Finite Group Databasebeta

← #190785

Subgroups — #190785

Order 512.

888 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#8724C211yesmaximal, characteristic
5#8723C211yesmaximal, characteristic
6#8591C211yesmaximal, characteristic, complemented
7#8590C211yesmaximal, characteristic, complemented
8#8587C211yesmaximal, characteristic, complemented
9#8586C211yesmaximal, characteristic, complemented
10#8343C211yesmaximal, characteristic
11#8343C211yesmaximal, characteristic
12#8236C211yesmaximal, characteristic, complemented
13#8235C211yesmaximal, characteristic, complemented
14#8233C211yesmaximal, characteristic, complemented
15#8232C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#433—21no—
18#2360—21no—
19#2360—21no—
20#2360C412yes—
21#2360C412yes—
22#2355—21no—
23#2355C412yes—
24Q8×C4⋊C4—21no—
25Q8×C4⋊C4C412yes—
26#2330—21no—
27#2330—21no—
28#2317—21no—
29C4×C4⋊Q8C2211yescharacteristic
30C4×C4⋊Q8C2211yescharacteristic
31C4×C4⋊Q8C2211yescharacteristic
32C4×C4⋊Q8C2211yescharacteristic
33C2×#384C2211yescharacteristic
34#2274—21no—
35C4×#117C2211yescharacteristic
36#1924—21no—
37#1924—21no—
38#1917—21no—
39#1917—21no—
40#1842C2211yescharacteristic
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1832—21no—
44#1831C2211yescharacteristic
45#1798—21no—
46#1797C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1768—21no—
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#533—22no—
72#533—22no—
73#533—22no—
74#533—22no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#125—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8C2×C412yes—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8C2×C412yes—
84C2×C4×Q8C2×C412yes—
85C2×C4×Q8C2×C412yes—
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2×C412yes—
92C22×C42C2311yescharacteristic
93#433—44no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116#391—21no—
117#391—21no—
118#391C2×C412yes—
119#391C2×C412yes—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389C2×C412yes—
131#389C2×C412yes—
132#387—22no—
133#387—22no—
134#387—22no—
135#387—22no—
136#385—22no—
137#385—22no—
138#384—22no—
139#384—21no—
140#384—21no—
141#384C2×C412yes—
142#384C2311yescharacteristic
143#384C2311yescharacteristic
144#382—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2×C412yes—
154C4×C4⋊C4C2×C412yes—
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C4⋊C4C2×C411yescharacteristic
157C4×C4⋊C4D411yescharacteristic
158C4×C4⋊C4D411yescharacteristic
159C4×C22⋊C4—22no—
160C4×C22⋊C4—22no—
161C4×C22⋊C4C2×C411yescharacteristic
162C4×C22⋊C4C2×C411yescharacteristic
163C4×C22⋊C4D411yescharacteristic
164C4×C22⋊C4D411yescharacteristic
165#376D411yescharacteristic
166#376D411yescharacteristic
167C43C2311yescharacteristic
168C43C2311yescharacteristic
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#333—22no—
180#333—22no—
181#333—22no—
182#333—22no—
183#332—22no—
184#332—22no—
185#332—22no—
186#332—22no—
187C22×Q8—21no—
188C22×Q8C22×C412yes—
189C23×C4—22no—
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8#4112yes—
206C4⋊Q8#4112yes—
207C4⋊Q8#4112yes—
208C4⋊Q8#4112yes—
209#125—24no—
210#125—24no—
211C22⋊Q8—24no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—81no—
220C4×Q8—81no—
221C4×Q8—81no—
222C4×Q8—81no—
223C4×Q8—82no—
224C4×Q8—82no—
225#117—24no—
226#117—24no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—81no—
250C2×C4⋊C4—81no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22×C412yes—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C412yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2411yescharacteristic
289C4⋊C8—21no—
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—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—42no—
306C4⋊C8—42no—
307C4⋊C8—42no—
308C4⋊C8—42no—
309C4⋊C8—81no—
310C4⋊C8—81no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—81no—
314C22⋊C8—81no—
315#97—42no—
316#97—81no—
317#97—81no—
318C4×C8—42no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95C22⋊C411yescharacteristic
328#95C22⋊C411yescharacteristic
329#95C22⋊C411yescharacteristic
330#95C22⋊C411yescharacteristic
331#95C22⋊C411yescharacteristic
332#95C22⋊C411yescharacteristic
333#95C22⋊C411yescharacteristic
334#95C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8#11714yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—44no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4AGL(1,ℤ/8)12yes—
379C22×C4#13712yes—
380C22×C4C23×C411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
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—41no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—44no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4#11714yes—
438C22⋊C4—28no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C42—28no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—82no—
469C42—82no—
470C42—82no—
471C42—82no—
472C42#13712yes—
473C42#13712yes—
474C42C2×#4111yescharacteristic
475C42C2×#4111yescharacteristic
476C42C2×#4111yescharacteristic
477C42C2×#4111yescharacteristic
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23#41712yes—
485C23C22×C22⋊C411yescharacteristic
486Q8—44no—
487Q8—44no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—164no—
547C2×C4#41712yes—
548C2×C4#41912yes—
549C2×C4#41912yes—
550C2×C4C2×#11711yescharacteristic
551C2×C4C2×#11711yescharacteristic
552C2×C4#52511yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4#52511yescharacteristic
555C2×C4#52511yescharacteristic
556C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
557C2×C4C2×#13711yescharacteristic
558C2×C4C2×#13711yescharacteristic
559C2×C4C2×#13711yescharacteristic
560C2×C4#57611yescharacteristic
561C2×C4#57711yescharacteristic
562C2×C4#57711yescharacteristic
563C2×C4#57811yescharacteristic
564C8—81no—
565C8—81no—
566C8—81no—
567C8—81no—
568C8—81no—
569C8—81no—
570C8—81no—
571C8—81no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22#228011yescharacteristic
580C22#288811yescharacteristic
581C22#288911yescharacteristic
582C22C2×#41911yescharacteristic
583C22C2×#41711yescharacteristic
584C22#289711yescharacteristic
585C22#289811yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—44no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—44no—
601C4—82no—
602C4—82no—
603C4#311211yescharacteristic
604C4#311211yescharacteristic
605C4#311311yescharacteristic
606C4#311311yescharacteristic
607C4#311911yescharacteristic
608C4#312011yescharacteristic
609C4#312411yescharacteristic
610C4#312511yescharacteristic
611C2—42no—
612C2#1776011yescharacteristic
613C2#1776111yescharacteristic
614C2#1776211yescharacteristic
615C2#1776311yescharacteristic
616C2#2073111yescharacteristic
617C2#2074011yescharacteristic
618C2#2074611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
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
6. #8591C2semidirect
7. #8590C2semidirect
8. #8587C2semidirect
9. #8586C2semidirect
12. #8236C2semidirect
13. #8235C2semidirect
14. #8233C2semidirect
15. #8232C2semidirect

Maximal subgroups

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

subgroupindex
#342142
#219262
#208122
#87242
#87232
#85912
#85902
#85872
#85862
#83432
#82362
#82352
#82332
#82322