Finite Group Databasebeta

← #192647

Subgroups — #192647

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34186C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic, complemented
3#20824C211yesmaximal, characteristic, complemented
4#20732C211yesmaximal, characteristic
5C2×#1490C211yesmaximal, characteristic
6#8897C212yesmaximal, complemented
7#8893C212yesmaximal, complemented
8#8889C212yesmaximal, complemented
9#8885C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic, complemented
11#7896C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2357—22no—
17#2357—22no—
18#2354—21no—
19#2354—21no—
20#2354C2212yes—
21#2354C2212yes—
22#2312—22no—
23#2309—21no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2280—22no—
28#2274—21no—
29#1948C2212yescomplemented
30#1947C2212yescomplemented
31#1854—22no—
32#1847—21no—
33#1845C2211yescharacteristic, complemented
34#1844C2211yescharacteristic, complemented
35#1836—21no—
36#1834C2211yescharacteristic, complemented
37#1834C2211yescharacteristic, complemented
38#1823—21no—
39#1809—22no—
40#1802C2212yes—
41#1802C2212yes—
42#1786—21no—
43#1776—22no—
44C4×#102—22no—
45#1490C418yescomplemented
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48#1296C2212yescomplemented
49#1296C2212yescomplemented
50#1279C2212yescomplemented
51#1279C2212yescomplemented
52C23×D4D411yescharacteristic
53C24×C4—21no—
54C22×M16—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C414yes—
60C2×C4×D4D412yes—
61C2×C4×D4Q814yes—
62C2×C4×D4C2312yes—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C42C2311yescharacteristic
71#432—22no—
72#432C2×C414yes—
73#431—22no—
74#431—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#102—22no—
82C2×#102—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C4×M16—22no—
86C4×M16C2×C414yes—
87C2×#97—21no—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#390—22no—
93#390—22no—
94#390—22no—
95#390—22no—
96#388—24no—
97#388—24no—
98#388—24no—
99#388—24no—
100#384D411yescharacteristic
101#384D411yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#382—21no—
107#382—21no—
108#382—21no—
109#381—24no—
110#380—21no—
111#380—21no—
112#380—22no—
113#380—22no—
114#380—21no—
115#380—21no—
116#380D412yes—
117#380D412yes—
118#379—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—24no—
130C4×C22⋊C4—24no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137D4⋊C8C2×C418yes—
138D4⋊C8C2×C418yes—
139C25—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4C4⋊C414yes—
145C22×D4C2×D412yes—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C2×M16—24no—
157C2×M16—42no—
158C2×M16—42no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4C4⋊C418yes—
171C4×D4C22×C414yes—
172C4×D4C2×Q814yes—
173#117—44no—
174#117—44no—
175#117—82no—
176#117—82no—
177#117—82no—
178#117—82no—
179#117—82no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C4⋊C414yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—41no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—41no—
218C2×C42C22×C412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×Q812yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#107—44no—
227#107—82no—
228#106—44no—
229#106—82no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8C22×C414yes—
234#102—44no—
235#102—44no—
236C22⋊C8—24no—
237C22⋊C8—42no—
238#97—24no—
239#97—24no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—42no—
243C4×C8C22×C412yes—
244C4×C8C22×C412yes—
245#95—24no—
246#95—24no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—41no—
252#95—41no—
253C24—22no—
254C24—24no—
255C24—42no—
256C24—42no—
257C24—81no—
258C24—81no—
259C24—81no—
260C24—81no—
261C24#12011yescharacteristic
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—48no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—48no—
272C2×D4C2×C4⋊C414yes—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—88no—
295C22×C4—88no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C2×C4⋊C412yes—
301C22×C4C4×D412yes—
302C22×C4C4×D412yes—
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4C22⋊Q812yes—
307C22×C4C22⋊Q812yes—
308C22×C4#12312yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312M16—48no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—24no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4C2×C4⋊C414yes—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—28no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C22⋊C4—84no—
357C22⋊C4—82no—
358C42—44no—
359C42—44no—
360C42—82no—
361C42—84no—
362C42—84no—
363C42C2×C4⋊C412yes—
364C42C4×D412yes—
365C42C4×D412yes—
366C42C22⋊Q812yes—
367C42C22⋊Q812yes—
368C42#12312yes—
369C42C23×C411yescharacteristic
370C42C22×Q811yescharacteristic
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—24no—
380C23—48no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—84no—
388C23—84no—
389C23—84no—
390C23—84no—
391C23—84no—
392C23—84no—
393C23#38512yes—
394C23C2×#12011yescharacteristic
395D4—24no—
396D4—24no—
397D4—48no—
398D4—42no—
399D4—42no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—88no—
434C2×C4—84no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4#38512yes—
442C2×C4#38512yes—
443C2×C4#38512yes—
444C2×C4C22×C4⋊C411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×C22⋊Q811yescharacteristic
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4C2×#12311yescharacteristic
450C2×C4#57512yes—
451C2×C4#57512yes—
452C2×C4#57612yes—
453C2×C4#57712yes—
454C8—24no—
455C8—42no—
456C8—44no—
457C8—82no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—816no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22C2×#38511yescharacteristic
474C22#300511yescharacteristic
475C22#300511yescharacteristic
476C22#300811yescharacteristic
477C22#300911yescharacteristic
478C22#301811yescharacteristic
479C22#301811yescharacteristic
480C4—22no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—22no—
485C4—44no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—161no—
491C4—161no—
492C4#291112yes—
493C4#294612yes—
494C4#308112yes—
495C4#309312yes—
496C2—42no—
497C2—84no—
498C2#1861411yescharacteristic
499C2#1861511yescharacteristic
500C2#1861611yescharacteristic
501C2#1861711yescharacteristic
502C2#2084511yescharacteristic
503C2#2086311yescharacteristic
504C2#2087611yescharacteristic
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. #21988C2semidirect
3. #20824C2semidirect
6. #8897C2semidirect
7. #8893C2semidirect
8. #8889C2semidirect
9. #8885C2semidirect
10. #8046C2semidirect
11. #7896C2semidirect
29. #1948C22semidirect
30. #1947C22semidirect
33. #1845C22semidirect
34. #1844C22semidirect
36. #1834C22semidirect
37. #1834C22semidirect
45. #1490C4semidirect
48. #1296C22semidirect
49. #1296C22semidirect
50. #1279C22semidirect
51. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#341862
#219882
#208242
#207322
C2×#14902
#88972
#88932
#88892
#88852
#80462
#78962