Finite Group Databasebeta

← #206157

Subgroups — #206157

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3210C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21410C211yesmaximal, characteristic
4#21409C211yesmaximal, characteristic, complemented
5#21401C211yesmaximal, characteristic
6#21318C211yesmaximal, characteristic, complemented
7C2×C8×M16C211yesmaximal, characteristic, complemented
8#9835C211yesmaximal, characteristic, complemented
9#9830C211yesmaximal, characteristic, complemented
10#9829C211yesmaximal, characteristic, complemented
11#9826C211yesmaximal, characteristic, complemented
12#9825C211yesmaximal, characteristic, complemented
13#9822C211yesmaximal, characteristic, complemented
14#9821C211yesmaximal, characteristic, complemented
15#9820C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3210—24no—
19#3210C418yescomplemented
20C2×#494C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#464—22no—
25C2×#452—22no—
26C2×#431C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#1991—22no—
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1990C2211yescharacteristic
42#1957C2211yescharacteristic
43#1957C2211yescharacteristic
44#1957C2211yescharacteristic
45#1957C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1949—22no—
49#1895—22no—
50#1894—22no—
51#1893—22no—
52#1854—22no—
53#1845C2211yescharacteristic
54#1844C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57#1763—22no—
58#1762—22no—
59C4×#102—22no—
60#1735—22no—
61C8×M16—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C22×#41—21no—
67C22×Q16—22no—
68C22×SD16—22no—
69C22×M16D412yes—
70C23×C8D412yes—
71#533—22no—
72#533—22no—
73#533—22no—
74#533C2×C414yes—
75#532—22no—
76#532—22no—
77#532—22no—
78#532C2×C414yes—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#127C2311yescharacteristic
83C2×#125—21no—
84C2×#124—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×#121—21no—
88C2×#121—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×#117—21no—
94C2×#117—21no—
95C2×#117—21no—
96C22×C42C2311yescharacteristic
97#494—22no—
98#494C2×C414yes—
99#492—22no—
100#492C2×C414yes—
101#487—22no—
102#487C2×C414yes—
103#486—22no—
104#486C2×C414yes—
105#464—216no—
106#452—216no—
107#431C2×C414yes—
108#431D414yes—
109C2×#107—22no—
110C2×#106—22no—
111C2×C4⋊C8D412yes—
112C2×#103—22no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#102—22no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—22no—
120C2×C22⋊C8D412yes—
121C4×M16—22no—
122C4×M16—22no—
123C2×#97D412yes—
124C2×C4×C8—22no—
125C2×C4×C8—22no—
126C2×C4×C8D412yes—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#382—21no—
138#382—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#322—22no—
160#322—22no—
161C82—22no—
162C82—22no—
163C2×#41—24no—
164C2×#41—42no—
165C2×#41—82no—
166C2×#41—82no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—41no—
174C23×C4—41no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C2×Q16—28no—
179C2×Q16—82no—
180C2×Q16—82no—
181C2×SD16—28no—
182C2×SD16—82no—
183C2×SD16—82no—
184C2×M16—24no—
185C2×M16—24no—
186C2×M16—24no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8—42no—
190C22×C8C2×D412yes—
191C22×C8C2×D412yes—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C22×C8C2×D412yes—
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⋊Q8C22×C412yes—
206C4⋊Q8C22×C412yes—
207C4⋊Q8C22×C412yes—
208#127—21no—
209#127—21no—
210#127—21no—
211#127C22×C412yes—
212#125—24no—
213#125—42no—
214#124—24no—
215#124—42no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220#121—24no—
221#121—24no—
222#121—42no—
223#121—42no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—42no—
227C4×Q8—42no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—42no—
231C4×D4—42no—
232#117—24no—
233#117—42no—
234#117—44no—
235#117—82no—
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—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42C22×C412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42#4112yes—
270C2×C42#4112yes—
271C2×C42C2411yescharacteristic
272#107—82no—
273#106—82no—
274C4⋊C8C22⋊C418yes—
275#103—28no—
276#103—44no—
277#103—44no—
278#102—28no—
279#102—44no—
280#102—44no—
281#102—44no—
282#102—44no—
283C22⋊C8C22⋊C418yes—
284#97—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×C8C22×C412yes—
295C4×C8C22×C412yes—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298#95—41no—
299#95—41no—
300#95—41no—
301#95—41no—
302C24—41no—
303C24#12711yescharacteristic
304#41—88no—
305#41—88no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—44no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—82no—
332C2×D4—82no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4C2×C22⋊C412yes—
350C22×C4C4×D412yes—
351C22×C4C4×D412yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#12412yes—
355C22×C4#12711yescharacteristic
356C22×C4#12711yescharacteristic
357C22×C4#12711yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361Q16—88no—
362SD16—88no—
363M16—216no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—48no—
378C2×C8C2×C22⋊C414yes—
379C2×C8C2×C22⋊C414yes—
380C2×C8#13514yes—
381C2×C8#13514yes—
382C2×C8#13714yes—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C42—44no—
415C42—82no—
416C42—82no—
417C42C2×C22⋊C412yes—
418C42C4×D412yes—
419C42C4×D412yes—
420C42#12112yes—
421C42#12112yes—
422C42#12412yes—
423C42C23×C411yescharacteristic
424C42C22×D411yescharacteristic
425C42C2×#4111yescharacteristic
426C42C2×#4111yescharacteristic
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—42no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23#39012yes—
441C23C2×#12711yescharacteristic
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—82no—
446Q8—82no—
447Q8—82no—
448D4—44no—
449D4—44no—
450D4—44no—
451D4—88no—
452D4—82no—
453D4—82no—
454D4—82no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—88no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4#39012yes—
490C2×C4#39012yes—
491C2×C4#39012yes—
492C2×C4C22×C22⋊C411yescharacteristic
493C2×C4C2×C4×D411yescharacteristic
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12411yescharacteristic
498C2×C4C2×#13512yes—
499C2×C4C2×#13512yes—
500C2×C4C2×#13712yes—
501C2×C4#57512yes—
502C2×C4#57612yes—
503C2×C4#57712yes—
504C8—24no—
505C8—24no—
506C8—24no—
507C8—24no—
508C8—24no—
509C8—42no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22C2×#39011yescharacteristic
525C22#314711yescharacteristic
526C22#314711yescharacteristic
527C22#315211yescharacteristic
528C22#315311yescharacteristic
529C22#315411yescharacteristic
530C22#315511yescharacteristic
531C4—22no—
532C4—22no—
533C4—22no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—161no—
538C4—161no—
539C4—161no—
540C4—161no—
541C4#289712yes—
542C4C4×#13512yes—
543C4C4×#13712yes—
544C4#305012yes—
545C4#305612yes—
546C4#313612yes—
547C2—42no—
548C2—162no—
549C2#1964211yescharacteristic
550C2#1964411yescharacteristic
551C2#1964611yescharacteristic
552C2#1965011yescharacteristic
553C2#2120611yescharacteristic
554C2#2122411yescharacteristic
555C2#2123711yescharacteristic
subgroupG/H
center, socleC23C2×#127
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
2. #21984C2semidirect
4. #21409C2semidirect
6. #21318C2semidirect
7. C2×C8×M16C2semidirect
8. #9835C2semidirect
9. #9830C2semidirect
10. #9829C2semidirect
11. #9826C2semidirect
12. #9825C2semidirect
13. #9822C2semidirect
14. #9821C2semidirect
15. #9820C2semidirect
19. #3210C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32102
#219842
#214102
#214092
#214012
#213182
C2×C8×M162
#98352
#98302
#98292
#98262
#98252
#98222
#98212
#98202