Finite Group Databasebeta

← #482116

Subgroups — #482116

Order 512.

1001 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#11899C212yesmaximal, complemented
5#11898C212yesmaximal, complemented
6#11897C214yesmaximal, complemented
7#11715C214yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C2×#431—21no—
11C2×#410—21no—
12#2582—21no—
13#2581C412yescomplemented
14#2580C2212yes—
15#2579—21no—
16#2579—21no—
17#2538—21no—
18#2487C411yescharacteristic, complemented
19#2487C2211yescharacteristic
20#2409C411yescharacteristic, complemented
21#2409C2211yescharacteristic
22#2396C412yescomplemented
23#2396C2212yes—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26#2297—21no—
27#2296—21no—
28#2285—21no—
29#2285C411yescharacteristic, complemented
30#1799—21no—
31#1799—21no—
32#1799C2212yes—
33#1799C2212yes—
34#1740C2212yes—
35#1461—21no—
36#1460C2212yes—
37#1459—21no—
38#1459C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1352C2218yes—
42#1351C2218yes—
43C22×#41D411yescharacteristic
44C23×Q8C2×C411yescharacteristic
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C4×#41—41no—
48C4×#41—41no—
49C4×#41—41no—
50C4×#41—41no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—41no—
54C2×C4×Q8—41no—
55C2×C4×Q8—41no—
56C2×C4×Q8—41no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C2×#117—22no—
69C2×#117C2×C411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C42—21no—
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78#431—22no—
79#431—22no—
80#431—41no—
81#431—41no—
82C2×C4⋊C8—41no—
83#410—44no—
84C2×#99—22no—
85C2×#99—22no—
86C2×C22⋊C8—22no—
87C2×C4×C8—41no—
88#400C2×C412yes—
89#400C2×C412yes—
90#399C2×C412yes—
91#399C2312yes—
92#398—22no—
93#398—22no—
94#396—22no—
95#396—22no—
96#395C2×C412yes—
97#395C2×C412yes—
98#393C2×C412yes—
99#393C2312yes—
100#391—22no—
101#391—21no—
102#391—21no—
103#390—22no—
104#389—22no—
105#389—22no—
106#389—21no—
107#389—21no—
108#387—22no—
109#387—22no—
110#386—21no—
111#386—21no—
112#386—21no—
113#386—21no—
114#385—22no—
115#385—22no—
116#385—21no—
117#385—21no—
118#385—21no—
119#385—21no—
120#384—21no—
121#384—41no—
122#382—41no—
123#382C2×C412yes—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—22no—
128#380—22no—
129#380C2×C411yescharacteristic
130#380C2×C411yescharacteristic
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—41no—
135C4×C4⋊C4—41no—
136C4×C4⋊C4C2×C412yes—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139#342—22no—
140#342C2314yes—
141#324—22no—
142#324—22no—
143#323—22no—
144#323C2314yes—
145C2×#41—41no—
146C2×#41—41no—
147C2×#41—42no—
148C2×#41SD1614yes—
149C2×#41Q1614yes—
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
170C22×C8—42no—
171C22⋊Q8—44no—
172C22⋊Q8—44no—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—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—
199#117—41no—
200#117—41no—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—44no—
206#117—44no—
207#117—44no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
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—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
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—
280C4×C8—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—42no—
292#95—42no—
293#95C22×C412yes—
294#95C22×C412yes—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297C24C22×D411yescharacteristic
298#41—44no—
299#41—44no—
300#41—44no—
301#41—44no—
302C2×Q8—24no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—44no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—44no—
309C2×Q8—48no—
310C2×Q8—44no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8#10314yes—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4#10314yes—
321C2×D4C2×SD1612yes—
322C2×D4C2×SD1612yes—
323C2×D4C2×Q1612yes—
324C2×D4C2×Q1612yes—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
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—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—82no—
360C22×C4—81no—
361C22×C4#10314yes—
362C22×C4#10314yes—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×SD1612yes—
374C22×C4C2×Q1612yes—
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C2×C8—44no—
378C2×C8—84no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—84no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C42—21no—
417C42—21no—
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—44no—
435C42—42no—
436C42—42no—
437C42—44no—
438C42—84no—
439C42#9911yescharacteristic
440C42#9911yescharacteristic
441C42#9911yescharacteristic
442C42#9911yescharacteristic
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23C2×#10312yes—
456C23C2×#10312yes—
457C23C22×C22⋊C411yescharacteristic
458C23C22×SD1611yescharacteristic
459C23C22×Q1611yescharacteristic
460Q8—44no—
461Q8—44no—
462Q8—44no—
463Q8—44no—
464Q8—44no—
465D4—44no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—42no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4C2×#9911yescharacteristic
528C2×C4C2×#9911yescharacteristic
529C2×C4C2×#9911yescharacteristic
530C2×C4C2×#9911yescharacteristic
531C2×C4C2×#9911yescharacteristic
532C2×C4C2×#9911yescharacteristic
533C2×C4C2×#10312yes—
534C2×C4C2×#10312yes—
535C2×C4C2×#10312yes—
536C2×C4C2×#10312yes—
537C2×C4#42112yes—
538C8—84no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—81no—
557C22—81no—
558C22C22×#9911yescharacteristic
559C22C22×#10311yescharacteristic
560C22C2×#42111yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—22no—
565C4—22no—
566C4—22no—
567C4—44no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—164no—
578C4#152012yes—
579C4#152012yes—
580C4#212012yes—
581C2—21no—
582C2—21no—
583C2—41no—
584C2—42no—
585C2—41no—
586C2—42no—
587C2C2×#152011yescharacteristic
588C2C2×#212011yescharacteristic
589C2#3335811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #11899C2semidirect
5. #11898C2semidirect
6. #11897C2semidirect
7. #11715C2semidirect
9. C22×C4×Q8C4semidirect
13. #2581C4semidirect
18. #2487C4semidirect
20. #2409C4semidirect
22. #2396C4semidirect
29. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#349522
#206322
#173492
#118992
#118982
#118972
#117152