Finite Group Databasebeta

← #481754

Subgroups — #481754

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#11788C212yesmaximal, complemented
5#11777C212yesmaximal, complemented
6#11755C214yesmaximal, complemented
7#11732C214yesmaximal, complemented
8C2×#533C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#99—21no—
11C2×#407—21no—
12#2848—21no—
13#2848—21no—
14#2633C412yescomplemented
15#2633C2212yes—
16#2547C411yescharacteristic, complemented
17#2547C2211yescharacteristic
18#2542C411yescharacteristic, complemented
19#2542C2211yescharacteristic
20#2536—21no—
21#2536—21no—
22#2502—21no—
23#2482C412yescomplemented
24#2482C2212yes—
25#2435—21no—
26#2326—21no—
27#2326—21no—
28#2283C411yescharacteristic, complemented
29C2×C4×C22⋊C4—21no—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34C2×#342C2212yes—
35#1465—21no—
36#1465C2212yes—
37#1463—21no—
38#1462C2212yes—
39#1459—21no—
40#1459C2212yes—
41#1352C2218yescomplemented
42#1344C2218yescomplemented
43C23×D4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C22×M16—41no—
47#533—41no—
48#533—41no—
49#533—41no—
50#533—41no—
51C2×C4⋊Q8C2×C411yescharacteristic, complemented
52C2×C4⋊Q8C2×C411yescharacteristic, complemented
53C2×#125C2×C412yescomplemented
54C2×#124—21no—
55C2×#124—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×D4—41no—
65C2×C4×D4—41no—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117C2×C412yes—
77C22×C4⋊C4D411yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2311yescharacteristic
81C22×C42—41no—
82C22×C42C2×C411yescharacteristic
83C2×#99—28no—
84C2×#99—22no—
85C2×#99—22no—
86#407—22no—
87#407—22no—
88#407—41no—
89#407—41no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—41no—
92#399C2×C412yes—
93#399C2×C412yescomplemented
94#399C2×C412yescomplemented
95#399C2312yes—
96#398—22no—
97#398—22no—
98#397C2×C412yes—
99#397C2×C412yescomplemented
100#397C2×C412yescomplemented
101#397C2312yes—
102#394—22no—
103#394—22no—
104#391—21no—
105#391—21no—
106#390—21no—
107#390—21no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#384—21no—
119#384—21no—
120#383C2×C412yescomplemented
121#382—21no—
122#381C2×C412yes—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128#379—22no—
129#379—22no—
130C4×C4⋊C4C2×C411yescharacteristic, complemented
131C4×C4⋊C4C2×C411yescharacteristic, complemented
132C4×C22⋊C4—21no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—41no—
136C4×C22⋊C4—41no—
137#376—21no—
138C2×#95—41no—
139#342—22no—
140#342C2314yes—
141#324—22no—
142#324—22no—
143#323—22no—
144#323C2314yes—
145C25C22⋊C411yescharacteristic
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8—21no—
149C22×D4—24no—
150C22×D4—41no—
151C22×D4—41no—
152C22×D4—42no—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22⋊C411yescharacteristic
155C22×D4C2×D411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—42no—
166C23×C4—82no—
167C23×C4—82no—
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—84no—
184C22×C8—42no—
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189#125—42no—
190#125—42no—
191#124—44no—
192#124—44no—
193C22⋊Q8—42no—
194C22⋊Q8—42no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C22⋊Q8—82no—
200C22⋊Q8—82no—
201C22⋊Q8—82no—
202C22⋊Q8—82no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
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—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—41no—
223#117—42no—
224#117—42no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22×C411yescharacteristic
264C2×C22⋊C4C22×C411yescharacteristic
265C2×C22⋊C4C22×C411yescharacteristic
266C2×C22⋊C4C2×D411yescharacteristic
267C2×C22⋊C4C2×D411yescharacteristic
268C2×C22⋊C4C2411yescharacteristic
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—84no—
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283#99—28no—
284#99—28no—
285C22⋊C8—24no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—42no—
297#95—42no—
298#95C22×C412yes—
299#95C22×C412yes—
300C24—21no—
301C24—22no—
302C24—21no—
303C24—22no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—41no—
308C24—42no—
309C24—42no—
310C24—81no—
311C24—81no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C24C22×D411yescharacteristic
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
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—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4#9911yescharacteristic
393C22×C4#9911yescharacteristic
394C22×C4#9911yescharacteristic
395C22×C4#9911yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406M16—84no—
407M16—84no—
408C2×C8—44no—
409C2×C8—84no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C22⋊C4—28no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—44no—
459C42—42no—
460C42—42no—
461C42—84no—
462C42—84no—
463C23—24no—
464C23—24no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23C2×#9911yescharacteristic
501C23C2×#9911yescharacteristic
502C23C2×#10412yes—
503C23C2×#10412yes—
504C23C22×C22⋊C411yescharacteristic
505C23#57512yes—
506Q8—82no—
507Q8—84no—
508Q8—82no—
509D4—44no—
510D4—44no—
511D4—44no—
512D4—44no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
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—41no—
528C2×C4—41no—
529C2×C4—41no—
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—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—48no—
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—42no—
561C2×C4—42no—
562C2×C4—42no—
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—81no—
573C2×C4—84no—
574C2×C4—816no—
575C2×C4—84no—
576C2×C4C2×#9911yescharacteristic
577C2×C4C2×#9911yescharacteristic
578C2×C4C2×#9911yescharacteristic
579C2×C4C2×#9911yescharacteristic
580C2×C4C2×#10412yes—
581C8—84no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—21no—
590C22—22no—
591C22—22no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—48no—
598C22—42no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—84no—
612C22—81no—
613C22—84no—
614C22—84no—
615C22#212312yes—
616C22C22×#9911yescharacteristic
617C22#289611yescharacteristic
618C22C22×#10411yescharacteristic
619C4—21no—
620C4—21no—
621C4—21no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—82no—
631C4—82no—
632C4—88no—
633C4—82no—
634C4—82no—
635C4#212712yes—
636C2—21no—
637C2—21no—
638C2—22no—
639C2—42no—
640C2—41no—
641C2—42no—
642C2—42no—
643C2—81no—
644C2#1754211yescharacteristic
645C2C2×#212311yescharacteristic
646C2C2×#212711yescharacteristic
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. #11788C2semidirect
5. #11777C2semidirect
6. #11755C2semidirect
7. #11732C2semidirect
8. C2×#533C4semidirect
14. #2633C4semidirect
16. #2547C4semidirect
18. #2542C4semidirect
23. #2482C4semidirect
28. #2283C4semidirect
41. #1352C22semidirect
42. #1344C22semidirect
51. C2×C4⋊Q8C2×C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
53. C2×#125C2×C4semidirect
93. #399C2×C4semidirect
94. #399C2×C4semidirect
99. #397C2×C4semidirect
100. #397C2×C4semidirect
120. #383C2×C4semidirect
130. C4×C4⋊C4C2×C4semidirect
131. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#352402
C2×C4×#992
#173622
#117882
#117772
#117552
#117322