Finite Group Databasebeta

← #199063

Subgroups — #199063

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3205C211yesmaximal, characteristic
2#34210C211yesmaximal, characteristic, complemented
3#21209C211yesmaximal, characteristic, complemented
4C2×#1545C211yesmaximal, characteristic, complemented
5#17574C211yesmaximal, characteristic, complemented
6#9299C218yesmaximal, complemented
7#7957C211yesmaximal, characteristic, complemented
8#7854C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×#127C2211yescharacteristic
11C2×#518—21no—
12#3205—22no—
13#3205—22no—
14#3205C414yescomplemented
15#3205C2214yes—
16C2×#493C412yescomplemented
17C2×#486C412yescomplemented
18C2×#459—22no—
19C2×#447—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#421—21no—
24C2×#421—21no—
25#2366—22no—
26#2366—22no—
27D4×C22⋊C4—22no—
28D4×C22⋊C4—22no—
29#2328C2214yescomplemented
30#2309—21no—
31C4×#127—22no—
32#2285C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1963C2214yescomplemented
35#1962—22no—
36#1545—22no—
37#1545C2214yescomplemented
38C2×#331—22no—
39#1519—22no—
40#1517C2214yescomplemented
41#1502—22no—
42#1286C2214yescomplemented
43#1274C2214yescomplemented
44C22×#41—21no—
45C23×D4—21no—
46C23×D4—21no—
47C23×D4D412yes—
48C22×D8—24no—
49C22×M16—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532C2×C412yescomplemented
57#532C2312yes—
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×#127C2×C411yescharacteristic, complemented
68C2×#127D412yes—
69C2×#127D412yes—
70C2×#127D412yes—
71C2×#127C2312yes—
72C2×#124—21no—
73C2×#121—21no—
74C2×#121—21no—
75#518—44no—
76#518—44no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×#117—21no—
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C22⋊C4—24no—
87C22×C42C2311yescharacteristic
88#493—24no—
89#493C2×C418yescomplemented
90#486—24no—
91#486C2×C418yescomplemented
92#459—28no—
93#459—28no—
94#447—28no—
95#447—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—
112#421—44no—
113#421—44no—
114C2×#104—24no—
115C2×#104—24no—
116C2×#102—24no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2×C412yescomplemented
121#390—24no—
122#390—24no—
123#390—24no—
124#390—24no—
125#386—28no—
126#386—28no—
127#384—21no—
128#382C2312yes—
129#380—21no—
130#380D412yes—
131#379—24no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4D412yes—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138#376C2312yes—
139C43D412yes—
140#331—28no—
141#327—28no—
142C25—22no—
143C2×#41—22no—
144C2×#41—22no—
145C2×#41—41no—
146C2×#41—41no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—42no—
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—41no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—84no—
176C22×D4—84no—
177C22×D4C22⋊C414yes—
178C22×D4C22⋊C414yes—
179C22×D4C2×D414yes—
180C23×C4—28no—
181C23×C4—41no—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C2×D8—216no—
186C2×D8—88no—
187C2×M16—22no—
188C2×M16—41no—
189C2×M16—42no—
190C2×M16—42no—
191C22×C8—22no—
192C22×C8—41no—
193C22×C8—41no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C22×C412yes—
198#127—21no—
199#127—22no—
200#127—22no—
201#127—22no—
202#127—22no—
203#127—21no—
204#127—21no—
205#127—22no—
206#127—22no—
207#127—22no—
208#127—22no—
209#127—22no—
210#127—22no—
211#127—22no—
212#127—22no—
213#127C22⋊C414yes—
214#127C22⋊C414yes—
215#127C22×C412yes—
216#127C22×C414yes—
217#127C2×D414yes—
218#124—22no—
219#124—22no—
220#124—41no—
221#124—41no—
222#121—22no—
223#121—22no—
224#121—22no—
225#121—22no—
226#121—41no—
227#121—41no—
228#121—41no—
229#121—41no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—48no—
235C4×D4—416no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—48no—
241C4×D4—88no—
242C4×D4—88no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—42no—
247#117—42no—
248#117D814yescomplemented
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—416no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42C2411yescharacteristic
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—41no—
288C4⋊C8—41no—
289C4⋊C8—41no—
290#104—48no—
291#104—48no—
292#102—216no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—41no—
297C4×C8—22no—
298C4×C8—22no—
299C4×C8—41no—
300C4×C8C22×C414yes—
301#95—22no—
302#95—22no—
303#95—22no—
304#95C2×D414yes—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—44no—
309C24—44no—
310C24—41no—
311C24—84no—
312C24—84no—
313C24#12011yescharacteristic
314#41—42no—
315#41—42no—
316#41—42no—
317#41—42no—
318#41—44no—
319#41—44no—
320#41—82no—
321#41—82no—
322#41—82no—
323#41—82no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—82no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4C2×C22⋊C418yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—416no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4C4×D411yescharacteristic
399C22×C4#12011yescharacteristic
400C22×C4#12011yescharacteristic
401C22×C4#12011yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12311yescharacteristic
407C22×C4#12311yescharacteristic
408C22×C4C2×D812yes—
409C22×C4#13512yes—
410C22×C4AGL(1,ℤ/8)12yes—
411C22×C4AGL(1,ℤ/8)12yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415D8—816no—
416M16—42no—
417M16—42no—
418M16—82no—
419C2×C8—24no—
420C2×C8—44no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—82no—
424C2×C8—82no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C22⋊C4—24no—
436C22⋊C4—416no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—44no—
449C42—44no—
450C42—44no—
451C42C2×C22⋊C411yescharacteristic
452C42C2×C22⋊C411yescharacteristic
453C42C4×D411yescharacteristic
454C42C4×D411yescharacteristic
455C42C4×D411yescharacteristic
456C42C4×D411yescharacteristic
457C42#12111yescharacteristic
458C42#12111yescharacteristic
459C42#12111yescharacteristic
460C42#12111yescharacteristic
461C42#12311yescharacteristic
462C42#12311yescharacteristic
463C42C2×D814yescomplemented
464C42C23×C411yescharacteristic
465C42C22×D411yescharacteristic
466C42C2×#4111yescharacteristic
467C42C2×#4111yescharacteristic
468C23—28no—
469C23—41no—
470C23—41no—
471C23—48no—
472C23—48no—
473C23—44no—
474C23—44no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—84no—
479C23—84no—
480C23—84no—
481C23—84no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—168no—
486C23—168no—
487C23#38611yescharacteristic
488C23#38611yescharacteristic
489C23#44712yes—
490C23#44912yes—
491C23C2×#12011yescharacteristic
492Q8—42no—
493Q8—42no—
494Q8—82no—
495D4—48no—
496D4—48no—
497D4—48no—
498D4—44no—
499D4—48no—
500D4—48no—
501D4—48no—
502D4—48no—
503D4—44no—
504D4—42no—
505D4—44no—
506D4—42no—
507D4—44no—
508D4—44no—
509D4—84no—
510D4—84no—
511D4—84no—
512D4—82no—
513D4—84no—
514D4—82no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—82no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—82no—
542C2×C4—84no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—1616no—
546C2×C4#38611yescharacteristic
547C2×C4#38611yescharacteristic
548C2×C4#38611yescharacteristic
549C2×C4#38611yescharacteristic
550C2×C4#38611yescharacteristic
551C2×C4#38611yescharacteristic
552C2×C4C4×D812yes—
553C2×C4C4×D812yes—
554C2×C4#44212yes—
555C2×C4#44212yes—
556C2×C4#44712yes—
557C2×C4#44912yes—
558C2×C4#45912yes—
559C2×C4#45912yes—
560C2×C4#46312yes—
561C2×C4#46312yes—
562C2×C4C22×C22⋊C411yescharacteristic
563C2×C4C2×C4×D411yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12311yescharacteristic
568C2×C4C22×D811yescharacteristic
569C2×C4C2×#13511yescharacteristic
570C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
571C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
572C8—44no—
573C8—82no—
574C8—82no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—48no—
584C22—48no—
585C22—48no—
586C22—88no—
587C22—88no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—168no—
592C22—168no—
593C22—164no—
594C22#187912yes—
595C22#187912yes—
596C22C2×#38611yescharacteristic
597C22C2×#44711yescharacteristic
598C22C2×#44911yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—44no—
606C4—82no—
607C4—84no—
608C4—82no—
609C4#187912yes—
610C4#187912yes—
611C4#190412yes—
612C4#191112yes—
613C4C2×C4×D811yescharacteristic
614C4C2×#44211yescharacteristic
615C4C2×#45911yescharacteristic
616C4C2×#46311yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—88no—
622C2—162no—
623C2C2×#187911yescharacteristic
624C2C2×#190411yescharacteristic
625C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3205C2
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. #34210C2semidirect
3. #21209C2semidirect
4. C2×#1545C2semidirect
5. #17574C2semidirect
6. #9299C2semidirect
7. #7957C2semidirect
8. #7854C2semidirect
14. #3205C4semidirect
16. C2×#493C4semidirect
17. C2×#486C4semidirect
29. #2328C22semidirect
34. #1963C22semidirect
37. #1545C22semidirect
40. #1517C22semidirect
42. #1286C22semidirect
43. #1274C22semidirect
56. #532C2×C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
67. C2×#127C2×C4semidirect
89. #493C2×C4semidirect
91. #486C2×C4semidirect
101. #431C2×C4semidirect
120. C2×C4×C8C2×C4semidirect
248. #117D8semidirect
463. C42C2×D8semidirect

Maximal subgroups

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

subgroupindex
C2×#32052
#342102
#212092
C2×#15452
#175742
#92992
#79572
#78542