Finite Group Databasebeta

← #206288

Subgroups — #206288

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3205C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21863C211yesmaximal, characteristic, complemented
4#21805C211yesmaximal, characteristic, complemented
5#21803C211yesmaximal, characteristic, complemented
6#21803C211yesmaximal, characteristic, complemented
7C2×#1454C211yesmaximal, characteristic, complemented
8#9838C211yesmaximal, characteristic, complemented
9#9837C211yesmaximal, characteristic, complemented
10#9832C211yesmaximal, characteristic, complemented
11#9832C211yesmaximal, characteristic, complemented
12#9830C211yesmaximal, characteristic, complemented
13#9827C211yesmaximal, characteristic, complemented
14#9824C211yesmaximal, characteristic, complemented
15#9822C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×#127C2211yescharacteristic
18#3205—24no—
19#3205C418yescomplemented
20C2×#493C2211yescharacteristic
21C2×#493C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#459—22no—
25C2×#447—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#1990C2211yescharacteristic, complemented
32#1990C2211yescharacteristic, complemented
33#1983—22no—
34#1982—22no—
35#1982—22no—
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1957C2211yescharacteristic
43#1957C2211yescharacteristic
44#1957C2211yescharacteristic, complemented
45#1957C2211yescharacteristic, complemented
46#1957C2211yescharacteristic, complemented
47#1957C2211yescharacteristic, complemented
48#1956C2211yescharacteristic, complemented
49#1956C2211yescharacteristic, complemented
50#1949—22no—
51#1875—22no—
52#1854—22no—
53#1845C2211yescharacteristic, complemented
54#1844C2211yescharacteristic, complemented
55#1834C2211yescharacteristic, complemented
56#1834C2211yescharacteristic, complemented
57#1804—22no—
58#1788—22no—
59#1786—22no—
60#1786—22no—
61#1454—24no—
62C2×#322C2211yescharacteristic, complemented
63C2×#322C2211yescharacteristic, complemented
64C2×#322C2211yescharacteristic, complemented
65C2×C82C2211yescharacteristic, complemented
66C22×#41—21no—
67C23×D4—21no—
68C23×D4—21no—
69C23×D4—21no—
70C22×D8—22no—
71C22×D8—22no—
72#532—22no—
73#532—22no—
74#532—22no—
75#532C2×C414yes—
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×#127C2311yescharacteristic
87C2×#124—21no—
88C2×#121—21no—
89C2×#121—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×#117—21no—
93C2×#117—21no—
94C22×C42C2311yescharacteristic
95#493—22no—
96#493—22no—
97#493C2×C414yes—
98#493C2×C414yes—
99#486—22no—
100#486—22no—
101#486C2×C414yes—
102#486C2×C414yes—
103#459—216no—
104#447—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×#102—22no—
129C2×#102—22no—
130C2×#102—22no—
131C2×#102—22no—
132C2×C22⋊C8D412yes—
133C2×C22⋊C8D412yes—
134C2×C22⋊C8D412yes—
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141#384C2311yescharacteristic
142#384C2311yescharacteristic
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#382—21no—
146#382—21no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
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—
164C2×#41—24no—
165C2×#41—42no—
166C2×#41—82no—
167C2×#41—82no—
168C22×Q8—21no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—41no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—82no—
189C22×D4—82no—
190C22×D4—82no—
191C22×D4—82no—
192C22×D4—82no—
193C22×D4—82no—
194C23×C4—41no—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C2×D8—28no—
199C2×D8—28no—
200C2×D8—82no—
201C2×D8—82no—
202C2×D8—82no—
203C2×D8—82no—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8C22×C412yes—
214#127—24no—
215#127—24no—
216#127—24no—
217#127—21no—
218#127—21no—
219#127—21no—
220#127—21no—
221#127—21no—
222#127—21no—
223#127—21no—
224#127—21no—
225#127—21no—
226#127—42no—
227#127—42no—
228#127—42no—
229#127C22×C412yes—
230#127C22×C412yes—
231#127C22×C412yes—
232#124—24no—
233#124—42no—
234#121—24no—
235#121—24no—
236#121—42no—
237#121—42no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—42no—
241C4×D4—42no—
242#117—82no—
243#117—82no—
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—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—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⋊C8—24no—
275C4⋊C8—24no—
276C4⋊C8C22⋊C418yes—
277#103—82no—
278#103—82no—
279#102—28no—
280#102—28no—
281#102—82no—
282#102—82no—
283#102—82no—
284#102—82no—
285#102—82no—
286#102—82no—
287#102—82no—
288#102—82no—
289#102—82no—
290#102—82no—
291#102—82no—
292#102—82no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295C22⋊C8C22⋊C418yes—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8C22×C412yes—
305C4×C8C22×C412yes—
306C4×C8C2×D412yes—
307C4×C8C2×D412yes—
308#95—41no—
309#95—41no—
310#95—41no—
311#95—41no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—44no—
316C24—82no—
317C24—82no—
318C24—82no—
319C24—82no—
320C24—82no—
321C24—82no—
322C24#12711yescharacteristic
323#41—88no—
324#41—88no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—82no—
382C2×D4—82no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—44no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4C2×C22⋊C412yes—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4#12412yes—
403C22×C4#12711yescharacteristic
404C22×C4#12711yescharacteristic
405C22×C4#12711yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409D8—88no—
410D8—88no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8C2×C22⋊C414yes—
428C2×C8C2×C22⋊C414yes—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C42—82no—
450C42—82no—
451C42C2×C22⋊C412yes—
452C42C4×D412yes—
453C42C4×D412yes—
454C42#12112yes—
455C42#12112yes—
456C42#12412yes—
457C42C23×C411yescharacteristic
458C42C22×D411yescharacteristic
459C42C2×#4111yescharacteristic
460C42C2×#4111yescharacteristic
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—88no—
471C23—88no—
472C23—88no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—164no—
492C23—164no—
493C23—164no—
494C23—164no—
495C23#39012yes—
496C23C2×#12711yescharacteristic
497Q8—44no—
498Q8—82no—
499D4—44no—
500D4—44no—
501D4—44no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—88no—
509D4—88no—
510D4—88no—
511D4—88no—
512D4—82no—
513D4—82no—
514D4—82no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—82no—
519D4—82no—
520D4—82no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—42no—
534C2×C4—88no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4#39012yes—
550C2×C4#39012yes—
551C2×C4#39012yes—
552C2×C4C22×C22⋊C411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×D411yescharacteristic
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×#12111yescharacteristic
557C2×C4C2×#12411yescharacteristic
558C2×C4#57612yes—
559C2×C4#57612yes—
560C2×C4#57612yes—
561C2×C4#57612yes—
562C2×C4#57712yes—
563C2×C4#57712yes—
564C8—24no—
565C8—24no—
566C8—42no—
567C8—42no—
568C8—42no—
569C8—42no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—1616no—
587C22—161no—
588C22—161no—
589C22—161no—
590C22—161no—
591C22—161no—
592C22—161no—
593C22—161no—
594C22—161no—
595C22—161no—
596C22—161no—
597C22—161no—
598C22—161no—
599C22C2×#39011yescharacteristic
600C22#315411yescharacteristic
601C22#315411yescharacteristic
602C22#315511yescharacteristic
603C22#315511yescharacteristic
604C22#315511yescharacteristic
605C22#315511yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—82no—
610C4—161no—
611C4—161no—
612C4—161no—
613C4—161no—
614C4#289712yes—
615C4#294612yes—
616C4#294812yes—
617C4#305812yes—
618C4#306212yes—
619C4#313612yes—
620C2—42no—
621C2—162no—
622C2—162no—
623C2—162no—
624C2#1973511yescharacteristic
625C2#1973511yescharacteristic
626C2#1973711yescharacteristic
627C2#1974311yescharacteristic
628C2#2120611yescharacteristic
629C2#2123311yescharacteristic
630C2#2123511yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3205C2
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. #21863C2semidirect
4. #21805C2semidirect
5. #21803C2semidirect
6. #21803C2semidirect
7. C2×#1454C2semidirect
8. #9838C2semidirect
9. #9837C2semidirect
10. #9832C2semidirect
11. #9832C2semidirect
12. #9830C2semidirect
13. #9827C2semidirect
14. #9824C2semidirect
15. #9822C2semidirect
19. #3205C4semidirect
31. #1990C22semidirect
32. #1990C22semidirect
44. #1957C22semidirect
45. #1957C22semidirect
46. #1957C22semidirect
47. #1957C22semidirect
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×#32052
#219842
#218632
#218052
#218032
C2×#14542
#98382
#98372
#98322
#98302
#98272
#98242
#98222