Finite Group Databasebeta

← #192654

Subgroups — #192654

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34194C211yesmaximal, characteristic
2#21962C211yesmaximal, characteristic, complemented
3#20760C211yesmaximal, characteristic, complemented
4#20738C211yesmaximal, characteristic
5C22×D4⋊C8C211yesmaximal, characteristic
6#8897C212yesmaximal, complemented
7#8894C212yesmaximal, complemented
8#8893C212yesmaximal, complemented
9#8890C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#7898C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2354C2212yes—
17#2352—22no—
18D4×C4⋊C4C2212yes—
19#2345—21no—
20#2342—22no—
21D4×C22⋊C4—21no—
22#2319—21no—
23#2318—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2283C2211yescharacteristic
27#2280—21no—
28C2×#380—21no—
29#2275—21no—
30#1949—22no—
31#1948C2212yescomplemented
32#1947C2212yescomplemented
33#1835—22no—
34#1834C2212yescomplemented
35#1834C2212yescomplemented
36#1824—21no—
37#1808—21no—
38#1807—21no—
39#1802C2212yes—
40#1802C2212yes—
41#1786—21no—
42#1762—21no—
43#1762—21no—
44#1735—22no—
45C2×D4⋊C8C418yescomplemented
46C2×D4⋊C8C2211yescharacteristic
47C2×D4⋊C8C2211yescharacteristic
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2211yescharacteristic
50#1296C2212yescomplemented
51#1296C2212yescomplemented
52#1279C2212yescomplemented
53#1279C2212yescomplemented
54C23×D4D411yescharacteristic
55C24×C4—21no—
56C23×C8—21no—
57C23×C8—21no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C414yes—
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4Q814yes—
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—22no—
70C2×#117—21no—
71C2×#117D412yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432—22no—
79C2×#107—22no—
80C2×#106—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8C2×C414yescomplemented
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#102—22no—
88C2×#102—22no—
89C2×C22⋊C8—21no—
90C2×#97—21no—
91C2×C4×C8—21no—
92C2×C4×C8—22no—
93C2×C4×C8C2×C414yescomplemented
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#390—22no—
97#389—22no—
98#388—24no—
99#388—24no—
100#387—24no—
101#386—24no—
102#385—24no—
103#384C2312yes—
104#384C2312yes—
105#383—21no—
106#382—21no—
107#382D412yes—
108#382D412yes—
109#381—22no—
110#381—21no—
111#380—22no—
112#380—22no—
113#380—21no—
114#380—21no—
115#380—22no—
116#380—22no—
117#380—21no—
118#380—21no—
119#380D411yescharacteristic
120#380D411yescharacteristic
121#379—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2312yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—21no—
132#376—21no—
133C2×#95—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—24no—
139#340—24no—
140D4⋊C8C2×C414yes—
141D4⋊C8C2×C414yes—
142D4⋊C8C2×C414yescomplemented
143D4⋊C8C2×C414yescomplemented
144C25—21no—
145C22×D4—24no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—42no—
149C22×D4C4⋊C414yes—
150C22×D4C2×D411yescharacteristic
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—24no—
171C22×C8—22no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—82no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4C4⋊C414yes—
188C4×D4C4⋊C414yes—
189C4×D4C22×C412yes—
190C4×D4C22×C412yes—
191C4×D4C2×Q812yes—
192C4×D4C2×Q812yes—
193#117—44no—
194#117—44no—
195#117—82no—
196#117—82no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C4⋊C414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D414yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—84no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—41no—
231C2×C42C22×C412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×Q812yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42C2411yescharacteristic
239#107—82no—
240#106—82no—
241C4⋊C8—24no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8C22×C412yes—
247C4⋊C8C22×C412yes—
248#102—44no—
249#102—44no—
250C22⋊C8—42no—
251#97—42no—
252C4×C8—24no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8C22×C412yes—
256C4×C8C22×C412yes—
257#95—24no—
258#95—24no—
259#95—21no—
260#95—22no—
261#95—21no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—41no—
267#95—41no—
268C24—21no—
269C24—21no—
270C24—24no—
271C24—44no—
272C24—42no—
273C24—42no—
274C24—82no—
275C24#12011yescharacteristic
276C2×D4—28no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—44no—
286C2×D4C2×C4⋊C412yes—
287C2×D4C2×C4⋊C412yes—
288C2×D4#13514yes—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—48no—
312C22×C4—42no—
313C22×C4—88no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4C2×C4⋊C412yes—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4C22⋊Q812yes—
328C22×C4C22⋊Q812yes—
329C22×C4#12312yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—48no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—84no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—84no—
366C4⋊C4C2×C4⋊C412yes—
367C4⋊C4C2×C4⋊C412yes—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—48no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C22⋊C4—82no—
381C42—42no—
382C42—42no—
383C42—82no—
384C42—82no—
385C42—84no—
386C42C2×C4⋊C412yes—
387C42C4×D412yes—
388C42C4×D412yes—
389C42C22⋊Q812yes—
390C42C22⋊Q812yes—
391C42#12312yes—
392C42C23×C411yescharacteristic
393C42C22×Q811yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C23—22no—
397C23—22no—
398C23—24no—
399C23—22no—
400C23—22no—
401C23—48no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23—48no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23—84no—
411C23—88no—
412C23—88no—
413C23#38512yes—
414C23#45412yes—
415C23#45512yes—
416C23C2×#12011yescharacteristic
417D4—24no—
418D4—22no—
419D4—22no—
420D4—22no—
421D4—48no—
422D4—42no—
423D4#42714yes—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—168no—
464C2×C4#38512yes—
465C2×C4#38512yes—
466C2×C4#38512yes—
467C2×C4C22×C4⋊C411yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C2×C22⋊Q811yescharacteristic
471C2×C4C2×C22⋊Q811yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4C2×#13512yes—
474C2×C4#57512yes—
475C2×C4#57612yes—
476C2×C4#57712yes—
477C8—42no—
478C8—42no—
479C8—42no—
480C8—42no—
481C8—82no—
482C22—22no—
483C22—22no—
484C22—24no—
485C22—24no—
486C22—24no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—816no—
495C22—82no—
496C22—82no—
497C22#186014yes—
498C22C2×#38511yescharacteristic
499C22#300212yes—
500C22#301212yes—
501C22C2×#45511yescharacteristic
502C22C2×#45411yescharacteristic
503C4—22no—
504C4—21no—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—84no—
511C4—82no—
512C4—82no—
513C4—162no—
514C4C2×#42711yescharacteristic
515C4#291111yescharacteristic
516C4#294811yescharacteristic
517C4#294911yescharacteristic
518C4#308112yes—
519C4#309112yes—
520C2—24no—
521C2—44no—
522C2—82no—
523C2#1862812yes—
524C2#1863812yes—
525C2#2084311yescharacteristic
526C2C2×#186011yescharacteristic
527C2#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. #21962C2semidirect
3. #20760C2semidirect
6. #8897C2semidirect
7. #8894C2semidirect
8. #8893C2semidirect
9. #8890C2semidirect
10. #8044C2semidirect
11. #7898C2semidirect
31. #1948C22semidirect
32. #1947C22semidirect
34. #1834C22semidirect
35. #1834C22semidirect
45. C2×D4⋊C8C4semidirect
50. #1296C22semidirect
51. #1296C22semidirect
52. #1279C22semidirect
53. #1279C22semidirect
83. C2×C4⋊C8C2×C4semidirect
93. C2×C4×C8C2×C4semidirect
142. D4⋊C8C2×C4semidirect
143. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341942
#219622
#207602
#207382
C22×D4⋊C82
#88972
#88942
#88932
#88902
#80442
#78982