Finite Group Databasebeta

← #183128

Subgroups — #183128

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34126C211yesmaximal, characteristic
2#20550C211yesmaximal, characteristic, complemented
3#20462C211yesmaximal, characteristic, complemented
4C2×#1486C211yesmaximal, characteristic
5C2×#1478C211yesmaximal, characteristic
6#8183C211yesmaximal, characteristic, complemented
7#8181C211yesmaximal, characteristic, complemented
8#8179C211yesmaximal, characteristic, complemented
9#8177C211yesmaximal, characteristic, complemented
10#8174C211yesmaximal, characteristic, complemented
11#8172C211yesmaximal, characteristic, complemented
12#8170C211yesmaximal, characteristic, complemented
13C4×D4⋊C8C211yesmaximal, characteristic, complemented
14#7899C211yesmaximal, characteristic, complemented
15#7894C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2357C412yes—
22#2353—22no—
23#2349C412yes—
24#2345C412yes—
25#2342—22no—
26D4×C22⋊C4C412yes—
27#2293C2211yescharacteristic
28#2293C2211yescharacteristic
29#2290—21no—
30#2290—21no—
31#2283C2211yescharacteristic
32#2280—21no—
33C42×D4C2211yescharacteristic
34C42×D4C2211yescharacteristic
35C4×#117C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37#1770—21no—
38C8×C4⋊C4—21no—
39#1768—22no—
40#1767C2211yescharacteristic, complemented
41#1767C2211yescharacteristic, complemented
42C4×C4⋊C8C2211yescharacteristic, complemented
43C4×C4⋊C8C2211yescharacteristic, complemented
44#1753—21no—
45#1753—21no—
46#1750—22no—
47#1744C2211yescharacteristic, complemented
48#1744C2211yescharacteristic, complemented
49C4×#97C2211yescharacteristic, complemented
50C42×C8C2211yescharacteristic, complemented
51#1486C418yescomplemented
52#1478C418yescomplemented
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1279C2211yescharacteristic, complemented
62#1279C2211yescharacteristic, complemented
63#1279C2211yescharacteristic, complemented
64#1279C2211yescharacteristic, complemented
65#1279C2211yescharacteristic, complemented
66#1279C2211yescharacteristic, complemented
67#1279C2211yescharacteristic, complemented
68#1279C2211yescharacteristic, complemented
69C23×D4C2×C411yescharacteristic
70C24×C4—21no—
71C22×M16—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4C2×C414yes—
77C2×C4×D4C2×C414yes—
78C2×C4×D4C2×C412yes—
79C2×C4×D4C2×C412yes—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×C4×D4C2311yescharacteristic
84C2×#117—21no—
85C2×#117—21no—
86C2×#117D412yes—
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#431—22no—
94#431—22no—
95#431—22no—
96#431C2×C414yescomplemented
97#431C2×C414yescomplemented
98#431C2×C414yescomplemented
99#423—22no—
100#423C2×C414yescomplemented
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
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×#97—22no—
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115C2×C4×C8—22no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#390C2×C412yes—
123#390C2×C412yes—
124#389C2×C412yes—
125#389C2×C412yes—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#387—22no—
131#387—22no—
132#386—22no—
133#386—22no—
134#385—22no—
135#385—22no—
136#383—21no—
137#382C2×C412yes—
138#381—21no—
139#381—22no—
140#380—22no—
141#380—22no—
142#380C2×C412yes—
143#379—22no—
144C4×C4⋊C4C2×C412yes—
145C4×C4⋊C4C2×C412yes—
146C4×C4⋊C4C2×C412yes—
147C4×C4⋊C4D411yescharacteristic
148C4×C4⋊C4D411yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C22⋊C4—21no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4C2×C412yes—
161C4×C22⋊C4D411yescharacteristic
162C4×C22⋊C4D411yescharacteristic
163#376—21no—
164#376D412yes—
165C2×#95—22no—
166C43C2311yescharacteristic
167C43C2311yescharacteristic
168#336—22no—
169#336—22no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174D4⋊C8C2×C414yes—
175D4⋊C8C2×C414yes—
176D4⋊C8C2×C414yes—
177D4⋊C8C2×C414yes—
178D4⋊C8C2×C414yescomplemented
179D4⋊C8C2×C414yescomplemented
180D4⋊C8C2×C414yescomplemented
181D4⋊C8C2×C414yescomplemented
182C25—21no—
183C22×D4—24no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4—42no—
187C22×D4C4214yes—
188C22×D4C22×C411yescharacteristic
189C22×D4C22×C411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—81no—
200C23×C4—81no—
201C23×C4—81no—
202C23×C4—81no—
203C23×C4C22×C411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C2×M16—24no—
207C2×M16—81no—
208C2×M16—81no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4C4214yes—
232C4×D4C4214yes—
233C4×D4C22×C412yes—
234C4×D4C22×C412yes—
235C4×D4C22×C412yes—
236C4×D4C22×C412yes—
237#117—44no—
238#117—81no—
239#117—81no—
240#117—81no—
241#117—81no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C4214yes—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4C22⋊C412yes—
268C2×C22⋊C4C22⋊C412yes—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—81no—
274C2×C42—81no—
275C2×C42—81no—
276C2×C42—81no—
277C2×C42C22×C412yes—
278C2×C42C22×C412yes—
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C412yes—
281C2×C42C22×C412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D412yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4112yes—
291C2×C42#4112yes—
292C2×C42C2411yescharacteristic
293C4⋊C8—24no—
294C4⋊C8—24no—
295C4⋊C8—24no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8C22×C412yes—
301C4⋊C8C22×C412yes—
302C4⋊C8C22×C412yes—
303C4⋊C8C22×C412yes—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307#97—44no—
308#97—81no—
309#97—81no—
310C4×C8—24no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—44no—
316C4×C8—81no—
317C4×C8—81no—
318C4×C8C22×C412yes—
319C4×C8C22×C412yes—
320C4×C8C22×C412yes—
321C4×C8C22×C412yes—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—21no—
330#95—21no—
331#95C22⋊C412yes—
332#95C22⋊C412yes—
333#95C22⋊C412yes—
334#95C22⋊C412yes—
335C24—21no—
336C24—21no—
337C24—24no—
338C24—44no—
339C24—44no—
340C24—81no—
341C24—81no—
342C24C2×C22⋊C411yescharacteristic
343C2×D4—28no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4C2×C4212yes—
354C2×D4C2×C4212yes—
355C2×D4#13114yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—44no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4C2×C4212yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C412yes—
406C22×C4#11712yes—
407C22×C4C4×D412yes—
408C22×C4C4×D412yes—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414M16—84no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4C2×C4212yes—
447C4⋊C4C2×C4212yes—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—24no—
476C42—42no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—82no—
482C42—82no—
483C42C2×C4212yes—
484C42#11712yes—
485C42C4×D412yes—
486C42C4×D412yes—
487C42C4×D412yes—
488C42C4×D412yes—
489C42C23×C411yescharacteristic
490C42C23×C411yescharacteristic
491C42C2×#4111yescharacteristic
492C42C2×#4111yescharacteristic
493C23—24no—
494C23—22no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—48no—
501C23—48no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—84no—
510C23—88no—
511C23—88no—
512C23C4×C22⋊C412yes—
513C23C22×C22⋊C411yescharacteristic
514D4—22no—
515D4—22no—
516D4—22no—
517D4—24no—
518D4—48no—
519D4—42no—
520D4#40514yes—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
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×C4—42no—
565C2×C4—42no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—88no—
569C2×C4—84no—
570C2×C4—88no—
571C2×C4C4×C22⋊C412yes—
572C2×C4C4×C22⋊C412yes—
573C2×C4C4×C22⋊C412yes—
574C2×C4#42114yes—
575C2×C4C22×C4211yescharacteristic
576C2×C4C2×#11711yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C2×C4×D411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4C2×#13112yes—
582C2×C4#56812yes—
583C2×C4#57611yescharacteristic
584C2×C4#57611yescharacteristic
585C2×C4#57711yescharacteristic
586C2×C4#57711yescharacteristic
587C8—42no—
588C8—42no—
589C8—42no—
590C8—42no—
591C8—42no—
592C8—42no—
593C8—42no—
594C8—42no—
595C22—22no—
596C22—24no—
597C22—22no—
598C22—22no—
599C22—44no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—816no—
609C22—82no—
610C22—82no—
611C22C2×C4×C22⋊C411yescharacteristic
612C22#288012yes—
613C22#289711yescharacteristic
614C22#289711yescharacteristic
615C22C2×#42112yes—
616C4—24no—
617C4—22no—
618C4—21no—
619C4—21no—
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—82no—
628C4—82no—
629C4C2×#40511yescharacteristic
630C4#287311yescharacteristic
631C4#292412yes—
632C4#294611yescharacteristic
633C4#294611yescharacteristic
634C4#294811yescharacteristic
635C4#294911yescharacteristic
636C2—42no—
637C2—44no—
638C2—82no—
639C2#1752112yes—
640C2#1752212yes—
641C2#2037911yescharacteristic
642C2C4×#42111yescharacteristic
643C2#2041611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20550C2semidirect
3. #20462C2semidirect
6. #8183C2semidirect
7. #8181C2semidirect
8. #8179C2semidirect
9. #8177C2semidirect
10. #8174C2semidirect
11. #8172C2semidirect
12. #8170C2semidirect
13. C4×D4⋊C8C2semidirect
14. #7899C2semidirect
15. #7894C2semidirect
40. #1767C22semidirect
41. #1767C22semidirect
42. C4×C4⋊C8C22semidirect
43. C4×C4⋊C8C22semidirect
47. #1744C22semidirect
48. #1744C22semidirect
49. C4×#97C22semidirect
50. C42×C8C22semidirect
51. #1486C4semidirect
52. #1478C4semidirect
61. #1279C22semidirect
62. #1279C22semidirect
63. #1279C22semidirect
64. #1279C22semidirect
65. #1279C22semidirect
66. #1279C22semidirect
67. #1279C22semidirect
68. #1279C22semidirect
96. #431C2×C4semidirect
97. #431C2×C4semidirect
98. #431C2×C4semidirect
100. #423C2×C4semidirect
178. D4⋊C8C2×C4semidirect
179. D4⋊C8C2×C4semidirect
180. D4⋊C8C2×C4semidirect
181. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341262
#205502
#204622
C2×#14862
C2×#14782
#81832
#81812
#81792
#81772
#81742
#81722
#81702
C4×D4⋊C82
#78992
#78942