Finite Group Databasebeta

← #198943

Subgroups — #198943

Order 512.

1376 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#60940C211yesmaximal, characteristic
2#60144C211yesmaximal, characteristic, complemented
3#21201C211yesmaximal, characteristic, complemented
4#17704C211yesmaximal, characteristic, complemented
5C2×#1517C211yesmaximal, characteristic, complemented
6#9293C214yesmaximal, complemented
7#9291C214yesmaximal, complemented
8C2×#1286C211yesmaximal, characteristic, complemented
9#7860C211yesmaximal, characteristic, complemented
10C2×#568—21no—
11C2×#532C2211yescharacteristic
12C2×#532C2211yescharacteristic
13#3245—22no—
14#3226C412yescomplemented
15#3226C2212yes—
16#3205C412yescomplemented
17#3205C2212yes—
18#3186—22no—
19#3169C414yescomplemented
20#3100—22no—
21#3100—22no—
22#2935—22no—
23#2935—22no—
24D4×M16—22no—
25D4×M16—22no—
26#2932C2214yes—
27#2931—22no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C2×#417—21no—
32C2×#417—21no—
33#2879—21no—
34C2×C4×M16C2211yescharacteristic
35C2×#382C2211yescharacteristic
36#1961C2212yescomplemented
37#1960C2212yescomplemented
38#1958—22no—
39#1547—22no—
40#1546C2212yescomplemented
41#1545C2212yescomplemented
42#1540—22no—
43#1517—22no—
44#1517C2214yes—
45C2×#327—22no—
46#1286C2214yescomplemented
47#1275C2214yescomplemented
48C22×#41—21no—
49C22×#41—21no—
50#568—44no—
51#568—44no—
52C2×#131—24no—
53C2×#131—24no—
54C22×M16—24no—
55C22×M16—21no—
56C22×M16D411yescharacteristic
57C22×M16D411yescharacteristic
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67#532—21no—
68#532C2×C412yes—
69#532C2×C412yes—
70#532C2×C412yes—
71#532D412yes—
72#532D412yes—
73#532C2312yes—
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#127C2×C411yescharacteristic
77C2×#127C2311yescharacteristic
78C2×#124—21no—
79C2×#124—21no—
80C2×#121—21no—
81C2×#121—21no—
82C2×#121—21no—
83C2×#121D412yes—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4D412yes—
88C22×C4⋊C4—21no—
89C22×C42C2311yescharacteristic
90#500C2×C414yescomplemented
91#493C2×C414yescomplemented
92#486C2×C414yescomplemented
93#486C2×C414yescomplemented
94#480—24no—
95#480—24no—
96#474—24no—
97#474—24no—
98#466—28no—
99#459—24no—
100#459—24no—
101#447—24no—
102#447—24no—
103C4×D8—28no—
104#436—24no—
105#436—24no—
106#435—28no—
107#435—28no—
108C8×D4—24no—
109C8×D4—24no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431C2×C412yes—
116#431C2312yes—
117#431C2312yes—
118#423—21no—
119#423—21no—
120#423D412yes—
121#423C2312yes—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8D412yes—
126#417—44no—
127#417—44no—
128C2×#103—22no—
129C2×#103—22no—
130C2×#102—22no—
131C2×#102—22no—
132#408—24no—
133#408—24no—
134#407—24no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C4×M16—21no—
138C4×M16—21no—
139C4×M16—21no—
140C4×M16—21no—
141C2×#97C2312yes—
142C2×C4×C8—21no—
143C2×C4×C8C2×C412yes—
144C2×C4×C8D412yes—
145#382—21no—
146#382—21no—
147#382—21no—
148#382C2312yes—
149C2×#95—21no—
150C2×#95—21no—
151#332—24no—
152#331—24no—
153#327—28no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—41no—
161C2×#41—41no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41—42no—
168C2×#41—42no—
169C2×#41—42no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×D4—22no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—41no—
179C22×D4C22⋊C412yes—
180C23×C4—22no—
181C23×C4—41no—
182C23×C4—41no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4#4111yescharacteristic
186C2×D8—28no—
187#131—48no—
188#131—416no—
189#131—48no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—28no—
194C2×M16—22no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—416no—
201C2×M16—42no—
202C2×M16—42no—
203C2×M16D814yescomplemented
204C22×C8—28no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—41no—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8C22⋊C412yes—
216C4⋊Q8C22×C412yes—
217C4⋊Q8C22×C412yes—
218#127—21no—
219#127—21no—
220#127—21no—
221#127—21no—
222#127—21no—
223#127C22⋊C412yes—
224#127C22×C412yes—
225#127C22×C412yes—
226#124—22no—
227#124—22no—
228#124—22no—
229#124—22no—
230#124—22no—
231#124—22no—
232#124—41no—
233#124—41no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—22no—
239#121—22no—
240#121—22no—
241#121—22no—
242#121—22no—
243#121—22no—
244#121—41no—
245#121—41no—
246#121—41no—
247#121—41no—
248#121C22⋊C414yes—
249#121C2×D414yes—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—41no—
263C4×D4—41no—
264C4×D4C22⋊C414yes—
265C4×D4C2×D414yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4C22⋊C412yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42#4111yescharacteristic
289C2×C42C2411yescharacteristic
290#107—28no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—41no—
298C4⋊C8C2×D414yes—
299#103—44no—
300#103—44no—
301#102—28no—
302#102—28no—
303#102—44no—
304#102—44no—
305C22⋊C8—28no—
306C22⋊C8—28no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—41no—
311#97—22no—
312#97—22no—
313C4×C8—22no—
314C4×C8—22no—
315C4×C8—22no—
316C4×C8—22no—
317C4×C8—41no—
318C4×C8C22×C414yes—
319#95—22no—
320#95—22no—
321#95—41no—
322#95—41no—
323C24—22no—
324C24—41no—
325C24#12111yescharacteristic
326#41—42no—
327#41—42no—
328#41—42no—
329#41—42no—
330#41—44no—
331#41—44no—
332#41—44no—
333#41—44no—
334#41—44no—
335#41—44no—
336#41—44no—
337#41—44no—
338#41—44no—
339#41—44no—
340#41—82no—
341#41—82no—
342#41—82no—
343#41—82no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4C2×C22⋊C414yes—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4#12011yescharacteristic
407C22×C4#12011yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12311yescharacteristic
414C22×C4#12311yescharacteristic
415C22×C4C2×D812yes—
416C22×C4#13512yes—
417C22×C4AGL(1,ℤ/8)12yes—
418C22×C4AGL(1,ℤ/8)12yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422D8—88no—
423M16—24no—
424M16—24no—
425M16—416no—
426M16—42no—
427M16—42no—
428M16—44no—
429M16—42no—
430M16—42no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—416no—
439C2×C8—44no—
440C2×C8—42no—
441C2×C8—82no—
442C2×C8—82no—
443C2×C8C2×D814yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—82no—
454C4⋊C4C2×C22⋊C414yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C42C2×C22⋊C411yescharacteristic
464C42C2×C22⋊C411yescharacteristic
465C42C4×D411yescharacteristic
466C42C4×D411yescharacteristic
467C42C4×D411yescharacteristic
468C42C4×D411yescharacteristic
469C42#12011yescharacteristic
470C42#12011yescharacteristic
471C42#12111yescharacteristic
472C42#12111yescharacteristic
473C42#12311yescharacteristic
474C42#12311yescharacteristic
475C42C23×C411yescharacteristic
476C42C22×D411yescharacteristic
477C42C22×D411yescharacteristic
478C42C2×#4111yescharacteristic
479C23—24no—
480C23—41no—
481C23—41no—
482C23—44no—
483C23—42no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—81no—
489C23—81no—
490C23#38611yescharacteristic
491C23#38611yescharacteristic
492C23#45912yes—
493C23#46312yes—
494C23C2×#12111yescharacteristic
495Q8—44no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—81no—
501Q8—81no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—48no—
507D4—44no—
508D4—44no—
509D4—44no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—82no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—84no—
519D4—84no—
520D4—81no—
521D4—81no—
522C2×C4—24no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—164no—
548C2×C4#38611yescharacteristic
549C2×C4#38611yescharacteristic
550C2×C4#38611yescharacteristic
551C2×C4#38611yescharacteristic
552C2×C4#38611yescharacteristic
553C2×C4#38611yescharacteristic
554C2×C4C4×D812yes—
555C2×C4C4×D812yes—
556C2×C4#44212yes—
557C2×C4#44212yes—
558C2×C4#44712yes—
559C2×C4#44712yes—
560C2×C4#44912yes—
561C2×C4#44912yes—
562C2×C4#45912yes—
563C2×C4#46312yes—
564C2×C4C22×C22⋊C411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×#12011yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12311yescharacteristic
570C2×C4C22×D811yescharacteristic
571C2×C4C2×#13511yescharacteristic
572C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
573C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
574C8—24no—
575C8—44no—
576C8—44no—
577C8—42no—
578C8—82no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—42no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—164no—
594C22—162no—
595C22—162no—
596C22#187912yes—
597C22#187912yes—
598C22C2×#38611yescharacteristic
599C22C2×#45911yescharacteristic
600C22C2×#46311yescharacteristic
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—44no—
610C4—82no—
611C4—162no—
612C4#187912yes—
613C4#187912yes—
614C4C2×C4×D811yescharacteristic
615C4C2×#44211yescharacteristic
616C4C2×#44711yescharacteristic
617C4C2×#44911yescharacteristic
618C2—21no—
619C2—22no—
620C2—21no—
621C2—22no—
622C2—84no—
623C2—162no—
624C2C2×#187911yescharacteristic
625C2#2099511yescharacteristic
626C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#60940C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ 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. #21201C2semidirect
4. #17704C2semidirect
5. C2×#1517C2semidirect
6. #9293C2semidirect
7. #9291C2semidirect
8. C2×#1286C2semidirect
9. #7860C2semidirect
14. #3226C4semidirect
16. #3205C4semidirect
19. #3169C4semidirect
36. #1961C22semidirect
37. #1960C22semidirect
40. #1546C22semidirect
41. #1545C22semidirect
46. #1286C22semidirect
47. #1275C22semidirect
90. #500C2×C4semidirect
91. #493C2×C4semidirect
92. #486C2×C4semidirect
93. #486C2×C4semidirect
203. C2×M16D8semidirect

Maximal subgroups

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

subgroupindex
#609402
#601442
#212012
#177042
C2×#15172
#92932
#92912
C2×#12862
#78602