Finite Group Databasebeta

← #188408

Subgroups — #188408

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34126C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic, complemented
3C4×#424C211yesmaximal, characteristic, complemented
4C2×#1490C211yesmaximal, characteristic
5C22×D4⋊C8C211yesmaximal, characteristic
6#8569C212yesmaximal, complemented
7#8567C212yesmaximal, complemented
8#8565C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10#7898C211yesmaximal, characteristic, complemented
11#7895C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#424—21no—
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2357C412yes—
19#2353—22no—
20#2349C412yes—
21#2345C411yescharacteristic
22#2345C411yescharacteristic
23#2342—21no—
24#2342—21no—
25D4×C22⋊C4C411yescharacteristic
26D4×C22⋊C4C411yescharacteristic
27#2293C2212yes—
28#2290—22no—
29#2283C2211yescharacteristic
30#2280—21no—
31C42×D4C2212yes—
32C4×#117C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34#1940—22no—
35#1850—21no—
36#1850—21no—
37#1842C2212yescomplemented
38#1831C2212yescomplemented
39#1768—22no—
40C4×C4⋊C8C2212yescomplemented
41C4×C4⋊C8C2212yescomplemented
42#1729—21no—
43#1490—24no—
44C2×D4⋊C8—24no—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2211yescharacteristic
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2211yescharacteristic
51#1279C2212yescomplemented
52#1279C2212yescomplemented
53#1279C2212yescomplemented
54#1279C2212yescomplemented
55C23×D4C2×C411yescharacteristic
56C24×C4—21no—
57C22×M16—21no—
58C23×C8—21no—
59C23×C8—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D414yes—
69C2×C4×D4Q814yes—
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—22no—
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—22no—
80C22×C42C2311yescharacteristic
81#432—22no—
82#432—22no—
83#424—28no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×C22⋊C8—21no—
95C4×M16—22no—
96C4×M16—22no—
97C2×#97—21no—
98C2×#97—21no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8—21no—
102C2×C4×C8C2312yes—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#390C2×C412yes—
106#390C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#387—22no—
114#387—22no—
115#386—22no—
116#386—22no—
117#385—22no—
118#385—22no—
119#383—21no—
120#382C2×C412yes—
121#381—21no—
122#381—22no—
123#380—21no—
124#380—21no—
125#380—22no—
126#380C2×C411yescharacteristic
127#380C2×C411yescharacteristic
128#379—21no—
129#379—21no—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4D412yes—
134C4×C4⋊C4C2312yes—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4C2×C411yescharacteristic
144C4×C22⋊C4C2×C411yescharacteristic
145C4×C22⋊C4D411yescharacteristic
146C4×C22⋊C4D411yescharacteristic
147#376—21no—
148#376D412yes—
149C2×#95—22no—
150C43C2312yes—
151#336—22no—
152#336—22no—
153#336—22no—
154#336—22no—
155D4⋊C8—24no—
156D4⋊C8—24no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161C25—21no—
162C22×D4—24no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—42no—
166C22×D4C4⋊C414yes—
167C22×D4C22×C411yescharacteristic
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—82no—
179C23×C4—82no—
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C2×M16—24no—
184C2×M16—42no—
185C2×M16—42no—
186C22×C8—24no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—82no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4C4⋊C414yes—
212C4×D4C4⋊C414yes—
213C4×D4C2×D412yes—
214C4×D4C2×D412yes—
215C4×D4C2×Q812yes—
216C4×D4C2×Q812yes—
217#117—24no—
218#117—42no—
219#117—82no—
220#117—82no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C4⋊C414yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C22⋊C4C22⋊C411yescharacteristic
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—82no—
256C2×C42—82no—
257C2×C42C22×C412yes—
258C2×C42C22×C412yes—
259C2×C42C22×C411yescharacteristic
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×Q812yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270C2×C42C2411yescharacteristic
271C4⋊C8—28no—
272C4⋊C8—28no—
273C4⋊C8—22no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—22no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—42no—
280C4⋊C8—82no—
281C4⋊C8—82no—
282C22⋊C8—42no—
283#97—42no—
284#97—42no—
285C4×C8—22no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—22no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—42no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95C22⋊C412yes—
302#95C22⋊C412yes—
303#95C22⋊C412yes—
304#95C22⋊C412yes—
305C24—21no—
306C24—21no—
307C24—24no—
308C24—44no—
309C24—42no—
310C24—42no—
311C24—82no—
312C24C2×C22⋊C411yescharacteristic
313C2×D4—28no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—44no—
321C2×D4—48no—
322C2×D4—44no—
323C2×D4C2×C4⋊C412yes—
324C2×D4C2×C4⋊C412yes—
325C2×D4#13114yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—88no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C4⋊C412yes—
367C22×C4#11712yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4C22⋊Q812yes—
371C22×C4C22⋊Q812yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375M16—48no—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—48no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—84no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4C2×C4⋊C412yes—
406C4⋊C4C2×C4⋊C412yes—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—44no—
422C22⋊C4—82no—
423C22⋊C4—82no—
424C22⋊C4#10414yes—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—24no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—82no—
438C42—82no—
439C42C2×C4⋊C412yes—
440C42#11712yes—
441C42#12112yes—
442C42#12112yes—
443C42C22⋊Q812yes—
444C42C22⋊Q812yes—
445C42C22×D411yescharacteristic
446C42C22×Q811yescharacteristic
447C42C2×#4111yescharacteristic
448C42C2×#4111yescharacteristic
449C23—22no—
450C23—22no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—48no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—48no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—84no—
464C23—88no—
465C23—88no—
466C23#38012yes—
467C23C2×#10412yes—
468C23#42112yes—
469C23C22×C22⋊C411yescharacteristic
470D4—22no—
471D4—22no—
472D4—22no—
473D4—24no—
474D4—48no—
475D4—42no—
476D4#42414yescomplemented
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—84no—
522C2×C4—88no—
523C2×C4#38012yes—
524C2×C4#38012yes—
525C2×C4#38012yes—
526C2×C4C2×#10414yes—
527C2×C4C22×C4⋊C411yescharacteristic
528C2×C4C2×#11711yescharacteristic
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×#13112yes—
534C2×C4#56812yes—
535C2×C4#57612yes—
536C2×C4#57712yes—
537C8—44no—
538C8—42no—
539C8—42no—
540C8—81no—
541C8—82no—
542C8—81no—
543C22—24no—
544C22—24no—
545C22—22no—
546C22—22no—
547C22—24no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—816no—
556C22—82no—
557C22—82no—
558C22#180614yes—
559C22C2×#38011yescharacteristic
560C22#288012yes—
561C22#289712yes—
562C22C22×#10411yescharacteristic
563C22C2×#42111yescharacteristic
564C4—24no—
565C4—22no—
566C4—22no—
567C4—21no—
568C4—21no—
569C4—22no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—82no—
577C4C2×#42411yescharacteristic
578C4#290511yescharacteristic
579C4#292412yes—
580C4#304111yescharacteristic
581C4#304211yescharacteristic
582C4#308112yes—
583C2—24no—
584C2—44no—
585C2—82no—
586C2#1751712yes—
587C2#1752112yes—
588C2#2062811yescharacteristic
589C2#2065011yescharacteristic
590C2C2×#180611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21976C2semidirect
3. C4×#424C2semidirect
6. #8569C2semidirect
7. #8567C2semidirect
8. #8565C2semidirect
9. #8563C2semidirect
10. #7898C2semidirect
11. #7895C2semidirect
37. #1842C22semidirect
38. #1831C22semidirect
40. C4×C4⋊C8C22semidirect
41. C4×C4⋊C8C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect
53. #1279C22semidirect
54. #1279C22semidirect
476. D4#424semidirect

Maximal subgroups

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

subgroupindex
#341262
#219762
C4×#4242
C2×#14902
C22×D4⋊C82
#85692
#85672
#85652
#85632
#78982
#78952