Finite Group Databasebeta

← #192770

Subgroups — #192770

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21964C211yesmaximal, characteristic, complemented
4#20738C211yesmaximal, characteristic
5C2×#1486C211yesmaximal, characteristic
6#8894C212yesmaximal, complemented
7#8890C212yesmaximal, complemented
8#8886C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8042C211yesmaximal, characteristic, complemented
11#7900C211yesmaximal, characteristic, complemented
12C2×#518—21no—
13C22×C4×D4C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
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#1949—22no—
30#1948—21no—
31#1948C2212yescomplemented
32#1947—21no—
33#1947C2212yescomplemented
34#1854—22no—
35#1845C2211yescharacteristic, complemented
36#1844C2211yescharacteristic, complemented
37#1834C2211yescharacteristic, complemented
38#1834C2211yescharacteristic, complemented
39#1824—21no—
40#1809—22no—
41#1802C2212yes—
42#1802C2212yes—
43#1786—21no—
44#1762—22no—
45#1486C418yescomplemented
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#432—22no—
79#431—22no—
80#431C2×C414yescomplemented
81C2×#107—22no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—22no—
85#423—22no—
86#423C2×C414yescomplemented
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#102—22no—
92C2×#102—22no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—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—81no—
155C23×C4—81no—
156C23×C4—81no—
157C23×C4—81no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—24no—
162C2×M16—81no—
163C2×M16—81no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—48no—
176C4×D4—48no—
177C4×D4—44no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4C4⋊C418yes—
182C4×D4C22×C414yes—
183C4×D4C2×Q814yes—
184#117—44no—
185#117—44no—
186#117—82no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
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—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C4⋊C414yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—416no—
224C2×C22⋊C4—41no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—41no—
228C2×C42C22×C412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×Q812yes—
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236#107—82no—
237#107—82no—
238#106—82no—
239#106—82no—
240C4⋊C8—24no—
241C4⋊C8—24no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8C22×C414yes—
246#102—44no—
247#102—44no—
248C22⋊C8—24no—
249C22⋊C8—42no—
250#97—42no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8C22×C412yes—
254C4×C8C22×C412yes—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—41no—
262#95—41no—
263C24—24no—
264C24—24no—
265C24—22no—
266C24—24no—
267C24—81no—
268C24—81no—
269C24—81no—
270C24—81no—
271C24#12011yescharacteristic
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—216no—
275C2×D4—216no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4C2×C4⋊C414yes—
281C2×D4AGL(1,ℤ/8)14yes—
282C2×D4#13714yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—28no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4C2×C4⋊C412yes—
321C22×C4C4×D412yes—
322C22×C4C4×D412yes—
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4C22⋊Q812yes—
327C22×C4C22⋊Q812yes—
328C22×C4#12312yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332M16—84no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
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—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4C2×C4⋊C414yes—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—28no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—82no—
382C42—44no—
383C42—42no—
384C42—42no—
385C42—82no—
386C42C2×C4⋊C412yes—
387C42C4×D412yes—
388C42C4×D412yes—
389C42C22⋊Q812yes—
390C42C22⋊Q812yes—
391C42#12312yes—
392C42C23×C411yescharacteristic
393C42C22×Q811yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C23—216no—
397C23—21no—
398C23—24no—
399C23—21no—
400C23—24no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23—84no—
414C23—84no—
415C23—84no—
416C23#38512yes—
417C23C2×#12011yescharacteristic
418D4—216no—
419D4—24no—
420D4—24no—
421D4—24no—
422D4#42818yes—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—88no—
459C2×C4—84no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—168no—
467C2×C4—168no—
468C2×C4#38512yes—
469C2×C4#38512yes—
470C2×C4#38512yes—
471C2×C4C22×C4⋊C411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4C2×#12311yescharacteristic
477C2×C4C2×AGL(1,ℤ/8)12yes—
478C2×C4C2×#13712yes—
479C2×C4#57611yescharacteristic
480C2×C4#57611yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57711yescharacteristic
483C8—42no—
484C8—42no—
485C8—44no—
486C8—82no—
487C22—22no—
488C22—28no—
489C22—22no—
490C22—22no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—41no—
495C22—41no—
496C22—44no—
497C22—44no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—816no—
502C22C2×#38511yescharacteristic
503C22#300311yescharacteristic
504C22#300411yescharacteristic
505C22#301211yescharacteristic
506C22#301211yescharacteristic
507C22#302111yescharacteristic
508C22#302211yescharacteristic
509C4—22no—
510C4—21no—
511C4—21no—
512C4—22no—
513C4—22no—
514C4—44no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—161no—
520C4—161no—
521C4C2×#42812yes—
522C4#294811yescharacteristic
523C4#294911yescharacteristic
524C4#308111yescharacteristic
525C4#308111yescharacteristic
526C4#309212yes—
527C2—42no—
528C2—48no—
529C2#1883811yescharacteristic
530C2#1883911yescharacteristic
531C2#1884611yescharacteristic
532C2#1884711yescharacteristic
533C2#2084411yescharacteristic
534C2#2086511yescharacteristic
535C2#2087911yescharacteristic
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. #21984C2semidirect
3. #21964C2semidirect
6. #8894C2semidirect
7. #8890C2semidirect
8. #8886C2semidirect
9. #8882C2semidirect
10. #8042C2semidirect
11. #7900C2semidirect
31. #1948C22semidirect
33. #1947C22semidirect
35. #1845C22semidirect
36. #1844C22semidirect
37. #1834C22semidirect
38. #1834C22semidirect
45. #1486C4semidirect
48. #1296C22semidirect
49. #1296C22semidirect
50. #1279C22semidirect
51. #1279C22semidirect
80. #431C2×C4semidirect
86. #423C2×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
#219842
#219642
#207382
C2×#14862
#88942
#88902
#88862
#88822
#80422
#79002