Finite Group Databasebeta

← #192754

Subgroups — #192754

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3#21170C211yesmaximal, characteristic, complemented
4#20732C211yesmaximal, characteristic
5C2×#1486C211yesmaximal, characteristic
6#8893C214yesmaximal, complemented
7#8889C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#117—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16#2354—22no—
17#2354—22no—
18#2354—22no—
19#2354C2214yes—
20C2×#390—21no—
21C2×#388—22no—
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#380—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1949—22no—
28#1948—22no—
29#1948C2212yescomplemented
30#1947C2212yescomplemented
31#1853—22no—
32#1844C2212yescomplemented
33#1834C2212yescomplemented
34#1823—21no—
35#1802C2212yes—
36#1802C2212yes—
37#1786—21no—
38C4×#102—22no—
39#1736—21no—
40#1486C418yescomplemented
41C2×D4⋊C8C2214yes—
42#1296C2214yescomplemented
43#1279C2214yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C22×M16—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C414yes—
52C2×C4×D4D412yes—
53C2×C4×D4Q814yes—
54C2×C4×D4C2312yes—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—21no—
58C2×#117—44no—
59C2×#117—44no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D412yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—22no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431—22no—
74#431—22no—
75#431C2×C414yes—
76#428—44no—
77C2×#107—22no—
78C2×#106—22no—
79C2×#106—24no—
80#423—22no—
81#423C2×C414yes—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×#102—22no—
87C2×#102—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2312yes—
92#390—24no—
93#390—24no—
94#388—28no—
95#388—28no—
96#384—21no—
97#384—21no—
98#384D412yes—
99#384C2311yescharacteristic
100#384C2311yescharacteristic
101#384C2312yes—
102#382—21no—
103#380—24no—
104#380—24no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C2×#95—22no—
116C2×#95—21no—
117#340—22no—
118#340—22no—
119#340—24no—
120#340—22no—
121#340—22no—
122D4⋊C8C2×C4116yescomplemented
123C25—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—42no—
127C22×D4—42no—
128C22×D4C4⋊C414yes—
129C22×D4C2×D412yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×M16—24no—
146C2×M16—42no—
147C2×M16—42no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4C4⋊C418yes—
159C4×D4C22×C414yes—
160C4×D4C2×Q814yes—
161#117—48no—
162#117—416no—
163#117—48no—
164#117—42no—
165#117—42no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C4⋊C414yes—
189C2×C4⋊C4SD1614yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D414yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—84no—
203C2×C42—28no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—48no—
207C2×C42—44no—
208C2×C42—41no—
209C2×C42C22×C412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q812yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42C2411yescharacteristic
217#107—44no—
218#106—44no—
219#106—84no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8C22×C414yes—
226#102—44no—
227#102—44no—
228C22⋊C8—24no—
229C22⋊C8—42no—
230C4×C8—22no—
231C4×C8—42no—
232C4×C8C22×C414yes—
233#95—28no—
234#95—22no—
235#95—22no—
236#95—41no—
237#95—41no—
238C24—22no—
239C24—24no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24#12011yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—48no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—48no—
257C2×D4C2×C4⋊C414yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4C2×C4⋊C412yes—
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4C22⋊Q812yes—
300C22×C4C22⋊Q812yes—
301C22×C4#12312yes—
302C22×C4C2×SD1612yes—
303C22×C4#13512yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4AGL(1,ℤ/8)12yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—48no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—82no—
341C4⋊C4C2×C4⋊C414yes—
342C4⋊C4C2×SD1614yes—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—28no—
346C22⋊C4—416no—
347C22⋊C4—88no—
348C22⋊C4—82no—
349C42—44no—
350C42—416no—
351C42—44no—
352C42—82no—
353C42C2×C4⋊C412yes—
354C42C4×D412yes—
355C42C4×D412yes—
356C42C22⋊Q812yes—
357C42C22⋊Q812yes—
358C42#12312yes—
359C42C23×C411yescharacteristic
360C42C22×Q811yescharacteristic
361C42C2×#4111yescharacteristic
362C42C2×#4111yescharacteristic
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—24no—
368C23—48no—
369C23—48no—
370C23—48no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—84no—
378C23—84no—
379C23—84no—
380C23—84no—
381C23#38512yes—
382C23#44912yes—
383C23#45012yes—
384C23C2×#12011yescharacteristic
385D4—24no—
386D4—24no—
387D4—48no—
388D4—42no—
389D4—42no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—416no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—88no—
420C2×C4—84no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4#38512yes—
428C2×C4#38512yes—
429C2×C4#38512yes—
430C2×C4C4×SD1614yes—
431C2×C4#44214yes—
432C2×C4#44912yes—
433C2×C4#45012yes—
434C2×C4#47614yes—
435C2×C4#47814yes—
436C2×C4C22×C4⋊C411yescharacteristic
437C2×C4C2×C4×D411yescharacteristic
438C2×C4C2×C4×D411yescharacteristic
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4C2×C22⋊Q811yescharacteristic
441C2×C4C2×#12311yescharacteristic
442C2×C4C22×SD1611yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4#57512yes—
447C2×C4#57512yes—
448C8—44no—
449C8—44no—
450C8—82no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—816no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22#186414yes—
466C22C2×#38511yescharacteristic
467C22C2×#45011yescharacteristic
468C22C2×#44911yescharacteristic
469C22#300511yescharacteristic
470C22#300511yescharacteristic
471C22#301911yescharacteristic
472C22#302011yescharacteristic
473C4—22no—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—44no—
478C4—44no—
479C4—84no—
480C4—82no—
481C4—161no—
482C4—161no—
483C4#186414yes—
484C4#291112yes—
485C4C2×C4×SD1611yescharacteristic
486C4C2×#44211yescharacteristic
487C4C2×#47611yescharacteristic
488C4C2×#47811yescharacteristic
489C4#309312yes—
490C2—24no—
491C2—84no—
492C2#1875711yescharacteristic
493C2#1876211yescharacteristic
494C2#1876711yescharacteristic
495C2#1877211yescharacteristic
496C2#2084511yescharacteristic
497C2#2086411yescharacteristic
498C2C2×#186411yescharacteristic
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. #21981C2semidirect
3. #21170C2semidirect
6. #8893C2semidirect
7. #8889C2semidirect
8. #8043C2semidirect
9. #7892C2semidirect
29. #1948C22semidirect
30. #1947C22semidirect
32. #1844C22semidirect
33. #1834C22semidirect
40. #1486C4semidirect
42. #1296C22semidirect
43. #1279C22semidirect
122. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
#219812
#211702
#207322
C2×#14862
#88932
#88892
#80432
#78922