Finite Group Databasebeta

← #205580

Subgroups — #205580

Order 512.

1123 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2C4×#405C211yesmaximal, characteristic, complemented
3#17793C211yesmaximal, characteristic, complemented
4#17648C211yesmaximal, characteristic, complemented
5#17647C211yesmaximal, characteristic
6#17545C211yesmaximal, characteristic, complemented
7#17544C211yesmaximal, characteristic
8#9761C218yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3167—21no—
12#3167—21no—
13#3167—21no—
14#3167—21no—
15#3167C414yescomplemented
16#3167C2214yes—
17C2×#498C411yescharacteristic, complemented
18C2×#492C411yescharacteristic, complemented
19C2×#487C411yescharacteristic, complemented
20C2×#486C411yescharacteristic, complemented
21C2×#465—22no—
22C2×C4×SD16—22no—
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25C2×#405—22no—
26C2×C4×M16C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1726C2214yescomplemented
30C4×#97—21no—
31C42×C8—21no—
32#1568C2214yescomplemented
33#1567—21no—
34#1566—21no—
35#1564—22no—
36#1531C2214yescomplemented
37#1530C2214yes—
38#1528—22no—
39#1527—22no—
40#1526—22no—
41#1525—22no—
42#1508—22no—
43#1507—22no—
44#1506C2214yescomplemented
45#1505C2214yes—
46#1503—22no—
47#1502—22no—
48C22×#41—21no—
49C22×SD16—22no—
50C22×M16—21no—
51C23×C8D412yes—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533C2×C412yes—
59#533C2312yes—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532C2×C412yes—
67#532C2312yes—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4⋊Q8C2×C411yescharacteristic
71C2×#127C2×C411yescharacteristic
72C2×#125—21no—
73C2×#124—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×#121—21no—
77C2×#121—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×#117—21no—
83C2×#117—21no—
84C22×C42C2311yescharacteristic
85#498—21no—
86#498—21no—
87#498C2×C414yes—
88#492—21no—
89#492—21no—
90#492C2×C414yes—
91#487—21no—
92#487—21no—
93#487C2×C414yes—
94#486—21no—
95#486—21no—
96#486C2×C414yes—
97#465—24no—
98#465—24no—
99#465—24no—
100#465—24no—
101C4×SD16—24no—
102C4×SD16—24no—
103C4×SD16—24no—
104C4×SD16—24no—
105#432—21no—
106#432—21no—
107#432C2312yes—
108#432C2312yes—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113C2×#106—22no—
114#423—21no—
115#423—21no—
116C2×#103—22no—
117C2×#102—22no—
118#407—21no—
119#407—21no—
120#405—216no—
121C4×M16—21no—
122C4×M16—21no—
123C4×M16—21no—
124C4×M16—21no—
125C4×M16D412yes—
126C4×M16D412yes—
127C2×#97—24no—
128C2×#97C2312yes—
129C2×C4×C8—24no—
130C2×C4×C8—21no—
131C2×C4×C8C2×C411yescharacteristic
132C2×C4×C8C2×C411yescharacteristic
133C2×C4×C8C2×C412yes—
134C2×C4×C8D411yescharacteristic
135C2×C4×C8D411yescharacteristic
136C2×C4×C8D412yes—
137C2×C4×C8D411yescharacteristic
138C2×C4×C8D411yescharacteristic
139C2×C4×C8C2312yes—
140C4×C4⋊C4—21no—
141C4×C4⋊C4D412yes—
142C4×C22⋊C4—21no—
143C4×C22⋊C4D412yes—
144#376C2312yes—
145C43—21no—
146#335—24no—
147#334—24no—
148#330—28no—
149#329—28no—
150#328—28no—
151#327—28no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—84no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—41no—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C2×SD16—24no—
169C2×SD16—24no—
170C2×SD16—48no—
171C2×M16—28no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—41no—
175C2×M16—41no—
176C22×C8—22no—
177C22×C8—28no—
178C22×C8C22⋊C412yes—
179C22×C8C22⋊C412yes—
180C22×C8C22⋊C412yes—
181C22×C8C22⋊C412yes—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C22×C412yes—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C412yes—
196#127—21no—
197#127—21no—
198#127—21no—
199#127C22×C412yes—
200#125—22no—
201#125—22no—
202#125—41no—
203#125—41no—
204#124—22no—
205#124—22no—
206#124—41no—
207#124—41no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—41no—
221#121—41no—
222#121—41no—
223#121—41no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—41no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240#117—28no—
241#117—22no—
242#117—22no—
243#117—41no—
244#117—41no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—41no—
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42C2411yescharacteristic
274#106—24no—
275#106—24no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280#103—24no—
281#103—24no—
282#102—24no—
283#102—24no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—41no—
287C22⋊C8—41no—
288#97—216no—
289#97—22no—
290#97—22no—
291#97—22no—
292#97C2×D414yes—
293C4×C8—216no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—41no—
297C4×C8—41no—
298C4×C8C22⋊C412yes—
299C4×C8C22⋊C412yes—
300C4×C8C22⋊C411yescharacteristic
301C4×C8C22⋊C411yescharacteristic
302C4×C8C22⋊C411yescharacteristic
303C4×C8C22⋊C411yescharacteristic
304C4×C8C22×C412yes—
305C4×C8C22×C412yes—
306C4×C8C2×D412yes—
307C4×C8C2×D412yes—
308#95—22no—
309#95C2×D414yes—
310C24—41no—
311C24#12111yescharacteristic
312#41—84no—
313#41—84no—
314#41—84no—
315#41—84no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—44no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—44no—
334C2×D4—82no—
335C2×D4—82no—
336C2×D4—84no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—84no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12411yescharacteristic
359C22×C4#12411yescharacteristic
360C22×C4#12711yescharacteristic
361C22×C4#12711yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365SD16—48no—
366SD16—48no—
367M16—28no—
368M16—28no—
369M16—82no—
370M16—82no—
371C2×C8—28no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8#10418yes—
381C2×C8C2×C22⋊C412yes—
382C2×C8C2×C22⋊C412yes—
383C2×C8C2×C22⋊C412yes—
384C2×C8C2×C22⋊C412yes—
385C4⋊C4—28no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C22⋊C4—28no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C42—28no—
410C42—44no—
411C42—42no—
412C42—42no—
413C42—44no—
414C42C2×C22⋊C411yescharacteristic
415C42C2×C22⋊C411yescharacteristic
416C42C4×D411yescharacteristic
417C42C4×D411yescharacteristic
418C42C4×D411yescharacteristic
419C42C4×D411yescharacteristic
420C42#12111yescharacteristic
421C42#12111yescharacteristic
422C42#12411yescharacteristic
423C42#12411yescharacteristic
424C42#12711yescharacteristic
425C42#12711yescharacteristic
426C42C23×C411yescharacteristic
427C42C22×D411yescharacteristic
428C42C22×D411yescharacteristic
429C42C2×#4111yescharacteristic
430C23—22no—
431C23—22no—
432C23—41no—
433C23—42no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23#39011yescharacteristic
438C23#39011yescharacteristic
439C23C2×#12111yescharacteristic
440Q8—44no—
441Q8—82no—
442Q8—82no—
443Q8—82no—
444Q8—82no—
445Q8—82no—
446D4—44no—
447D4—82no—
448D4—82no—
449D4—84no—
450D4—84no—
451D4—82no—
452D4—82no—
453D4—82no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—28no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—84no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—84no—
481C2×C4#39011yescharacteristic
482C2×C4#39011yescharacteristic
483C2×C4#39011yescharacteristic
484C2×C4#39011yescharacteristic
485C2×C4#39011yescharacteristic
486C2×C4#39011yescharacteristic
487C2×C4C2×#10414yes—
488C2×C4#47114yes—
489C2×C4C22×C22⋊C411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×#12411yescharacteristic
494C2×C4C2×#12711yescharacteristic
495C8—28no—
496C8—22no—
497C8—22no—
498C8—82no—
499C8—82no—
500C8C2×#10414yes—
501C8C2×#10414yes—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—44no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—164no—
514C22#196212yes—
515C22C2×#39011yescharacteristic
516C22C2×#47112yes—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—48no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4#196212yes—
529C4#197612yes—
530C4#197712yes—
531C4#197712yes—
532C4C22×#10412yes—
533C2—22no—
534C2—22no—
535C2—41no—
536C2—162no—
537C2C2×#196211yescharacteristic
538C2C2×#197611yescharacteristic
539C2C2×#197711yescharacteristic
subgroupG/H
centerC2×C4#390
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. C4×#405C2semidirect
3. #17793C2semidirect
4. #17648C2semidirect
6. #17545C2semidirect
8. #9761C2semidirect
15. #3167C4semidirect
17. C2×#498C4semidirect
18. C2×#492C4semidirect
19. C2×#487C4semidirect
20. C2×#486C4semidirect
29. #1726C22semidirect
32. #1568C22semidirect
36. #1531C22semidirect
44. #1506C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
C4×#4052
#177932
#176482
#176472
#175452
#175442
#97612