Finite Group Databasebeta

← #186629

Subgroups — #186629

Order 512.

2590 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic, complemented
3#20732C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic, complemented
5C2×#1548C211yesmaximal, characteristic, complemented
6C2×#1486C211yesmaximal, characteristic
7C2×#1479C211yesmaximal, characteristic, complemented
8#8450C218yesmaximal, complemented
9C2×#554—24no—
10C2×#554—22no—
11C2×#554—24no—
12C2×#554—22no—
13C2×#554—24no—
14C2×#554—22no—
15C2×#554C418yescomplemented
16C2×#554C2214yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125—21no—
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—22no—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4—21no—
23C22×C4×D4C2211yescharacteristic
24C23×C4⋊C4—21no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419—21no—
29#2285C2211yescharacteristic
30#1981—22no—
31#1956C2214yescomplemented
32#1823—21no—
33#1805—22no—
34#1803C2214yescomplemented
35#1802C2214yes—
36#1787—22no—
37#1786—21no—
38C4×#102—22no—
39#1736—21no—
40#1548—24no—
41C2×#333C2214yescomplemented
42#1486—24no—
43#1479—24no—
44C2×Q8⋊C8C2214yescomplemented
45C2×D4⋊C8C2214yes—
46C23×Q8D411yescharacteristic
47C23×Q8D412yes—
48C23×D4D411yescharacteristic
49C24×C4—22no—
50C24×C4—21no—
51C22×M16—21no—
52#554—216no—
53#554—216no—
54#554—216no—
55#554—216no—
56#554—28no—
57#554—216no—
58#554—28no—
59#554—216no—
60#554—28no—
61#554C2×C4116yes—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8C2×C414yes—
67C2×C4⋊Q8D412yes—
68C2×C4⋊Q8D414yes—
69C2×C4⋊Q8C2312yes—
70C2×#125—28no—
71C2×#125—24no—
72C2×#125—28no—
73C2×#125—24no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—28no—
82C2×C4×Q8—22no—
83C2×C4×Q8—21no—
84C2×C4×Q8—22no—
85C2×C4×Q8—21no—
86C2×C4×Q8C2×C414yes—
87C2×C4×Q8D414yes—
88C2×C4×Q8D412yes—
89C2×C4×Q8C2312yes—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4C2×C414yes—
99C2×C4×D4D414yes—
100C2×C4×D4D412yes—
101C2×C4×D4C2312yes—
102C2×#117—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C22⋊C4—22no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
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#419—44no—
129C2×#103—24no—
130C2×#103—28no—
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—
145Q8⋊C8—28no—
146D4⋊C8—28no—
147C25—21no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—84no—
157C22×Q8C22⋊C414yes—
158C22×Q8C22⋊C418yes—
159C22×Q8C2×D412yes—
160C22×Q8C2×D414yes—
161C22×D4—28no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4C22⋊C414yes—
165C22×D4C2×D412yes—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—28no—
171C23×C4—216no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—88no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C2×M16—42no—
185C2×M16—42no—
186C2×M16—42no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192C4⋊Q8—24no—
193C4⋊Q8—24no—
194C4⋊Q8—24no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8C22⋊C418yes—
198C4⋊Q8C22×C414yes—
199C4⋊Q8C2×D414yes—
200#125—216no—
201#125—232no—
202#125—216no—
203C22⋊Q8—216no—
204C22⋊Q8—232no—
205C22⋊Q8—232no—
206C22⋊Q8—264no—
207C22⋊Q8—216no—
208C22⋊Q8—232no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8C22⋊C418yes—
217C4×Q8C22×C414yes—
218C4×Q8C2×D414yes—
219C4×D4—216no—
220C4×D4—232no—
221C4×D4—216no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4C22⋊C418yes—
230C4×D4C22×C414yes—
231C4×D4C2×D414yes—
232#117—42no—
233#117—42no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—232no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C22⋊C414yes—
261C2×C4⋊C4C22⋊C414yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—41no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—41no—
276C2×C42C22×C412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2411yescharacteristic
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—42no—
289C4⋊C8—42no—
290#103—48no—
291#103—88no—
292#102—48no—
293C22⋊C8—42no—
294C22⋊C8—42no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—42no—
298#95—41no—
299#95—41no—
300C24—28no—
301C24—22no—
302C24—24no—
303C24—44no—
304C24—42no—
305C24—42no—
306C24#12011yescharacteristic
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—28no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—28no—
316C2×Q8—28no—
317C2×Q8—48no—
318C2×Q8—42no—
319C2×Q8—44no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—84no—
323C2×Q8—88no—
324C2×Q8—88no—
325C2×Q8C2×C22⋊C414yes—
326C2×Q8C2×C22⋊C418yes—
327C2×D4—28no—
328C2×D4—216no—
329C2×D4—216no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4C2×C22⋊C414yes—
334C2×D4#13718yes—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—232no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—44no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—42no—
366C22×C4—832no—
367C22×C4—816no—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4#12012yes—
372C22×C4#12011yescharacteristic
373C22×C4#12011yescharacteristic
374C22×C4#12012yes—
375C22×C4#12012yes—
376C22×C4#12011yescharacteristic
377C22×C4#13512yes—
378C22×C4#13512yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4#13712yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384M16—84no—
385C2×C8—44no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—44no—
392C4⋊C4—232no—
393C4⋊C4—232no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—28no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4C2×C22⋊C414yes—
412C4⋊C4C2×C22⋊C414yes—
413C22⋊C4—232no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—28no—
417C22⋊C4—82no—
418C42—24no—
419C42—24no—
420C42—28no—
421C42—82no—
422C42C2×C22⋊C412yes—
423C42C2×C22⋊C412yes—
424C42C2×C22⋊C412yes—
425C42#12012yes—
426C42#12012yes—
427C42#12012yes—
428C42C23×C411yescharacteristic
429C42C22×D411yescharacteristic
430C42C22×D411yescharacteristic
431C42C22×D411yescharacteristic
432C23—22no—
433C23—22no—
434C23—216no—
435C23—28no—
436C23—24no—
437C23—48no—
438C23—416no—
439C23—416no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—48no—
444C23#37912yes—
445C23#44912yes—
446C23#45212yes—
447C23C2×#12011yescharacteristic
448Q8—24no—
449Q8—24no—
450Q8—48no—
451Q8—42no—
452Q8—42no—
453Q8—48no—
454Q8—84no—
455Q8—88no—
456D4—216no—
457D4—24no—
458D4—28no—
459D4#41918yescomplemented
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—48no—
472C2×C4—416no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—416no—
486C2×C4—832no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4#37912yes—
491C2×C4#37912yes—
492C2×C4#37912yes—
493C2×C4#41614yes—
494C2×C4#41714yes—
495C2×C4#44914yes—
496C2×C4#44912yes—
497C2×C4#45214yes—
498C2×C4#45212yes—
499C2×C4C22×C22⋊C411yescharacteristic
500C2×C4C22×C22⋊C411yescharacteristic
501C2×C4C22×C22⋊C411yescharacteristic
502C2×C4C2×#12011yescharacteristic
503C2×C4C2×#12011yescharacteristic
504C2×C4C2×#12011yescharacteristic
505C2×C4C2×#13511yescharacteristic
506C2×C4C2×#13511yescharacteristic
507C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
508C2×C4C2×#13714yes—
509C2×C4C2×#13711yescharacteristic
510C8—82no—
511C8—82no—
512C8—82no—
513C22—24no—
514C22—24no—
515C22—22no—
516C22—22no—
517C22—28no—
518C22—432no—
519C22—48no—
520C22—48no—
521C22—44no—
522C22—44no—
523C22#178814yes—
524C22C2×#37911yescharacteristic
525C22C2×#44911yescharacteristic
526C22C2×#45211yescharacteristic
527C22#300412yes—
528C22#301912yes—
529C4—22no—
530C4—21no—
531C4—21no—
532C4—22no—
533C4—44no—
534C4—44no—
535C4—44no—
536C4—48no—
537C4—162no—
538C4#178814yes—
539C4C2×#41611yescharacteristic
540C4C2×#41912yes—
541C4C2×#41711yescharacteristic
542C4C2×#44911yescharacteristic
543C4C2×#45211yescharacteristic
544C4#300412yes—
545C2—24no—
546C2—48no—
547C2#1875212yes—
548C2#1875312yes—
549C2C2×#178811yescharacteristic
550C2#2059911yescharacteristic
551C2#2061611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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
4. #20647C2semidirect
5. C2×#1548C2semidirect
7. C2×#1479C2semidirect
8. #8450C2semidirect
15. C2×#554C4semidirect
31. #1956C22semidirect
34. #1803C22semidirect
41. C2×#333C22semidirect
44. C2×Q8⋊C8C22semidirect
459. D4#419semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
#218042
#207322
#206472
C2×#15482
C2×#14862
C2×#14792
#84502