Finite Group Databasebeta

← #496423

Subgroups — #496423

Order 512.

1020 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#35349C211yesmaximal, characteristic
2#20729C211yesmaximal, characteristic
3#20724C211yesmaximal, characteristic
4#13297C212yesmaximal, complemented
5#13295C212yesmaximal, complemented
6#13294C212yesmaximal, complemented
7#13293C212yesmaximal, complemented
8#12854C212yesmaximal
9#12852C212yesmaximal
10C22×C22⋊Q8C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2921—21no—
13#2579C411yescharacteristic, complemented
14#2579C411yescharacteristic, complemented
15#2579C2211yescharacteristic
16#2579C2211yescharacteristic
17#2487C411yescharacteristic, complemented
18#2487C2211yescharacteristic
19#2450—21no—
20#2450—21no—
21#2449—21no—
22#2449—21no—
23#2438—21no—
24#2438—21no—
25#2437—21no—
26#2437—21no—
27#2409C411yescharacteristic, complemented
28#2409C2211yescharacteristic
29#2315C411yescharacteristic, complemented
30#2315C411yescharacteristic, complemented
31#2312—21no—
32#2308C2211yescharacteristic
33C2×#386C2211yescharacteristic
34C2×#385C411yescharacteristic, complemented
35#1822—21no—
36#1819—21no—
37#1800—21no—
38#1800C2212yes—
39#1799—21no—
40#1799C2212yes—
41#1783C2212yes—
42#1781C2212yes—
43C4×#100—21no—
44C4×#100C2212yes—
45C4×#99—21no—
46C4×#99C2212yes—
47#1390C2214yes—
48#1390C2214yes—
49#1390C2214yes—
50#1390C2214yes—
51C23×Q8C2×C411yescharacteristic
52C23×D4C2311yescharacteristic
53C24×C4C2×C411yescharacteristic
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—44no—
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⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84#433—44no—
85#423—22no—
86C2×#100—22no—
87C2×#100C2314yes—
88C2×#99—22no—
89C2×#99C2314yes—
90#407—22no—
91C4×M16—22no—
92#400—21no—
93#400—21no—
94#399—21no—
95#399—21no—
96#398—21no—
97#398—21no—
98#398—21no—
99#398—21no—
100#396—21no—
101#396—21no—
102#396—21no—
103#396—21no—
104#395—21no—
105#395—21no—
106#393—21no—
107#393—21no—
108#391—21no—
109#391—21no—
110#389—21no—
111#389—21no—
112#388—44no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#386—21no—
118#386—21no—
119#386—21no—
120#386—21no—
121#386C2×C411yescharacteristic
122#386C2×C411yescharacteristic
123#386C2×C411yescharacteristic
124#386C2×C411yescharacteristic
125#386C2311yescharacteristic
126#386C2311yescharacteristic
127#386C2311yescharacteristic
128#386C2311yescharacteristic
129#385—21no—
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—21no—
139#385—21no—
140#385—21no—
141#385C2×C411yescharacteristic
142#385C2×C411yescharacteristic
143#385C2×C411yescharacteristic
144#385C2×C411yescharacteristic
145#385C2×C411yescharacteristic
146#385C2×C411yescharacteristic
147#385C2×C411yescharacteristic
148#385C2×C411yescharacteristic
149#381—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—22no—
155#380C2×C411yescharacteristic
156#380C2×C411yescharacteristic
157#380C2×C411yescharacteristic
158#380C2×C411yescharacteristic
159#379—21no—
160#379—22no—
161C4×C22⋊C4—22no—
162C2×#95—21no—
163C2×#95—21no—
164#343—22no—
165#343—22no—
166#342—22no—
167#342—22no—
168C25C22×C411yescharacteristic
169C25C2×D411yescharacteristic
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—42no—
173C22×Q8—82no—
174C22×D4—24no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4C22×C411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C22×D4C2411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—42no—
186C23×C4—82no—
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C2×M16—24no—
201C22⋊Q8—44no—
202C22⋊Q8—44no—
203C22⋊Q8—44no—
204C22⋊Q8—44no—
205C22⋊Q8—82no—
206C22⋊Q8—82no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215#117—24no—
216#117—42no—
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—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—24no—
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—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—82no—
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—42no—
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C4⋊C8—44no—
289#100—28no—
290#100—44no—
291#99—28no—
292#99—44no—
293C22⋊C8—44no—
294#97—42no—
295C4×C8—42no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
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—42no—
317#95—42no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—41no—
324C24—81no—
325C24—81no—
326C24—81no—
327C24—81no—
328C24C2×C22⋊C411yescharacteristic
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24C23×C411yescharacteristic
332C24C22×D411yescharacteristic
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—81no—
338C2×Q8—81no—
339C2×Q8—84no—
340C2×Q8—84no—
341C2×D4—22no—
342C2×D4—22no—
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—416no—
353C2×D42−1+414yes—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—44no—
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—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414M16—44no—
415M16—44no—
416C2×C8—44no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—162no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
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—42no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C22⋊C4—162no—
470C22⋊C4—162no—
471C22⋊C4#9914yes—
472C42—22no—
473C42—22no—
474C42—41no—
475C42—41no—
476C42—162no—
477C23—28no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—41no—
490C23—41no—
491C23—44no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—162no—
508C23C2×#9912yes—
509C23#41012yes—
510C23C22×C22⋊C411yescharacteristic
511C23#51912yes—
512C23#53612yes—
513Q8—84no—
514Q8—84no—
515D4—44no—
516D4—44no—
517D4—44no—
518D4—44no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
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—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
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—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—161no—
579C2×C4—161no—
580C2×C4—161no—
581C2×C4—161no—
582C2×C4—162no—
583C2×C4C2×#9912yes—
584C2×C4C2×#9912yes—
585C2×C4#53612yes—
586C8—84no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—81no—
613C22—81no—
614C22—88no—
615C22—164no—
616C22#231011yescharacteristic
617C22C22×#9911yescharacteristic
618C22C2×#41011yescharacteristic
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—164no—
638C2—21no—
639C2—21no—
640C2—42no—
641C2—42no—
642C2—42no—
643C2—42no—
644C2—81no—
645C2#2182511yescharacteristic
646C2#3334811yescharacteristic
647C2#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. #13297C2semidirect
5. #13295C2semidirect
6. #13294C2semidirect
7. #13293C2semidirect
10. C22×C22⋊Q8C4semidirect
13. #2579C4semidirect
14. #2579C4semidirect
17. #2487C4semidirect
27. #2409C4semidirect
29. #2315C4semidirect
30. #2315C4semidirect
34. C2×#385C4semidirect

Maximal subgroups

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

subgroupindex
#353492
#207292
#207242
#132972
#132952
#132942
#132932
#128542
#128522