Finite Group Databasebeta

← #205331

Subgroups — #205331

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3185C211yesmaximal, characteristic
2#21394C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4C4⋊C4×M16C211yesmaximal, characteristic
5C2×#1569C211yesmaximal, characteristic
6C2×#1561C211yesmaximal, characteristic
7C2×#1559C211yesmaximal, characteristic
8#9748C211yesmaximal, characteristic, complemented
9#9745C211yesmaximal, characteristic, complemented
10#9742C211yesmaximal, characteristic, complemented
11#9741C211yesmaximal, characteristic, complemented
12#9737C211yesmaximal, characteristic, complemented
13#9736C211yesmaximal, characteristic, complemented
14#9733C211yesmaximal, characteristic, complemented
15#9729C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3185—24no—
19#3185C418yescomplemented
20C2×#498C2211yescharacteristic
21C2×#494C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#470—21no—
25C2×#470—21no—
26C2×#441—21no—
27C2×#441—21no—
28C2×#432C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C2×C4×M16—21no—
32C2×C4×M16C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1989—22no—
35#1988C2211yescharacteristic
36#1988C2211yescharacteristic
37#1988C2211yescharacteristic
38#1988C2211yescharacteristic
39#1987C2211yescharacteristic
40#1987C2211yescharacteristic
41#1981C2211yescharacteristic
42#1981C2211yescharacteristic
43#1872—22no—
44#1771—21no—
45#1771—21no—
46#1770C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1763—22no—
51#1736—22no—
52#1728—21no—
53#1569—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1561—24no—
59#1559—24no—
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#332C2211yescharacteristic
67C2×#332C2211yescharacteristic
68C22×Q16—21no—
69C22×Q16—21no—
70C22×M16—21no—
71C22×M16—21no—
72C22×M16D411yescharacteristic
73C22×M16D411yescharacteristic
74#533—22no—
75#533—22no—
76#533—22no—
77#533—22no—
78#533—22no—
79#533—22no—
80#533C2×C414yes—
81#533C2×C414yes—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#125—21no—
87C2×#125—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×Q8—21no—
96C2×#117—21no—
97C2×#117—21no—
98C22×C4⋊C4D412yes—
99C22×C42—21no—
100C22×C42C2311yescharacteristic
101#498—22no—
102#498C2×C414yescomplemented
103#494—22no—
104#494C2×C414yescomplemented
105#487—22no—
106#487—22no—
107#487C2×C414yescomplemented
108#487C2×C414yescomplemented
109#470—28no—
110#470—28no—
111#441—28no—
112#441—28no—
113#432—22no—
114#432—22no—
115C2×#106—21no—
116C2×#106—21no—
117#423—22no—
118#423—22no—
119#423—22no—
120#423D414yes—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×#103—21no—
130C2×#103—21no—
131C2×#103—21no—
132C2×#103—21no—
133C2×#103—22no—
134C2×#103—22no—
135C2×C22⋊C8—21no—
136C4×M16—44no—
137C4×M16C2×C414yes—
138C4×M16D414yes—
139C2×#97—21no—
140C2×#97—21no—
141C2×#97—21no—
142C2×#97D411yescharacteristic
143C2×#97D411yescharacteristic
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4D412yes—
151C4×C4⋊C4C2311yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153C2×#95—21no—
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#332—22no—
171#332—22no—
172#332—22no—
173#332—22no—
174#332—22no—
175#332—22no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C23×C4—22no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4#4111yescharacteristic
184C2×Q16—24no—
185C2×Q16—24no—
186C2×Q16—42no—
187C2×Q16—42no—
188C2×Q16—42no—
189C2×Q16—42no—
190C2×M16—24no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—41no—
194C2×M16—41no—
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—44no—
198C2×M16—42no—
199C2×M16C22⋊C414yes—
200C2×M16C22⋊C414yes—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
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⋊Q8C22×C412yes—
224C4⋊Q8C22×C412yes—
225C4⋊Q8C22×C412yes—
226C4⋊Q8C22×C412yes—
227#125—24no—
228#125—24no—
229#125—42no—
230#125—42no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8—42no—
236C22⋊Q8—42no—
237C22⋊Q8—42no—
238C22⋊Q8—42no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—42no—
244C4×Q8—42no—
245C4×Q8—42no—
246C4×Q8—42no—
247#117—24no—
248#117—24no—
249#117—42no—
250#117—42no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42C22×C412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D412yes—
282C2×C42#4111yescharacteristic
283C2×C42#4112yes—
284C2×C42C2411yescharacteristic
285#106—24no—
286#106—24no—
287C4⋊C8—24no—
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—21no—
298C4⋊C8—42no—
299C4⋊C8C2×D412yes—
300C4⋊C8C2×D412yes—
301#103—24no—
302#103—24no—
303#103—24no—
304#103—24no—
305#103—44no—
306#103—44no—
307C22⋊C8—42no—
308#97—42no—
309#97—42no—
310#97—42no—
311#97C22⋊C414yes—
312#97C22⋊C414yes—
313C4×C8—42no—
314C4×C8—42no—
315C4×C8C22×C412yes—
316C4×C8C22×C412yes—
317C4×C8C2×D412yes—
318C4×C8C2×D412yes—
319#95—22no—
320#95—21no—
321#95—21no—
322C24#12111yescharacteristic
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—42no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—82no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—82no—
358C22×C4C2×C22⋊C412yes—
359C22×C4C4×D412yes—
360C22×C4C4×D412yes—
361C22×C4#12112yes—
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12412yes—
366C22×C4#12712yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
370Q16—48no—
371Q16—48no—
372M16—28no—
373M16—44no—
374M16—48no—
375M16—84no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8C2×C22⋊C412yes—
397C2×C8C2×C22⋊C412yes—
398C2×C8C2×C22⋊C412yes—
399C2×C8C2×C22⋊C412yes—
400C2×C8AGL(1,ℤ/8)12yes—
401C2×C8#13712yes—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—44no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4#13714yes—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C42—44no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—44no—
435C42—44no—
436C42C2×C22⋊C412yes—
437C42C4×D412yes—
438C42C4×D412yes—
439C42#12112yes—
440C42#12412yes—
441C42#12712yes—
442C42C23×C411yescharacteristic
443C42C22×D411yescharacteristic
444C42C22×D411yescharacteristic
445C42C2×#4111yescharacteristic
446C23—21no—
447C23—21no—
448C23—22no—
449C23—22no—
450C23#39012yes—
451C23C2×#12111yescharacteristic
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—44no—
457Q8—82no—
458Q8—82no—
459Q8—81no—
460Q8—81no—
461Q8—81no—
462Q8—81no—
463Q8—81no—
464Q8—81no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—88no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#39012yes—
497C2×C4#39012yes—
498C2×C4#39012yes—
499C2×C4C22×C22⋊C411yescharacteristic
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C2×C4×D411yescharacteristic
502C2×C4C2×#12111yescharacteristic
503C2×C4C2×#12411yescharacteristic
504C2×C4C2×#12711yescharacteristic
505C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
506C2×C4C2×#13711yescharacteristic
507C2×C4C2×#13712yes—
508C2×C4#57511yescharacteristic
509C2×C4#57511yescharacteristic
510C2×C4#57512yes—
511C8—22no—
512C8—22no—
513C8—22no—
514C8—42no—
515C8—42no—
516C8—81no—
517C8—81no—
518C8—81no—
519C8—81no—
520C8#41714yes—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—42no—
525C22—42no—
526C22C2×#39011yescharacteristic
527C22#303611yescharacteristic
528C22#303711yescharacteristic
529C22#305612yes—
530C22#306912yes—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—44no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4C2×#41711yescharacteristic
543C4#289611yescharacteristic
544C4C4×#13712yes—
545C4#303611yescharacteristic
546C4#303711yescharacteristic
547C4#315211yescharacteristic
548C4#315311yescharacteristic
549C2—42no—
550C2#1930712yes—
551C2#1931112yes—
552C2#2123711yescharacteristic
553C2#2125011yescharacteristic
554C2#2125811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
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
8. #9748C2semidirect
9. #9745C2semidirect
10. #9742C2semidirect
11. #9741C2semidirect
12. #9737C2semidirect
13. #9736C2semidirect
14. #9733C2semidirect
15. #9729C2semidirect
19. #3185C4semidirect
102. #498C2×C4semidirect
104. #494C2×C4semidirect
107. #487C2×C4semidirect
108. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31852
#213942
#213122
C4⋊C4×M162
C2×#15692
C2×#15612
C2×#15592
#97482
#97452
#97422
#97412
#97372
#97362
#97332
#97292