Finite Group Databasebeta

← #195480

Subgroups — #195480

Order 512.

1407 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#60144C211yesmaximal, characteristic, complemented
3C2×#1980C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic, complemented
5#17485C211yesmaximal, characteristic, complemented
6#9053C218yesmaximal, complemented
7#7901C211yesmaximal, characteristic, complemented
8#7882C211yesmaximal, characteristic, complemented
9C2×#568—21no—
10C2×#532C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3245—22no—
13#3226C412yescomplemented
14#3226C412yescomplemented
15#3205C412yescomplemented
16#3205C412yescomplemented
17#3186—22no—
18#3169C2214yes—
19#3100—22no—
20#3100—22no—
21#2935—22no—
22#2935C2214yescomplemented
23D4×M16—22no—
24D4×M16—22no—
25#2932—22no—
26#2931—22no—
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C22×#104—21no—
31C22×#104—21no—
32#2879—21no—
33C2×C4×M16C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1980—22no—
36#1980C2214yescomplemented
37#1502—22no—
38#1502C2214yescomplemented
39C2×#327—22no—
40#1492—22no—
41#1486—22no—
42#1480C2214yescomplemented
43#1279C2214yescomplemented
44#1278C2214yescomplemented
45C22×#41—21no—
46C22×#41—21no—
47#568—44no—
48#568—44no—
49C2×#131—24no—
50C2×#131—24no—
51C22×M16—24no—
52C22×M16—21no—
53C22×M16D411yescharacteristic
54C22×M16D411yescharacteristic
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532C2×C412yes—
66#532C2×C412yes—
67#532D412yes—
68#532D412yes—
69#532C2312yes—
70#532C2312yes—
71C2×C4⋊Q8C2×C411yescharacteristic
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×#127C2×C411yescharacteristic
74C2×#127C2×C411yescharacteristic
75C2×#124—21no—
76C2×#124—21no—
77C2×#121—21no—
78C2×#121—21no—
79C2×#121—21no—
80C2×#121D412yes—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4D412yes—
85C22×C4⋊C4—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#500C2×C414yescomplemented
89#493C2×C414yescomplemented
90#486C2×C414yescomplemented
91#486C2×C414yescomplemented
92#480—24no—
93#480—24no—
94#474—24no—
95#474—24no—
96#466—28no—
97#459—24no—
98#459—24no—
99#447—24no—
100#447—24no—
101C4×D8—28no—
102#436—24no—
103#436—24no—
104#435—28no—
105#435—28no—
106C8×D4—24no—
107C8×D4—24no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431C2×C412yes—
114#431C2×C412yes—
115#431C2312yes—
116#423—21no—
117#423—21no—
118#423D412yes—
119#423D412yes—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2312yes—
124C2×#104—24no—
125C2×#104—24no—
126C2×#104—44no—
127C2×#104—44no—
128#408—24no—
129#408—24no—
130#407—24no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C4×M16—21no—
134C4×M16—21no—
135C4×M16D412yes—
136C4×M16C2312yes—
137C2×#97—21no—
138C2×C4×C8—21no—
139C2×C4×C8—21no—
140C2×C4×C8C2312yes—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2312yes—
145C2×#95—21no—
146#327—28no—
147D4⋊C8—28no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—42no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—42no—
160C2×#41—42no—
161C2×#41—42no—
162C2×#41—42no—
163C2×#41—42no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—41no—
173C22×D4C22⋊C412yes—
174C23×C4—22no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4#4111yescharacteristic
180C2×D8—28no—
181#131—416no—
182#131—48no—
183#131—48no—
184C2×M16—21no—
185C2×M16—22no—
186C2×M16—28no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—21no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—416no—
193C2×M16—42no—
194C2×M16—42no—
195C2×M16D814yescomplemented
196C2×M16C2×D412yes—
197C2×M16C2×D412yes—
198C22×C8—21no—
199C22×C8—28no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—41no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C22⋊C412yes—
210C4⋊Q8C22×C412yes—
211C4⋊Q8C22×C412yes—
212#127—21no—
213#127—21no—
214#127—21no—
215#127—21no—
216#127—21no—
217#127C22⋊C412yes—
218#127C22×C412yes—
219#127C22×C412yes—
220#124—22no—
221#124—22no—
222#124—22no—
223#124—22no—
224#124—22no—
225#124—22no—
226#124—41no—
227#124—41no—
228#121—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—41no—
239#121—41no—
240#121—41no—
241#121—41no—
242#121C22⋊C414yes—
243#121C2×D414yes—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—41no—
255C4×D4—41no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4C22⋊C414yes—
259C4×D4C2×D414yes—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4C22⋊C412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C42—22no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—41no—
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42C2411yescharacteristic
287#107—28no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—41no—
295C4⋊C8C2×D414yes—
296#104—48no—
297#104—48no—
298#102—28no—
299#102—28no—
300C22⋊C8—28no—
301C22⋊C8—28no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—41no—
306#97—22no—
307#97—22no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—22no—
311C4×C8—22no—
312C4×C8—41no—
313C4×C8C22×C414yes—
314#95—22no—
315#95—41no—
316C24—22no—
317C24—41no—
318C24#12111yescharacteristic
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323#41—44no—
324#41—44no—
325#41—44no—
326#41—44no—
327#41—44no—
328#41—44no—
329#41—42no—
330#41—42no—
331#41—42no—
332#41—42no—
333#41—44no—
334#41—44no—
335#41—82no—
336#41—82no—
337#41—82no—
338#41—82no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—41no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4C2×C22⋊C414yes—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—81no—
387C22×C4—82no—
388C22×C4—82no—
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#12011yescharacteristic
398C22×C4#12011yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12311yescharacteristic
405C22×C4#12311yescharacteristic
406C22×C4C2×D812yes—
407C22×C4#13512yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4AGL(1,ℤ/8)12yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C2×#4111yescharacteristic
413D8—88no—
414M16—22no—
415M16—24no—
416M16—22no—
417M16—22no—
418M16—416no—
419M16—44no—
420M16—44no—
421M16—42no—
422M16C2×D814yes—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—416no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—82no—
434C2×C8—82no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4C2×C22⋊C414yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C42—42no—
455C42—42no—
456C42—82no—
457C42C2×C22⋊C411yescharacteristic
458C42C2×C22⋊C411yescharacteristic
459C42C4×D411yescharacteristic
460C42C4×D411yescharacteristic
461C42C4×D411yescharacteristic
462C42C4×D411yescharacteristic
463C42#12011yescharacteristic
464C42#12011yescharacteristic
465C42#12111yescharacteristic
466C42#12111yescharacteristic
467C42#12311yescharacteristic
468C42#12311yescharacteristic
469C42C23×C411yescharacteristic
470C42C22×D411yescharacteristic
471C42C22×D411yescharacteristic
472C42C2×#4111yescharacteristic
473C23—24no—
474C23—41no—
475C23—41no—
476C23—44no—
477C23—42no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23#38611yescharacteristic
484C23#38611yescharacteristic
485C23#45912yes—
486C23#46312yes—
487C23C2×#12111yescharacteristic
488Q8—44no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—82no—
494D4—44no—
495D4—44no—
496D4—44no—
497D4—48no—
498D4—44no—
499D4—44no—
500D4—44no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—44no—
506D4—44no—
507D4—82no—
508D4—82no—
509D4—82no—
510D4—84no—
511D4—82no—
512D4—82no—
513D4—82no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—41no—
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—44no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—82no—
534C2×C4—88no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—84no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4#38611yescharacteristic
541C2×C4#38611yescharacteristic
542C2×C4#38611yescharacteristic
543C2×C4#38611yescharacteristic
544C2×C4#38611yescharacteristic
545C2×C4#38611yescharacteristic
546C2×C4C4×D812yes—
547C2×C4C4×D812yes—
548C2×C4#44212yes—
549C2×C4#44212yes—
550C2×C4#44712yes—
551C2×C4#44712yes—
552C2×C4#44912yes—
553C2×C4#44912yes—
554C2×C4#45912yes—
555C2×C4#46312yes—
556C2×C4C22×C22⋊C411yescharacteristic
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C2×#12011yescharacteristic
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C2×C4C22×D811yescharacteristic
563C2×C4C2×#13511yescharacteristic
564C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
565C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
566C8—24no—
567C8—44no—
568C8—44no—
569C8—42no—
570C8—82no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—164no—
586C22—164no—
587C22#187912yes—
588C22#187912yes—
589C22C2×#38611yescharacteristic
590C22C2×#45911yescharacteristic
591C22C2×#46311yescharacteristic
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—44no—
600C4—84no—
601C4—82no—
602C4#187912yes—
603C4#187912yes—
604C4C2×C4×D811yescharacteristic
605C4C2×#44211yescharacteristic
606C4#295612yes—
607C4C2×#44711yescharacteristic
608C4C2×#44911yescharacteristic
609C2—21no—
610C2—22no—
611C2—22no—
612C2—21no—
613C2—84no—
614C2—162no—
615C2#2091811yescharacteristic
616C2C2×#187911yescharacteristic
617C2#2094711yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#60940C2
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. #60144C2semidirect
3. C2×#1980C2semidirect
4. C2×#1502C2semidirect
5. #17485C2semidirect
6. #9053C2semidirect
7. #7901C2semidirect
8. #7882C2semidirect
13. #3226C4semidirect
14. #3226C4semidirect
15. #3205C4semidirect
16. #3205C4semidirect
22. #2935C22semidirect
36. #1980C22semidirect
38. #1502C22semidirect
42. #1480C22semidirect
43. #1279C22semidirect
44. #1278C22semidirect
88. #500C2×C4semidirect
89. #493C2×C4semidirect
90. #486C2×C4semidirect
91. #486C2×C4semidirect
195. C2×M16D8semidirect

Maximal subgroups

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

subgroupindex
#609402
#601442
C2×#19802
C2×#15022
#174852
#90532
#79012
#78822