Finite Group Databasebeta

← #200861

Subgroups — #200861

Order 512.

1391 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3138C211yesmaximal, characteristic
2#34364C211yesmaximal, characteristic
3#17741C211yesmaximal, characteristic
4#17584C211yesmaximal, characteristic, complemented
5#17562C211yesmaximal, characteristic, complemented
6#9425C218yesmaximal, complemented
7#7974C211yesmaximal, characteristic, complemented
8#7973C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×#127C2211yescharacteristic
11#3138—22no—
12#3138—22no—
13#3138C414yescomplemented
14#3138C2214yes—
15C2×#490—22no—
16C2×#486C412yescomplemented
17C2×#486C412yescomplemented
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#418—22no—
21C2×C4×M16C2211yescharacteristic
22#2380C2214yes—
23#2366—22no—
24D4×C22⋊C4—22no—
25D4×C22⋊C4—22no—
26#2328—22no—
27#2328—22no—
28#2308—22no—
29#2283C2211yescharacteristic
30#2283C2211yescharacteristic
31#1555C2214yes—
32#1545—22no—
33#1545—22no—
34#1518—22no—
35#1515C2214yescomplemented
36#1515C2214yescomplemented
37#1511—22no—
38#1508—22no—
39#1503—22no—
40#1288C2214yescomplemented
41#1288C2214yescomplemented
42C23×D4—21no—
43C23×D4—21no—
44C23×D4D412yes—
45C22×M16—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yescomplemented
53#533C2312yes—
54C2×C4⋊Q8C2×C411yescharacteristic, complemented
55C2×C4⋊Q8C2×C411yescharacteristic, complemented
56C2×#127—21no—
57C2×#127—21no—
58C2×#127—21no—
59C2×#127—21no—
60C2×#127—21no—
61C2×#127—21no—
62C2×#127C2×C412yes—
63C2×#127C2×C412yes—
64C2×#127D412yes—
65C2×#127D412yes—
66C2×#127D412yes—
67C2×#127C2312yes—
68C2×#125—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×D4—24no—
74C2×C4×D4—24no—
75C2×#117—21no—
76C22×C22⋊C4—24no—
77C22×C42C2311yescharacteristic
78#490—28no—
79#490—28no—
80#486—24no—
81#486—24no—
82#486C2×C418yescomplemented
83#486C2×C418yescomplemented
84#432—21no—
85#432—21no—
86#432C2312yes—
87#432C2312yes—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#423—21no—
93#423—21no—
94#418—28no—
95#418—28no—
96C2×#102—24no—
97C2×#102—24no—
98#407—21no—
99#407—21no—
100C4×M16—21no—
101C4×M16—21no—
102C4×M16—21no—
103C4×M16—21no—
104C4×M16C2×C412yescomplemented
105C4×M16C2312yes—
106C2×#97—21no—
107C2×C4×C8C2×C412yescomplemented
108#390—24no—
109#390—24no—
110#386—28no—
111#386—28no—
112#386—28no—
113#386—28no—
114#383C2312yes—
115#383C2312yes—
116#382—21no—
117#382D412yes—
118#381—21no—
119#381D412yes—
120#380—21no—
121#380—21no—
122#379—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—21no—
126C4×C22⋊C4D412yes—
127#376—21no—
128#376—21no—
129#331—28no—
130#328—28no—
131#328—28no—
132C25—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—48no—
151C22×D4—84no—
152C22×D4—84no—
153C22×D4C22⋊C414yes—
154C22×D4C22⋊C414yes—
155C22×D4C2×D414yes—
156C23×C4—28no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—44no—
163C2×M16—41no—
164C2×M16—41no—
165C22×C8—22no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C412yes—
174#127—22no—
175#127—22no—
176#127—22no—
177#127—22no—
178#127—22no—
179#127—22no—
180#127—22no—
181#127—22no—
182#127—22no—
183#127—22no—
184#127—22no—
185#127—22no—
186#127C22⋊C414yes—
187#127C22⋊C414yes—
188#127C22×C414yes—
189#127C2×D414yes—
190#125—22no—
191#125—22no—
192#125—41no—
193#125—41no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×D4—88no—
211C4×D4—88no—
212#117—22no—
213#117—22no—
214#117—41no—
215#117—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—88no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—41no—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42C2411yescharacteristic
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250#102—216no—
251#102—216no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—41no—
255C22⋊C8—41no—
256#97—22no—
257#97—22no—
258#97—22no—
259#97—22no—
260C4×C8—22no—
261C4×C8—41no—
262C4×C8—41no—
263C4×C8C22×C414yes—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95C2×D414yes—
271#95C2×D414yes—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—48no—
276C24—84no—
277C24—84no—
278C24#12011yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—84no—
302C2×D4—84no—
303C2×D4—84no—
304C2×D4—84no—
305C2×D4—84no—
306C2×D4—84no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—82no—
313C2×D4—84no—
314C2×D4—82no—
315C2×D4C2×C22⋊C418yes—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12311yescharacteristic
340C22×C4#12311yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344M16—44no—
345M16—44no—
346M16—82no—
347M16—82no—
348C2×C8—24no—
349C2×C8—44no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—88no—
371C22⋊C4—88no—
372C22⋊C4—88no—
373C42—44no—
374C42—42no—
375C42—42no—
376C42—44no—
377C42C2×C22⋊C411yescharacteristic
378C42C2×C22⋊C411yescharacteristic
379C42C4×D411yescharacteristic
380C42C4×D411yescharacteristic
381C42C4×D411yescharacteristic
382C42C4×D411yescharacteristic
383C42#12111yescharacteristic
384C42#12111yescharacteristic
385C42#12111yescharacteristic
386C42#12111yescharacteristic
387C42#12311yescharacteristic
388C42#12311yescharacteristic
389C42C23×C411yescharacteristic
390C42C22×D411yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C23—22no—
394C23—22no—
395C23—28no—
396C23—41no—
397C23—44no—
398C23—48no—
399C23—44no—
400C23—48no—
401C23—48no—
402C23—48no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23—84no—
407C23—1616no—
408C23#38611yescharacteristic
409C23#38611yescharacteristic
410C23C2×#12011yescharacteristic
411Q8—82no—
412Q8—82no—
413Q8—82no—
414Q8—82no—
415D4—48no—
416D4—48no—
417D4—48no—
418D4—48no—
419D4—48no—
420D4—84no—
421D4—84no—
422D4—84no—
423D4—84no—
424D4—84no—
425D4—84no—
426D4—84no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—1616no—
455C2×C4#38611yescharacteristic
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4#45514yes—
462C2×C4C22×C22⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12311yescharacteristic
468C8—44no—
469C8—82no—
470C8—82no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—44no—
479C22—48no—
480C22—48no—
481C22—48no—
482C22—88no—
483C22—88no—
484C22—168no—
485C22—168no—
486C22#190412yes—
487C22C2×#38611yescharacteristic
488C22C2×#45512yes—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—84no—
499C4—84no—
500C4—82no—
501C4—82no—
502C4#188512yes—
503C4#188512yes—
504C4#190412yes—
505C2—22no—
506C2—22no—
507C2—41no—
508C2—88no—
509C2C2×#188511yescharacteristic
510C2C2×#190411yescharacteristic
511C2C2×#190411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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
4. #17584C2semidirect
5. #17562C2semidirect
6. #9425C2semidirect
7. #7974C2semidirect
8. #7973C2semidirect
13. #3138C4semidirect
16. C2×#486C4semidirect
17. C2×#486C4semidirect
35. #1515C22semidirect
36. #1515C22semidirect
40. #1288C22semidirect
41. #1288C22semidirect
52. #533C2×C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
82. #486C2×C4semidirect
83. #486C2×C4semidirect
104. C4×M16C2×C4semidirect
107. C2×C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31382
#343642
#177412
#175842
#175622
#94252
#79742
#79732