Finite Group Databasebeta

← #482051

Subgroups — #482051

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35272C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic, complemented
3#17349C211yesmaximal, characteristic, complemented
4#11877C214yesmaximal, complemented
5#11873C214yesmaximal, complemented
6#11693C214yesmaximal, complemented
7C2×#535—21no—
8C22×#127C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10C2×#431—21no—
11C2×#410—21no—
12#2856—21no—
13#2856—21no—
14#2619C412yescomplemented
15#2619C2212yes—
16#2548—21no—
17#2548C412yescomplemented
18#2541—21no—
19#2541C2212yes—
20#2408—21no—
21#2408—21no—
22#2408C412yescomplemented
23#2408C2212yes—
24#2309—21no—
25#2285—21no—
26#2285C411yescharacteristic, complemented
27#1799—21no—
28#1799—21no—
29#1799C2212yes—
30#1799C2212yescomplemented
31#1740C2212yescomplemented
32#1461—21no—
33#1460C2212yes—
34#1459—21no—
35#1459C2212yescomplemented
36#1458—21no—
37#1458C2212yescomplemented
38#1343C22116yescomplemented
39C22×#41D411yescharacteristic
40C23×D4—21no—
41C23×D4—21no—
42C23×D4C2×C412yes—
43#535—44no—
44#535—44no—
45C2×#127—22no—
46C2×#127—21no—
47C2×#127—21no—
48C2×#127—41no—
49C2×#127—41no—
50C2×#127—41no—
51C2×#127—41no—
52C2×#127C2×C412yes—
53C2×#127C2×C412yes—
54C2×#124—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#120—22no—
62C2×#120—22no—
63C4×#41—41no—
64C4×#41—41no—
65C4×#41—41no—
66C4×#41—41no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76C2×#117—22no—
77C2×#117C2×C411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83#431—22no—
84#431—22no—
85#431—41no—
86#431—41no—
87C2×C4⋊C8—41no—
88#410—44no—
89C2×#99—22no—
90C2×#99—22no—
91C2×C22⋊C8—22no—
92C2×C4×C8—41no—
93#394—22no—
94#394—22no—
95#394C2×C414yes—
96#394C2×C414yes—
97#392—22no—
98#392—22no—
99#392C2×C414yes—
100#392C2314yes—
101#390—22no—
102#390—22no—
103#390—22no—
104#390—22no—
105#390—22no—
106#390—22no—
107#390—22no—
108#390—22no—
109#384—21no—
110#384—41no—
111#382—41no—
112#382C2×C412yes—
113#380—21no—
114#380—22no—
115#380—22no—
116#380C2×C412yes—
117#379—22no—
118#379—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—41no—
121C4×C4⋊C4—41no—
122C4×C4⋊C4C2×C412yes—
123#342—22no—
124#342C2314yes—
125#324—22no—
126#324—22no—
127#323—22no—
128#323C2314yes—
129C25—22no—
130C2×#41—41no—
131C2×#41—41no—
132C2×#41—42no—
133C2×#41D814yes—
134C2×#41SD1614yes—
135C22×Q8C22⋊C411yescharacteristic
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—44no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—44no—
151C22×D4—84no—
152C22×D4—84no—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22×C414yes—
155C22×D4C2×D411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C22×C8—42no—
167#127—42no—
168#127—42no—
169#127—42no—
170#127—42no—
171#127—42no—
172#127—42no—
173#127—44no—
174#127—44no—
175#127—41no—
176#127—41no—
177#127—41no—
178#127—41no—
179#124—48no—
180#121—48no—
181#121—44no—
182#121—44no—
183#121—84no—
184#121—84no—
185#121—84no—
186#121—84no—
187#120—48no—
188#120—44no—
189#120—44no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—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—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206#117—41no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—44no—
213#117—44no—
214#117—44no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22×C411yescharacteristic
238C2×C22⋊C4C2×D411yescharacteristic
239C2×C22⋊C4C2×D411yescharacteristic
240C2×C22⋊C4C2411yescharacteristic
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C4⋊C8—42no—
266C4⋊C8—42no—
267#99—44no—
268#99—44no—
269C22⋊C8—24no—
270C22⋊C8—42no—
271C4×C8—42no—
272C4×C8—42no—
273#95—22no—
274#95—42no—
275#95—42no—
276#95C22×C414yes—
277C24—24no—
278C24—44no—
279C24—44no—
280C24—44no—
281C24—84no—
282C24—84no—
283C24C2×C22⋊C411yescharacteristic
284C24C2×C22⋊C411yescharacteristic
285C24C22×D411yescharacteristic
286#41—44no—
287#41—44no—
288#41—44no—
289#41—44no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8#10214yes—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—48no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—88no—
320C2×D4—88no—
321C2×D4—84no—
322C2×D4—88no—
323C2×D4—88no—
324C2×D4—88no—
325C2×D4—88no—
326C2×D4—84no—
327C2×D4—84no—
328C2×D4—84no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4#10214yes—
332C2×D4C2×D812yes—
333C2×D4C2×D812yes—
334C2×D4C2×SD1612yes—
335C2×D4C2×SD1612yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
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—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4#10214yes—
366C22×C4#10214yes—
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×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×D812yes—
378C22×C4C2×SD1612yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C2×C8—44no—
382C2×C8—84no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—44no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—84no—
407C22⋊C4—84no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
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—44no—
429C42—84no—
430C42#9911yescharacteristic
431C42#9911yescharacteristic
432C42#9911yescharacteristic
433C42#9911yescharacteristic
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—88no—
454C23—88no—
455C23—88no—
456C23—168no—
457C23—168no—
458C23C2×#10212yes—
459C23C2×#10212yes—
460C23C22×C22⋊C411yescharacteristic
461C23C22×D811yescharacteristic
462C23C22×SD1611yescharacteristic
463Q8—44no—
464D4—44no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—84no—
470D4—84no—
471D4—88no—
472D4—88no—
473D4—88no—
474D4—84no—
475D4—84no—
476D4—84no—
477D4—84no—
478D4—164no—
479D4—164no—
480D4—168no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
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×#10212yes—
534C2×C4C2×#10212yes—
535C2×C4C2×#10212yes—
536C2×C4C2×#10212yes—
537C2×C4#42112yes—
538C8—84no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—21no—
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—
558C22—84no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—1616no—
566C22C22×#9911yescharacteristic
567C22C22×#10211yescharacteristic
568C22C2×#42111yescharacteristic
569C4—21no—
570C4—21no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—84no—
581C4—164no—
582C4#151912yes—
583C4#151912yes—
584C4#212212yes—
585C4#212812yes—
586C2—21no—
587C2—21no—
588C2—41no—
589C2—42no—
590C2—41no—
591C2—42no—
592C2—164no—
593C2C2×#151911yescharacteristic
594C2C2×#212211yescharacteristic
595C2C2×#212811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35272C2
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
2. #20632C2semidirect
3. #17349C2semidirect
4. #11877C2semidirect
5. #11873C2semidirect
6. #11693C2semidirect
8. C22×#127C4semidirect
14. #2619C4semidirect
17. #2548C4semidirect
22. #2408C4semidirect
26. #2285C4semidirect
30. #1799C22semidirect
31. #1740C22semidirect
35. #1459C22semidirect
37. #1458C22semidirect
38. #1343C22semidirect

Maximal subgroups

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

subgroupindex
#352722
#206322
#173492
#118772
#118732
#116932