Finite Group Databasebeta

← #205427

Subgroups — #205427

Order 512.

910 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3231C211yesmaximal, characteristic
2#21892C211yesmaximal, characteristic
3#21837C211yesmaximal, characteristic
4#20545C211yesmaximal, characteristic
5C2×#1574C211yesmaximal, characteristic
6C2×#1550C211yesmaximal, characteristic
7C2×#1542C211yesmaximal, 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×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32#2283C2211yescharacteristic
33#1991—22no—
34#1988C2211yescharacteristic
35#1988C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1987C2211yescharacteristic
39#1987C2211yescharacteristic
40#1981C2211yescharacteristic
41#1981C2211yescharacteristic
42#1894—22no—
43#1825—22no—
44#1805—22no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1768—21no—
50#1768—21no—
51#1752—21no—
52C8×C22⋊C4—21no—
53#1574—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1550—24no—
59#1542—24no—
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#333C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#332C2211yescharacteristic
67C2×#332C2211yescharacteristic
68C22×M16—21no—
69C23×C8—21no—
70#533—22no—
71#533—22no—
72#533—22no—
73#533—22no—
74#533—22no—
75#533—22no—
76#533C2×C414yes—
77#533C2×C414yes—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#125—21no—
83C2×#125—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8—21no—
92C2×#117—21no—
93C2×#117—21no—
94C22×C42C2311yescharacteristic
95#500—22no—
96#500C2×C414yes—
97#498—22no—
98#498C2×C414yes—
99#487—22no—
100#487—22no—
101#487C2×C414yes—
102#487C2×C414yes—
103#485—28no—
104#485—28no—
105#479—28no—
106#479—28no—
107#431—22no—
108#431—22no—
109#431C2×C414yes—
110#431D414yes—
111C2×#107—21no—
112C2×#107—21no—
113C2×#106—21no—
114C2×#106—21no—
115#423—22no—
116#423—22no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8D414yes—
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⋊C8D411yescharacteristic
137C2×C22⋊C8D411yescharacteristic
138C2×#97—21no—
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#383—21no—
146#382D412yes—
147#381—21no—
148#380D412yes—
149C4×C4⋊C4C2311yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C22⋊C4—21no—
152#376—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×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×M16—42no—
183C2×M16—82no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—24no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—81no—
192C22×C8—81no—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
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⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C22×C412yes—
210C4⋊Q8C22×C412yes—
211C4⋊Q8C22×C412yes—
212C4⋊Q8C22×C412yes—
213#125—24no—
214#125—24no—
215#125—42no—
216#125—42no—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220C22⋊Q8—24no—
221C22⋊Q8—42no—
222C22⋊Q8—42no—
223C22⋊Q8—42no—
224C22⋊Q8—42no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×Q8—42no—
233#117—24no—
234#117—24no—
235#117—42no—
236#117—42no—
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—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42C22×C412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D412yes—
263C2×C42#4111yescharacteristic
264C2×C42#4112yes—
265C2×C42C2411yescharacteristic
266#107—24no—
267#107—24no—
268#106—24no—
269#106—24no—
270C4⋊C8—24no—
271C4⋊C8—21no—
272C4⋊C8—21no—
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—42no—
282C4⋊C8C22⋊C414yes—
283C4⋊C8C22⋊C414yes—
284C4⋊C8C2×D412yes—
285C4⋊C8C2×D412yes—
286#103—24no—
287#103—24no—
288#103—24no—
289#103—24no—
290#103—82no—
291#103—82no—
292C22⋊C8—42no—
293C22⋊C8C22⋊C414yes—
294C22⋊C8C22⋊C414yes—
295#97—81no—
296#97—81no—
297C4×C8—42no—
298C4×C8—81no—
299C4×C8—81no—
300C4×C8C22×C412yes—
301C4×C8C22×C412yes—
302C4×C8C2×D412yes—
303C4×C8C2×D412yes—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—21no—
308#95—21no—
309C24#12111yescharacteristic
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—82no—
325C2×Q8—82no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4C2×C22⋊C412yes—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4#12112yes—
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12412yes—
347C22×C4#12712yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351M16—84no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—84no—
375C2×C8C2×C22⋊C412yes—
376C2×C8C2×C22⋊C412yes—
377C2×C8C2×C22⋊C412yes—
378C2×C8C2×C22⋊C412yes—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
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—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—44no—
411C42—44no—
412C42C2×C22⋊C412yes—
413C42C4×D412yes—
414C42C4×D412yes—
415C42#12112yes—
416C42#12412yes—
417C42#12712yes—
418C42C23×C411yescharacteristic
419C42C22×D411yescharacteristic
420C42C22×D411yescharacteristic
421C42C2×#4111yescharacteristic
422C23—21no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23#39012yes—
427C23C2×#12111yescharacteristic
428Q8—42no—
429Q8—42no—
430Q8—44no—
431Q8—82no—
432Q8—82no—
433Q8—82no—
434Q8—82no—
435Q8—81no—
436Q8—81no—
437Q8—81no—
438Q8—81no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4#39012yes—
469C2×C4#39012yes—
470C2×C4#39012yes—
471C2×C4C22×C22⋊C411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×#12411yescharacteristic
476C2×C4C2×#12711yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57711yescharacteristic
479C2×C4#57712yes—
480C2×C4#57811yescharacteristic
481C2×C4#57811yescharacteristic
482C2×C4#57812yes—
483C8—22no—
484C8—22no—
485C8—22no—
486C8—22no—
487C8—42no—
488C8—42no—
489C8—81no—
490C8—81no—
491C8—81no—
492C8—81no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—42no—
497C22—42no—
498C22C2×#39011yescharacteristic
499C22#304611yescharacteristic
500C22#304611yescharacteristic
501C22#306312yes—
502C22#306712yes—
503C4—21no—
504C4—21no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—82no—
514C4#289811yescharacteristic
515C4#289811yescharacteristic
516C4#294712yes—
517C4#304211yescharacteristic
518C4#304411yescharacteristic
519C4#315411yescharacteristic
520C4#315611yescharacteristic
521C2—42no—
522C2#1958512yes—
523C2#1958712yes—
524C2#2123411yescharacteristic
525C2#2124811yescharacteristic
526C2#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

Maximal subgroups

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

subgroupindex
C2×#32312
#218922
#218372
#205452
C2×#15742
C2×#15502
C2×#15422
#97482
#97452
#97412
#97392
#97372
#97362
#97302
#97292