Finite Group Databasebeta

← #182845

Subgroups — #182845

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34225C211yesmaximal, characteristic
2#20549C211yesmaximal, characteristic
3#20431C211yesmaximal, characteristic
4C2×#1561C211yesmaximal, characteristic
5C2×#1552C211yesmaximal, characteristic
6#8237C211yesmaximal, characteristic, complemented
7#8237C211yesmaximal, characteristic, complemented
8#8236C211yesmaximal, characteristic, complemented
9#8236C211yesmaximal, characteristic, complemented
10#8233C211yesmaximal, characteristic, complemented
11#8233C211yesmaximal, characteristic, complemented
12C4×#332C211yesmaximal, characteristic, complemented
13C4×#332C211yesmaximal, characteristic, complemented
14#8045C211yesmaximal, characteristic
15#8042C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#405—21no—
22#2370C412yes—
23#2370C412yes—
24#2362—21no—
25#2362—21no—
26#2356—21no—
27#2351—21no—
28#2347C412yes—
29#2346C412yes—
30#2323—21no—
31#2323—21no—
32#2320—21no—
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37#2285C2211yescharacteristic
38#2276—21no—
39C4×#117C2211yescharacteristic
40#1771—21no—
41#1771—21no—
42#1768—21no—
43#1768—21no—
44#1767C2211yescharacteristic
45#1767C2211yescharacteristic
46#1767C2211yescharacteristic
47#1767C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52#1753—21no—
53#1753—21no—
54#1729—21no—
55#1561C418yescomplemented
56#1552C418yescomplemented
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#1296C2211yescharacteristic
72#1296C2211yescharacteristic
73C22×M16—21no—
74C22×M16—21no—
75C23×C8—21no—
76#533—22no—
77#533—22no—
78#533C2×C414yes—
79#533C2×C414yes—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#125C2×C412yes—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×#117—21no—
92C2×#117C2×C412yes—
93C2×#117D411yescharacteristic
94C2×#117D411yescharacteristic
95C22×C42C2311yescharacteristic
96#432—22no—
97#432C2×C414yes—
98#423—22no—
99#423—22no—
100#423C2×C414yes—
101#423C2×C414yes—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8C2×C414yes—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C22⋊C8—21no—
116#405—44no—
117C2×#97—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126#391—21no—
127#391—21no—
128#391—21no—
129#391—21no—
130#389—22no—
131#389—22no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389C2×C412yes—
137#389C2×C412yes—
138#389C2×C412yes—
139#389C2×C412yes—
140#388—22no—
141#388—22no—
142#388—22no—
143#388—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#383—22no—
151#382—21no—
152#382—22no—
153#380—21no—
154#380C2×C412yes—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4C2×C412yes—
164C4×C4⋊C4C2×C412yes—
165C4×C4⋊C4C2×C412yes—
166C4×C4⋊C4C2×C412yes—
167C4×C4⋊C4C2×C412yes—
168C4×C4⋊C4C2×C411yescharacteristic
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4D411yescharacteristic
172C4×C22⋊C4C2×C411yescharacteristic
173C4×C22⋊C4C2×C411yescharacteristic
174C4×C22⋊C4D411yescharacteristic
175C4×C22⋊C4D411yescharacteristic
176#376—22no—
177#376D411yescharacteristic
178#376D411yescharacteristic
179C43C2311yescharacteristic
180C43C2311yescharacteristic
181#336—21no—
182#336—21no—
183#336—21no—
184#336—21no—
185#336—21no—
186#336—21no—
187#336—21no—
188#336—21no—
189#332C2×C414yes—
190#332C2×C414yes—
191#332C2×C414yes—
192#332C2×C414yes—
193#332C2×C414yes—
194#332C2×C414yes—
195#332C2×C414yes—
196#332C2×C414yes—
197C22×Q8—21no—
198C22×Q8—21no—
199C23×C4C22×C411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C2×M16—24no—
203C2×M16—24no—
204C2×M16—42no—
205C2×M16—42no—
206C2×M16—81no—
207C2×M16—81no—
208C22×C8—24no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—42no—
218C22×C8—42no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8C22×C412yes—
224C4⋊Q8C22×C412yes—
225C4⋊Q8C22×C412yes—
226C4⋊Q8C22×C412yes—
227#125—24no—
228#125C4218yes—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—42no—
238C4×Q8—42no—
239C4×Q8—42no—
240C4×Q8—42no—
241C4×Q8—81no—
242C4×Q8—81no—
243C4×Q8—81no—
244C4×Q8—81no—
245#117—24no—
246#117—22no—
247#117—22no—
248#117—41no—
249#117—41no—
250#117—81no—
251#117—81no—
252#117C4218yes—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22×C412yes—
270C2×C4⋊C4C22×C412yes—
271C2×C4⋊C4C22×C412yes—
272C2×C4⋊C4C22×C412yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C412yes—
291C2×C42C22×C412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42#4112yes—
303C2×C42#4112yes—
304C2×C42C2411yescharacteristic
305C4⋊C8—24no—
306C4⋊C8—24no—
307C4⋊C8—24no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8C22×C412yes—
317C4⋊C8C22×C412yes—
318C4⋊C8C22×C412yes—
319C4⋊C8C22×C412yes—
320C4⋊C8C22×C412yes—
321C4⋊C8C22×C412yes—
322C4⋊C8C22×C412yes—
323C4⋊C8C22×C412yes—
324C22⋊C8—24no—
325#97—24no—
326#97—42no—
327#97—42no—
328#97—81no—
329#97—81no—
330C4×C8—42no—
331C4×C8—42no—
332C4×C8—81no—
333C4×C8—81no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352C24C2×C22⋊C411yescharacteristic
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—41no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4C2×C4212yes—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4#11712yes—
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4C4×D412yes—
395C22×C4C4×D412yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399M16—48no—
400M16—84no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—48no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4C2×C4214yes—
453C4⋊C4C2×C4214yes—
454C22⋊C4—28no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—82no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—28no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—44no—
480C42—44no—
481C42—82no—
482C42—82no—
483C42—82no—
484C42C2×C4212yes—
485C42#11712yes—
486C42C4×D412yes—
487C42C4×D412yes—
488C42C4×D412yes—
489C42C4×D412yes—
490C42AGL(1,ℤ/8)12yes—
491C42#13712yes—
492C42C23×C411yescharacteristic
493C42C23×C411yescharacteristic
494C42C2×#4111yescharacteristic
495C42C2×#4111yescharacteristic
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C4×C22⋊C412yes—
503C23C22×C22⋊C411yescharacteristic
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
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—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4C4×C22⋊C412yes—
565C2×C4C4×C22⋊C412yes—
566C2×C4C4×C22⋊C412yes—
567C2×C4#41712yes—
568C2×C4#41712yes—
569C2×C4C22×C4211yescharacteristic
570C2×C4C2×#11711yescharacteristic
571C2×C4C2×C4×D411yescharacteristic
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
576C2×C4C2×#13711yescharacteristic
577C2×C4#57511yescharacteristic
578C2×C4#57511yescharacteristic
579C2×C4#57611yescharacteristic
580C2×C4#57711yescharacteristic
581C2×C4#57711yescharacteristic
582C2×C4#57811yescharacteristic
583C8—42no—
584C8—42no—
585C8—42no—
586C8—42no—
587C8—42no—
588C8—42no—
589C8—42no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22C2×C4×C22⋊C411yescharacteristic
599C22#288911yescharacteristic
600C22#288911yescharacteristic
601C22C2×#41711yescharacteristic
602C22#289611yescharacteristic
603C22#289711yescharacteristic
604C22#289811yescharacteristic
605C4—22no—
606C4—22no—
607C4—21no—
608C4—22no—
609C4—21no—
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—44no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4C4×AGL(1,ℤ/8)11yescharacteristic
623C4C4×AGL(1,ℤ/8)11yescharacteristic
624C4C4×#13711yescharacteristic
625C4C4×#13711yescharacteristic
626C4#294611yescharacteristic
627C4#294711yescharacteristic
628C4#295111yescharacteristic
629C4#295211yescharacteristic
630C2—42no—
631C2#1776311yescharacteristic
632C2#1776511yescharacteristic
633C2#1776711yescharacteristic
634C2#1776911yescharacteristic
635C2#2039911yescharacteristic
636C2C4×#41711yescharacteristic
637C2#2041511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×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. #8237C2semidirect
7. #8237C2semidirect
8. #8236C2semidirect
9. #8236C2semidirect
10. #8233C2semidirect
11. #8233C2semidirect
12. C4×#332C2semidirect
13. C4×#332C2semidirect
55. #1561C4semidirect
56. #1552C4semidirect

Maximal subgroups

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

subgroupindex
#342252
#205492
#204312
C2×#15612
C2×#15522
#82372
#82362
#82332
C4×#3322
#80452
#80422