Finite Group Databasebeta

← #182839

Subgroups — #182839

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 632 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×#1559C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#8237C211yesmaximal, characteristic, complemented
7#8236C211yesmaximal, characteristic, complemented
8#8235C211yesmaximal, characteristic, complemented
9#8235C211yesmaximal, characteristic, complemented
10#8233C211yesmaximal, characteristic, complemented
11#8232C211yesmaximal, characteristic, complemented
12C4×#332C211yesmaximal, characteristic, complemented
13C4×#333C211yesmaximal, characteristic, complemented
14#8044C211yesmaximal, characteristic
15#8041C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#405—21no—
22#2370—21no—
23#2370—21no—
24#2362C412yes—
25#2362C412yes—
26#2356C412yes—
27#2351C412yes—
28#2347—21no—
29#2346—21no—
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#1559C418yescomplemented
56#1550C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
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×#125—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8C2×C412yes—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8C2×C412yes—
89C2×C4×Q8C2×C412yes—
90C2×C4×Q8C2×C412yes—
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#391C2×C412yes—
129#391C2×C412yes—
130#389—22no—
131#389—22no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
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—22no—
152#382C2×C412yes—
153#380—21no—
154#380—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
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#333C2×C414yes—
190#333C2×C414yes—
191#333C2×C414yes—
192#333C2×C414yes—
193#332C2×C414yes—
194#332C2×C414yes—
195#332C2×C414yes—
196#332C2×C414yes—
197C22×Q8—21no—
198C22×Q8C22×C412yes—
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#125—24no—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8C4218yes—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—42no—
237C4×Q8—42no—
238C4×Q8—42no—
239C4×Q8—42no—
240C4×Q8—82no—
241C4×Q8—82no—
242C4×Q8C4218yes—
243#117—24no—
244#117—24no—
245#117—22no—
246#117—22no—
247#117—41no—
248#117—41no—
249#117—82no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C412yes—
288C2×C42C22×C412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42#4112yes—
300C2×C42#4112yes—
301C2×C42C2411yescharacteristic
302C4⋊C8—24no—
303C4⋊C8—24no—
304C4⋊C8—24no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8C22×C412yes—
314C4⋊C8C22×C412yes—
315C4⋊C8C22×C412yes—
316C4⋊C8C22×C412yes—
317C4⋊C8C22×C412yes—
318C4⋊C8C22×C412yes—
319C4⋊C8C22×C412yes—
320C4⋊C8C22×C412yes—
321C22⋊C8—24no—
322#97—24no—
323#97—42no—
324#97—42no—
325#97—81no—
326#97—81no—
327C4×C8—42no—
328C4×C8—42no—
329C4×C8—81no—
330C4×C8—81no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349C24C2×C22⋊C411yescharacteristic
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8C2×C4214yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4C2×C4212yes—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
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×C4#11712yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4C4×D412yes—
390C22×C4C4×D412yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394M16—48no—
395M16—84no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—48no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
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—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—44no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4C2×C4214yes—
449C22⋊C4—28no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—82no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—28no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—44no—
475C42—44no—
476C42—82no—
477C42—82no—
478C42—82no—
479C42C2×C4212yes—
480C42#11712yes—
481C42C4×D412yes—
482C42C4×D412yes—
483C42C4×D412yes—
484C42C4×D412yes—
485C42AGL(1,ℤ/8)12yes—
486C42#13712yes—
487C42C23×C411yescharacteristic
488C42C23×C411yescharacteristic
489C42C2×#4111yescharacteristic
490C42C2×#4111yescharacteristic
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23C4×C22⋊C412yes—
498C23C22×C22⋊C411yescharacteristic
499Q8—44no—
500Q8—44no—
501Q8—42no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
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—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
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—24no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4C4×C22⋊C412yes—
558C2×C4C4×C22⋊C412yes—
559C2×C4C4×C22⋊C412yes—
560C2×C4#41712yes—
561C2×C4#41712yes—
562C2×C4C22×C4211yescharacteristic
563C2×C4C2×#11711yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
569C2×C4C2×#13711yescharacteristic
570C2×C4#57511yescharacteristic
571C2×C4#57511yescharacteristic
572C2×C4#57711yescharacteristic
573C2×C4#57711yescharacteristic
574C2×C4#57811yescharacteristic
575C2×C4#57811yescharacteristic
576C8—42no—
577C8—42no—
578C8—42no—
579C8—42no—
580C8—42no—
581C8—42no—
582C8—42no—
583C8—42no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×C22⋊C411yescharacteristic
592C22#288811yescharacteristic
593C22#288911yescharacteristic
594C22C2×#41711yescharacteristic
595C22#289611yescharacteristic
596C22#289811yescharacteristic
597C22#289811yescharacteristic
598C4—22no—
599C4—22no—
600C4—21no—
601C4—22no—
602C4—21no—
603C4—22no—
604C4—22no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—44no—
612C4—44no—
613C4—82no—
614C4—82no—
615C4C4×AGL(1,ℤ/8)11yescharacteristic
616C4C4×AGL(1,ℤ/8)11yescharacteristic
617C4C4×#13711yescharacteristic
618C4C4×#13711yescharacteristic
619C4#294711yescharacteristic
620C4#294711yescharacteristic
621C4#294911yescharacteristic
622C4#295011yescharacteristic
623C2—42no—
624C2#1776311yescharacteristic
625C2#1776411yescharacteristic
626C2#1776711yescharacteristic
627C2#1776811yescharacteristic
628C2#2039711yescharacteristic
629C2C4×#41711yescharacteristic
630C2#2041711yescharacteristic
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. #8236C2semidirect
8. #8235C2semidirect
9. #8235C2semidirect
10. #8233C2semidirect
11. #8232C2semidirect
12. C4×#332C2semidirect
13. C4×#333C2semidirect
55. #1559C4semidirect
56. #1550C4semidirect

Maximal subgroups

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

subgroupindex
#342252
#205492
#204312
C2×#15592
C2×#15502
#82372
#82362
#82352
#82332
#82322
C4×#3322
C4×#3332
#80442
#80412