Finite Group Databasebeta

← #192643

Subgroups — #192643

Order 512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic, complemented
3#20824C211yesmaximal, characteristic, complemented
4#20732C211yesmaximal, characteristic
5C2×#1490C211yesmaximal, characteristic
6#8894C212yesmaximal, complemented
7#8890C212yesmaximal, complemented
8#8886C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic, complemented
11#7896C211yesmaximal, characteristic, complemented
12C2×#518—21no—
13C22×C4×D4C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2349—22no—
18#2349—22no—
19D4×C4⋊C4—21no—
20D4×C4⋊C4—21no—
21D4×C4⋊C4C2212yes—
22D4×C4⋊C4C2212yes—
23#2316—21no—
24#2315—22no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#380—22no—
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—
55#518—48no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C414yes—
65C2×C4×D4D412yes—
66C2×C4×D4Q814yes—
67C2×C4×D4C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—24no—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432C2×C414yescomplemented
79#431—22no—
80#431—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#102—22no—
88C2×#102—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C4×M16—22no—
92C4×M16C2×C414yescomplemented
93C2×#97—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#385—24no—
103#385—24no—
104#385—24no—
105#385—24no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—21no—
113#382—21no—
114#382—21no—
115#380—24no—
116#380—24no—
117#380—21no—
118#380—21no—
119#380—22no—
120#380—22no—
121#380—21no—
122#380—21no—
123#380D412yes—
124#380D412yes—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4C2312yes—
131C2×#95—24no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138D4⋊C8C2×C418yescomplemented
139D4⋊C8C2×C418yescomplemented
140C25—21no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4C4⋊C414yes—
146C22×D4C2×D412yes—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×M16—24no—
160C2×M16—42no—
161C2×M16—42no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—48no—
174C4×D4—48no—
175C4×D4—44no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—44no—
179C4×D4C4⋊C418yes—
180C4×D4C22×C414yes—
181C4×D4C2×Q814yes—
182#117—44no—
183#117—44no—
184#117—82no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
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⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C4⋊C414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—416no—
222C2×C22⋊C4—41no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—41no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×Q812yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234#107—44no—
235#107—82no—
236#106—44no—
237#106—82no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8C22×C414yes—
242#102—44no—
243#102—44no—
244C22⋊C8—24no—
245C22⋊C8—42no—
246#97—24no—
247#97—24no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—42no—
251C4×C8C22×C412yes—
252C4×C8C22×C412yes—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—41no—
260#95—41no—
261C24—24no—
262C24—24no—
263C24—22no—
264C24—24no—
265C24—81no—
266C24—81no—
267C24—81no—
268C24—81no—
269C24#12011yescharacteristic
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—216no—
273C2×D4—216no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4C2×C4⋊C414yes—
279C2×D4AGL(1,ℤ/8)14yes—
280C2×D4#13714yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—88no—
305C22×C4—88no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4C2×C4⋊C412yes—
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4C22⋊Q812yes—
321C22×C4C22⋊Q812yes—
322C22×C4#12312yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326M16—48no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4C2×C4⋊C414yes—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—28no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—82no—
372C42—44no—
373C42—42no—
374C42—42no—
375C42—82no—
376C42C2×C4⋊C412yes—
377C42C4×D412yes—
378C42C4×D412yes—
379C42C22⋊Q812yes—
380C42C22⋊Q812yes—
381C42#12312yes—
382C42C23×C411yescharacteristic
383C42C22×Q811yescharacteristic
384C42C2×#4111yescharacteristic
385C42C2×#4111yescharacteristic
386C23—216no—
387C23—21no—
388C23—24no—
389C23—21no—
390C23—24no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—24no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—84no—
401C23—84no—
402C23—84no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23#38512yes—
407C23C2×#12011yescharacteristic
408D4—216no—
409D4—24no—
410D4—24no—
411D4—24no—
412D4#42818yes—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—88no—
447C2×C4—84no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—168no—
455C2×C4—168no—
456C2×C4#38512yes—
457C2×C4#38512yes—
458C2×C4#38512yes—
459C2×C4C22×C4⋊C411yescharacteristic
460C2×C4C2×C4×D411yescharacteristic
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C2×C22⋊Q811yescharacteristic
463C2×C4C2×C22⋊Q811yescharacteristic
464C2×C4C2×#12311yescharacteristic
465C2×C4C2×AGL(1,ℤ/8)12yes—
466C2×C4C2×#13712yes—
467C2×C4#57612yes—
468C2×C4#57712yes—
469C8—24no—
470C8—42no—
471C8—44no—
472C8—82no—
473C22—22no—
474C22—28no—
475C22—22no—
476C22—22no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—41no—
481C22—44no—
482C22—41no—
483C22—41no—
484C22—44no—
485C22—41no—
486C22—44no—
487C22—816no—
488C22C2×#38511yescharacteristic
489C22#300311yescharacteristic
490C22#300411yescharacteristic
491C22#300811yescharacteristic
492C22#300911yescharacteristic
493C22#301811yescharacteristic
494C22#301811yescharacteristic
495C4—22no—
496C4—21no—
497C4—21no—
498C4—22no—
499C4—22no—
500C4—44no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—161no—
506C4—161no—
507C4C2×#42812yes—
508C4#294612yes—
509C4#308112yes—
510C4#309212yes—
511C2—42no—
512C2—48no—
513C2#1859811yescharacteristic
514C2#1859911yescharacteristic
515C2#1860011yescharacteristic
516C2#1860111yescharacteristic
517C2#2084411yescharacteristic
518C2#2086311yescharacteristic
519C2#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. #8894C2semidirect
7. #8890C2semidirect
8. #8886C2semidirect
9. #8882C2semidirect
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
78. #432C2×C4semidirect
92. C4×M16C2×C4semidirect
138. D4⋊C8C2×C4semidirect
139. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341812
#219882
#208242
#207322
C2×#14902
#88942
#88902
#88862
#88822
#80462
#78962