Finite Group Databasebeta

← #206432

Subgroups — #206432

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60946C211yesmaximal, characteristic
2#21856C211yesmaximal, characteristic, complemented
3#21815C211yesmaximal, characteristic, complemented
4C8×#117C211yesmaximal, characteristic, complemented
5#17817C211yesmaximal, characteristic, complemented
6#17473C211yesmaximal, characteristic, complemented
7C2×#1492C211yesmaximal, characteristic, complemented
8#9854C212yesmaximal, complemented
9#9852C212yesmaximal, complemented
10#9844C212yesmaximal, complemented
11#9843C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3247—22no—
15#3230C411yescharacteristic, complemented
16#3228C412yescomplemented
17#3227C411yescharacteristic, complemented
18#3208C411yescharacteristic, complemented
19#3207C412yescomplemented
20#3205C411yescharacteristic, complemented
21#3188—22no—
22#3169C2212yes—
23#3167C2212yes—
24#3106—22no—
25#3105—22no—
26#2920C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1994C2212yescomplemented
32#1980—22no—
33#1962C2212yes—
34#1962C2212yes—
35#1962C2212yescomplemented
36#1871—22no—
37#1806—22no—
38#1789—22no—
39C8×C4⋊C4—22no—
40C8×C22⋊C4—22no—
41#1744C2212yescomplemented
42C42×C8C2212yescomplemented
43#1574—22no—
44#1573—22no—
45#1567C2212yescomplemented
46#1566C2212yescomplemented
47#1494—22no—
48#1493C2211yescharacteristic, complemented
49#1493C2211yescharacteristic, complemented
50#1492—22no—
51#1492C2212yes—
52#1492C2212yes—
53#1492C2211yescharacteristic, complemented
54#1492C2211yescharacteristic, complemented
55#1476—22no—
56C2×D4⋊C8—22no—
57C22×#41—21no—
58C22×M16—21no—
59C23×C8—21no—
60#533C2×C411yescharacteristic
61#533C2×C411yescharacteristic
62#533C2311yescharacteristic
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532C2×C412yes—
70#532C2×C411yescharacteristic
71#532C2×C411yescharacteristic
72#532C2312yes—
73#532C2311yescharacteristic
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×#127C2×C411yescharacteristic
76#529—21no—
77#529—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82C2×#124—21no—
83#525—21no—
84#525—21no—
85#525—21no—
86C2×#121—21no—
87C2×#121—21no—
88C4×#41—21no—
89C4×#41—21no—
90C4×#41—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×#117—21no—
94C22×C42C2311yescharacteristic
95#500C2×C412yes—
96#498C2×C412yes—
97#493C2×C412yes—
98#492C2×C412yes—
99#487C2×C412yes—
100#486C2×C412yes—
101#486C2×C412yes—
102#486C2×C412yes—
103#484—22no—
104#482—24no—
105#481—22no—
106#478—24no—
107#477—22no—
108#476—22no—
109#466—24no—
110#465—24no—
111#463—24no—
112#460—22no—
113#459—22no—
114#450—22no—
115#449—24no—
116#447—22no—
117C4×SD16—24no—
118C4×D8—24no—
119#432—21no—
120#432—21no—
121#431—21no—
122#431—21no—
123#431C2×C411yescharacteristic
124#431C2×C412yes—
125#431C2×C411yescharacteristic
126#431D412yes—
127#431D412yes—
128#423—21no—
129#423—21no—
130#423—21no—
131#423C2311yescharacteristic
132#423C2312yes—
133#423C2311yescharacteristic
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×C4⋊C8D412yes—
137C2×#104—24no—
138C2×#104—24no—
139C2×#104—24no—
140#407—21no—
141C2×C22⋊C8D412yes—
142C4×M16—21no—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8—24no—
147C2×C4×C8D412yes—
148C2×C4×C8D412yes—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2312yes—
152C4×C4⋊C4—21no—
153C4×C4⋊C4D412yes—
154C4×C22⋊C4—21no—
155C4×C22⋊C4D412yes—
156#376—21no—
157C43C2312yes—
158#339—24no—
159#336—24no—
160#335—24no—
161#334—24no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164D4⋊C8—24no—
165D4⋊C8—24no—
166D4⋊C8—22no—
167D4⋊C8—22no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—22no—
171C2×#41—21no—
172C2×#41—22no—
173C2×#41—41no—
174C2×#41—41no—
175C2×#41—82no—
176C2×#41—82no—
177C22×Q8—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—41no—
182C23×C4—41no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4#4111yescharacteristic
186C2×SD16—24no—
187C2×D8—24no—
188C2×M16—22no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—41no—
192C2×M16—82no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—21no—
197C22×C8—44no—
198C22×C8—41no—
199C22×C8C2×D414yes—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8C22×C412yes—
204C4⋊Q8C22×C411yescharacteristic
205C4⋊Q8C22×C411yescharacteristic
206C4⋊Q8C22×C411yescharacteristic
207#127—21no—
208#127—21no—
209#127—21no—
210#127C22×C412yes—
211#127C22×C411yescharacteristic
212#126—41no—
213#126—41no—
214#125—21no—
215#125—21no—
216#125—21no—
217#124—21no—
218#124—22no—
219#124—21no—
220#124—22no—
221#124—21no—
222#124—41no—
223#124—41no—
224#123—41no—
225#123—41no—
226#123—41no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233#121—22no—
234#121—21no—
235#121—22no—
236#121—21no—
237#121—21no—
238#121—21no—
239#121—22no—
240#121—21no—
241#121—22no—
242#121—21no—
243#121—41no—
244#121—41no—
245#121—41no—
246#121—41no—
247#120—41no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×D4—22no—
255C4×D4—21no—
256C4×D4—22no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274#117—28no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—41no—
279#117—41no—
280#117—41no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—41no—
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D414yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42#4111yescharacteristic
301C2×C42#4111yescharacteristic
302C2×C42C2411yescharacteristic
303#107—24no—
304#106—24no—
305C4⋊C8—22no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—22no—
310C4⋊C8—21no—
311C4⋊C8—41no—
312C4⋊C8—41no—
313C4⋊C8C22⋊C412yes—
314C4⋊C8C22⋊C412yes—
315C4⋊C8C22⋊C414yes—
316#104—84no—
317#104—84no—
318#104—84no—
319#103—24no—
320#102—24no—
321#102—24no—
322#102—24no—
323C22⋊C8—41no—
324C22⋊C8—41no—
325C22⋊C8C22⋊C412yes—
326C22⋊C8C22⋊C414yes—
327C22⋊C8C22⋊C412yes—
328#97—41no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—48no—
334C4×C8—41no—
335C4×C8C22×C412yes—
336C4×C8C22×C412yes—
337C4×C8C2×D412yes—
338C4×C8C2×D412yes—
339C4×C8C2×D414yes—
340#95—22no—
341#95—22no—
342C24—41no—
343C24#12111yescharacteristic
344#41—42no—
345#41—44no—
346#41—44no—
347#41—42no—
348#41—44no—
349#41—82no—
350#41—84no—
351#41—84no—
352#41—82no—
353#41—82no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—82no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—44no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—84no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4#12111yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12411yescharacteristic
403C22×C4#12411yescharacteristic
404C22×C4#12711yescharacteristic
405C22×C4#12711yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409SD16—84no—
410D8—84no—
411M16—42no—
412M16—44no—
413M16—42no—
414M16—82no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—42no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—42no—
425C2×C8—82no—
426C2×C8C2×C22⋊C414yes—
427C2×C8C2×C22⋊C414yes—
428C4⋊C4—28no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C22⋊C4—28no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—42no—
450C22⋊C4—41no—
451C22⋊C4—42no—
452C22⋊C4—41no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C42—28no—
460C42—24no—
461C42—24no—
462C42—44no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42C2×C22⋊C411yescharacteristic
467C42C2×C22⋊C411yescharacteristic
468C42C4×D411yescharacteristic
469C42C4×D411yescharacteristic
470C42C4×D411yescharacteristic
471C42C4×D411yescharacteristic
472C42#12111yescharacteristic
473C42#12111yescharacteristic
474C42#12411yescharacteristic
475C42#12411yescharacteristic
476C42#12711yescharacteristic
477C42#12711yescharacteristic
478C42C23×C411yescharacteristic
479C42C22×D411yescharacteristic
480C42C22×D411yescharacteristic
481C42C2×#4111yescharacteristic
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23#39011yescharacteristic
491C23#39011yescharacteristic
492C23C2×#12111yescharacteristic
493Q8—42no—
494Q8—44no—
495Q8—42no—
496Q8—42no—
497Q8—82no—
498D4—42no—
499D4—44no—
500D4—44no—
501D4—42no—
502D4—44no—
503D4—42no—
504D4—82no—
505D4—82no—
506D4—84no—
507D4—82no—
508D4—82no—
509D4—84no—
510D4—82no—
511D4—82no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4#39011yescharacteristic
546C2×C4#39011yescharacteristic
547C2×C4#39011yescharacteristic
548C2×C4#39011yescharacteristic
549C2×C4#39011yescharacteristic
550C2×C4#39011yescharacteristic
551C2×C4#44314yes—
552C2×C4C22×C22⋊C411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×D411yescharacteristic
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×#12411yescharacteristic
557C2×C4C2×#12711yescharacteristic
558C2×C4#57612yes—
559C2×C4#57712yes—
560C8—24no—
561C8—24no—
562C8—24no—
563C8—42no—
564C8—81no—
565C8—82no—
566C8—81no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—44no—
575C22—81no—
576C22—82no—
577C22—82no—
578C22—81no—
579C22—81no—
580C22—82no—
581C22—164no—
582C22#197412yes—
583C22C2×#39011yescharacteristic
584C22#306212yes—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—48no—
593C4—81no—
594C4—82no—
595C4—81no—
596C4—81no—
597C4#197412yes—
598C4#290112yes—
599C4#294612yes—
600C4C2×#44312yes—
601C4#315411yescharacteristic
602C4#315511yescharacteristic
603C2—22no—
604C2—22no—
605C2—41no—
606C2—161no—
607C2—162no—
608C2#2121011yescharacteristic
609C2#2123511yescharacteristic
610C2C2×#197411yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60946C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21856C2semidirect
3. #21815C2semidirect
4. C8×#117C2semidirect
5. #17817C2semidirect
6. #17473C2semidirect
7. C2×#1492C2semidirect
8. #9854C2semidirect
9. #9852C2semidirect
10. #9844C2semidirect
11. #9843C2semidirect
15. #3230C4semidirect
16. #3228C4semidirect
17. #3227C4semidirect
18. #3208C4semidirect
19. #3207C4semidirect
20. #3205C4semidirect
31. #1994C22semidirect
35. #1962C22semidirect
41. #1744C22semidirect
42. C42×C8C22semidirect
45. #1567C22semidirect
46. #1566C22semidirect
48. #1493C22semidirect
49. #1493C22semidirect
53. #1492C22semidirect
54. #1492C22semidirect

Maximal subgroups

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

subgroupindex
#609462
#218562
#218152
C8×#1172
#178172
#174732
C2×#14922
#98542
#98522
#98442
#98432