Finite Group Databasebeta

← #6506464

Subgroups — #6506464

Order 512.

2380 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic, complemented
2C2×#2432C211yesmaximal, characteristic
3#29110C212yesmaximal, complemented
4#29106C212yesmaximal, complemented
5#27284C218yesmaximal, complemented
6#27272C218yesmaximal, complemented
7#25861C212yesmaximal, complemented
8#25859C212yesmaximal, complemented
9#21765C212yesmaximal, complemented
10#20601C212yesmaximal
11#20524C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C2×#491C2214yescomplemented
17C2×#488C2214yescomplemented
18C2×#417C2214yes—
19C2×#417C2214yes—
20C2×#407C2212yes—
21#2851C2212yes—
22#2851C2212yes—
23#2646—22no—
24#2569—24no—
25#2442C2214yescomplemented
26#2442C2214yescomplemented
27#2432—24no—
28#2432C2218yes—
29C2×#394—22no—
30#2315—22no—
31C2×#391C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#385—22no—
34C2×#380C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2074C2214yescomplemented
37#2073C2214yescomplemented
38#2043C2218yescomplemented
39#2042C2218yescomplemented
40#2026C2214yescomplemented
41#2025C2214yescomplemented
42#1960C2214yescomplemented
43#1958C2214yescomplemented
44#1936C2218yescomplemented
45#1933C2218yescomplemented
46#1894C2218yescomplemented
47#1872C2214yescomplemented
48#1872C2214yescomplemented
49#1823C2214yes—
50#1788C2218yes—
51#1787C2214yes—
52#1787C2214yes—
53#1786C2214yes—
54#1753C2218yescomplemented
55#1749C2212yes—
56C4×C22⋊C8C2212yes—
57C22×#41—22no—
58C22×#41—28no—
59C22×#41—44no—
60C22×#41—44no—
61C22×#41C2314yes—
62C23×Q8C2311yescharacteristic
63C23×Q8C2311yescharacteristic
64C23×D4—21no—
65C23×D4C2311yescharacteristic
66C24×C4—21no—
67C24×C4C2311yescharacteristic
68C22×M16C2314yes—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—44no—
71C2×C4⋊Q8C2314yes—
72C2×#125C2312yes—
73C2×#125C2312yes—
74C2×#124—22no—
75C2×#124—44no—
76C2×#124C2314yes—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8C2314yes—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4D412yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#491—48no—
93#488—48no—
94#417—48no—
95#417—48no—
96C2×#103C2318yes—
97C2×#103C2318yes—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×#102C2318yes—
101C2×#102C2318yes—
102#407—28no—
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C2×#97C2314yes—
106C2×C4×C8C2314yes—
107#400—24no—
108#400—24no—
109#398—24no—
110#398—24no—
111#394—28no—
112#394—28no—
113#391—22no—
114#391C2314yes—
115#390—22no—
116#390C2314yes—
117#389C2314yes—
118#389C2314yes—
119#387—24no—
120#387—24no—
121#385—24no—
122#385—24no—
123#385—24no—
124#385—24no—
125#385—28no—
126#385—28no—
127#382C2312yes—
128#382C2312yes—
129#380—22no—
130#380—22no—
131#380—24no—
132#380—24no—
133#380C2314yes—
134#380C2314yes—
135C4×C22⋊C4D414yes—
136C4×C22⋊C4C2314yes—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95D412yes—
140#340C2318yes—
141#340C2318yes—
142#336C2314yes—
143#336C2314yes—
144C25—21no—
145C25C2×D411yescharacteristic
146C2×#41—28no—
147C2×#41—28no—
148C2×#41—28no—
149C2×#41—28no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41—48no—
153C2×#41—48no—
154C2×#41—464no—
155C2×#41—48no—
156C22×Q8—24no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8C2412yes—
165C22×Q8C2412yes—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—21no—
169C22×D4—24no—
170C22×D4—42no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4C2412yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4#4118yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C2×M16—28no—
194C2×M16—44no—
195C2×M16—44no—
196C22×C8C2418yes—
197C4⋊Q8—28no—
198C4⋊Q8—44no—
199C4⋊Q8—48no—
200#125—44no—
201#125—44no—
202#124—28no—
203#124—44no—
204#124—48no—
205C22⋊Q8—28no—
206C22⋊Q8—28no—
207C22⋊Q8—48no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4C2412yes—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2414yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—48no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4C2×D414yes—
236C2×C22⋊C4C2×D414yes—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2412yes—
242C2×C42C2412yes—
243#103—216no—
244#103—48no—
245#103—48no—
246#103—48no—
247#102—216no—
248#102—48no—
249C22⋊C8—28no—
250C22⋊C8—28no—
251#97—44no—
252C4×C8—44no—
253#95—24no—
254#95—24no—
255#95—28no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95C2×D414yes—
261#95C2×D414yes—
262C24—24no—
263C24—24no—
264C24—22no—
265C24—41no—
266C24—41no—
267C24—42no—
268C24—42no—
269C24—44no—
270C24—42no—
271C24—44no—
272C24#12414yes—
273C24C22×D412yes—
274C24C22×D411yescharacteristic
275#41—232no—
276#41—416no—
277#41—416no—
278#41—416no—
279#41—464no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—416no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—88no—
294C2×Q8H(5,𝔽2)14yes—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—22no—
299C2×D4—416no—
300C2×D4—48no—
301C2×D4—42no—
302C2×D4—48no—
303C2×D4—432no—
304C2×D4—48no—
305C2×D4—42no—
306C2×D4H(5,𝔽2)14yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—416no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—432no—
332C22×C4—42no—
333C22×C4—48no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4#12414yes—
346C22×C4#12418yes—
347C22×C4C22×D414yes—
348C22×C4C22×D414yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C2×#4114yes—
352C22×C4C2×#4114yes—
353C22×C4C2×#4114yes—
354C22×C4C2511yescharacteristic
355M16—88no—
356C2×C8—28no—
357C2×C8—28no—
358C2×C8—28no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—416no—
363C4⋊C4—42no—
364C4⋊C4—48no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—416no—
374C22⋊C4—416no—
375C22⋊C4—48no—
376C22⋊C4—48no—
377C22⋊C4—48no—
378C42—24no—
379C42—24no—
380C42—42no—
381C42—42no—
382C23—22no—
383C23—21no—
384C23—21no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—28no—
392C23—24no—
393C23—44no—
394C23—44no—
395C23—48no—
396C23—416no—
397C23—44no—
398C23—42no—
399C23—42no—
400C23—44no—
401C23—84no—
402C23—84no—
403C23—88no—
404C23—84no—
405C23—84no—
406C23C2×#12414yes—
407C23C2×#12412yes—
408C23#57514yes—
409C23C23×D411yescharacteristic
410Q8—28no—
411Q8—48no—
412Q8—44no—
413Q8—48no—
414Q8—84no—
415D4—28no—
416D4—48no—
417D4—432no—
418D4—44no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—432no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—164no—
455C2×C4—164no—
456C2×C4C2×#12414yes—
457C2×C4C2×#12412yes—
458C2×C4C22×#4112yes—
459C2×C4C2×H(5,𝔽2)12yes—
460C2×C4C2×H(5,𝔽2)12yes—
461C8—48no—
462C22—22no—
463C22—24no—
464C22—24no—
465C22—22no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—48no—
475C22—88no—
476C22—816no—
477C22—84no—
478C22—82no—
479C22—82no—
480C22#314214yes—
481C22C22×#12411yescharacteristic
482C22#348411yescharacteristic
483C22#348411yescharacteristic
484C22C2×#57512yes—
485C22#358612yes—
486C4—24no—
487C4—22no—
488C4—44no—
489C4—44no—
490C4—44no—
491C4—82no—
492C4—84no—
493C4—84no—
494C4—82no—
495C4#349014yes—
496C2—24no—
497C2—44no—
498C2—84no—
499C2#3536711yescharacteristic
500C2C2×#314211yescharacteristic
501C2#6062011yescharacteristic
502C2#6157512yes—
503C2#6158312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #35020C2semidirect
3. #29110C2semidirect
4. #29106C2semidirect
5. #27284C2semidirect
6. #27272C2semidirect
7. #25861C2semidirect
8. #25859C2semidirect
9. #21765C2semidirect
11. #20524C2semidirect
16. C2×#491C22semidirect
17. C2×#488C22semidirect
25. #2442C22semidirect
26. #2442C22semidirect
36. #2074C22semidirect
37. #2073C22semidirect
38. #2043C22semidirect
39. #2042C22semidirect
40. #2026C22semidirect
41. #2025C22semidirect
42. #1960C22semidirect
43. #1958C22semidirect
44. #1936C22semidirect
45. #1933C22semidirect
46. #1894C22semidirect
47. #1872C22semidirect
48. #1872C22semidirect
54. #1753C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#24322
#291102
#291062
#272842
#272722
#258612
#258592
#217652
#206012
#205242