Finite Group Databasebeta

← #206286

Subgroups — #206286

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3208C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21864C211yesmaximal, characteristic, complemented
4#21835C211yesmaximal, characteristic, complemented
5#21803C211yesmaximal, characteristic
6#21803C211yesmaximal, characteristic
7C2×#1454C211yesmaximal, characteristic, complemented
8#9842C211yesmaximal, characteristic, complemented
9#9838C211yesmaximal, characteristic, complemented
10#9830C211yesmaximal, characteristic, complemented
11#9829C211yesmaximal, characteristic, complemented
12#9829C211yesmaximal, characteristic, complemented
13#9827C211yesmaximal, characteristic, complemented
14#9822C211yesmaximal, characteristic, complemented
15#9820C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×#127C2211yescharacteristic
18#3208—24no—
19#3208C418yescomplemented
20C2×#492C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#460—22no—
25C2×#450—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#1990C2211yescharacteristic, complemented
32#1990C2211yescharacteristic, complemented
33#1990C2211yescharacteristic, complemented
34#1990C2211yescharacteristic, complemented
35#1990C2211yescharacteristic, complemented
36#1990C2211yescharacteristic, complemented
37#1989—22no—
38#1982—22no—
39#1982—22no—
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1957C2211yescharacteristic
43#1957C2211yescharacteristic
44#1957C2211yescharacteristic
45#1957C2211yescharacteristic
46#1957C2211yescharacteristic
47#1957C2211yescharacteristic
48#1956C2211yescharacteristic, complemented
49#1956C2211yescharacteristic, complemented
50#1949—22no—
51#1873—22no—
52#1854—22no—
53#1845C2211yescharacteristic, complemented
54#1844C2211yescharacteristic, complemented
55#1834C2211yescharacteristic, complemented
56#1834C2211yescharacteristic, complemented
57#1825—22no—
58#1805—22no—
59#1786—22no—
60#1786—22no—
61#1454—24no—
62C2×#322C2211yescharacteristic, complemented
63C2×#322C2211yescharacteristic, complemented
64C2×#322C2211yescharacteristic, complemented
65C2×C82C2211yescharacteristic, complemented
66C23×D4—21no—
67C23×D4—21no—
68C23×D4—21no—
69C22×SD16—22no—
70C22×SD16—22no—
71#533—22no—
72#533—22no—
73#533—22no—
74#533C2×C414yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#127—22no—
78C2×#127—21no—
79C2×#127—21no—
80C2×#127—21no—
81C2×#127—22no—
82C2×#127—22no—
83C2×#127C2×C414yes—
84C2×#127C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#125—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×#117—21no—
92C2×#117—21no—
93C2×#117—21no—
94C22×C42C2311yescharacteristic
95#492—22no—
96#492—22no—
97#492C2×C414yes—
98#492C2×C414yes—
99#486—22no—
100#486—22no—
101#486C2×C414yes—
102#486C2×C414yes—
103#460—216no—
104#450—216no—
105#431—22no—
106#431—22no—
107#431—22no—
108#431—22no—
109#431C2×C414yes—
110#431D414yes—
111C2×#107—22no—
112C2×#106—22no—
113C2×C4⋊C8D412yes—
114C2×C4⋊C8D412yes—
115C2×C4⋊C8D412yes—
116C2×#103—22no—
117C2×#103—22no—
118C2×#102—22no—
119C2×#102—22no—
120C2×#102—22no—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×#102—22no—
127C2×#102—22no—
128C2×C22⋊C8D412yes—
129C2×C22⋊C8D412yes—
130C2×C22⋊C8D412yes—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#382—21no—
142#382—21no—
143#380—21no—
144#380—21no—
145#380—21no—
146#380—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
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—
161#322—22no—
162C82—22no—
163C25—41no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—82no—
182C22×D4—82no—
183C22×D4—82no—
184C22×D4—82no—
185C22×D4—82no—
186C22×D4—82no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C2×SD16—28no—
191C2×SD16—28no—
192C2×SD16—82no—
193C2×SD16—82no—
194C2×SD16—82no—
195C2×SD16—82no—
196C22×C8C2×D412yes—
197C22×C8C2×D412yes—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C22×C412yes—
209C4⋊Q8C22×C412yes—
210#127—24no—
211#127—24no—
212#127—24no—
213#127—21no—
214#127—21no—
215#127—21no—
216#127—21no—
217#127—21no—
218#127—21no—
219#127—42no—
220#127—42no—
221#127—42no—
222#127C22×C412yes—
223#127C22×C412yes—
224#125—24no—
225#125—42no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—42no—
229C22⋊Q8—42no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—42no—
233C4×Q8—42no—
234#117—24no—
235#117—42no—
236#117—82no—
237#117—82no—
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×C42—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42C22×C412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4112yes—
268C2×C42#4112yes—
269C2×C42C2411yescharacteristic
270#107—82no—
271#106—82no—
272C4⋊C8—24no—
273C4⋊C8—24no—
274C4⋊C8C22⋊C418yes—
275#103—82no—
276#103—82no—
277#102—28no—
278#102—28no—
279#102—82no—
280#102—82no—
281#102—82no—
282#102—82no—
283#102—82no—
284#102—82no—
285#102—82no—
286#102—82no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8C22⋊C418yes—
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×C8C22×C412yes—
299C4×C8C22×C412yes—
300C4×C8C2×D412yes—
301C4×C8C2×D412yes—
302#95—41no—
303#95—41no—
304#95—41no—
305#95—41no—
306C24—41no—
307C24—41no—
308C24—44no—
309C24—82no—
310C24—82no—
311C24—82no—
312C24—82no—
313C24—82no—
314C24—82no—
315C24#12711yescharacteristic
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—88no—
352C2×D4—88no—
353C2×D4—88no—
354C2×D4—82no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—88no—
361C2×D4—88no—
362C2×D4—88no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4C2×C22⋊C412yes—
376C22×C4C4×D412yes—
377C22×C4C4×D412yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12412yes—
381C22×C4#12711yescharacteristic
382C22×C4#12711yescharacteristic
383C22×C4#12711yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387SD16—88no—
388SD16—88no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8C2×C22⋊C414yes—
406C2×C8C2×C22⋊C414yes—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C42—44no—
433C42—82no—
434C42—82no—
435C42C2×C22⋊C412yes—
436C42C4×D412yes—
437C42C4×D412yes—
438C42#12112yes—
439C42#12112yes—
440C42#12412yes—
441C42C23×C411yescharacteristic
442C42C22×D411yescharacteristic
443C42C2×#4111yescharacteristic
444C42C2×#4111yescharacteristic
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—42no—
452C23—42no—
453C23—88no—
454C23—88no—
455C23—88no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—164no—
469C23—164no—
470C23—164no—
471C23—164no—
472C23#39012yes—
473C23C2×#12711yescharacteristic
474Q8—44no—
475Q8—44no—
476Q8—82no—
477Q8—82no—
478D4—44no—
479D4—44no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—44no—
484D4—88no—
485D4—88no—
486D4—88no—
487D4—82no—
488D4—82no—
489D4—82no—
490D4—82no—
491D4—82no—
492D4—82no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4#39012yes—
524C2×C4#39012yes—
525C2×C4#39012yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12411yescharacteristic
532C2×C4#57612yes—
533C2×C4#57612yes—
534C2×C4#57712yes—
535C2×C4#57712yes—
536C2×C4#57712yes—
537C2×C4#57712yes—
538C8—24no—
539C8—24no—
540C8—42no—
541C8—42no—
542C8—42no—
543C8—42no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—1616no—
558C22—161no—
559C22—161no—
560C22—161no—
561C22—161no—
562C22—161no—
563C22—161no—
564C22—161no—
565C22—161no—
566C22C2×#39011yescharacteristic
567C22#315411yescharacteristic
568C22#315411yescharacteristic
569C22#315411yescharacteristic
570C22#315411yescharacteristic
571C22#315511yescharacteristic
572C22#315511yescharacteristic
573C4—22no—
574C4—22no—
575C4—22no—
576C4—82no—
577C4—82no—
578C4—161no—
579C4—161no—
580C4—161no—
581C4—161no—
582C4#289712yes—
583C4#294612yes—
584C4#294912yes—
585C4#305712yes—
586C4#306212yes—
587C4#313612yes—
588C2—42no—
589C2—162no—
590C2—162no—
591C2#1973311yescharacteristic
592C2#1973511yescharacteristic
593C2#1973711yescharacteristic
594C2#1973711yescharacteristic
595C2#2120611yescharacteristic
596C2#2123111yescharacteristic
597C2#2123511yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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
3. #21864C2semidirect
4. #21835C2semidirect
7. C2×#1454C2semidirect
8. #9842C2semidirect
9. #9838C2semidirect
10. #9830C2semidirect
11. #9829C2semidirect
12. #9829C2semidirect
13. #9827C2semidirect
14. #9822C2semidirect
15. #9820C2semidirect
19. #3208C4semidirect
31. #1990C22semidirect
32. #1990C22semidirect
33. #1990C22semidirect
34. #1990C22semidirect
35. #1990C22semidirect
36. #1990C22semidirect
48. #1956C22semidirect
49. #1956C22semidirect
53. #1845C22semidirect
54. #1844C22semidirect
55. #1834C22semidirect
56. #1834C22semidirect
62. C2×#322C22semidirect
63. C2×#322C22semidirect
64. C2×#322C22semidirect
65. C2×C82C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32082
#219842
#218642
#218352
#218032
C2×#14542
#98422
#98382
#98302
#98292
#98272
#98222
#98202