Finite Group Databasebeta

← #199060

Subgroups — #199060

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3208C211yesmaximal, characteristic
2#34210C211yesmaximal, characteristic
3#21329C211yesmaximal, characteristic, complemented
4C2×#1545C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic, complemented
6#9295C218yesmaximal, complemented
7#7957C211yesmaximal, characteristic, complemented
8#7854C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×#127C2211yescharacteristic
11C2×#518—21no—
12#3208—22no—
13#3208—22no—
14#3208C414yescomplemented
15#3208C2214yes—
16C2×#492C412yescomplemented
17C2×#486C412yescomplemented
18C2×#460—22no—
19C2×#450—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23#2366—22no—
24#2366—22no—
25D4×C22⋊C4—22no—
26D4×C22⋊C4—22no—
27#2328C2214yes—
28#2309—21no—
29C4×#127—22no—
30#2285C2211yescharacteristic
31C4×#117C2211yescharacteristic
32#1995C2214yescomplemented
33#1994—22no—
34#1733—21no—
35#1733—21no—
36#1545—22no—
37#1545C2214yes—
38C2×#331—22no—
39#1520—22no—
40#1518C2214yescomplemented
41#1503—22no—
42#1286C2214yescomplemented
43#1274C2214yescomplemented
44C23×D4—21no—
45C23×D4—21no—
46C23×D4D412yes—
47C22×SD16—24no—
48C22×M16—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533C2×C412yescomplemented
56#533C2312yes—
57C2×C4⋊Q8C2×C411yescharacteristic, complemented
58C2×C4⋊Q8C2×C411yescharacteristic, complemented
59C2×#127—21no—
60C2×#127—21no—
61C2×#127—21no—
62C2×#127—21no—
63C2×#127—21no—
64C2×#127—21no—
65C2×#127C2×C412yes—
66C2×#127C2×C412yes—
67C2×#127D412yes—
68C2×#127D412yes—
69C2×#127D412yes—
70C2×#127C2312yes—
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74#518—44no—
75#518—44no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C22⋊C4—24no—
87C22×C42C2311yescharacteristic
88#492—24no—
89#492C2×C418yescomplemented
90#486—24no—
91#486C2×C418yescomplemented
92#460—28no—
93#460—28no—
94#450—28no—
95#450—28no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431C2×C412yescomplemented
102#431C2312yes—
103#431C2312yes—
104#423—21no—
105#423—21no—
106#423—21no—
107#423C2312yes—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×#102—24no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2×C412yescomplemented
117#390—24no—
118#390—24no—
119#390—24no—
120#390—24no—
121#386—28no—
122#386—28no—
123#384—21no—
124#382C2312yes—
125#380—21no—
126#380D412yes—
127#379—24no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134#376C2312yes—
135C43D412yes—
136#339—24no—
137#339—24no—
138#331—28no—
139#328—28no—
140C25—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—84no—
163C22×D4—84no—
164C22×D4C22⋊C414yes—
165C22×D4C22⋊C414yes—
166C22×D4C2×D414yes—
167C23×C4—28no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C2×SD16—216no—
172C2×SD16—88no—
173C2×M16—22no—
174C2×M16—41no—
175C2×M16—42no—
176C2×M16—42no—
177C22×C8—22no—
178C22×C8—41no—
179C22×C8—41no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22×C412yes—
187C4⋊Q8C22×C412yes—
188#127—22no—
189#127—22no—
190#127—22no—
191#127—22no—
192#127—22no—
193#127—22no—
194#127—22no—
195#127—22no—
196#127—22no—
197#127—22no—
198#127—22no—
199#127—22no—
200#127C22⋊C414yes—
201#127C22⋊C414yes—
202#127C22×C414yes—
203#127C2×D414yes—
204#125—22no—
205#125—22no—
206#125—41no—
207#125—41no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—41no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×D4—48no—
225C4×D4—416no—
226C4×D4—48no—
227C4×D4—88no—
228C4×D4—88no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—41no—
235#117—41no—
236#117—42no—
237#117—42no—
238#117SD1614yescomplemented
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—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—416no—
257C2×C22⋊C4—41no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—41no—
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42C2411yescharacteristic
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—41no—
282C4⋊C8—41no—
283C4⋊C8—41no—
284#102—216no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—41no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—41no—
292C4×C8C22×C414yes—
293#95—22no—
294#95—22no—
295#95—22no—
296#95C2×D414yes—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—44no—
301C24—44no—
302C24—84no—
303C24—84no—
304C24#12011yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—44no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—82no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—84no—
340C2×D4—84no—
341C2×D4—84no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4C2×C22⋊C418yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—416no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—88no—
358C22×C4—88no—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12011yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12311yescharacteristic
373C22×C4#12311yescharacteristic
374C22×C4C2×SD1612yes—
375C22×C4#13512yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4#13712yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381SD16—816no—
382M16—42no—
383M16—42no—
384M16—82no—
385C2×C8—24no—
386C2×C8—44no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—82no—
390C2×C8—82no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C22⋊C4—24no—
410C22⋊C4—416no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—42no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—44no—
421C42—44no—
422C42—42no—
423C42—42no—
424C42—44no—
425C42C2×C22⋊C411yescharacteristic
426C42C2×C22⋊C411yescharacteristic
427C42C4×D411yescharacteristic
428C42C4×D411yescharacteristic
429C42C4×D411yescharacteristic
430C42C4×D411yescharacteristic
431C42#12111yescharacteristic
432C42#12111yescharacteristic
433C42#12111yescharacteristic
434C42#12111yescharacteristic
435C42#12311yescharacteristic
436C42#12311yescharacteristic
437C42C2×SD1614yescomplemented
438C42C23×C411yescharacteristic
439C42C22×D411yescharacteristic
440C42C2×#4111yescharacteristic
441C42C2×#4111yescharacteristic
442C23—28no—
443C23—41no—
444C23—41no—
445C23—48no—
446C23—48no—
447C23—44no—
448C23—44no—
449C23—48no—
450C23—48no—
451C23—84no—
452C23—84no—
453C23—84no—
454C23—84no—
455C23—168no—
456C23—168no—
457C23#38611yescharacteristic
458C23#38611yescharacteristic
459C23#45012yes—
460C23#45212yes—
461C23C2×#12011yescharacteristic
462Q8—44no—
463Q8—82no—
464Q8—82no—
465Q8—82no—
466D4—48no—
467D4—48no—
468D4—48no—
469D4—48no—
470D4—48no—
471D4—48no—
472D4—48no—
473D4—44no—
474D4—44no—
475D4—44no—
476D4—44no—
477D4—84no—
478D4—84no—
479D4—84no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—1616no—
506C2×C4#38611yescharacteristic
507C2×C4#38611yescharacteristic
508C2×C4#38611yescharacteristic
509C2×C4#38611yescharacteristic
510C2×C4#38611yescharacteristic
511C2×C4#38611yescharacteristic
512C2×C4C4×SD1612yes—
513C2×C4C4×SD1612yes—
514C2×C4#44012yes—
515C2×C4#44012yes—
516C2×C4#45012yes—
517C2×C4#45212yes—
518C2×C4#46112yes—
519C2×C4#46112yes—
520C2×C4#46312yes—
521C2×C4#46312yes—
522C2×C4C22×C22⋊C411yescharacteristic
523C2×C4C2×C4×D411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×#12111yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4C2×#12311yescharacteristic
528C2×C4C22×SD1611yescharacteristic
529C2×C4C2×#13511yescharacteristic
530C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
531C2×C4C2×#13711yescharacteristic
532C8—44no—
533C8—82no—
534C8—82no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—42no—
542C22—42no—
543C22—48no—
544C22—48no—
545C22—48no—
546C22—88no—
547C22—88no—
548C22—168no—
549C22—168no—
550C22#187712yes—
551C22#187712yes—
552C22C2×#38611yescharacteristic
553C22C2×#45011yescharacteristic
554C22C2×#45211yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—44no—
562C4—82no—
563C4—84no—
564C4—82no—
565C4—82no—
566C4#187712yes—
567C4#187712yes—
568C4#190412yes—
569C4#191112yes—
570C4C2×C4×SD1611yescharacteristic
571C4C2×#44011yescharacteristic
572C4C2×#46111yescharacteristic
573C4C2×#46311yescharacteristic
574C2—21no—
575C2—21no—
576C2—22no—
577C2—22no—
578C2—88no—
579C2C2×#187711yescharacteristic
580C2C2×#190411yescharacteristic
581C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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
3. #21329C2semidirect
5. #17575C2semidirect
6. #9295C2semidirect
7. #7957C2semidirect
8. #7854C2semidirect
14. #3208C4semidirect
16. C2×#492C4semidirect
17. C2×#486C4semidirect
32. #1995C22semidirect
40. #1518C22semidirect
42. #1286C22semidirect
43. #1274C22semidirect
55. #533C2×C4semidirect
57. C2×C4⋊Q8C2×C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
89. #492C2×C4semidirect
91. #486C2×C4semidirect
101. #431C2×C4semidirect
116. C2×C4×C8C2×C4semidirect
238. #117SD16semidirect
437. C42C2×SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32082
#342102
#213292
C2×#15452
#175752
#92952
#79572
#78542