Finite Group Databasebeta

← #205498

Subgroups — #205498

Order 512.

1003 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3131C211yesmaximal, characteristic
2#20473C211yesmaximal, characteristic, complemented
3#17793C211yesmaximal, characteristic, complemented
4#17545C211yesmaximal, characteristic, complemented
5#17544C211yesmaximal, characteristic
6#17464C211yesmaximal, characteristic, complemented
7#17463C211yesmaximal, characteristic
8#9754C218yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3131—21no—
12#3131—21no—
13#3131—21no—
14#3131—21no—
15#3131C414yescomplemented
16#3131C2214yes—
17C2×#488—22no—
18C2×#487C411yescharacteristic, complemented
19C2×#487C411yescharacteristic, complemented
20C2×#486C411yescharacteristic, complemented
21C2×#486C411yescharacteristic, complemented
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#416—22no—
26C22×C4×C8C2211yescharacteristic
27C4×#117C2211yescharacteristic
28#1770C2214yescomplemented
29#1752—22no—
30#1744—21no—
31C4×#97—21no—
32#1568—22no—
33#1567—21no—
34#1566—21no—
35#1564C2214yescomplemented
36#1508—22no—
37#1507—22no—
38#1506—22no—
39#1505—22no—
40#1503C2214yescomplemented
41#1502C2214yes—
42#1485—22no—
43#1484—22no—
44#1481C2214yescomplemented
45#1480C2214yes—
46#1479—22no—
47#1478—22no—
48C22×#41—21no—
49C23×C8D412yes—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533C2×C412yes—
57#533C2312yes—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532C2×C412yes—
65#532C2312yes—
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×#127C2×C411yescharacteristic
70C2×#125—21no—
71C2×#124—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×#117—21no—
81C2×#117—21no—
82C22×C42C2311yescharacteristic
83#488—24no—
84#488—24no—
85#488—24no—
86#488—24no—
87#487—21no—
88#487—21no—
89#487—21no—
90#487—21no—
91#487C2×C414yes—
92#487C2×C414yes—
93#486—21no—
94#486—21no—
95#486—21no—
96#486—21no—
97#486C2×C414yes—
98#486C2×C414yes—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431D412yes—
108#431D412yes—
109#431C2312yes—
110#431C2312yes—
111#423—21no—
112#423—21no—
113C2×C4⋊C8C2312yes—
114#416—24no—
115#416—24no—
116#416—24no—
117#416—24no—
118C2×#103—22no—
119C2×#103—22no—
120C2×#102—22no—
121C2×#102—22no—
122#407—21no—
123#407—21no—
124C2×C22⋊C8—21no—
125C4×M16—21no—
126C4×M16—21no—
127C2×#97—24no—
128C2×C4×C8—21no—
129C2×C4×C8C2×C411yescharacteristic
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C412yes—
132C2×C4×C8D411yescharacteristic
133C2×C4×C8D411yescharacteristic
134C2×C4×C8D411yescharacteristic
135C2×C4×C8D411yescharacteristic
136C2×C4×C8D412yes—
137C2×C4×C8C2312yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2312yes—
140C4×C22⋊C4—21no—
141C4×C22⋊C4D412yes—
142#376D412yes—
143C43—21no—
144#336—24no—
145#335—24no—
146#334—24no—
147#328—28no—
148#327—28no—
149Q8⋊C8—28no—
150D4⋊C8—28no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—48no—
154C2×#41—41no—
155C2×#41—41no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—41no—
163C23×C4—41no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4#4111yescharacteristic
167C2×M16—41no—
168C2×M16—41no—
169C22×C8—22no—
170C22×C8—44no—
171C22×C8C22⋊C412yes—
172C22×C8C22⋊C412yes—
173C22×C8C22⋊C412yes—
174C22×C8C22⋊C412yes—
175C22×C8C2×D412yes—
176C22×C8C2×D412yes—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22×C412yes—
187C4⋊Q8C22×C412yes—
188C4⋊Q8C22×C412yes—
189#127—21no—
190#127—21no—
191#127—21no—
192#127C22×C412yes—
193#125—22no—
194#125—22no—
195#125—41no—
196#125—41no—
197#124—22no—
198#124—22no—
199#124—41no—
200#124—41no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209#121—22no—
210#121—22no—
211#121—22no—
212#121—22no—
213#121—41no—
214#121—41no—
215#121—41no—
216#121—41no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233#117—22no—
234#117—22no—
235#117—44no—
236#117—41no—
237#117—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—41no—
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42C2411yescharacteristic
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—41no—
271C4⋊C8—41no—
272C4⋊C8C2×D414yes—
273#103—24no—
274#103—24no—
275#103—24no—
276#103—24no—
277#102—24no—
278#102—24no—
279#102—24no—
280#102—24no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—41no—
286C22⋊C8—41no—
287#97—48no—
288#97—41no—
289#97—41no—
290C4×C8—22no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8C22⋊C411yescharacteristic
295C4×C8C22⋊C411yescharacteristic
296C4×C8C22⋊C411yescharacteristic
297C4×C8C22⋊C411yescharacteristic
298C4×C8C22⋊C412yes—
299C4×C8C22⋊C412yes—
300C4×C8C22×C412yes—
301C4×C8C22×C412yes—
302C4×C8C2×D412yes—
303C4×C8C2×D412yes—
304#95—22no—
305#95C2×D414yes—
306C24—41no—
307C24#12111yescharacteristic
308#41—48no—
309#41—48no—
310#41—84no—
311#41—84no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—44no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—44no—
320C2×Q8—82no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—44no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—44no—
331C2×D4—82no—
332C2×D4—84no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—84no—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4C4×D411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12411yescharacteristic
355C22×C4#12411yescharacteristic
356C22×C4#12711yescharacteristic
357C22×C4#12711yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361M16—82no—
362M16—82no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—44no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8C2×C22⋊C412yes—
374C2×C8C2×C22⋊C412yes—
375C2×C8C2×C22⋊C412yes—
376C2×C8C2×C22⋊C412yes—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C22⋊C4—44no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C42—44no—
402C42—44no—
403C42—42no—
404C42—44no—
405C42—42no—
406C42C2×C22⋊C411yescharacteristic
407C42C2×C22⋊C411yescharacteristic
408C42C4×D411yescharacteristic
409C42C4×D411yescharacteristic
410C42C4×D411yescharacteristic
411C42C4×D411yescharacteristic
412C42#12111yescharacteristic
413C42#12111yescharacteristic
414C42#12411yescharacteristic
415C42#12411yescharacteristic
416C42#12711yescharacteristic
417C42#12711yescharacteristic
418C42C23×C411yescharacteristic
419C42C22×D411yescharacteristic
420C42C22×D411yescharacteristic
421C42C2×#4111yescharacteristic
422C23—22no—
423C23—22no—
424C23—41no—
425C23—42no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23#39011yescharacteristic
430C23#39011yescharacteristic
431C23C2×#12111yescharacteristic
432Q8—44no—
433Q8—44no—
434Q8—82no—
435Q8—82no—
436Q8—82no—
437Q8—82no—
438D4—44no—
439D4—44no—
440D4—82no—
441D4—82no—
442D4—84no—
443D4—84no—
444D4—82no—
445D4—82no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—84no—
473C2×C4#39011yescharacteristic
474C2×C4#39011yescharacteristic
475C2×C4#39011yescharacteristic
476C2×C4#39011yescharacteristic
477C2×C4#39011yescharacteristic
478C2×C4#39011yescharacteristic
479C2×C4#44414yes—
480C2×C4C22×C22⋊C411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×#12411yescharacteristic
485C2×C4C2×#12711yescharacteristic
486C2×C4#57514yes—
487C8—22no—
488C8—22no—
489C8—22no—
490C8—22no—
491C8—44no—
492C8—82no—
493C8—82no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—44no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—164no—
506C22#197512yes—
507C22C2×#39011yescharacteristic
508C22#304012yes—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—84no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4#197512yes—
521C4#197612yes—
522C4#289612yes—
523C4C2×#44412yes—
524C4#304012yes—
525C2—22no—
526C2—22no—
527C2—41no—
528C2—162no—
529C2C2×#197511yescharacteristic
530C2C2×#197611yescharacteristic
531C2#2126111yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radicalC2×#532C22
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. #20473C2semidirect
3. #17793C2semidirect
4. #17545C2semidirect
6. #17464C2semidirect
8. #9754C2semidirect
15. #3131C4semidirect
18. C2×#487C4semidirect
19. C2×#487C4semidirect
20. C2×#486C4semidirect
21. C2×#486C4semidirect
28. #1770C22semidirect
35. #1564C22semidirect
40. #1503C22semidirect
44. #1481C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31312
#204732
#177932
#175452
#175442
#174642
#174632
#97542