Finite Group Databasebeta

← #204815

Subgroups — #204815

Order 512.

1082 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35298C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic, complemented
3#20470C211yesmaximal, characteristic, complemented
4C2×#1553C211yesmaximal, characteristic
5C2×#1552C211yesmaximal, characteristic, complemented
6C2×#1544C211yesmaximal, characteristic, complemented
7C2×#1540C211yesmaximal, characteristic
8#9710C211yesmaximal, characteristic, complemented
9#9708C211yesmaximal, characteristic, complemented
10#9704C211yesmaximal, characteristic, complemented
11#9703C211yesmaximal, characteristic, complemented
12#9700C211yesmaximal, characteristic, complemented
13#9698C211yesmaximal, characteristic, complemented
14#9696C211yesmaximal, characteristic, complemented
15#9692C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2868C2211yescharacteristic
23#2862C2211yescharacteristic
24#2858C2211yescharacteristic
25#2858C2211yescharacteristic
26#2620—21no—
27#2620—21no—
28#2566C414yes—
29#2545—22no—
30#2500C414yes—
31#2432—22no—
32#2326—21no—
33#2326—21no—
34C4×#117C2211yescharacteristic
35#1939—22no—
36#1850—22no—
37#1842C2211yescharacteristic
38#1842C2211yescharacteristic
39#1831C2211yescharacteristic
40#1831C2211yescharacteristic
41#1767C2211yescharacteristic
42#1767C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45#1750—21no—
46#1750—21no—
47#1746—21no—
48#1726—21no—
49#1553—24no—
50#1552—24no—
51#1544—24no—
52#1540—24no—
53C2×#332C2211yescharacteristic
54C2×#332C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C2×#331C2211yescharacteristic
69C22×#41—21no—
70C23×C8—21no—
71#533—22no—
72#533—22no—
73#533—22no—
74#533—22no—
75#532—22no—
76#532—22no—
77#532—22no—
78#532—22no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×#127C2311yescharacteristic
84C2×#125C2×C412yes—
85C2×#124—21no—
86C2×#124—21no—
87C2×#124—21no—
88C2×#124—22no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×#121—21no—
92C2×#121C2×C412yes—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2×C412yes—
97C2×#117C2×C412yes—
98C2×#117D411yescharacteristic
99C2×#117D411yescharacteristic
100C22×C42C2311yescharacteristic
101#432—22no—
102#432D414yes—
103#431—22no—
104#431—22no—
105#431—22no—
106#431D414yes—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130#396—24no—
131#396—24no—
132#394—24no—
133#394—24no—
134#391—21no—
135#391—21no—
136#391—22no—
137#390—21no—
138#390—21no—
139#390—22no—
140#390C2×C414yes—
141#390C2×C414yes—
142#388—24no—
143#388—24no—
144#385—24no—
145#385—24no—
146#384—21no—
147#384—21no—
148#384C2×C414yes—
149#382C2×C414yes—
150C4×C4⋊C4C2×C412yes—
151C4×C4⋊C4D412yes—
152C4×C22⋊C4C2×C412yes—
153C4×C22⋊C4D412yes—
154#376D411yescharacteristic
155#376D411yescharacteristic
156C43C2311yescharacteristic
157C43C2311yescharacteristic
158#336—22no—
159#336—22no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#332—22no—
165#332—22no—
166#332—22no—
167#332—22no—
168#332—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#332—22no—
174#332—22no—
175#332—22no—
176#331—22no—
177#331—22no—
178#331—22no—
179#331—22no—
180C2×#41—42no—
181C2×#41—42no—
182C2×#41—82no—
183C2×#41—82no—
184C22×Q8—21no—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×D4—22no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4C22×C412yes—
191C23×C4—22no—
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—42no—
204C22×C8—82no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—82no—
218C4⋊Q8—82no—
219#127—21no—
220#127—21no—
221#127—21no—
222#127—21no—
223#125—24no—
224#125—24no—
225#124—42no—
226#124—42no—
227#124—42no—
228#124—42no—
229#124—42no—
230#124—42no—
231#124—82no—
232#124—82no—
233C22⋊Q8—42no—
234C22⋊Q8—42no—
235C22⋊Q8—42no—
236C22⋊Q8—42no—
237#121—24no—
238#121—24no—
239#121—42no—
240#121—42no—
241C4×Q8—42no—
242C4×Q8—42no—
243C4×Q8—42no—
244C4×Q8—42no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—42no—
248C4×D4—42no—
249#117—22no—
250#117—22no—
251#117—24no—
252#117—24no—
253#117—41no—
254#117—41no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4C22⋊C412yes—
263C2×C4⋊C4C22⋊C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4C22⋊C412yes—
275C2×C22⋊C4C22⋊C412yes—
276C2×C42—22no—
277C2×C42C22×C412yes—
278C2×C42C22×C412yes—
279C2×C42C22×C411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4112yes—
291C2×C42#4112yes—
292C2×C42C2411yescharacteristic
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—42no—
306C4⋊C8—82no—
307C4⋊C8—82no—
308C4⋊C8C2×D412yes—
309C4⋊C8C2×D412yes—
310C4⋊C8C2×D412yes—
311C4⋊C8C2×D412yes—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8—42no—
315#97—24no—
316#97—42no—
317#97—42no—
318C4×C8—24no—
319C4×C8—42no—
320C4×C8—42no—
321C4×C8—42no—
322#95—24no—
323#95—24no—
324#95C22⋊C412yes—
325#95C22⋊C412yes—
326#95C22⋊C412yes—
327#95C22⋊C412yes—
328C24—22no—
329C24C2×C22⋊C411yescharacteristic
330#41—88no—
331#41—84no—
332#41—84no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—84no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—84no—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C412yes—
381C22×C4C4×D412yes—
382C22×C4C4×D412yes—
383C22×C4#12112yes—
384C22×C4#12112yes—
385C22×C4#12412yes—
386C22×C4#12712yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—84no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C22⋊C4—24no—
430C22⋊C4—42no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—42no—
449C42—42no—
450C42—44no—
451C42C4×D412yes—
452C42C4×D412yes—
453C42#12112yes—
454C42#12112yes—
455C42#12412yes—
456C42#12712yes—
457C42#13512yes—
458C42#13512yes—
459C42C22×D411yescharacteristic
460C42C22×D411yescharacteristic
461C42C2×#4111yescharacteristic
462C42C2×#4111yescharacteristic
463C23—22no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—42no—
472C23#39012yes—
473C23C22×C22⋊C411yescharacteristic
474Q8—82no—
475Q8—82no—
476Q8—81no—
477Q8—81no—
478Q8—81no—
479Q8—81no—
480Q8—81no—
481Q8—81no—
482Q8—81no—
483Q8—81no—
484D4—82no—
485D4—82no—
486D4—88no—
487D4—82no—
488D4—81no—
489D4—81no—
490D4—81no—
491D4—81no—
492D4—81no—
493D4—81no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—88no—
527C2×C4#39012yes—
528C2×C4#39012yes—
529C2×C4#39012yes—
530C2×C4#41614yes—
531C2×C4C2×C4×D411yescharacteristic
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×#12111yescharacteristic
535C2×C4C2×#12411yescharacteristic
536C2×C4C2×#12711yescharacteristic
537C2×C4C2×#13511yescharacteristic
538C2×C4C2×#13511yescharacteristic
539C2×C4#57511yescharacteristic
540C2×C4#57511yescharacteristic
541C2×C4#57612yes—
542C2×C4#57712yes—
543C8—42no—
544C8—42no—
545C8—42no—
546C8—42no—
547C8—81no—
548C8—81no—
549C8—81no—
550C8—81no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—42no—
555C22—42no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—82no—
560C22—82no—
561C22C2×#39011yescharacteristic
562C22#288912yes—
563C22C2×#41611yescharacteristic
564C22#289611yescharacteristic
565C22#289712yes—
566C4—22no—
567C4—24no—
568C4—22no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4#303511yescharacteristic
579C4#303511yescharacteristic
580C4#303511yescharacteristic
581C4#303511yescharacteristic
582C4#313612yes—
583C4#315411yescharacteristic
584C4#315511yescharacteristic
585C2—42no—
586C2—162no—
587C2#1772712yes—
588C2#1776912yes—
589C2#2120611yescharacteristic
590C2#2122011yescharacteristic
591C2#2125711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21972C2semidirect
3. #20470C2semidirect
5. C2×#1552C2semidirect
6. C2×#1544C2semidirect
8. #9710C2semidirect
9. #9708C2semidirect
10. #9704C2semidirect
11. #9703C2semidirect
12. #9700C2semidirect
13. #9698C2semidirect
14. #9696C2semidirect
15. #9692C2semidirect

Maximal subgroups

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

subgroupindex
#352982
#219722
#204702
C2×#15532
C2×#15522
C2×#15442
C2×#15402
#97102
#97082
#97042
#97032
#97002
#96982
#96962
#96922