Finite Group Databasebeta

← #188412

Subgroups — #188412

Order 512.

1192 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#21974C211yesmaximal, characteristic, complemented
3#20809C211yesmaximal, characteristic, complemented
4C2×#1486C211yesmaximal, characteristic
5C2×#1478C211yesmaximal, characteristic
6#8569C212yesmaximal, complemented
7#8567C212yesmaximal, complemented
8#8565C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10#7899C211yesmaximal, characteristic, complemented
11#7894C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#2357C411yescharacteristic
18#2357C411yescharacteristic
19#2353—21no—
20#2353—21no—
21#2349C411yescharacteristic
22#2349C411yescharacteristic
23#2345C412yes—
24#2342—22no—
25D4×C22⋊C4C412yes—
26#2293C2212yes—
27#2290—22no—
28#2283C2211yescharacteristic
29#2280—21no—
30C42×D4C2212yes—
31C4×#117C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33#1939—22no—
34#1851—21no—
35#1851—21no—
36#1843—21no—
37#1842C2212yescomplemented
38#1832—21no—
39#1831C2212yescomplemented
40#1798—22no—
41C4×C4⋊C8C2212yescomplemented
42C4×C4⋊C8C2212yescomplemented
43#1486—24no—
44#1478—24no—
45C2×D4⋊C8C2211yescharacteristic
46C2×D4⋊C8C2211yescharacteristic
47C2×D4⋊C8C2211yescharacteristic
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2212yes—
50C2×D4⋊C8C2212yes—
51#1279C2212yescomplemented
52#1279C2212yescomplemented
53#1279C2212yescomplemented
54#1279C2212yescomplemented
55C23×D4C2×C411yescharacteristic
56C24×C4—21no—
57C22×M16—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D414yes—
67C2×C4×D4Q814yes—
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—22no—
71C2×#117D412yes—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C42—22no—
76C22×C42C2311yescharacteristic
77#431—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#423—22no—
84#423—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2311yescharacteristic
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2312yes—
102#390C2×C412yes—
103#390C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#386—22no—
113#386—22no—
114#385—22no—
115#385—22no—
116#383—21no—
117#382C2×C411yescharacteristic
118#382C2×C411yescharacteristic
119#381—21no—
120#381—21no—
121#381—21no—
122#380—22no—
123#380—21no—
124#380—21no—
125#380C2×C412yes—
126#379—22no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4D412yes—
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4C2×C412yes—
142C4×C22⋊C4D411yescharacteristic
143C4×C22⋊C4D411yescharacteristic
144#376—21no—
145#376D411yescharacteristic
146#376D411yescharacteristic
147C2×#95—21no—
148C2×#95—21no—
149C43C2312yes—
150#336—22no—
151#336—22no—
152#336—22no—
153#336—22no—
154D4⋊C8—22no—
155D4⋊C8—22no—
156D4⋊C8—22no—
157D4⋊C8—22no—
158D4⋊C8—24no—
159D4⋊C8—24no—
160C25—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4C4⋊C414yes—
168C22×D4C22×C411yescharacteristic
169C22×D4C22×C411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—82no—
180C23×C4—82no—
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C2×M16—42no—
185C2×M16—82no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—21no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4C4⋊C414yes—
207C4×D4C4⋊C414yes—
208C4×D4C2×D412yes—
209C4×D4C2×D412yes—
210C4×D4C2×Q812yes—
211C4×D4C2×Q812yes—
212#117—44no—
213#117—82no—
214#117—82no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C4⋊C414yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4C22⋊C412yes—
241C2×C22⋊C4C22⋊C412yes—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—82no—
246C2×C42—82no—
247C2×C42C22×C412yes—
248C2×C42C22×C412yes—
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×Q812yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42C2411yescharacteristic
261C4⋊C8—22no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—22no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C4⋊C8—42no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272C4⋊C8—82no—
273C4⋊C8—82no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—22no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—22no—
283C4×C8—42no—
284#95—21no—
285#95—21no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—21no—
290#95—21no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95C22⋊C411yescharacteristic
295#95C22⋊C411yescharacteristic
296#95C22⋊C412yes—
297#95C22⋊C412yes—
298#95C22⋊C411yescharacteristic
299#95C22⋊C411yescharacteristic
300C24—21no—
301C24—21no—
302C24—24no—
303C24—42no—
304C24—42no—
305C24—44no—
306C24—82no—
307C24C2×C22⋊C411yescharacteristic
308C2×D4—28no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—24no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4C2×C4⋊C412yes—
324C2×D4C2×C4⋊C412yes—
325C2×D4#13112yes—
326C2×D4#13112yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—88no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C4⋊C412yes—
369C22×C4#11712yes—
370C22×C4#12112yes—
371C22×C4#12112yes—
372C22×C4C22⋊Q812yes—
373C22×C4C22⋊Q812yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377M16—84no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4C2×C4⋊C412yes—
404C4⋊C4C2×C4⋊C412yes—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—24no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—82no—
433C42—82no—
434C42C2×C4⋊C412yes—
435C42#11712yes—
436C42#12112yes—
437C42#12112yes—
438C42C22⋊Q812yes—
439C42C22⋊Q812yes—
440C42C22×D411yescharacteristic
441C42C22×Q811yescharacteristic
442C42C2×#4111yescharacteristic
443C42C2×#4111yescharacteristic
444C23—22no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—48no—
453C23—48no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—84no—
461C23—88no—
462C23—88no—
463C23#38012yes—
464C23C22×C22⋊C411yescharacteristic
465D4—22no—
466D4—22no—
467D4—22no—
468D4—22no—
469D4—22no—
470D4—48no—
471D4—41no—
472D4—41no—
473D4#42414yes—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
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—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—88no—
518C2×C4—84no—
519C2×C4—88no—
520C2×C4#38012yes—
521C2×C4#38012yes—
522C2×C4#38012yes—
523C2×C4#42114yes—
524C2×C4C22×C4⋊C411yescharacteristic
525C2×C4C2×#11711yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×C22⋊Q811yescharacteristic
530C2×C4C2×#13111yescharacteristic
531C2×C4C2×#13111yescharacteristic
532C2×C4#56811yescharacteristic
533C2×C4#56811yescharacteristic
534C2×C4#57612yes—
535C2×C4#57712yes—
536C8—81no—
537C8—81no—
538C8—82no—
539C8—81no—
540C8—82no—
541C8—81no—
542C22—22no—
543C22—24no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—42no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—816no—
556C22—82no—
557C22—82no—
558C22C2×#38011yescharacteristic
559C22#288011yescharacteristic
560C22#288011yescharacteristic
561C22#289712yes—
562C22C2×#42112yes—
563C4—24no—
564C4—22no—
565C4—22no—
566C4—21no—
567C4—21no—
568C4—22no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—82no—
575C4—82no—
576C4C2×#42411yescharacteristic
577C4#290511yescharacteristic
578C4#292411yescharacteristic
579C4#292411yescharacteristic
580C4#304111yescharacteristic
581C4#304211yescharacteristic
582C4#308112yes—
583C2—42no—
584C2—44no—
585C2—82no—
586C2#1752112yes—
587C2#1752212yes—
588C2#2062811yescharacteristic
589C2#2065011yescharacteristic
590C2#2065411yescharacteristic
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. #21974C2semidirect
3. #20809C2semidirect
6. #8569C2semidirect
7. #8567C2semidirect
8. #8565C2semidirect
9. #8563C2semidirect
10. #7899C2semidirect
11. #7894C2semidirect
37. #1842C22semidirect
39. #1831C22semidirect
41. C4×C4⋊C8C22semidirect
42. C4×C4⋊C8C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect
53. #1279C22semidirect
54. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#341262
#219742
#208092
C2×#14862
C2×#14782
#85692
#85672
#85652
#85632
#78992
#78942