202C2×M16C22⋊C414yes—
203C2×M16C22⋊C414yes—
204C22×C8—24no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8C22×C412yes—
227C4⋊Q8C22×C412yes—
228C4⋊Q8C22×C412yes—
229#127—21no—
230#127—21no—
231#127—21no—
232#127C22×C412yes—
233#125—24no—
234#125—42no—
235#124—24no—
236#124—42no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—42no—
240C22⋊Q8—42no—
241#121—24no—
242#121—24no—
243#121—42no—
244#121—42no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—42no—
252C4×D4—42no—
253#117—24no—
254#117—42no—
255#117—42no—
256#117—42no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—41no—
276C2×C42C22×C412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D412yes—
281C2×C42#4111yescharacteristic
282C2×C42#4112yes—
283C2×C42C2411yescharacteristic
284#107—24no—
285#107—24no—
286C4⋊C8—24no—
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—21no—
296C4⋊C8—21no—
297C4⋊C8—42no—
298C4⋊C8C2×D412yes—
299C4⋊C8C2×D412yes—
300#103—24no—
301#103—24no—
302#102—24no—
303#102—24no—
304#102—82no—
305#102—82no—
306C22⋊C8—42no—
307#97—42no—
308#97—42no—
309#97—42no—
310#97C22⋊C414yes—
311#97C22⋊C414yes—
312C4×C8—42no—
313C4×C8—42no—
314C4×C8C22×C412yes—
315C4×C8C22×C412yes—
316C4×C8C2×D412yes—
317C4×C8C2×D412yes—
318#95—22no—
319#95—21no—
320#95—21no—
321C24—41no—
322C24#12111yescharacteristic
323#41—88no—
324#41—88no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—82no—
336C2×Q8—82no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—44no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—82no—
351C2×D4—84no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—44no—
365C22×C4—84no—
366C22×C4C2×C22⋊C412yes—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4#12112yes—
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12412yes—
374C22×C4#12712yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378SD16—48no—
379SD16—48no—
380M16—28no—
381M16—44no—
382M16—84no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—48no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8C2×C22⋊C412yes—
405C2×C8C2×C22⋊C412yes—
406C2×C8C2×C22⋊C412yes—
407C2×C8C2×C22⋊C412yes—
408C2×C8AGL(1,ℤ/8)12yes—
409C2×C8#13712yes—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—44no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—82no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—44no—
439C42C2×C22⋊C412yes—
440C42C4×D412yes—
441C42C4×D412yes—
442C42#12112yes—
443C42#12412yes—
444C42#12712yes—
445C42C23×C411yescharacteristic
446C42C22×D411yescharacteristic
447C42C22×D411yescharacteristic
448C42C2×#4111yescharacteristic
449C23—21no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—42no—
454C23—81no—
455C23—81no—
456C23—82no—
457C23—82no—
458C23#39012yes—
459C23C2×#12111yescharacteristic
460Q8—42no—
461Q8—42no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465Q8—82no—
466Q8—81no—
467Q8—81no—
468D4—42no—
469D4—42no—
470D4—44no—
471D4—82no—
472D4—82no—
473D4—88no—
474D4—81no—
475D4—81no—
476D4—81no—
477D4—81no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—88no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4#39012yes—
507C2×C4#39012yes—
508C2×C4#39012yes—
509C2×C4C22×C22⋊C411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12411yescharacteristic
514C2×C4C2×#12711yescharacteristic
515C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
516C2×C4C2×#13711yescharacteristic
517C2×C4#57511yescharacteristic
518C2×C4#57511yescharacteristic
519C2×C4#57712yes—
520C2×C4#57812yes—
521C8—22no—
522C8—22no—
523C8—22no—
524C8—42no—
525C8—42no—
526C8—81no—
527C8—81no—
528C8—81no—
529C8—81no—
530C8#41714yes—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—162no—
540C22—162no—
541C22C2×#39011yescharacteristic
542C22#303611yescharacteristic
543C22#303711yescharacteristic
544C22#306312yes—
545C22#306812yes—
546C4—22no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—22no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4C2×#41711yescharacteristic
557C4#289611yescharacteristic
558C4#294712yes—
559C4#303611yescharacteristic
560C4#303711yescharacteristic
561C4#315411yescharacteristic
562C4#315611yescharacteristic
563C2—42no—
564C2—162no—
565C2#1936912yes—
566C2#1937312yes—
567C2#2123411yescharacteristic
568C2#2124911yescharacteristic
569C2#2125811yescharacteristic
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
3. #21839C2semidirect
4. #20441C2semidirect
5. C2×#1569C2semidirect
7. C2×#1550C2semidirect
8. #9747C2semidirect
9. #9745C2semidirect
10. #9744C2semidirect
11. #9739C2semidirect
12. #9737C2semidirect
13. #9735C2semidirect
14. #9732C2semidirect
15. #9730C2semidirect
19. #3182C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31822
#218932
#218392
#204412
C2×#15692
C2×#15512
C2×#15502
#97472
#97452
#97442
#97392
#97372
#97352
#97322
#97302