Finite Group Databasebeta

← #482634

Subgroups — #482634

Order 512.

983 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34971C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#11898C212yesmaximal, complemented
5#11874C212yesmaximal, complemented
6#11823C214yesmaximal, complemented
7#11756C214yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10#2920—21no—
11C2×#410—21no—
12#2599C412yescomplemented
13#2599C2212yes—
14#2581—21no—
15#2536—21no—
16#2486C411yescharacteristic, complemented
17#2486C2211yescharacteristic
18#2484—21no—
19#2409C411yescharacteristic, complemented
20#2409C2211yescharacteristic
21#2406—21no—
22#2399C412yescomplemented
23#2398C2212yes—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×#123—21no—
27C4×#123—21no—
28#2285—21no—
29#2283C411yescharacteristic, complemented
30#1799—21no—
31#1799—21no—
32#1799C2212yes—
33#1799C2212yes—
34#1740C2212yes—
35#1465—21no—
36#1465C2212yes—
37#1464—21no—
38#1464C2212yes—
39#1461—21no—
40#1460C2212yes—
41#1351C2218yes—
42#1344C2218yes—
43C22×#41D411yescharacteristic
44C23×Q8C2×C411yescharacteristic
45C2×#123—21no—
46C2×#123—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C4×#41—41no—
50C4×#41—41no—
51C4×#41—41no—
52C4×#41—41no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—41no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—22no—
70C2×#117C2×C411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42—21no—
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#432—22no—
80#432—41no—
81#431—22no—
82#431—41no—
83#423—41no—
84#410—44no—
85C2×#99—22no—
86C2×#99—22no—
87#407—22no—
88C4×M16—41no—
89#400—22no—
90#398—22no—
91#398C2×C412yes—
92#398C2×C412yes—
93#397C2×C412yes—
94#397C2312yes—
95#395—22no—
96#395C2×C412yes—
97#395C2×C412yes—
98#394—22no—
99#393C2×C412yes—
100#393C2312yes—
101#391—21no—
102#391—21no—
103#389—21no—
104#389—21no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#386—21no—
112#386—21no—
113#385—22no—
114#385—22no—
115#385—21no—
116#385—21no—
117#384—41no—
118#383C2×C412yes—
119#382—21no—
120#382—41no—
121#381—21no—
122#381—21no—
123#381—21no—
124#380—21no—
125#380—21no—
126#380—22no—
127#380—22no—
128#380C2×C411yescharacteristic
129#380C2×C411yescharacteristic
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—41no—
134C4×C4⋊C4—41no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138#376C2×C412yes—
139#342—22no—
140#342C2314yes—
141#324—22no—
142#324—22no—
143#323—22no—
144#323C2314yes—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C22×Q8—21no—
151C22×Q8—41no—
152C22×Q8—41no—
153C22×Q8—44no—
154C22×Q8C22⋊C411yescharacteristic
155C22×Q8C22×C412yes—
156C22×D4C22⋊C411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×M16—42no—
171#123—44no—
172#123—42no—
173#123—42no—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208#117—44no—
209#117—44no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22⋊C411yescharacteristic
242C2×C22⋊C4C22×C411yescharacteristic
243C2×C22⋊C4C2×D411yescharacteristic
244C2×C22⋊C4C2×D411yescharacteristic
245C2×C22⋊C4C2411yescharacteristic
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C4⋊C8—42no—
275C4⋊C8—42no—
276#99—44no—
277#99—44no—
278C22⋊C8—24no—
279C22⋊C8—42no—
280#97—42no—
281C4×C8—42no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—42no—
294#95—42no—
295#95C22×C412yes—
296#95C22×C412yes—
297C24C2×C22⋊C411yescharacteristic
298C24C2×C22⋊C411yescharacteristic
299C24C22×D411yescharacteristic
300#41—44no—
301#41—44no—
302#41—44no—
303#41—44no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—44no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—44no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4AGL(1,ℤ/8)12yes—
326C2×D4#13712yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
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—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380M16—84no—
381C2×C8—44no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—44no—
437C42—44no—
438C42—42no—
439C42—42no—
440C42—84no—
441C42#9911yescharacteristic
442C42#9911yescharacteristic
443C42#9911yescharacteristic
444C42#9911yescharacteristic
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23#41712yes—
459C23C22×C22⋊C411yescharacteristic
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23C2×#13711yescharacteristic
462Q8—44no—
463Q8—44no—
464Q8—44no—
465Q8—44no—
466Q8—44no—
467D4—42no—
468D4—42no—
469D4—44no—
470D4—42no—
471D4—42no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4C2×#9911yescharacteristic
533C2×C4C2×#9911yescharacteristic
534C2×C4C2×#9911yescharacteristic
535C2×C4C2×#9911yescharacteristic
536C2×C4C2×#9911yescharacteristic
537C2×C4C2×#9911yescharacteristic
538C2×C4#41712yes—
539C2×C4#42112yes—
540C8—84no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—81no—
559C22—81no—
560C22C22×#9911yescharacteristic
561C22C2×#41711yescharacteristic
562C22C2×#42111yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—22no—
568C4—22no—
569C4—44no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—164no—
581C4#212012yes—
582C4#212912yes—
583C2—21no—
584C2—21no—
585C2—41no—
586C2—42no—
587C2—41no—
588C2—81no—
589C2#1759211yescharacteristic
590C2C2×#212011yescharacteristic
591C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11898C2semidirect
5. #11874C2semidirect
6. #11823C2semidirect
7. #11756C2semidirect
9. C22×C4×Q8C4semidirect
12. #2599C4semidirect
16. #2486C4semidirect
19. #2409C4semidirect
22. #2399C4semidirect
29. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#349712
#206322
#173662
#118982
#118742
#118232
#117562