Finite Group Databasebeta

← #481745

Subgroups — #481745

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35084C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#11785C212yesmaximal, complemented
5#11773C212yesmaximal, complemented
6#11759C214yesmaximal, complemented
7#11731C214yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10#2920—21no—
11C22×#99—21no—
12#2855—21no—
13#2848—21no—
14#2638C412yescomplemented
15#2638C2212yes—
16#2583C412yescomplemented
17#2561C2212yes—
18#2486C411yescharacteristic, complemented
19#2486C2211yescharacteristic
20#2483—21no—
21#2415—21no—
22#2415—21no—
23#2409C411yescharacteristic, complemented
24#2409C2211yescharacteristic
25#2405—21no—
26#2322—21no—
27#2322—21no—
28#2280C411yescharacteristic, complemented
29C2×C4×C22⋊C4—21no—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34C2×#342C2212yes—
35#1466—21no—
36#1466C2212yes—
37#1462—21no—
38#1462C2212yes—
39#1460—21no—
40#1460C2212yes—
41#1351C2218yescomplemented
42#1344C2218yescomplemented
43C23×Q8C2×C411yescharacteristic, complemented
44C23×D4D411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C2×#123—21no—
48C2×#123—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—41no—
54C2×C22⋊Q8—41no—
55C2×C22⋊Q8—41no—
56C2×C22⋊Q8—41no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yescomplemented
59C2×C4×D4—41no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4—41no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic, complemented
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4C2311yescharacteristic
79C22×C42—41no—
80C22×C42D411yescharacteristic
81#432—22no—
82#432—41no—
83#431—22no—
84#431—41no—
85#423—41no—
86C2×#99—28no—
87C2×#99—22no—
88C2×#99—22no—
89#407—22no—
90C4×M16—41no—
91#398C2×C412yescomplemented
92#398C2×C412yescomplemented
93#397—22no—
94#397—22no—
95#397C2×C412yes—
96#397C2312yes—
97#395C2×C412yescomplemented
98#395C2×C412yescomplemented
99#394—22no—
100#394—22no—
101#393C2×C412yes—
102#393C2312yes—
103#391—21no—
104#391—21no—
105#389—22no—
106#389—22no—
107#389—21no—
108#389—21no—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—22no—
113#386—21no—
114#386—21no—
115#385—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—21no—
120#385—21no—
121#381—22no—
122#381—21no—
123#381—21no—
124#381C2×C412yescomplemented
125#380—21no—
126#380—21no—
127#380—21no—
128#380C2×C411yescharacteristic, complemented
129#380C2×C411yescharacteristic, complemented
130#379—21no—
131#379—22no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—41no—
136C4×C22⋊C4—41no—
137C4×C22⋊C4C2×C412yes—
138C2×#95—41no—
139#342—22no—
140#342C2314yes—
141#323—22no—
142#323—22no—
143#323—22no—
144#323C2314yes—
145C25C22⋊C411yescharacteristic
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8—41no—
149C22×Q8—41no—
150C22×Q8—82no—
151C22×Q8C22×C412yes—
152C22×D4—24no—
153C22×D4—41no—
154C22×D4—41no—
155C22×D4—42no—
156C22×D4C22⋊C411yescharacteristic
157C22×D4C22⋊C411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—42no—
171C23×C4—82no—
172C23×C4—82no—
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C2×M16—42no—
187#123—44no—
188#123—44no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C22⋊Q8—82no—
200C22⋊Q8—82no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217#117—82no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22×C411yescharacteristic
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C22⋊C4C2×D411yescharacteristic
265C2×C22⋊C4C2411yescharacteristic
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—84no—
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C4⋊C8—42no—
279C4⋊C8—42no—
280#99—28no—
281#99—28no—
282C22⋊C8—24no—
283C22⋊C8—42no—
284#97—42no—
285C4×C8—42no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—42no—
296#95—42no—
297#95C22×C412yes—
298#95C22×C412yes—
299C24—21no—
300C24—22no—
301C24—21no—
302C24—22no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—41no—
307C24—42no—
308C24—42no—
309C24—81no—
310C24—81no—
311C24C2×C22⋊C411yescharacteristic
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315C24C22×D411yescharacteristic
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—82no—
322C2×Q8—84no—
323C2×Q8—84no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4#10414yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4#9911yescharacteristic
398C22×C4#9911yescharacteristic
399C22×C4#9911yescharacteristic
400C22×C4#9911yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411M16—84no—
412C2×C8—44no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—44no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C22⋊C4—28no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—44no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4—82no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—82no—
460C42—84no—
461C42—84no—
462C23—24no—
463C23—24no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23C2×#9911yescharacteristic
500C23C2×#9911yescharacteristic
501C23C2×#10412yes—
502C23C2×#10412yes—
503C23C22×C22⋊C411yescharacteristic
504C23#57512yes—
505Q8—82no—
506Q8—84no—
507Q8—82no—
508D4—44no—
509D4—44no—
510D4—44no—
511D4—44no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
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—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
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—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—816no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4C2×#9911yescharacteristic
578C2×C4C2×#9911yescharacteristic
579C2×C4C2×#9911yescharacteristic
580C2×C4C2×#9911yescharacteristic
581C2×C4C2×#10412yes—
582C8—84no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—21no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—48no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—81no—
610C22—81no—
611C22—84no—
612C22—81no—
613C22—81no—
614C22—84no—
615C22—84no—
616C22#212912yes—
617C22C22×#9911yescharacteristic
618C22#289611yescharacteristic
619C22C22×#10411yescharacteristic
620C4—21no—
621C4—21no—
622C4—21no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—82no—
632C4—82no—
633C4—88no—
634C4—82no—
635C4—82no—
636C4#212112yes—
637C2—21no—
638C2—21no—
639C2—22no—
640C2—42no—
641C2—41no—
642C2—42no—
643C2—42no—
644C2—81no—
645C2#1754211yescharacteristic
646C2C2×#212111yescharacteristic
647C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11785C2semidirect
5. #11773C2semidirect
6. #11759C2semidirect
7. #11731C2semidirect
8. C22×C22⋊Q8C4semidirect
14. #2638C4semidirect
16. #2583C4semidirect
18. #2486C4semidirect
23. #2409C4semidirect
28. #2280C4semidirect
41. #1351C22semidirect
42. #1344C22semidirect
43. C23×Q8C2×C4semidirect
58. C2×C22⋊Q8C2×C4semidirect
74. C22×C4⋊C4C2×C4semidirect
91. #398C2×C4semidirect
92. #398C2×C4semidirect
97. #395C2×C4semidirect
98. #395C2×C4semidirect
124. #381C2×C4semidirect
128. #380C2×C4semidirect
129. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350842
C2×C4×#992
#173642
#117852
#117732
#117592
#117312