Finite Group Databasebeta

← #215005

Subgroups — #215005

Order 512.

881 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35846C211yesmaximal, characteristic
2#20725C211yesmaximal, characteristic
3#17368C211yesmaximal, characteristic
4#9482C212yesmaximal, complemented
5#9412C212yesmaximal, complemented
6#9183C212yesmaximal, complemented
7#9011C212yesmaximal, complemented
8#8778C212yesmaximal, complemented
9#8365C212yesmaximal, complemented
10C2×#556C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2973C2212yes—
13#2638C411yescharacteristic, complemented
14#2638C2211yescharacteristic
15#2621C411yescharacteristic, complemented
16#2583C411yescharacteristic, complemented
17#2583C2211yescharacteristic
18#2581C411yescharacteristic, complemented
19#2546C411yescharacteristic, complemented
20#2484C411yescharacteristic, complemented
21#2484C2211yescharacteristic
22#2478—21no—
23#2478—21no—
24#2466—21no—
25#2466—21no—
26#2447—21no—
27#2432—21no—
28#2396C411yescharacteristic, complemented
29#2395C411yescharacteristic, complemented
30#2395C2211yescharacteristic
31#2365—21no—
32#2360—21no—
33#2357C2211yescharacteristic
34#2313C2212yes—
35#1826C2214yes—
36#1820—21no—
37#1799—21no—
38#1799—21no—
39#1782—21no—
40C4×#99C2212yes—
41C4×#99C2212yes—
42#1508C2214yes—
43#1467—21no—
44#1465C2212yes—
45#1464—21no—
46#1463C2212yes—
47#1459—21no—
48C2×#324—21no—
49#1297C2214yes—
50#1284C2214yes—
51C22×#41D411yescharacteristic
52#556—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#126—21no—
56C2×#126—21no—
57C2×#124—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2311yescharacteristic
61#520—44no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80#435—44no—
81#432—22no—
82C2×#99—22no—
83C2×#99—22no—
84#407C2314yes—
85C2×C22⋊C8—22no—
86#400C2×C411yescharacteristic
87#400C2×C411yescharacteristic
88#400C2×C411yescharacteristic
89#399C2×C411yescharacteristic
90#398—21no—
91#398—21no—
92#398C2×C411yescharacteristic
93#398C2×C411yescharacteristic
94#398C2×C411yescharacteristic
95#398C2×C411yescharacteristic
96#398C2×C411yescharacteristic
97#398C2×C411yescharacteristic
98#397—21no—
99#397—21no—
100#397—21no—
101#397—21no—
102#397—21no—
103#397C2×C411yescharacteristic
104#397C2×C411yescharacteristic
105#396—21no—
106#395C2×C411yescharacteristic
107#395C2×C411yescharacteristic
108#395C2×C411yescharacteristic
109#394—21no—
110#394—21no—
111#394—21no—
112#394—21no—
113#393C2×C411yescharacteristic
114#391—21no—
115#391—21no—
116#391—21no—
117#390—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#388—21no—
126#388—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—44no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#382C2×C411yescharacteristic
145#381—21no—
146#381—21no—
147#381—22no—
148#381C2×C411yescharacteristic
149#381C2311yescharacteristic
150#380—22no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4C2×C411yescharacteristic
160C4×C22⋊C4C2314yes—
161C4×C22⋊C4C2311yescharacteristic
162#342—22no—
163#342—22no—
164#339—44no—
165#328—44no—
166#324—22no—
167#324—22no—
168#324—22no—
169#323—22no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C22×Q8—21no—
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×D4C22⋊C411yescharacteristic
177C22×D4C2×D411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C2×M16—24no—
189C22×C8—42no—
190C4⋊Q8—42no—
191C4⋊Q8—44no—
192#126—44no—
193#126—44no—
194#124—44no—
195C22⋊Q8—24no—
196C22⋊Q8—42no—
197C22⋊Q8—44no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—44no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208#117—24no—
209#117—42no—
210#117—42no—
211#117—42no—
212#117—42no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
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—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
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—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22×C411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C22⋊C4C2×D411yescharacteristic
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—24no—
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C4⋊C8—42no—
279#99—44no—
280#99—44no—
281C22⋊C8—24no—
282C22⋊C8—24no—
283C22⋊C8—42no—
284#97—42no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
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—24no—
307#95—42no—
308#95—44no—
309#95C22×C411yescharacteristic
310#95C22×C411yescharacteristic
311#95C22×C411yescharacteristic
312#95C22×C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315C24C22×D411yescharacteristic
316#41—416no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—81no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—44no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
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×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4H(5,𝔽2)14yes—
376M16—84no—
377C2×C8—44no—
378C2×C8—162no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—81no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C4⋊C4—162no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—81no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C42—22no—
430C42—22no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—81no—
440C42—81no—
441C42—84no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23#42112yes—
456C23C22×C22⋊C411yescharacteristic
457C23#53412yes—
458Q8—44no—
459Q8—82no—
460Q8—84no—
461D4—44no—
462D4—44no—
463D4—44no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—161no—
524C2×C4—161no—
525C2×C4#42112yes—
526C2×C4#51812yes—
527C2×C4#53412yes—
528C8—84no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22#230811yescharacteristic
547C22C2×#42111yescharacteristic
548C22C2×#42111yescharacteristic
549C4—22no—
550C4—22no—
551C4—22no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—164no—
570C2—21no—
571C2—21no—
572C2—42no—
573C2—42no—
574C2—81no—
575C2#2177411yescharacteristic
576C2#2177711yescharacteristic
577C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9482C2semidirect
5. #9412C2semidirect
6. #9183C2semidirect
7. #9011C2semidirect
8. #8778C2semidirect
9. #8365C2semidirect
13. #2638C4semidirect
15. #2621C4semidirect
16. #2583C4semidirect
18. #2581C4semidirect
19. #2546C4semidirect
20. #2484C4semidirect
28. #2396C4semidirect
29. #2395C4semidirect

Maximal subgroups

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

subgroupindex
#358462
#207252
#173682
#94822
#94122
#91832
#90112
#87782
#83652