Finite Group Databasebeta

← #192648

Subgroups — #192648

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34194C211yesmaximal, characteristic
2#21962C211yesmaximal, characteristic, complemented
3#20797C211yesmaximal, characteristic, complemented
4#20738C211yesmaximal, characteristic
5C22×D4⋊C8C211yesmaximal, characteristic
6#8889C212yesmaximal, complemented
7#8886C212yesmaximal, complemented
8#8885C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#7898C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#427—22no—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2354C2212yes—
18#2352—22no—
19D4×C4⋊C4C2212yes—
20#2345—21no—
21#2342—22no—
22D4×C22⋊C4—21no—
23#2319—21no—
24#2318—22no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2283C2211yescharacteristic
28#2280—21no—
29C2×#380—21no—
30#2275—21no—
31#1949—22no—
32#1948C2212yescomplemented
33#1947C2212yescomplemented
34#1846—22no—
35#1845C2212yescomplemented
36#1844C2212yescomplemented
37#1824—21no—
38#1808—21no—
39#1807—21no—
40#1802C2212yes—
41#1802C2212yes—
42#1786—21no—
43#1762—21no—
44#1762—21no—
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—
79#427—48no—
80C2×#107—24no—
81C2×#107—22no—
82C2×#106—24no—
83C2×#106—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8C2×C414yescomplemented
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102—22no—
91C2×#102—22no—
92C2×C22⋊C8—21no—
93C2×#97—21no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96C2×C4×C8C2×C414yescomplemented
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#390—22no—
100#389—22no—
101#388—24no—
102#388—24no—
103#387—24no—
104#386—24no—
105#385—24no—
106#384C2312yes—
107#384C2312yes—
108#383—21no—
109#382—21no—
110#382D412yes—
111#382D412yes—
112#381—22no—
113#381—21no—
114#380—22no—
115#380—22no—
116#380—21no—
117#380—21no—
118#380—22no—
119#380—22no—
120#380—21no—
121#380—21no—
122#380D411yescharacteristic
123#380D411yescharacteristic
124#379—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—21no—
135#376—21no—
136C2×#95—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141D4⋊C8C2×C414yes—
142D4⋊C8C2×C414yes—
143D4⋊C8C2×C414yescomplemented
144D4⋊C8C2×C414yescomplemented
145C25—21no—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—42no—
150C22×D4C4⋊C414yes—
151C22×D4C2×D411yescharacteristic
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C22×C8—24no—
167C22×C8—24no—
168C22×C8—24no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—24no—
174C22×C8—22no—
175C22×C8—82no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—44no—
187C4×D4—44no—
188C4×D4C4⋊C414yes—
189C4×D4C4⋊C414yes—
190C4×D4C22×C412yes—
191C4×D4C22×C412yes—
192C4×D4C2×Q812yes—
193C4×D4C2×Q812yes—
194#117—44no—
195#117—44no—
196#117—82no—
197#117—82no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C4⋊C414yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—84no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—41no—
232C2×C42C22×C412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×Q812yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42C2411yescharacteristic
240#107—48no—
241#107—82no—
242#106—48no—
243#106—82no—
244C4⋊C8—24no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8C22×C412yes—
250C4⋊C8C22×C412yes—
251#102—44no—
252#102—44no—
253C22⋊C8—42no—
254#97—42no—
255C4×C8—24no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8C22×C412yes—
259C4×C8C22×C412yes—
260#95—24no—
261#95—24no—
262#95—21no—
263#95—22no—
264#95—21no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—41no—
270#95—41no—
271C24—21no—
272C24—21no—
273C24—24no—
274C24—44no—
275C24—42no—
276C24—42no—
277C24—82no—
278C24#12011yescharacteristic
279C2×D4—28no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—44no—
289C2×D4C2×C4⋊C412yes—
290C2×D4C2×C4⋊C412yes—
291C2×D4#13514yes—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—48no—
315C22×C4—42no—
316C22×C4—88no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4C2×C4⋊C412yes—
325C22×C4C4×D412yes—
326C22×C4C4×D412yes—
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12011yescharacteristic
330C22×C4C22⋊Q812yes—
331C22×C4C22⋊Q812yes—
332C22×C4#12312yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—216no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—84no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—84no—
369C4⋊C4C2×C4⋊C412yes—
370C4⋊C4C2×C4⋊C412yes—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—48no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—42no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C22⋊C4—82no—
384C42—42no—
385C42—42no—
386C42—82no—
387C42—82no—
388C42—84no—
389C42C2×C4⋊C412yes—
390C42C4×D412yes—
391C42C4×D412yes—
392C42C22⋊Q812yes—
393C42C22⋊Q812yes—
394C42#12312yes—
395C42C23×C411yescharacteristic
396C42C22×Q811yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C23—22no—
400C23—22no—
401C23—24no—
402C23—22no—
403C23—22no—
404C23—48no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—48no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—84no—
414C23—88no—
415C23—88no—
416C23#38512yes—
417C23#45312yes—
418C23#45612yes—
419C23C2×#12011yescharacteristic
420D4—24no—
421D4—22no—
422D4—22no—
423D4—22no—
424D4—48no—
425D4—42no—
426D4#42714yescomplemented
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—168no—
467C2×C4#38512yes—
468C2×C4#38512yes—
469C2×C4#38512yes—
470C2×C4C22×C4⋊C411yescharacteristic
471C2×C4C2×C4×D411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×C22⋊Q811yescharacteristic
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4C2×#12311yescharacteristic
476C2×C4C2×#13512yes—
477C2×C4#57512yes—
478C2×C4#57612yes—
479C2×C4#57712yes—
480C8—24no—
481C8—24no—
482C8—42no—
483C8—42no—
484C8—82no—
485C22—22no—
486C22—22no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—44no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—816no—
498C22—82no—
499C22—82no—
500C22#185914yes—
501C22C2×#38511yescharacteristic
502C22#300212yes—
503C22#301212yes—
504C22C2×#45311yescharacteristic
505C22C2×#45611yescharacteristic
506C4—22no—
507C4—21no—
508C4—21no—
509C4—22no—
510C4—22no—
511C4—42no—
512C4—42no—
513C4—84no—
514C4—82no—
515C4—82no—
516C4—162no—
517C4C2×#42711yescharacteristic
518C4#291111yescharacteristic
519C4#294811yescharacteristic
520C4#294911yescharacteristic
521C4#308112yes—
522C4#309112yes—
523C2—24no—
524C2—44no—
525C2—82no—
526C2#1861812yes—
527C2#1864812yes—
528C2#2084311yescharacteristic
529C2C2×#185911yescharacteristic
530C2#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. #20797C2semidirect
6. #8889C2semidirect
7. #8886C2semidirect
8. #8885C2semidirect
9. #8882C2semidirect
10. #8044C2semidirect
11. #7898C2semidirect
32. #1948C22semidirect
33. #1947C22semidirect
35. #1845C22semidirect
36. #1844C22semidirect
45. C2×D4⋊C8C4semidirect
50. #1296C22semidirect
51. #1296C22semidirect
52. #1279C22semidirect
53. #1279C22semidirect
86. C2×C4⋊C8C2×C4semidirect
96. C2×C4×C8C2×C4semidirect
143. D4⋊C8C2×C4semidirect
144. D4⋊C8C2×C4semidirect
426. D4#427semidirect

Maximal subgroups

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

subgroupindex
#341942
#219622
#207972
#207382
C22×D4⋊C82
#88892
#88862
#88852
#88822
#80442
#78982