Finite Group Databasebeta

← #205550

Subgroups — #205550

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3131C211yesmaximal, characteristic
2#20818C211yesmaximal, characteristic, complemented
3#17590C211yesmaximal, characteristic, complemented
4#17589C211yesmaximal, characteristic
5#17584C211yesmaximal, characteristic, complemented
6#17583C211yesmaximal, characteristic
7#17323C211yesmaximal, characteristic, complemented
8#9758C214yesmaximal, complemented
9#9757C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3131—21no—
13#3131—21no—
14#3131—21no—
15#3131—21no—
16#3131C412yescomplemented
17#3131C412yescomplemented
18#3131C2212yes—
19#3131C2212yes—
20C2×#488—21no—
21C2×#488—21no—
22C2×#487C411yescharacteristic, complemented
23C2×#487C411yescharacteristic, complemented
24C2×#486C411yescharacteristic, complemented
25C2×#486C411yescharacteristic, complemented
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#416—21no—
29C2×#416—21no—
30C22×C4×C8C2211yescharacteristic
31#2283C2211yescharacteristic
32#2283C2211yescharacteristic
33#1846C2212yescomplemented
34#1835C2212yescomplemented
35#1808—21no—
36#1807—21no—
37C4×#107—21no—
38C4×#106—21no—
39#1518C2214yescomplemented
40#1518C2214yescomplemented
41#1517C2214yes—
42#1517C2214yes—
43#1515—22no—
44#1514—22no—
45#1511—22no—
46#1510—22no—
47#1508—22no—
48#1507—22no—
49#1506—22no—
50#1505—22no—
51#1452C2212yescomplemented
52#1451—21no—
53#1451—21no—
54#1450C2212yescomplemented
55C8×M16—21no—
56C8×M16—21no—
57C22×#41—21no—
58C23×C8D411yescharacteristic
59C23×C8D411yescharacteristic
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66#533C2×C412yes—
67#533C2312yes—
68#532—21no—
69#532—21no—
70#532—21no—
71#532—21no—
72#532—21no—
73#532—21no—
74#532C2×C412yes—
75#532C2312yes—
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#127C2×C411yescharacteristic
80C2×#125—21no—
81C2×#124—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×#121—21no—
85C2×#121—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×#117—21no—
91C22×C42C2311yescharacteristic
92#488—22no—
93#488—22no—
94#488—22no—
95#488—22no—
96#488—22no—
97#488—22no—
98#488—22no—
99#488—22no—
100#487—21no—
101#487—21no—
102#487—21no—
103#487—21no—
104#487C2×C412yes—
105#487C2×C412yes—
106#487C2×C412yes—
107#487C2×C412yes—
108#486—21no—
109#486—21no—
110#486—21no—
111#486—21no—
112#486C2×C412yes—
113#486C2×C412yes—
114#486C2×C412yes—
115#486C2×C412yes—
116#432—21no—
117#432—21no—
118#432D412yes—
119#432D412yes—
120#431—21no—
121#431—21no—
122#431C2312yes—
123#431C2312yes—
124C2×#107—22no—
125C2×#106—22no—
126#423—21no—
127#423—21no—
128#416—22no—
129#416—22no—
130#416—22no—
131#416—22no—
132#416—22no—
133#416—22no—
134#416—22no—
135#416—22no—
136C2×#103—21no—
137C2×#103—21no—
138C2×#103—21no—
139C2×#103—21no—
140C2×#102—21no—
141C2×#102—21no—
142C2×#102—21no—
143C2×#102—21no—
144#407D412yes—
145#407D412yes—
146C4×M16—21no—
147C4×M16—21no—
148C2×C4×C8C2×C411yescharacteristic
149C2×C4×C8C2×C411yescharacteristic
150C2×C4×C8C2×C411yescharacteristic
151C2×C4×C8C2×C411yescharacteristic
152C2×C4×C8D411yescharacteristic
153C2×C4×C8D411yescharacteristic
154C2×C4×C8D411yescharacteristic
155C2×C4×C8D411yescharacteristic
156C2×C4×C8D411yescharacteristic
157C2×C4×C8D411yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160#383—21no—
161#383—21no—
162#382C2312yes—
163#382C2312yes—
164#381—21no—
165#381—21no—
166#380—21no—
167#380—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4—21no—
172#376—21no—
173#376—21no—
174#340—22no—
175#340—22no—
176#328—28no—
177#328—28no—
178#327—28no—
179#327—28no—
180#322—22no—
181#322—22no—
182#322—22no—
183C82—22no—
184C2×#41—22no—
185C2×#41—22no—
186C2×#41—44no—
187C2×#41—44no—
188C2×#41—41no—
189C2×#41—41no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—41no—
197C23×C4—41no—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C2×M16—22no—
202C2×M16—22no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8C22⋊C411yescharacteristic
206C22×C8C22⋊C411yescharacteristic
207C22×C8C22⋊C411yescharacteristic
208C22×C8C22⋊C411yescharacteristic
209C22×C8C22⋊C411yescharacteristic
210C22×C8C22⋊C411yescharacteristic
211C22×C8C22⋊C411yescharacteristic
212C22×C8C22⋊C411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8C22×C412yes—
227C4⋊Q8C22×C412yes—
228C4⋊Q8C22×C412yes—
229#127—21no—
230#127—21no—
231#127—21no—
232#127C22×C412yes—
233#125—22no—
234#125—22no—
235#125—41no—
236#125—41no—
237#124—22no—
238#124—22no—
239#124—41no—
240#124—41no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247C22⋊Q8—41no—
248C22⋊Q8—41no—
249#121—22no—
250#121—22no—
251#121—22no—
252#121—22no—
253#121—41no—
254#121—41no—
255#121—41no—
256#121—41no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—41no—
262C4×Q8—41no—
263C4×Q8—41no—
264C4×Q8—41no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4—41no—
273#117—22no—
274#117—22no—
275#117—41no—
276#117—41no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42—41no—
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42#4111yescharacteristic
305C2×C42#4111yescharacteristic
306C2×C42C2411yescharacteristic
307#107—44no—
308#106—44no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313#103—22no—
314#103—22no—
315#103—22no—
316#103—22no—
317#103—22no—
318#103—22no—
319#103—22no—
320#103—22no—
321#102—22no—
322#102—22no—
323#102—22no—
324#102—22no—
325#102—22no—
326#102—22no—
327#102—22no—
328#102—22no—
329C22⋊C8—22no—
330C22⋊C8—22no—
331C22⋊C8C2×D414yes—
332C22⋊C8C2×D414yes—
333#97—22no—
334#97—22no—
335C4×C8—22no—
336C4×C8—22no—
337C4×C8C22⋊C411yescharacteristic
338C4×C8C22⋊C411yescharacteristic
339C4×C8C22⋊C411yescharacteristic
340C4×C8C22⋊C411yescharacteristic
341C4×C8C22⋊C411yescharacteristic
342C4×C8C22⋊C411yescharacteristic
343C4×C8C22⋊C411yescharacteristic
344C4×C8C22⋊C411yescharacteristic
345C4×C8C22×C411yescharacteristic
346C4×C8C22×C411yescharacteristic
347C4×C8C22×C411yescharacteristic
348C4×C8C22×C411yescharacteristic
349C4×C8C2×D411yescharacteristic
350C4×C8C2×D411yescharacteristic
351C4×C8C2×D411yescharacteristic
352C4×C8C2×D411yescharacteristic
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—41no—
358#95—41no—
359#95—41no—
360#95—41no—
361C24—41no—
362C24#12711yescharacteristic
363#41—44no—
364#41—44no—
365#41—44no—
366#41—44no—
367#41—84no—
368#41—84no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—84no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—84no—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12411yescharacteristic
415C22×C4#12411yescharacteristic
416C22×C4#12711yescharacteristic
417C22×C4#12711yescharacteristic
418C22×C4#12711yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422M16—44no—
423M16—44no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8C2×C22⋊C411yescharacteristic
439C2×C8C2×C22⋊C411yescharacteristic
440C2×C8C2×C22⋊C411yescharacteristic
441C2×C8C2×C22⋊C411yescharacteristic
442C2×C8C2×C22⋊C411yescharacteristic
443C2×C8C2×C22⋊C411yescharacteristic
444C2×C8C2×C22⋊C411yescharacteristic
445C2×C8C2×C22⋊C411yescharacteristic
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C42—42no—
475C42—42no—
476C42—82no—
477C42—82no—
478C42C2×C22⋊C411yescharacteristic
479C42C2×C22⋊C411yescharacteristic
480C42C4×D411yescharacteristic
481C42C4×D411yescharacteristic
482C42C4×D411yescharacteristic
483C42C4×D411yescharacteristic
484C42#12111yescharacteristic
485C42#12111yescharacteristic
486C42#12111yescharacteristic
487C42#12111yescharacteristic
488C42#12411yescharacteristic
489C42#12411yescharacteristic
490C42C23×C411yescharacteristic
491C42C22×D411yescharacteristic
492C42C2×#4111yescharacteristic
493C42C2×#4111yescharacteristic
494C23—22no—
495C23—22no—
496C23—41no—
497C23—42no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23#39011yescharacteristic
502C23#39011yescharacteristic
503C23C2×#12711yescharacteristic
504Q8—42no—
505Q8—42no—
506Q8—82no—
507Q8—82no—
508Q8—82no—
509Q8—82no—
510Q8—82no—
511D4—42no—
512D4—42no—
513D4—82no—
514D4—82no—
515D4—82no—
516D4—82no—
517D4—84no—
518D4—84no—
519D4—82no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—84no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—84no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4#39011yescharacteristic
554C2×C4#39011yescharacteristic
555C2×C4#39011yescharacteristic
556C2×C4#39011yescharacteristic
557C2×C4#39011yescharacteristic
558C2×C4#39011yescharacteristic
559C2×C4C22×C22⋊C411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12411yescharacteristic
565C2×C4#57512yes—
566C2×C4#57512yes—
567C8—21no—
568C8—21no—
569C8—21no—
570C8—21no—
571C8—21no—
572C8—21no—
573C8—21no—
574C8—21no—
575C8—44no—
576C8—44no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—164no—
589C22#197612yes—
590C22C2×#39011yescharacteristic
591C22#315312yes—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—162no—
605C4—162no—
606C4#197612yes—
607C4#289611yescharacteristic
608C4#289611yescharacteristic
609C4#314012yes—
610C2—22no—
611C2—22no—
612C2—41no—
613C2—162no—
614C2#2120411yescharacteristic
615C2C2×#197611yescharacteristic
616C2C2×#197611yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20818C2semidirect
3. #17590C2semidirect
5. #17584C2semidirect
7. #17323C2semidirect
8. #9758C2semidirect
9. #9757C2semidirect
16. #3131C4semidirect
17. #3131C4semidirect
22. C2×#487C4semidirect
23. C2×#487C4semidirect
24. C2×#486C4semidirect
25. C2×#486C4semidirect
33. #1846C22semidirect
34. #1835C22semidirect
39. #1518C22semidirect
40. #1518C22semidirect
51. #1452C22semidirect
54. #1450C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31312
#208182
#175902
#175892
#175842
#175832
#173232
#97582
#97572