Finite Group Databasebeta

← #205452

Subgroups — #205452

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3207C211yesmaximal, characteristic
2#21393C211yesmaximal, characteristic, complemented
3#21257C211yesmaximal, characteristic, complemented
4#20550C211yesmaximal, characteristic, complemented
5C2×#1573C211yesmaximal, characteristic, complemented
6C2×#1557C211yesmaximal, characteristic, complemented
7C2×#1547C211yesmaximal, characteristic, complemented
8#9748C211yesmaximal, characteristic, complemented
9#9746C211yesmaximal, characteristic, complemented
10#9744C211yesmaximal, characteristic, complemented
11#9743C211yesmaximal, characteristic, complemented
12#9738C211yesmaximal, characteristic, complemented
13#9736C211yesmaximal, characteristic, complemented
14#9734C211yesmaximal, characteristic, complemented
15#9732C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3207—24no—
19#3207C418yescomplemented
20C2×#493C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#463—21no—
25C2×#463—21no—
26C2×#449—21no—
27C2×#449—21no—
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C2×#416—22no—
33C4×#117C2211yescharacteristic
34#1988C2211yescharacteristic
35#1988C2211yescharacteristic
36#1983—22no—
37#1982C2211yescharacteristic
38#1982C2211yescharacteristic
39#1982C2211yescharacteristic
40#1982C2211yescharacteristic
41#1981C2211yescharacteristic
42#1981C2211yescharacteristic
43#1875—22no—
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48#1767—21no—
49C4×C4⋊C8—21no—
50#1762—22no—
51#1753—21no—
52#1753—21no—
53#1573—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1557—24no—
59#1547—24no—
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#331C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C22×#41—21no—
69C22×#41—21no—
70C22×SD16—21no—
71C22×SD16—21no—
72C22×D8—21no—
73C22×D8—21no—
74C22×M16—21no—
75C23×C8—21no—
76#532—22no—
77#532—22no—
78#532—22no—
79#532—22no—
80#532—22no—
81#532—22no—
82#532C2×C414yes—
83#532C2×C414yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#127C2311yescharacteristic
88C2×#124—21no—
89C2×#124—21no—
90C2×#121—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×#121—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×#117D412yes—
99C22×C42C2311yescharacteristic
100#493—22no—
101#493C2×C414yes—
102#492—22no—
103#492C2×C414yes—
104#487—22no—
105#487C2×C414yes—
106#486—22no—
107#486C2×C414yes—
108#463—28no—
109#463—28no—
110#449—28no—
111#449—28no—
112#432—22no—
113#432—22no—
114#431C2×C414yes—
115#431D414yes—
116#423—22no—
117#423D414yes—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8D411yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130#416—48no—
131C2×#103—21no—
132C2×#103—21no—
133C2×#103—22no—
134C2×#103—22no—
135C2×#102—21no—
136C2×#102—21no—
137C2×#102—22no—
138C2×#102—22no—
139C2×#102—22no—
140C2×#102—22no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8D411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144C2×#97—21no—
145C2×#97—21no—
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C4×C4⋊C4D412yes—
152C4×C4⋊C4C2311yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156#376—21no—
157C43—21no—
158#340—22no—
159#340—22no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#335—22no—
165#335—22no—
166#334—22no—
167#334—22no—
168#332—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#331—22no—
173#331—22no—
174#331—22no—
175#331—22no—
176C2×#41—24no—
177C2×#41—24no—
178C2×#41—44no—
179C2×#41—44no—
180C2×#41—42no—
181C2×#41—42no—
182C2×#41—81no—
183C2×#41—81no—
184C2×#41—81no—
185C2×#41—81no—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—41no—
195C22×D4—41no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4#4111yescharacteristic
201C2×SD16—24no—
202C2×SD16—24no—
203C2×SD16—81no—
204C2×SD16—81no—
205C2×SD16—81no—
206C2×SD16—81no—
207C2×D8—24no—
208C2×D8—24no—
209C2×D8—81no—
210C2×D8—81no—
211C2×D8—81no—
212C2×D8—81no—
213C2×M16—24no—
214C2×M16—81no—
215C2×M16—81no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—44no—
223C22×C8—42no—
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8C22×C412yes—
235C4⋊Q8C22×C412yes—
236#127—21no—
237#127—21no—
238#127—21no—
239#127—21no—
240#127—21no—
241#127—21no—
242#127C22×C412yes—
243#127C22×C412yes—
244#124—24no—
245#124—24no—
246#124—42no—
247#124—42no—
248#121—24no—
249#121—24no—
250#121—24no—
251#121—24no—
252#121—42no—
253#121—42no—
254#121—42no—
255#121—42no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—42no—
264#117—24no—
265#117—42no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42C22×C412yes—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D412yes—
286C2×C42#4111yescharacteristic
287C2×C42#4112yes—
288C2×C42C2411yescharacteristic
289C4⋊C8—24no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—42no—
301C4⋊C8C22⋊C414yes—
302C4⋊C8C22⋊C414yes—
303C4⋊C8C2×D412yes—
304C4⋊C8C2×D412yes—
305#103—24no—
306#103—24no—
307#103—44no—
308#103—44no—
309#102—24no—
310#102—24no—
311#102—44no—
312#102—44no—
313#102—44no—
314#102—44no—
315C22⋊C8—42no—
316C22⋊C8C22⋊C414yes—
317C22⋊C8C22⋊C414yes—
318#97—42no—
319#97—81no—
320#97—81no—
321C4×C8—81no—
322C4×C8—81no—
323C4×C8C22×C412yes—
324C4×C8C22×C412yes—
325C4×C8C2×D412yes—
326C4×C8C2×D412yes—
327#95—22no—
328#95—21no—
329#95—21no—
330C24—41no—
331C24—41no—
332C24#12111yescharacteristic
333#41—416no—
334#41—84no—
335#41—84no—
336#41—84no—
337#41—84no—
338#41—84no—
339#41—84no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—84no—
379C2×D4—82no—
380C2×D4—82no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—84no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4C2×C22⋊C412yes—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4#12112yes—
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12412yes—
405C22×C4#12712yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409SD16—84no—
410SD16—84no—
411D8—84no—
412D8—84no—
413M16—84no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—84no—
437C2×C8C2×C22⋊C412yes—
438C2×C8C2×C22⋊C412yes—
439C2×C8C2×C22⋊C412yes—
440C2×C8C2×C22⋊C412yes—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4AGL(1,ℤ/8)14yes—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C42—44no—
459C42—44no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42C2×C22⋊C412yes—
465C42C4×D412yes—
466C42C4×D412yes—
467C42#12112yes—
468C42#12412yes—
469C42#12712yes—
470C42C23×C411yescharacteristic
471C42C22×D411yescharacteristic
472C42C22×D411yescharacteristic
473C42C2×#4111yescharacteristic
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23#39012yes—
489C23C2×#12111yescharacteristic
490Q8—42no—
491Q8—42no—
492Q8—44no—
493Q8—81no—
494Q8—81no—
495Q8—81no—
496Q8—81no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—44no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—82no—
507D4—88no—
508D4—88no—
509D4—81no—
510D4—81no—
511D4—81no—
512D4—81no—
513D4—81no—
514D4—81no—
515D4—81no—
516D4—81no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—88no—
540C2×C4—88no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4#39012yes—
545C2×C4#39012yes—
546C2×C4#39012yes—
547C2×C4C22×C22⋊C411yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×#12411yescharacteristic
552C2×C4C2×#12711yescharacteristic
553C2×C4C2×AGL(1,ℤ/8)12yes—
554C2×C4#57512yes—
555C2×C4#57611yescharacteristic
556C2×C4#57611yescharacteristic
557C2×C4#57711yescharacteristic
558C2×C4#57711yescharacteristic
559C8—22no—
560C8—22no—
561C8—22no—
562C8—22no—
563C8—42no—
564C8—42no—
565C8—81no—
566C8—81no—
567C8—81no—
568C8—81no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—42no—
573C22—42no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—162no—
581C22—162no—
582C22—162no—
583C22—162no—
584C22C2×#39011yescharacteristic
585C22#304511yescharacteristic
586C22#304511yescharacteristic
587C22#305512yes—
588C22#306712yes—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—44no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4#289711yescharacteristic
599C4#289711yescharacteristic
600C4C4×AGL(1,ℤ/8)12yes—
601C4#304111yescharacteristic
602C4#304311yescharacteristic
603C4#315111yescharacteristic
604C4#315311yescharacteristic
605C2—42no—
606C2—162no—
607C2—162no—
608C2#1948012yes—
609C2#1949212yes—
610C2#2123611yescharacteristic
611C2#2124811yescharacteristic
612C2#2126311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3207C2
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. #21393C2semidirect
3. #21257C2semidirect
4. #20550C2semidirect
5. C2×#1573C2semidirect
6. C2×#1557C2semidirect
7. C2×#1547C2semidirect
8. #9748C2semidirect
9. #9746C2semidirect
10. #9744C2semidirect
11. #9743C2semidirect
12. #9738C2semidirect
13. #9736C2semidirect
14. #9734C2semidirect
15. #9732C2semidirect
19. #3207C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32072
#213932
#212572
#205502
C2×#15732
C2×#15572
C2×#15472
#97482
#97462
#97442
#97432
#97382
#97362
#97342
#97322