Finite Group Databasebeta

← #196032

Subgroups — #196032

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic, complemented
2C2×#2325C211yesmaximal, characteristic
3#21742C211yesmaximal, characteristic, complemented
4C2×#1465C212yesmaximal, complemented
5#9086C212yesmaximal, complemented
6#9084C212yesmaximal, complemented
7#9081C212yesmaximal, complemented
8#9079C212yesmaximal, complemented
9#7909C212yesmaximal, complemented
10C22×#121C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13#2973—28no—
14C2×#435—24no—
15C2×#432C2212yes—
16C2×#407C2212yes—
17C2×#407C2212yes—
18C22×C22⋊C8C2212yes—
19#2325—24no—
20#2325—24no—
21C2×#388C411yescharacteristic
22C2×#388C2211yescharacteristic
23C2×#387C411yescharacteristic
24C2×#387C2211yescharacteristic
25C2×#386C411yescharacteristic
26C2×#386C411yescharacteristic
27C2×#386C411yescharacteristic
28C2×#386C2211yescharacteristic
29C2×#385C411yescharacteristic
30C2×#385C2211yescharacteristic
31C2×#380C411yescharacteristic
32C22×#95C2211yescharacteristic
33#1870—24no—
34#1817C2212yes—
35#1779C2212yes—
36#1465—28no—
37C2×#324C2214yes—
38C2×#323C2214yes—
39#1280C2214yes—
40#1280C2214yes—
41C23×D4C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C23×D4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C23×C8—22no—
52C2×#121—24no—
53C2×#121—24no—
54C2×#121—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4D414yes—
62C2×C4×D4D414yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2311yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2311yescharacteristic
80#435—232no—
81#432—28no—
82C2×C4⋊C8—22no—
83#407—24no—
84#407—24no—
85#407—24no—
86#407D418yescomplemented
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—28no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2314yes—
94C2×C22⋊C8C2314yes—
95C2×#97—22no—
96#388—24no—
97#388—22no—
98#388—22no—
99#387—24no—
100#387—22no—
101#387—22no—
102#386—24no—
103#386—22no—
104#386—22no—
105#386—24no—
106#386—24no—
107#385—22no—
108#385—22no—
109#385—24no—
110#380—24no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2311yescharacteristic
122C2×#95C2311yescharacteristic
123#336—24no—
124#324—28no—
125#323—28no—
126C25C22×C411yescharacteristic
127C25C22×C411yescharacteristic
128C25C22×C411yescharacteristic
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—81no—
141C22×D4—81no—
142C22×D4C22⋊C414yes—
143C22×D4M1614yescomplemented
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—42no—
153C23×C4—81no—
154C23×C4—81no—
155C23×C4—81no—
156C23×C4—81no—
157C23×C4—81no—
158C23×C4—81no—
159C23×C4—81no—
160C23×C4—81no—
161C23×C4—81no—
162C23×C4—81no—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D414yes—
179C23×C4#4114yes—
180C23×C4C2411yescharacteristic
181C2×M16—28no—
182C2×M16—44no—
183C2×M16—82no—
184C2×M16—82no—
185C2×M16—82no—
186C2×M16—82no—
187C22×C8—24no—
188C22×C8—28no—
189C22×C8—24no—
190C22×C8—82no—
191C22×C8—82no—
192#121—232no—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—28no—
196C4×D4—28no—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4C22⋊C414yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—81no—
232C2×C22⋊C4—81no—
233C2×C22⋊C4—81no—
234C2×C22⋊C4—81no—
235C2×C22⋊C4C22⋊C414yes—
236C2×C22⋊C4C22⋊C414yes—
237C2×C22⋊C4M1614yescomplemented
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C4⋊C8—28no—
245C22⋊C8—28no—
246C22⋊C8—28no—
247C22⋊C8—24no—
248C22⋊C8—24no—
249C22⋊C8—24no—
250C22⋊C8—44no—
251C22⋊C8—82no—
252C22⋊C8—82no—
253C22⋊C8C2×D418yes—
254#97—28no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24C2×C22⋊C412yes—
289C24C2×C22⋊C412yes—
290C24C4×D414yes—
291C24C2×M1612yes—
292C24C2×M1612yes—
293C24C2×M1612yes—
294C24C23×C411yescharacteristic
295C2×D4—28no—
296C2×D4—28no—
297C2×D4—28no—
298C2×D4—28no—
299C2×D4—44no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—84no—
309C2×D4—84no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—28no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—88no—
379C22×C4—88no—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C4×D414yes—
385C22×C4#12014yes—
386C22×C4#12114yes—
387C22×C4#12114yes—
388C22×C4#12314yes—
389C22×C4C2×M1612yes—
390C22×C4C2×M1612yes—
391C22×C4C2×M1612yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C2×#4112yes—
396M16—88no—
397M16—88no—
398C2×C8—28no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—44no—
404C2×C8—88no—
405C4⋊C4—28no—
406C4⋊C4—28no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4—44no—
417C22⋊C4—41no—
418C22⋊C4—42no—
419C22⋊C4—41no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C22⋊C4—82no—
423C22⋊C4—82no—
424C42—22no—
425C42—24no—
426C42—22no—
427C42—44no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—22no—
449C23—24no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23#38614yes—
482C23#40714yes—
483C23C22×C22⋊C411yescharacteristic
484C23C2×C4×D412yes—
485C23C22×M1611yescharacteristic
486C23C22×M1611yescharacteristic
487C23#56812yes—
488D4—416no—
489D4—416no—
490D4—88no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
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—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—84no—
545C2×C4—88no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4#38614yes—
555C2×C4C2×#12011yescharacteristic
556C2×C4C2×#12112yes—
557C2×C4C2×#12311yescharacteristic
558C8—48no—
559C8—84no—
560C22—22no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—48no—
575C22—48no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C2×#38611yescharacteristic
596C22C2×#40711yescharacteristic
597C22#287911yescharacteristic
598C22#288311yescharacteristic
599C22#288311yescharacteristic
600C22#288811yescharacteristic
601C22#288911yescharacteristic
602C22D4×M1612yes—
603C22D4×M1612yes—
604C4—21no—
605C4—22no—
606C4—21no—
607C4—24no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—81no—
618C2—22no—
619C2—22no—
620C2—44no—
621C2—44no—
622C2—82no—
623C2#1736711yescharacteristic
624C2#1736811yescharacteristic
625C2#1740511yescharacteristic
626C2#1740611yescharacteristic
627C2#2089811yescharacteristic
628C2#2098411yescharacteristic
629C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2973C2semidirect
3. #21742C2semidirect
4. C2×#1465C2semidirect
5. #9086C2semidirect
6. #9084C2semidirect
7. #9081C2semidirect
8. #9079C2semidirect
9. #7909C2semidirect
86. #407D4semidirect
143. C22×D4M16semidirect
237. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
C2×#23252
#217422
C2×#14652
#90862
#90842
#90812
#90792
#79092