Finite Group Databasebeta

← #496563

Subgroups — #496563

Order 512.

989 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35452C211yesmaximal, characteristic
2#20729C211yesmaximal, characteristic
3#20633C211yesmaximal, characteristic
4#13294C212yesmaximal, complemented
5#13217C212yesmaximal, complemented
6#13084C212yesmaximal
7#13084C212yesmaximal
8#12796C212yesmaximal, complemented
9#12794C212yesmaximal, complemented
10C22×#126C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2921—21no—
13C22×#99C2212yes—
14#2579C411yescharacteristic, complemented
15#2579C2211yescharacteristic
16#2486C411yescharacteristic, complemented
17#2486C2211yescharacteristic
18#2483C411yescharacteristic, complemented
19#2483C411yescharacteristic, complemented
20#2480—21no—
21#2480—21no—
22#2478—21no—
23#2478—21no—
24#2468—21no—
25#2468—21no—
26#2466—21no—
27#2466—21no—
28#2405C2211yescharacteristic
29#2405C2211yescharacteristic
30#2322C411yescharacteristic, complemented
31#2322C2211yescharacteristic
32#2315—21no—
33#2312C411yescharacteristic, complemented
34C4×#123C411yescharacteristic, complemented
35C4×#123C2211yescharacteristic
36C2×#352C2214yes—
37C2×#352C2214yes—
38#1822—21no—
39#1800—21no—
40#1800C2212yes—
41#1799—21no—
42#1799—21no—
43#1799C2212yes—
44#1799C2212yes—
45#1783C2212yes—
46C4×#100—21no—
47C4×#100C2212yes—
48C2×#342—21no—
49#1390C2214yes—
50#1390C2214yes—
51C23×D4C2311yescharacteristic
52C2×#126—21no—
53C2×#126—21no—
54C2×#126—21no—
55C2×#126—21no—
56C2×#126—44no—
57C2×#123C2×C411yescharacteristic
58C2×#123C2×C411yescharacteristic
59C2×#123C2311yescharacteristic
60C2×#123C2311yescharacteristic
61#518—44no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic, complemented
73C22×C4⋊C4C2×C411yescharacteristic, complemented
74C22×C4⋊C4C2×C411yescharacteristic, complemented
75C22×C22⋊C4—22no—
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42C2×C411yescharacteristic, complemented
83#433—44no—
84#423—22no—
85C2×#100—22no—
86C2×#100C2314yes—
87C2×#99—22no—
88C2×#99—44no—
89C2×#99C2314yes—
90#407—22no—
91C4×M16—22no—
92#398—21no—
93#398—21no—
94#398—21no—
95#398—21no—
96#397—21no—
97#397—21no—
98#397—21no—
99#397—21no—
100#397—21no—
101#397—21no—
102#396—21no—
103#396—21no—
104#394—21no—
105#394—21no—
106#394—21no—
107#394—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#388—21no—
113#388—21no—
114#388—21no—
115#388—21no—
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#386C2×C411yescharacteristic
127#386C2×C411yescharacteristic
128#386C2311yescharacteristic
129#386C2311yescharacteristic
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—44no—
139#385C2×C411yescharacteristic, complemented
140#385C2×C411yescharacteristic, complemented
141#385C2×C411yescharacteristic, complemented
142#385C2×C411yescharacteristic, complemented
143#381—21no—
144#381—21no—
145#381C2×C411yescharacteristic, complemented
146#381C2×C411yescharacteristic, complemented
147#381C2×C411yescharacteristic, complemented
148#381C2×C411yescharacteristic, complemented
149#380—21no—
150#380—21no—
151#380—22no—
152#380—22no—
153#380C2×C411yescharacteristic, complemented
154#380C2×C411yescharacteristic, complemented
155#379—21no—
156#379—21no—
157#379C2×C411yescharacteristic
158#379C2×C411yescharacteristic
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4C2×C411yescharacteristic, complemented
164C4×C22⋊C4C2×C411yescharacteristic, complemented
165#352—44no—
166#352—44no—
167#343—22no—
168#343—22no—
169#342—22no—
170#342—22no—
171C25C22×C411yescharacteristic
172C25C2×D411yescharacteristic
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4C22×C411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C22×D4C2411yescharacteristic
181C23×C4—42no—
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—24no—
196#126—44no—
197#126—44no—
198#126—44no—
199#126—44no—
200#123—22no—
201#123—22no—
202#123—22no—
203#123—22no—
204#123—42no—
205#123—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214#117—24no—
215#117—42no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22×C411yescharacteristic
276C2×C22⋊C4C22×C411yescharacteristic
277C2×C22⋊C4C22×C411yescharacteristic
278C2×C22⋊C4C22×C411yescharacteristic
279C2×C22⋊C4C22×C411yescharacteristic
280C2×C22⋊C4C22×C411yescharacteristic
281C2×C22⋊C4C2×D411yescharacteristic
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—42no—
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C4⋊C8—44no—
293#100—24no—
294#100—24no—
295#100—44no—
296#99—24no—
297#99—24no—
298#99—44no—
299C22⋊C8—44no—
300#97—42no—
301C4×C8—42no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—42no—
323#95—42no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—41no—
331C24—81no—
332C24—81no—
333C24—81no—
334C24—81no—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C23×C411yescharacteristic
339C24C22×D411yescharacteristic
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4H(5,𝔽2)12yes—
355C2×D42−1+412yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—84no—
395C22×C4—162no—
396C22×C4—162no—
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×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407M16—44no—
408M16—44no—
409C2×C8—44no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—162no—
431C4⋊C4—162no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—81no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4—84no—
468C22⋊C4#9914yes—
469C42—22no—
470C42—22no—
471C42—41no—
472C42—41no—
473C42—42no—
474C42—42no—
475C42—81no—
476C42—81no—
477C23—24no—
478C23—24no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—41no—
491C23—41no—
492C23—44no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—161no—
506C23—161no—
507C23—162no—
508C23—162no—
509C23—162no—
510C23C2×#9912yes—
511C23#41012yes—
512C23C22×C22⋊C411yescharacteristic
513C23#52012yes—
514C23#53511yescharacteristic
515C23#53611yescharacteristic
516D4—42no—
517D4—42no—
518D4—44no—
519D4—44no—
520D4—42no—
521D4—42no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
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—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—161no—
578C2×C4—161no—
579C2×C4—161no—
580C2×C4—161no—
581C2×C4—168no—
582C2×C4C2×#9912yes—
583C2×C4C2×#9912yes—
584C2×C4#53712yes—
585C8—84no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—21no—
592C22—22no—
593C22—21no—
594C22—22no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—42no—
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—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—84no—
616C22—84no—
617C22—164no—
618C22#231111yescharacteristic
619C22C22×#9911yescharacteristic
620C22C2×#41011yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—164no—
640C2—21no—
641C2—21no—
642C2—41no—
643C2—42no—
644C2—42no—
645C2—41no—
646C2—42no—
647C2—81no—
648C2#2187411yescharacteristic
649C2#3335011yescharacteristic
650C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13294C2semidirect
5. #13217C2semidirect
8. #12796C2semidirect
9. #12794C2semidirect
10. C22×#126C4semidirect
14. #2579C4semidirect
16. #2486C4semidirect
18. #2483C4semidirect
19. #2483C4semidirect
30. #2322C4semidirect
33. #2312C4semidirect
34. C4×#123C4semidirect
72. C22×C4⋊C4C2×C4semidirect
73. C22×C4⋊C4C2×C4semidirect
74. C22×C4⋊C4C2×C4semidirect
82. C22×C42C2×C4semidirect
139. #385C2×C4semidirect
140. #385C2×C4semidirect
141. #385C2×C4semidirect
142. #385C2×C4semidirect
145. #381C2×C4semidirect
146. #381C2×C4semidirect
147. #381C2×C4semidirect
148. #381C2×C4semidirect
153. #380C2×C4semidirect
154. #380C2×C4semidirect
163. C4×C22⋊C4C2×C4semidirect
164. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#354522
#207292
#206332
#132942
#132172
#130842
#127962
#127942