Finite Group Databasebeta

← #480951

Subgroups — #480951

Order 512.

839 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36184C211yesmaximal, characteristic
2#17417C212yesmaximal, complemented
3#11571C212yesmaximal, complemented
4#11550C212yesmaximal, complemented
5#11549C212yesmaximal, complemented
6#11530C211yesmaximal, characteristic, complemented
7#11513C211yesmaximal, characteristic, complemented
8#11511C211yesmaximal, characteristic, complemented
9#11476C211yesmaximal, characteristic, complemented
10#11447C211yesmaximal, characteristic, complemented
11C2×#1324C211yesmaximal, characteristic, complemented
12C2×#553C411yescharacteristic
13C2×#552C2211yescharacteristic
14#2924—22no—
15#2859C411yescharacteristic
16#2850C411yescharacteristic
17#2850C411yescharacteristic
18#2566C411yescharacteristic
19#2565C2211yescharacteristic
20#2553—22no—
21#2519—22no—
22#2502C411yescharacteristic
23#2501C2211yescharacteristic
24#2500C2211yescharacteristic
25#2499C411yescharacteristic
26#2425—22no—
27#2359—22no—
28C4×#124C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C42×D4C411yescharacteristic
32#1473—22no—
33#1470—22no—
34#1468—22no—
35#1467C2212yes—
36#1464C2212yes—
37#1463C2212yes—
38#1461C2212yes—
39#1458C2212yes—
40C2×#324C2212yes—
41#1325C2212yes—
42#1325C2212yes—
43#1324C2211yescharacteristic
44#1324C2211yescharacteristic
45#1324C2212yes—
46#1324C2212yes—
47#1324C2211yescharacteristic
48#1324C2211yescharacteristic
49#1318C2212yes—
50#1318C2212yes—
51#553—41no—
52#553—41no—
53#553—41no—
54#553—41no—
55#552—41no—
56#552—41no—
57#552—41no—
58#552—41no—
59C2×#126—21no—
60C2×#126D412yes—
61C2×#125C2×C411yescharacteristic
62C2×#125C2×C411yescharacteristic
63C2×#125D412yes—
64C2×#124C2×C411yescharacteristic
65C2×#123—22no—
66C2×#123—21no—
67C2×#123D412yes—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×#121—21no—
71C2×#121C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117D412yes—
80C2×#117C2311yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85#433—42no—
86#433—42no—
87#432—22no—
88#432—22no—
89#431—22no—
90#431—22no—
91#424—42no—
92#407—22no—
93C2×C22⋊C8—22no—
94#405—42no—
95#399—22no—
96#398—22no—
97#398—22no—
98#397—22no—
99#396—22no—
100#394—22no—
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#390C2×C411yescharacteristic
106#390C2311yescharacteristic
107#389—22no—
108#389—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#387—22no—
114#387—22no—
115#386—22no—
116#386—22no—
117#385—22no—
118#385—22no—
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#384C2×C411yescharacteristic
122#384C2311yescharacteristic
123#382—22no—
124#382C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#382C2311yescharacteristic
128#382C2311yescharacteristic
129#380—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4C2×C411yescharacteristic
137C4×C22⋊C4C2×C411yescharacteristic
138C2×#95—22no—
139C43C2×C411yescharacteristic
140#324—22no—
141#324—22no—
142#324—22no—
143#324—22no—
144#324C2312yes—
145#324C2312yes—
146#324C2312yes—
147#324C2312yes—
148#323—22no—
149#323—22no—
150C22×Q8C22×C411yescharacteristic
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22×C411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D412yes—
162C2×M16—42no—
163C22×C8—42no—
164#126—42no—
165#126—42no—
166#126—42no—
167#126—42no—
168#125—41no—
169#125—41no—
170#125—42no—
171#125—42no—
172#125—41no—
173#125—41no—
174#124—22no—
175#124—41no—
176#123—42no—
177#123—42no—
178#123—48no—
179#123—42no—
180#123—42no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185#121—41no—
186#121—41no—
187#121—42no—
188#121—42no—
189#121—41no—
190#121—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—44no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2411yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4C22⋊C412yes—
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C412yes—
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C22⋊C4C2×D412yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C4⋊C8—42no—
269C4⋊C8—42no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278#97—42no—
279#97—42no—
280C4×C8—42no—
281C4×C8—42no—
282#95—22no—
283#95—22no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293C24—21no—
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—82no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D412yes—
353M16—84no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—84no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C22⋊C4—24no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—44no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—42no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—82no—
401C22⋊C4#9911yescharacteristic
402C22⋊C4#9911yescharacteristic
403C22⋊C4#9911yescharacteristic
404C22⋊C4#9911yescharacteristic
405C42—21no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—42no—
417C42—42no—
418C42—41no—
419C42—42no—
420C42—41no—
421C42—41no—
422C42—81no—
423C42#13112yes—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23C2×#9911yescharacteristic
439C23C2×#9911yescharacteristic
440C23C22×C22⋊C411yescharacteristic
441Q8—81no—
442Q8—81no—
443D4—42no—
444D4—42no—
445D4—81no—
446D4—81no—
447D4—82no—
448D4—82no—
449D4—162no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—82no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4—161no—
500C2×C4—161no—
501C2×C4—161no—
502C2×C4—161no—
503C2×C4C2×#9911yescharacteristic
504C2×C4C2×#9911yescharacteristic
505C2×C4C2×#9911yescharacteristic
506C2×C4C2×#9911yescharacteristic
507C2×C4C2×#13111yescharacteristic
508C2×C4#56811yescharacteristic
509C8—82no—
510C8—82no—
511C8—82no—
512C8—82no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—81no—
526C22—81no—
527C22—82no—
528C22—82no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22#212612yes—
533C22#288011yescharacteristic
534C22C22×#9911yescharacteristic
535C22#288911yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—41no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—41no—
550C4—42no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—81no—
555C2—21no—
556C2—21no—
557C2—22no—
558C2—41no—
559C2—81no—
560C2—161no—
561C2#1741611yescharacteristic
562C2C2×#212611yescharacteristic
563C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#553C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#552 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17417C2semidirect
3. #11571C2semidirect
4. #11550C2semidirect
5. #11549C2semidirect
6. #11530C2semidirect
7. #11513C2semidirect
8. #11511C2semidirect
9. #11476C2semidirect
10. #11447C2semidirect
11. C2×#1324C2semidirect

Maximal subgroups

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

subgroupindex
#361842
#174172
#115712
#115502
#115492
#115302
#115132
#115112
#114762
#114472
C2×#13242