Finite Group Databasebeta

← #198795

Subgroups — #198795

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3207C211yesmaximal, characteristic
2#60236C211yesmaximal, characteristic, complemented
3#17544C211yesmaximal, characteristic, complemented
4#17544C211yesmaximal, characteristic, complemented
5#17478C211yesmaximal, characteristic, complemented
6#9281C218yesmaximal, complemented
7#7965C211yesmaximal, characteristic, complemented
8#7854C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3207—22no—
12#3207—22no—
13#3207C414yescomplemented
14#3207C2214yes—
15C2×#493C411yescharacteristic, complemented
16C2×#492C411yescharacteristic, complemented
17C2×#487C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#463—22no—
20C2×#449—22no—
21#2981C2214yescomplemented
22#2979—22no—
23#2975—21no—
24#2974—22no—
25#2972—21no—
26#2935—22no—
27#2931—22no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2880—22no—
33C4×#117C2211yescharacteristic
34#1507—22no—
35#1507—22no—
36#1505C2214yescomplemented
37#1505C2214yescomplemented
38#1502—22no—
39#1502—22no—
40#1496C2214yescomplemented
41#1492—22no—
42#1478—21no—
43#1478—21no—
44#1287C2214yescomplemented
45#1274C2214yescomplemented
46C22×#41—21no—
47C22×#41—21no—
48C22×SD16—22no—
49C22×D8—22no—
50C2×#131—24no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532C2×C412yes—
62#532C2×C412yes—
63#532D412yes—
64#532D412yes—
65#532C2312yes—
66#532C2312yes—
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×#127C2×C411yescharacteristic
70C2×#127C2×C411yescharacteristic
71C2×#124—21no—
72C2×#124—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121D412yes—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4D412yes—
81C2×#117—21no—
82C22×C42C2311yescharacteristic
83#493—22no—
84#493C2×C414yes—
85#492—22no—
86#492C2×C414yes—
87#487—22no—
88#487C2×C414yes—
89#486—22no—
90#486C2×C414yes—
91#463—28no—
92#463—28no—
93#449—28no—
94#449—28no—
95#436—24no—
96#436—24no—
97#436—24no—
98#435—28no—
99C8×D4—24no—
100C8×D4—24no—
101C8×D4—24no—
102#432—21no—
103#432—21no—
104#432C2312yes—
105#432C2312yes—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431C2×C412yes—
114#431D412yes—
115#431D411yescharacteristic
116#431D411yescharacteristic
117#431C2312yes—
118#431C2312yes—
119#423—21no—
120#423—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×#103—22no—
124C2×#102—22no—
125#408—24no—
126#408—28no—
127#407—24no—
128#407—21no—
129#407D412yes—
130C2×C22⋊C8—24no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C4×M16—21no—
134C4×M16—21no—
135C2×C4×C8C2×C411yescharacteristic
136C2×C4×C8C2×C411yescharacteristic
137C2×C4×C8D412yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#376C2312yes—
143C43—21no—
144#327—28no—
145#327—28no—
146D4⋊C8—24no—
147D4⋊C8—24no—
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—84no—
157C2×#41—84no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—41no—
167C22×D4C22⋊C412yes—
168C23×C4—22no—
169C23×C4—41no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173C2×SD16—28no—
174C2×SD16—84no—
175C2×D8—28no—
176C2×D8—84no—
177#131—88no—
178C2×M16—22no—
179C2×M16—28no—
180C2×M16—41no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—28no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8C22⋊C412yes—
191C4⋊Q8C22×C412yes—
192C4⋊Q8C22×C412yes—
193#127—21no—
194#127—21no—
195#127—21no—
196#127—21no—
197#127—21no—
198#127C22⋊C412yes—
199#127C22×C412yes—
200#127C22×C412yes—
201#124—22no—
202#124—22no—
203#124—22no—
204#124—22no—
205#124—22no—
206#124—22no—
207#124—41no—
208#124—41no—
209#121—22no—
210#121—22no—
211#121—22no—
212#121—22no—
213#121—22no—
214#121—22no—
215#121—22no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—41no—
220#121—41no—
221#121—41no—
222#121—41no—
223#121C22⋊C414yes—
224#121C2×D414yes—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4C22⋊C414yes—
240C4×D4C2×D414yes—
241#117—24no—
242#117—42no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4C22⋊C412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C42—22no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42C2411yescharacteristic
268C4⋊C8—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—41no—
277#103—28no—
278#102—28no—
279C22⋊C8—28no—
280C22⋊C8—28no—
281C22⋊C8—22no—
282C22⋊C8—21no—
283C22⋊C8—22no—
284C22⋊C8—21no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—41no—
289C22⋊C8C2×D414yes—
290#97—22no—
291#97—22no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—41no—
297C4×C8C22×C412yes—
298C4×C8C22×C412yes—
299C4×C8C2×D412yes—
300C4×C8C2×D412yes—
301#95—22no—
302#95—22no—
303C24—22no—
304C24—41no—
305C24#12111yescharacteristic
306#41—48no—
307#41—84no—
308#41—84no—
309#41—84no—
310#41—84no—
311#41—84no—
312#41—84no—
313#41—84no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—44no—
318C2×Q8—41no—
319C2×Q8—82no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4C2×C22⋊C414yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12311yescharacteristic
373C22×C4#12311yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4111yescharacteristic
377SD16—88no—
378D8—88no—
379M16—44no—
380M16—88no—
381M16—82no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—44no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—88no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—44no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4C2×C22⋊C414yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C42—24no—
411C42—42no—
412C42—82no—
413C42—82no—
414C42C2×C22⋊C411yescharacteristic
415C42C2×C22⋊C411yescharacteristic
416C42C4×D411yescharacteristic
417C42C4×D411yescharacteristic
418C42C4×D411yescharacteristic
419C42C4×D411yescharacteristic
420C42#12011yescharacteristic
421C42#12011yescharacteristic
422C42#12111yescharacteristic
423C42#12111yescharacteristic
424C42#12311yescharacteristic
425C42#12311yescharacteristic
426C42C23×C411yescharacteristic
427C42C22×D411yescharacteristic
428C42C22×D411yescharacteristic
429C42C2×#4111yescharacteristic
430C23—22no—
431C23—24no—
432C23—41no—
433C23—41no—
434C23—44no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23#38611yescharacteristic
442C23#38611yescharacteristic
443C23C2×#12111yescharacteristic
444Q8—44no—
445Q8—44no—
446Q8—82no—
447Q8—82no—
448D4—44no—
449D4—44no—
450D4—48no—
451D4—44no—
452D4—44no—
453D4—44no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—82no—
460D4—84no—
461D4—84no—
462D4—84no—
463D4—82no—
464D4—82no—
465D4—82no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#38611yescharacteristic
497C2×C4#38611yescharacteristic
498C2×C4#38611yescharacteristic
499C2×C4#38611yescharacteristic
500C2×C4#38611yescharacteristic
501C2×C4#38611yescharacteristic
502C2×C4#44312yes—
503C2×C4#44412yes—
504C2×C4C22×C22⋊C411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×#12011yescharacteristic
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×#12311yescharacteristic
510C8—42no—
511C8—42no—
512C8—42no—
513C8—42no—
514C8—44no—
515C8—82no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—164no—
532C22—164no—
533C22#187612yes—
534C22C2×#38611yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—44no—
544C4—84no—
545C4—82no—
546C4#187612yes—
547C4#190412yes—
548C4#191012yes—
549C4C2×#44311yescharacteristic
550C4C2×#44411yescharacteristic
551C2—22no—
552C2—41no—
553C2—41no—
554C2—84no—
555C2—162no—
556C2C2×#187611yescharacteristic
557C2C2×#190411yescharacteristic
558C2C2×#191011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3207C2
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. #60236C2semidirect
3. #17544C2semidirect
4. #17544C2semidirect
5. #17478C2semidirect
6. #9281C2semidirect
7. #7965C2semidirect
8. #7854C2semidirect
13. #3207C4semidirect
15. C2×#493C4semidirect
16. C2×#492C4semidirect
17. C2×#487C4semidirect
18. C2×#486C4semidirect
21. #2981C22semidirect
36. #1505C22semidirect
37. #1505C22semidirect
40. #1496C22semidirect
44. #1287C22semidirect
45. #1274C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32072
#602362
#175442
#174782
#92812
#79652
#78542