Finite Group Databasebeta

← #205462

Subgroups — #205462

Order 512.

944 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3231C211yesmaximal, characteristic
2#21385C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4#20535C211yesmaximal, characteristic
5C2×#1574C211yesmaximal, characteristic
6C2×#1559C211yesmaximal, characteristic
7C2×#1554C211yesmaximal, characteristic
8#9748C211yesmaximal, characteristic, complemented
9#9745C211yesmaximal, characteristic, complemented
10#9741C211yesmaximal, characteristic, complemented
11#9739C211yesmaximal, characteristic, complemented
12#9737C211yesmaximal, characteristic, complemented
13#9736C211yesmaximal, characteristic, complemented
14#9730C211yesmaximal, characteristic, complemented
15#9729C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3231—24no—
19#3231C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#485—21no—
25C2×#485—21no—
26C2×#479—21no—
27C2×#479—21no—
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1989—22no—
34#1988C2211yescharacteristic
35#1988C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1987C2211yescharacteristic
39#1987C2211yescharacteristic
40#1981C2211yescharacteristic
41#1981C2211yescharacteristic
42#1872—22no—
43#1771—21no—
44#1771—21no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#103—22no—
50#1749—21no—
51C4×C22⋊C8—21no—
52#1736—22no—
53#1574—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1559—24no—
59#1554—24no—
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#333C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#332C2211yescharacteristic
67C2×#332C2211yescharacteristic
68C22×M16—21no—
69#533—22no—
70#533—22no—
71#533—22no—
72#533—22no—
73#533—22no—
74#533—22no—
75#533C2×C414yes—
76#533C2×C414yes—
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×#125—21no—
82C2×#125—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×#117—21no—
92C2×#117—21no—
93C22×C4⋊C4D412yes—
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96#500—22no—
97#500C2×C414yescomplemented
98#498—22no—
99#498C2×C414yescomplemented
100#487—22no—
101#487—22no—
102#487C2×C414yescomplemented
103#487C2×C414yescomplemented
104#485—28no—
105#485—28no—
106#479—28no—
107#479—28no—
108#432—22no—
109#432D414yes—
110#431—22no—
111#431—22no—
112#431C2×C414yes—
113#431D414yes—
114C2×#107—21no—
115C2×#107—21no—
116C2×#106—21no—
117C2×#106—21no—
118#423—22no—
119#423—22no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8D411yescharacteristic
122C2×C4⋊C8D411yescharacteristic
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—
136C2×C22⋊C8—21no—
137C2×C22⋊C8D411yescharacteristic
138C2×C22⋊C8D411yescharacteristic
139C2×#97—21no—
140C2×#97—21no—
141C2×#97—21no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4D412yes—
150C4×C4⋊C4C2311yescharacteristic
151C4×C4⋊C4C2311yescharacteristic
152C2×#95—21no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
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#333—22no—
170#333—22no—
171#332—22no—
172#332—22no—
173#332—22no—
174#332—22no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C23×C4—22no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4111yescharacteristic
183C2×M16—44no—
184C2×M16—42no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8C22×C412yes—
208C4⋊Q8C22×C412yes—
209C4⋊Q8C22×C412yes—
210C4⋊Q8C22×C412yes—
211#125—24no—
212#125—24no—
213#125—42no—
214#125—42no—
215C22⋊Q8—24no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219C22⋊Q8—42no—
220C22⋊Q8—42no—
221C22⋊Q8—42no—
222C22⋊Q8—42no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—42no—
228C4×Q8—42no—
229C4×Q8—42no—
230C4×Q8—42no—
231#117—24no—
232#117—24no—
233#117—42no—
234#117—42no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42—81no—
260C2×C42—81no—
261C2×C42C22×C412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D412yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42#4111yescharacteristic
267C2×C42#4112yes—
268C2×C42C2411yescharacteristic
269#107—24no—
270#107—24no—
271#106—24no—
272#106—24no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—42no—
284C4⋊C8C22⋊C414yes—
285C4⋊C8C22⋊C414yes—
286C4⋊C8C2×D412yes—
287C4⋊C8C2×D412yes—
288#103—24no—
289#103—24no—
290#103—24no—
291#103—24no—
292#103—44no—
293#103—44no—
294C22⋊C8—24no—
295C22⋊C8—42no—
296C22⋊C8C22⋊C414yes—
297C22⋊C8C22⋊C414yes—
298#97—24no—
299#97—81no—
300#97—81no—
301C4×C8—42no—
302C4×C8—81no—
303C4×C8—81no—
304C4×C8C22×C412yes—
305C4×C8C22×C412yes—
306C4×C8C2×D412yes—
307C4×C8C2×D412yes—
308#95—22no—
309#95—21no—
310#95—21no—
311C24#12111yescharacteristic
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—82no—
327C2×Q8—82no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—82no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4#12112yes—
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12412yes—
352C22×C4#12712yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356M16—84no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8C2×C22⋊C412yes—
380C2×C8C2×C22⋊C412yes—
381C2×C8C2×C22⋊C412yes—
382C2×C8C2×C22⋊C412yes—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—44no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4#13714yes—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—44no—
415C42—44no—
416C42—82no—
417C42C2×C22⋊C412yes—
418C42C4×D412yes—
419C42C4×D412yes—
420C42#12112yes—
421C42#12412yes—
422C42#12712yes—
423C42C23×C411yescharacteristic
424C42C22×D411yescharacteristic
425C42C22×D411yescharacteristic
426C42C2×#4111yescharacteristic
427C23—21no—
428C23—21no—
429C23—22no—
430C23—22no—
431C23#39012yes—
432C23C2×#12111yescharacteristic
433Q8—42no—
434Q8—42no—
435Q8—44no—
436Q8—82no—
437Q8—82no—
438Q8—82no—
439Q8—82no—
440Q8—81no—
441Q8—81no—
442Q8—81no—
443Q8—81no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—88no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4#39012yes—
475C2×C4#39012yes—
476C2×C4#39012yes—
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×#12411yescharacteristic
482C2×C4C2×#12711yescharacteristic
483C2×C4C2×#13712yes—
484C2×C4#57512yes—
485C2×C4#57711yescharacteristic
486C2×C4#57711yescharacteristic
487C2×C4#57811yescharacteristic
488C2×C4#57811yescharacteristic
489C8—22no—
490C8—22no—
491C8—22no—
492C8—22no—
493C8—42no—
494C8—42no—
495C8—81no—
496C8—81no—
497C8—81no—
498C8—81no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22C2×#39011yescharacteristic
505C22#304211yescharacteristic
506C22#304411yescharacteristic
507C22#305612yes—
508C22#306712yes—
509C4—21no—
510C4—21no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—44no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4#289811yescharacteristic
521C4#289811yescharacteristic
522C4C4×#13712yes—
523C4#304611yescharacteristic
524C4#304611yescharacteristic
525C4#315211yescharacteristic
526C4#315311yescharacteristic
527C2—42no—
528C2#1946612yes—
529C2#1947812yes—
530C2#2123711yescharacteristic
531C2#2124811yescharacteristic
532C2#2126211yescharacteristic
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. #9741C2semidirect
11. #9739C2semidirect
12. #9737C2semidirect
13. #9736C2semidirect
14. #9730C2semidirect
15. #9729C2semidirect
19. #3231C4semidirect
97. #500C2×C4semidirect
99. #498C2×C4semidirect
102. #487C2×C4semidirect
103. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32312
#213852
#213122
#205352
C2×#15742
C2×#15592
C2×#15542
#97482
#97452
#97412
#97392
#97372
#97362
#97302
#97292