Finite Group Databasebeta

← #205318

Subgroups — #205318

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3185C211yesmaximal, characteristic
2#21894C211yesmaximal, characteristic
3#21839C211yesmaximal, characteristic
4#20441C211yesmaximal, characteristic
5C2×#1569C211yesmaximal, characteristic
6C2×#1552C211yesmaximal, characteristic
7C2×#1550C211yesmaximal, 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
30C22×C4⋊C8C2211yescharacteristic
31C2×#405—21no—
32C2×C4×M16C2211yescharacteristic
33#2285C2211yescharacteristic
34#1992—22no—
35#1988C2211yescharacteristic
36#1988C2211yescharacteristic
37#1988C2211yescharacteristic
38#1988C2211yescharacteristic
39#1987C2211yescharacteristic
40#1987C2211yescharacteristic
41#1981C2211yescharacteristic
42#1981C2211yescharacteristic
43#1894—22no—
44#1825—22no—
45#1803—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#1552—24no—
59#1550—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×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73C23×C8—21no—
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—
98C2×#117—21no—
99C22×C42C2311yescharacteristic
100#498—22no—
101#498C2×C414yes—
102#494—22no—
103#494C2×C414yes—
104#487—22no—
105#487—22no—
106#487C2×C414yes—
107#487C2×C414yes—
108#470—28no—
109#470—28no—
110#441—28no—
111#441—28no—
112#432—22no—
113#432—22no—
114C2×#106—21no—
115C2×#106—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×#103—21no—
131C2×#103—21no—
132C2×#103—22no—
133C2×#103—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#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×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4111yescharacteristic
183C2×Q16—24no—
184C2×Q16—24no—
185C2×Q16—42no—
186C2×Q16—42no—
187C2×Q16—42no—
188C2×Q16—42no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—41no—
192C2×M16—41no—
193C2×M16—42no—
194C2×M16—82no—
195C2×M16C22⋊C414yes—
196C2×M16C22⋊C414yes—
197C22×C8—24no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C4⋊Q8—21no—
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⋊Q8C22×C412yes—
223C4⋊Q8C22×C412yes—
224C4⋊Q8C22×C412yes—
225C4⋊Q8C22×C412yes—
226#125—24no—
227#125—24no—
228#125—42no—
229#125—42no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—42no—
235C22⋊Q8—42no—
236C22⋊Q8—42no—
237C22⋊Q8—42no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—42no—
243C4×Q8—42no—
244C4×Q8—42no—
245C4×Q8—42no—
246#117—24no—
247#117—24no—
248#117—42no—
249#117—42no—
250#117—42no—
251#117—42no—
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—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
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—22no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42C22×C412yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42#4111yescharacteristic
281C2×C42#4112yes—
282C2×C42C2411yescharacteristic
283#106—24no—
284#106—24no—
285C4⋊C8—24no—
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—21no—
296C4⋊C8—42no—
297C4⋊C8C2×D412yes—
298C4⋊C8C2×D412yes—
299#103—24no—
300#103—24no—
301#103—24no—
302#103—24no—
303#103—82no—
304#103—82no—
305C22⋊C8—42no—
306#97—42no—
307#97—42no—
308#97—42no—
309#97C22⋊C414yes—
310#97C22⋊C414yes—
311C4×C8—42no—
312C4×C8—42no—
313C4×C8C22×C412yes—
314C4×C8C22×C412yes—
315C4×C8C2×D412yes—
316C4×C8C2×D412yes—
317#95—22no—
318#95—21no—
319#95—21no—
320C24#12111yescharacteristic
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—82no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C4×D412yes—
355C22×C4C4×D412yes—
356C22×C4#12112yes—
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12412yes—
361C22×C4#12712yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365Q16—48no—
366Q16—48no—
367M16—28no—
368M16—44no—
369M16—84no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—48no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8C2×C22⋊C412yes—
392C2×C8C2×C22⋊C412yes—
393C2×C8C2×C22⋊C412yes—
394C2×C8C2×C22⋊C412yes—
395C2×C8AGL(1,ℤ/8)12yes—
396C2×C8#13712yes—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—44no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—82no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—44no—
431C42—44no—
432C42C2×C22⋊C412yes—
433C42C4×D412yes—
434C42C4×D412yes—
435C42#12112yes—
436C42#12412yes—
437C42#12712yes—
438C42C23×C411yescharacteristic
439C42C22×D411yescharacteristic
440C42C22×D411yescharacteristic
441C42C2×#4111yescharacteristic
442C23—21no—
443C23—21no—
444C23—22no—
445C23—22no—
446C23#39012yes—
447C23C2×#12111yescharacteristic
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—44no—
453Q8—82no—
454Q8—82no—
455Q8—81no—
456Q8—81no—
457Q8—81no—
458Q8—81no—
459Q8—81no—
460Q8—81no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#39012yes—
492C2×C4#39012yes—
493C2×C4#39012yes—
494C2×C4C22×C22⋊C411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12411yescharacteristic
499C2×C4C2×#12711yescharacteristic
500C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
501C2×C4C2×#13711yescharacteristic
502C2×C4#57511yescharacteristic
503C2×C4#57511yescharacteristic
504C2×C4#57712yes—
505C2×C4#57812yes—
506C8—22no—
507C8—22no—
508C8—22no—
509C8—42no—
510C8—42no—
511C8—81no—
512C8—81no—
513C8—81no—
514C8—81no—
515C8#41714yes—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—42no—
520C22—42no—
521C22C2×#39011yescharacteristic
522C22#303611yescharacteristic
523C22#303711yescharacteristic
524C22#306312yes—
525C22#306912yes—
526C4—21no—
527C4—21no—
528C4—22no—
529C4—22no—
530C4—22no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4C2×#41711yescharacteristic
538C4#289611yescharacteristic
539C4#294712yes—
540C4#303611yescharacteristic
541C4#303711yescharacteristic
542C4#315411yescharacteristic
543C4#315611yescharacteristic
544C2—42no—
545C2#1937112yes—
546C2#1937512yes—
547C2#2123411yescharacteristic
548C2#2125011yescharacteristic
549C2#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

Maximal subgroups

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

subgroupindex
C2×#31852
#218942
#218392
#204412
C2×#15692
C2×#15522
C2×#15502
#97482
#97452
#97422
#97412
#97372
#97362
#97332
#97292