Finite Group Databasebeta

← #214948

Subgroups — #214948

Order 512.

1021 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36010C211yesmaximal, characteristic
2#20392C211yesmaximal, characteristic
3#17367C211yesmaximal, characteristic
4#9477C212yesmaximal, complemented
5#9407C212yesmaximal, complemented
6#9183C212yesmaximal, complemented
7#9011C212yesmaximal, complemented
8#8776C212yesmaximal, complemented
9#8363C212yesmaximal, complemented
10C2×#552C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2973C2212yes—
13C22×#99—21no—
14#2638C2211yescharacteristic
15#2633C411yescharacteristic, complemented
16#2626C411yescharacteristic, complemented
17#2611C411yescharacteristic, complemented
18#2599C411yescharacteristic, complemented
19#2564C411yescharacteristic, complemented
20#2561C2211yescharacteristic
21#2486C411yescharacteristic, complemented
22#2482C2211yescharacteristic
23#2480—21no—
24#2475—21no—
25#2466—21no—
26#2460—21no—
27#2419—21no—
28#2405—21no—
29#2398C411yescharacteristic, complemented
30#2395C2211yescharacteristic
31C2×#397C411yescharacteristic, complemented
32#2359—21no—
33#2353C2211yescharacteristic
34#2342—21no—
35#2273C2212yes—
36#1826C2214yescomplemented
37C4×#99—21no—
38C4×#99—21no—
39C4×#99C2212yes—
40C4×#99C2212yes—
41C2×#342—21no—
42#1508C2214yescomplemented
43#1466—21no—
44#1465C2212yes—
45#1464—21no—
46#1462C2212yes—
47#1459—21no—
48C2×#323—21no—
49#1297C2214yescomplemented
50#1284C2214yescomplemented
51C23×D4D411yescharacteristic
52#552—44no—
53C2×#126—21no—
54C2×#125—21no—
55C2×#125C2×C411yescharacteristic, complemented
56C2×#123—21no—
57C2×#123—21no—
58C2×C22⋊Q8C2×C411yescharacteristic, complemented
59C2×C22⋊Q8C2311yescharacteristic
60#518—44no—
61C2×C4×Q8C2×C411yescharacteristic, complemented
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117C2×C411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic, complemented
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4C2311yescharacteristic
81#435—44no—
82#432—22no—
83C2×#99—28no—
84C2×#99—22no—
85C2×#99—22no—
86#407C2314yes—
87C2×C22⋊C8—22no—
88#400—21no—
89#400C2×C411yescharacteristic, complemented
90#399C2×C411yescharacteristic
91#399C2×C411yescharacteristic
92#399C2×C411yescharacteristic, complemented
93#398—21no—
94#398C2×C411yescharacteristic, complemented
95#398C2×C411yescharacteristic, complemented
96#397—21no—
97#397—21no—
98#397—21no—
99#397—21no—
100#397C2×C411yescharacteristic
101#397C2×C411yescharacteristic
102#397C2×C411yescharacteristic
103#397C2×C411yescharacteristic
104#397C2×C411yescharacteristic, complemented
105#397C2×C411yescharacteristic, complemented
106#396—21no—
107#396—21no—
108#395C2×C411yescharacteristic, complemented
109#394—21no—
110#394—21no—
111#394—21no—
112#394—21no—
113#393C2×C411yescharacteristic
114#393C2×C411yescharacteristic
115#393C2×C411yescharacteristic, complemented
116#389—21no—
117#389—21no—
118#388—21no—
119#388—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#386—21no—
132#386—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#383C2×C411yescharacteristic, complemented
142#381—21no—
143#381—21no—
144#381C2311yescharacteristic
145#380—21no—
146#380—21no—
147#380C2×C411yescharacteristic, complemented
148#379—21no—
149#379—21no—
150#379—21no—
151C4×C4⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—22no—
157C4×C22⋊C4—44no—
158C4×C22⋊C4C2314yes—
159C4×C22⋊C4C2311yescharacteristic
160#376C2×C411yescharacteristic, complemented
161C2×#95—21no—
162C2×#95C2×C411yescharacteristic, complemented
163#342—22no—
164#342—22no—
165#339—44no—
166#328—44no—
167#324—22no—
168#323—22no—
169#323—22no—
170#323—22no—
171C25C22⋊C411yescharacteristic
172C25C2×D411yescharacteristic
173C22×Q8C22×C411yescharacteristic
174C22×D4—24no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C22×D4C2×D411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—42no—
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2411yescharacteristic
193C2×M16—24no—
194C22×C8—42no—
195#126—44no—
196#125—42no—
197#125—44no—
198#123—44no—
199#123—44no—
200C22⋊Q8—24no—
201C22⋊Q8—42no—
202C4×Q8—42no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211#117—24no—
212#117—42no—
213#117—42no—
214#117—44no—
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—42no—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
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—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22×C411yescharacteristic
268C2×C22⋊C4C22×C411yescharacteristic
269C2×C22⋊C4C22×C411yescharacteristic
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C22⋊C4C2×D411yescharacteristic
272C2×C22⋊C4C2×D411yescharacteristic
273C2×C22⋊C4C2×D411yescharacteristic
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—42no—
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C4⋊C8—42no—
286#99—28no—
287#99—28no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8—42no—
291#97—42no—
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—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—24no—
312#95—42no—
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317C24—22no—
318C24—22no—
319C24—22no—
320C24—41no—
321C24—41no—
322C24—81no—
323C24—81no—
324C24—81no—
325C24—81no—
326C24#10414yes—
327C24C2×C22⋊C411yescharacteristic
328C24C2×C22⋊C411yescharacteristic
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24C22×D411yescharacteristic
332C2×Q8—22no—
333C2×Q8—81no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—48no—
347C2×D4—48no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—162no—
383C22×C4—162no—
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4H(5,𝔽2)14yes—
395M16—84no—
396C2×C8—44no—
397C2×C8—162no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—28no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C22⋊C4—81no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—42no—
456C42—42no—
457C42—81no—
458C42—81no—
459C42—84no—
460C42—162no—
461C23—24no—
462C23—24no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—41no—
474C23—41no—
475C23—44no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—82no—
485C23—82no—
486C23—84no—
487C23—161no—
488C23—161no—
489C23—162no—
490C23—162no—
491C23—162no—
492C23C2×#10412yes—
493C23C2×#10412yes—
494C23C2×#10412yes—
495C23#42112yes—
496C23C22×C22⋊C411yescharacteristic
497C23#53412yes—
498Q8—82no—
499D4—44no—
500D4—44no—
501D4—44no—
502D4—44no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—48no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—161no—
557C2×C4—161no—
558C2×C4—161no—
559C2×C4—161no—
560C2×C4—162no—
561C2×C4—168no—
562C2×C4#51812yes—
563C2×C4#53412yes—
564C8—84no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—84no—
593C22—84no—
594C22—164no—
595C22#230811yescharacteristic
596C22C22×#10411yescharacteristic
597C22C2×#42111yescharacteristic
598C4—22no—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—88no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—81no—
617C2—21no—
618C2—21no—
619C2—42no—
620C2—42no—
621C2—42no—
622C2—42no—
623C2—42no—
624C2#2177311yescharacteristic
625C2#2177711yescharacteristic
626C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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. #9477C2semidirect
5. #9407C2semidirect
6. #9183C2semidirect
7. #9011C2semidirect
8. #8776C2semidirect
9. #8363C2semidirect
15. #2633C4semidirect
16. #2626C4semidirect
17. #2611C4semidirect
18. #2599C4semidirect
19. #2564C4semidirect
21. #2486C4semidirect
29. #2398C4semidirect
31. C2×#397C4semidirect
36. #1826C22semidirect
42. #1508C22semidirect
49. #1297C22semidirect
50. #1284C22semidirect
55. C2×#125C2×C4semidirect
58. C2×C22⋊Q8C2×C4semidirect
61. C2×C4×Q8C2×C4semidirect
74. C22×C4⋊C4C2×C4semidirect
89. #400C2×C4semidirect
92. #399C2×C4semidirect
94. #398C2×C4semidirect
95. #398C2×C4semidirect
104. #397C2×C4semidirect
105. #397C2×C4semidirect
108. #395C2×C4semidirect
115. #393C2×C4semidirect
141. #383C2×C4semidirect
147. #380C2×C4semidirect
160. #376C2×C4semidirect
162. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#360102
#203922
#173672
#94772
#94072
#91832
#90112
#87762
#83632