Finite Group Databasebeta

← #195475

Subgroups — #195475

Order 512.

1393 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2C2×#2935C211yesmaximal, characteristic, complemented
3#21252C211yesmaximal, characteristic, complemented
4#17574C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic, complemented
6#9053C218yesmaximal, complemented
7#7893C211yesmaximal, characteristic, complemented
8#7882C211yesmaximal, characteristic, complemented
9C22×#131—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#2935—22no—
23#2935—22no—
24#2935C2214yescomplemented
25C2×#436—21no—
26C2×#435—22no—
27C2×C8×D4—21no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×#421—21no—
32C2×#421—21no—
33C2×#408—21no—
34C2×C4×M16C2211yescharacteristic
35#2285C2211yescharacteristic
36#1980—22no—
37#1980C2214yescomplemented
38#1519—22no—
39#1517—22no—
40#1502C2214yescomplemented
41#1480—21no—
42#1480—21no—
43#1480C2214yescomplemented
44C2×D4⋊C8—22no—
45#1279C2214yescomplemented
46#1278C2214yescomplemented
47C22×#41—21no—
48C22×#41—21no—
49C2×#131—24no—
50C2×#131—24no—
51C2×#131—44no—
52C2×#131—44no—
53C22×M16—22no—
54C22×M16D411yescharacteristic
55C22×M16D411yescharacteristic
56C23×C8—22no—
57C23×C8—21no—
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#532D412yes—
71#532D412yes—
72#532C2312yes—
73#532C2312yes—
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#127C2×C411yescharacteristic
77C2×#127C2×C411yescharacteristic
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—
88C2×#117—21no—
89C22×C42C2311yescharacteristic
90#500C2×C414yes—
91#493C2×C414yes—
92#486C2×C414yes—
93#486C2×C414yes—
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#431C2×C412yes—
117#431C2312yes—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8D411yescharacteristic
124C2×C4⋊C8D411yescharacteristic
125C2×C4⋊C8D412yes—
126C2×C4⋊C8C2312yes—
127#421—44no—
128#421—44no—
129C2×#104—24no—
130C2×#104—24no—
131#408—24no—
132#408—24no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8—22no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C4×M16—21no—
138C4×M16—21no—
139C4×M16D412yes—
140C4×M16C2312yes—
141C2×#97—21no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8C2312yes—
145#384—21no—
146#382—21no—
147#380—21no—
148#380—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4C2312yes—
151#327—28no—
152D4⋊C8—24no—
153D4⋊C8—24no—
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×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185C2×D8—28no—
186#131—48no—
187#131—416no—
188#131—48no—
189C2×M16—21no—
190C2×M16—22no—
191C2×M16—28no—
192C2×M16—21no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—48no—
196C2×M16D814yescomplemented
197C2×M16C2×D412yes—
198C2×M16C2×D412yes—
199C22×C8—21no—
200C22×C8—28no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—48no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—41no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8C22⋊C412yes—
217C4⋊Q8C22×C412yes—
218C4⋊Q8C22×C412yes—
219#127—21no—
220#127—21no—
221#127—21no—
222#127—21no—
223#127—21no—
224#127C22⋊C412yes—
225#127C22×C412yes—
226#127C22×C412yes—
227#124—22no—
228#124—22no—
229#124—22no—
230#124—22no—
231#124—22no—
232#124—22no—
233#124—41no—
234#124—41no—
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—22no—
245#121—41no—
246#121—41no—
247#121—41no—
248#121—41no—
249#121C22⋊C414yes—
250#121C2×D414yes—
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—22no—
261C4×D4—41no—
262C4×D4—41no—
263C4×D4—41no—
264C4×D4—41no—
265C4×D4C22⋊C414yes—
266C4×D4C2×D414yes—
267#117—42no—
268#117—42no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4C22⋊C412yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C42—22no—
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42C2411yescharacteristic
294#107—28no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—21no—
298C4⋊C8—22no—
299C4⋊C8—21no—
300C4⋊C8—22no—
301C4⋊C8—22no—
302C4⋊C8—41no—
303C4⋊C8C2×D412yes—
304C4⋊C8C2×D412yes—
305#104—48no—
306#104—48no—
307#102—28no—
308#102—28no—
309C22⋊C8—28no—
310C22⋊C8—28no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—41no—
315#97—22no—
316#97—22no—
317C4×C8—22no—
318C4×C8—22no—
319C4×C8—22no—
320C4×C8—22no—
321C4×C8—41no—
322C4×C8C22×C414yes—
323#95—22no—
324#95—41no—
325C24—22no—
326C24—41no—
327C24#12111yescharacteristic
328#41—42no—
329#41—42no—
330#41—42no—
331#41—42no—
332#41—44no—
333#41—44no—
334#41—44no—
335#41—44no—
336#41—44no—
337#41—44no—
338#41—42no—
339#41—42no—
340#41—42no—
341#41—42no—
342#41—44no—
343#41—44no—
344#41—82no—
345#41—82no—
346#41—82no—
347#41—82no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—41no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
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—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—81no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C4×D411yescharacteristic
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4#12011yescharacteristic
404C22×C4#12011yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12311yescharacteristic
411C22×C4#12311yescharacteristic
412C22×C4C2×D812yes—
413C22×C4#13512yes—
414C22×C4AGL(1,ℤ/8)12yes—
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419D8—88no—
420M16—22no—
421M16—24no—
422M16—22no—
423M16—22no—
424M16—416no—
425M16—42no—
426M16C2×D814yes—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—416no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—44no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—82no—
442C2×C8—82no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—44no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4C2×C22⋊C414yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—82no—
463C42—42no—
464C42—42no—
465C42C2×C22⋊C411yescharacteristic
466C42C2×C22⋊C411yescharacteristic
467C42C4×D411yescharacteristic
468C42C4×D411yescharacteristic
469C42C4×D411yescharacteristic
470C42C4×D411yescharacteristic
471C42#12011yescharacteristic
472C42#12011yescharacteristic
473C42#12111yescharacteristic
474C42#12111yescharacteristic
475C42#12311yescharacteristic
476C42#12311yescharacteristic
477C42C23×C411yescharacteristic
478C42C22×D411yescharacteristic
479C42C22×D411yescharacteristic
480C42C2×#4111yescharacteristic
481C23—24no—
482C23—41no—
483C23—41no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23#38611yescharacteristic
492C23#38611yescharacteristic
493C23#45912yes—
494C23#46312yes—
495C23C2×#12111yescharacteristic
496Q8—44no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—82no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—48no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—44no—
514D4—44no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—84no—
519D4—82no—
520D4—82no—
521D4—82no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
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—82no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#38611yescharacteristic
548C2×C4#38611yescharacteristic
549C2×C4#38611yescharacteristic
550C2×C4#38611yescharacteristic
551C2×C4#38611yescharacteristic
552C2×C4#38611yescharacteristic
553C2×C4C4×D812yes—
554C2×C4C4×D812yes—
555C2×C4#44212yes—
556C2×C4#44212yes—
557C2×C4#44712yes—
558C2×C4#44712yes—
559C2×C4#44912yes—
560C2×C4#44912yes—
561C2×C4#45912yes—
562C2×C4#46312yes—
563C2×C4C22×C22⋊C411yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×#12011yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12311yescharacteristic
569C2×C4C22×D811yescharacteristic
570C2×C4C2×#13511yescharacteristic
571C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
572C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
573C8—24no—
574C8—44no—
575C8—42no—
576C8—42no—
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—88no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—164no—
595C22#187912yes—
596C22#187912yes—
597C22C2×#38611yescharacteristic
598C22C2×#45911yescharacteristic
599C22C2×#46311yescharacteristic
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—44no—
608C4—82no—
609C4—162no—
610C4#187912yes—
611C4#187912yes—
612C4C2×C4×D811yescharacteristic
613C4C2×#44211yescharacteristic
614C4#295512yes—
615C4C2×#44711yescharacteristic
616C4C2×#44911yescharacteristic
617C2—21no—
618C2—22no—
619C2—22no—
620C2—21no—
621C2—84no—
622C2—162no—
623C2#2091711yescharacteristic
624C2C2×#187911yescharacteristic
625C2#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. C2×#2935C2semidirect
3. #21252C2semidirect
4. #17574C2semidirect
5. C2×#1480C2semidirect
6. #9053C2semidirect
7. #7893C2semidirect
8. #7882C2semidirect
13. #3226C4semidirect
14. #3226C4semidirect
15. #3205C4semidirect
16. #3205C4semidirect
24. #2935C22semidirect
37. #1980C22semidirect
40. #1502C22semidirect
43. #1480C22semidirect
45. #1279C22semidirect
46. #1278C22semidirect
196. C2×M16D8semidirect

Maximal subgroups

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

subgroupindex
#609402
C2×#29352
#212522
#175742
C2×#14802
#90532
#78932
#78822