Finite Group Databasebeta

← #186631

Subgroups — #186631

Order 512.

2654 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#555C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic, complemented
3#20732C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic, complemented
5C2×#1548C211yesmaximal, characteristic, complemented
6C2×#1486C211yesmaximal, characteristic, complemented
7C2×#1478C211yesmaximal, characteristic, complemented
8#8452C218yesmaximal, complemented
9C2×#555—24no—
10C2×#555—22no—
11C2×#555—24no—
12C2×#555—24no—
13C2×#555—22no—
14C2×#555—22no—
15C2×#555C418yescomplemented
16C2×#555C2214yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124—22no—
19C22×#121—21no—
20C22×#121—21no—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×#117—21no—
24C22×#117—21no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#418—21no—
29#2285C2211yescharacteristic
30#1981—22no—
31#1956C2214yescomplemented
32#1823—21no—
33#1804—22no—
34#1802C2214yescomplemented
35#1802C2214yescomplemented
36#1787—22no—
37#1786—21no—
38C4×#102—22no—
39#1736—21no—
40#1548—24no—
41C2×#333C2214yescomplemented
42#1486—24no—
43#1478—24no—
44C2×D4⋊C8C2214yescomplemented
45C2×D4⋊C8C2214yescomplemented
46C23×Q8D412yes—
47C23×D4—22no—
48C23×D4D411yescharacteristic
49C23×D4D411yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C22×M16—21no—
53#555—216no—
54#555—216no—
55#555—216no—
56#555—216no—
57#555—28no—
58#555—216no—
59#555—216no—
60#555—28no—
61#555—28no—
62#555C2×C4116yes—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8C2×C414yes—
68C2×C4⋊Q8D412yes—
69C2×C4⋊Q8D414yes—
70C2×C4⋊Q8C2312yes—
71C2×#124—28no—
72C2×#124—216no—
73C2×#124—216no—
74C2×#124—28no—
75C2×#121—24no—
76C2×#121—24no—
77C2×#121—28no—
78C2×#121—28no—
79C2×#121—28no—
80C2×#121—28no—
81C2×#121—24no—
82C2×#121—24no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—22no—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4C2×C414yes—
92C2×C4×D4C2×C414yes—
93C2×C4×D4D414yes—
94C2×C4×D4D414yes—
95C2×C4×D4D412yes—
96C2×C4×D4D412yes—
97C2×C4×D4C2312yes—
98C2×C4×D4C2312yes—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—28no—
102C2×#117—28no—
103C2×#117—28no—
104C2×#117—28no—
105C2×#117—24no—
106C2×#117—24no—
107C2×#117—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C22⋊C4—22no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—22no—
117C22×C42C2311yescharacteristic
118#431—22no—
119#431—22no—
120#431—22no—
121#431—22no—
122#423—22no—
123#423—22no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128#418—44no—
129C2×#103—28no—
130C2×#102—24no—
131C2×#102—24no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8C2312yes—
136#384C2312yes—
137#382—21no—
138#380—21no—
139#380—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142#340—24no—
143#340—24no—
144#333—28no—
145D4⋊C8—28no—
146D4⋊C8—28no—
147C25—21no—
148C25—21no—
149C22×Q8—22no—
150C22×Q8—24no—
151C22×Q8—84no—
152C22×Q8C22⋊C418yes—
153C22×Q8C2×D414yes—
154C22×D4—28no—
155C22×D4—216no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—44no—
165C22×D4—88no—
166C22×D4C22⋊C414yes—
167C22×D4C22⋊C414yes—
168C22×D4C2×D412yes—
169C22×D4C2×D412yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—48no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C4⋊Q8—24no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8C22⋊C418yes—
203C4⋊Q8C22×C414yes—
204C4⋊Q8C2×D414yes—
205#124—232no—
206#124—264no—
207#124—232no—
208#121—216no—
209#121—216no—
210#121—232no—
211#121—232no—
212#121—216no—
213#121—216no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4C22⋊C418yes—
229C4×D4C22⋊C418yes—
230C4×D4C22×C414yes—
231C4×D4C22×C414yes—
232C4×D4C2×D414yes—
233C4×D4C2×D414yes—
234#117—216no—
235#117—216no—
236#117—232no—
237#117—232no—
238#117—216no—
239#117—216no—
240#117—42no—
241#117—42no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C22⋊C414yes—
258C2×C4⋊C4C22⋊C414yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—41no—
274C2×C42—28no—
275C2×C42—216no—
276C2×C42—41no—
277C2×C42C22×C412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2411yescharacteristic
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291#103—88no—
292#102—48no—
293#102—48no—
294C22⋊C8—42no—
295C22⋊C8—42no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8—42no—
299#95—41no—
300#95—41no—
301C24—22no—
302C24—22no—
303C24—24no—
304C24—24no—
305C24—44no—
306C24—44no—
307C24—44no—
308C24—44no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24#12011yescharacteristic
314C2×Q8—24no—
315C2×Q8—28no—
316C2×Q8—48no—
317C2×Q8—84no—
318C2×Q8—88no—
319C2×Q8—88no—
320C2×Q8C2×C22⋊C418yes—
321C2×D4—232no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—48no—
335C2×D4—44no—
336C2×D4—48no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4—44no—
340C2×D4—42no—
341C2×D4—832no—
342C2×D4—816no—
343C2×D4C2×C22⋊C414yes—
344C2×D4C2×C22⋊C414yes—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—42no—
359C22×C4—48no—
360C22×C4—416no—
361C22×C4—416no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4#12012yes—
382C22×C4#12011yescharacteristic
383C22×C4#12011yescharacteristic
384C22×C4#12012yes—
385C22×C4#12012yes—
386C22×C4#12011yescharacteristic
387C22×C4#13512yes—
388C22×C4#13512yes—
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4#13712yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394M16—84no—
395C2×C8—44no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—44no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4C2×C22⋊C414yes—
417C4⋊C4C2×C22⋊C414yes—
418C22⋊C4—232no—
419C22⋊C4—232no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—82no—
427C42—232no—
428C42—82no—
429C42C2×C22⋊C412yes—
430C42C2×C22⋊C412yes—
431C42C2×C22⋊C412yes—
432C42#12012yes—
433C42#12012yes—
434C42#12012yes—
435C42C23×C411yescharacteristic
436C42C22×D411yescharacteristic
437C42C22×D411yescharacteristic
438C42C22×D411yescharacteristic
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—48no—
444C23—48no—
445C23—48no—
446C23—48no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—48no—
456C23—48no—
457C23—88no—
458C23—88no—
459C23—88no—
460C23—88no—
461C23#37912yes—
462C23#44912yes—
463C23#45212yes—
464C23C2×#12011yescharacteristic
465Q8—48no—
466Q8—84no—
467Q8—88no—
468D4—24no—
469D4—24no—
470D4—24no—
471D4—24no—
472D4—48no—
473D4—42no—
474D4—42no—
475D4—44no—
476D4—48no—
477D4—832no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—416no—
489C2×C4—48no—
490C2×C4—416no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—48no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—48no—
502C2×C4—416no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4#37912yes—
507C2×C4#37912yes—
508C2×C4#37912yes—
509C2×C4#41614yes—
510C2×C4#41714yes—
511C2×C4#44914yes—
512C2×C4#44912yes—
513C2×C4#45214yes—
514C2×C4#45212yes—
515C2×C4C22×C22⋊C411yescharacteristic
516C2×C4C22×C22⋊C411yescharacteristic
517C2×C4C22×C22⋊C411yescharacteristic
518C2×C4C2×#12011yescharacteristic
519C2×C4C2×#12011yescharacteristic
520C2×C4C2×#12011yescharacteristic
521C2×C4C2×#13511yescharacteristic
522C2×C4C2×#13511yescharacteristic
523C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
524C2×C4C2×#13711yescharacteristic
525C2×C4#57514yes—
526C8—82no—
527C8—82no—
528C8—82no—
529C22—24no—
530C22—24no—
531C22—22no—
532C22—22no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—816no—
540C22—816no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22#178814yes—
546C22C2×#37911yescharacteristic
547C22C2×#44911yescharacteristic
548C22C2×#45211yescharacteristic
549C22#300512yes—
550C22#301912yes—
551C4—22no—
552C4—21no—
553C4—21no—
554C4—22no—
555C4—44no—
556C4—44no—
557C4—48no—
558C4—162no—
559C4#178814yes—
560C4C2×#41611yescharacteristic
561C4C2×#41711yescharacteristic
562C4#289612yes—
563C4C2×#44911yescharacteristic
564C4C2×#45211yescharacteristic
565C4#300512yes—
566C2—24no—
567C2—84no—
568C2—84no—
569C2#1877212yes—
570C2#1877312yes—
571C2C2×#178811yescharacteristic
572C2#2060011yescharacteristic
573C2#2061611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#555C2
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. #21804C2semidirect
3. #20732C2semidirect
4. #20646C2semidirect
5. C2×#1548C2semidirect
6. C2×#1486C2semidirect
7. C2×#1478C2semidirect
8. #8452C2semidirect
15. C2×#555C4semidirect
31. #1956C22semidirect
34. #1802C22semidirect
35. #1802C22semidirect
41. C2×#333C22semidirect
44. C2×D4⋊C8C22semidirect
45. C2×D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5552
#218042
#207322
#206462
C2×#15482
C2×#14862
C2×#14782
#84522