Finite Group Databasebeta

← #4125864

Subgroups — #4125864

Order 512.

2667 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17229C211yesmaximal, characteristic, complemented
2#16247C211yesmaximal, characteristic, complemented
3#16012C214yesmaximal, complemented
4#15658C214yesmaximal, complemented
5#15432C214yesmaximal, complemented
6#15283C212yesmaximal
7#14253C212yesmaximal, complemented
8#14134C211yesmaximal, characteristic
9#14086C214yesmaximal, complemented
10#14012C214yesmaximal, complemented
11#13873C214yesmaximal, complemented
12C22×#120C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14C2×#399C2212yes—
15C2×#397C2212yes—
16C2×#394C2214yes—
17C2×#390C2212yes—
18C2×#388C2214yes—
19C2×#388C2214yes—
20C2×#388C2214yescomplemented
21C2×#387C2214yescomplemented
22C2×#386C2214yes—
23C2×#380C2214yes—
24C2×#379C2214yes—
25C2×C4×C22⋊C4C2214yes—
26#1445C2214yescomplemented
27#1444C2214yescomplemented
28#1444C2214yescomplemented
29#1443C2214yescomplemented
30#1443C2214yescomplemented
31#1442C2214yescomplemented
32#1440C2214yes—
33#1440C2214yes—
34#1440C2211yescharacteristic
35#1440C2211yescharacteristic
36#1440C2214yescomplemented
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yes—
40#1439C2214yescomplemented
41#1438C2212yes—
42#1438C2212yes—
43#1438C2214yes—
44#1437C2212yes—
45#1437C2214yes—
46#1437C2214yes—
47#1437C2212yes—
48#1437C2214yescomplemented
49#1436C2214yescomplemented
50#1436C2214yescomplemented
51#1434C2214yescomplemented
52#1434C2214yescomplemented
53C22⋊#95C2214yes—
54C22⋊#95C2214yes—
55C22⋊#95C2214yes—
56C4×#95C2214yescomplemented
57C26C2311yescharacteristic
58C23×D4C2312yes—
59C23×D4C2311yescharacteristic
60C24×C4C2314yes—
61C2×#120—24no—
62C2×#120—44no—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C22⋊C4—216no—
68C22×C22⋊C4—44no—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2311yescharacteristic
83C22×C42C2314yes—
84C22×C42C2314yes—
85C22×C42C2312yes—
86#399—44no—
87#397—28no—
88#394—48no—
89#390—44no—
90#388—216no—
91#388—48no—
92#388—48no—
93#387—48no—
94#386—216no—
95#380—48no—
96#379—48no—
97C4×C22⋊C4—48no—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2312yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2312yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C25—22no—
119C25—24no—
120C25—28no—
121C25—44no—
122C25—44no—
123C25C2411yescharacteristic
124C25C2411yescharacteristic
125C25C2412yes—
126C22×D4—28no—
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—24no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—44no—
133C22×D4—44no—
134C23×C4—28no—
135C23×C4—28no—
136C23×C4—28no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4C2414yes—
142C23×C4C2414yes—
143C23×C4C2414yes—
144C23×C4C2411yescharacteristic
145C23×C4C2414yes—
146C23×C4C2414yes—
147C23×C4C2412yes—
148C23×C4C2414yes—
149#120—416no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—432no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4#41116yes—
201C2×C22⋊C4#4114yes—
202C2×C42—28no—
203C2×C42—28no—
204C2×C42—44no—
205C2×C42—44no—
206C2×C42—44no—
207C2×C42—42no—
208C2×C42—44no—
209C2×C42—44no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—28no—
214#95—24no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—42no—
230C24—22no—
231C24—24no—
232C24—24no—
233C24—24no—
234C24—22no—
235C24—24no—
236C24—24no—
237C24—28no—
238C24—24no—
239C24—24no—
240C24—22no—
241C24—24no—
242C24—24no—
243C24—21no—
244C24—24no—
245C24—22no—
246C24—24no—
247C24—416no—
248C24—416no—
249C24—416no—
250C24—48no—
251C24—48no—
252C24—88no—
253C24—88no—
254C24—88no—
255C24—84no—
256C24—88no—
257C24C2×#4112yes—
258C24C2×#4118yes—
259C24H(5,𝔽2)14yes—
260C24C2511yescharacteristic
261C2×D4—48no—
262C2×D4—416no—
263C2×D4—44no—
264C2×D4—48no—
265C2×D4—88no—
266C2×D4—84no—
267C2×D4—84no—
268C2×D4—88no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—416no—
316C22×C4—416no—
317C22×C4—416no—
318C22×C4—88no—
319C22×C4—88no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4C2×#4114yes—
323C22×C4C2×#4118yes—
324C22×C4C2×#4118yes—
325C22×C4H(5,𝔽2)18yes—
326C4⋊C4—416no—
327C4⋊C4—84no—
328C4⋊C4—88no—
329C4⋊C4—88no—
330C22⋊C4—232no—
331C22⋊C4—44no—
332C22⋊C4—416no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—416no—
336C22⋊C4—416no—
337C22⋊C4—48no—
338C22⋊C4—88no—
339C22⋊C4—88no—
340C22⋊C4—88no—
341C22⋊C4—88no—
342C22⋊C4—88no—
343C22⋊C4—88no—
344C42—88no—
345C42—88no—
346C42—84no—
347C23—28no—
348C23—28no—
349C23—28no—
350C23—24no—
351C23—28no—
352C23—28no—
353C23—28no—
354C23—28no—
355C23—24no—
356C23—28no—
357C23—28no—
358C23—22no—
359C23—24no—
360C23—432no—
361C23—42no—
362C23—44no—
363C23—42no—
364C23—42no—
365C23—44no—
366C23—44no—
367C23—42no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—44no—
378C23—44no—
379C23—42no—
380C23—44no—
381C23—44no—
382C23—41no—
383C23—44no—
384C23—42no—
385C23—44no—
386C23—44no—
387C23—42no—
388C23—44no—
389C23—44no—
390C23—832no—
391C23—88no—
392C23—832no—
393C23—832no—
394C23#55118yes—
395C23C22×#4114yes—
396C23C22×#4111yescharacteristic
397C23C2×H(5,𝔽2)12yes—
398C23C2×H(5,𝔽2)14yes—
399C23#58514yes—
400D4—816no—
401D4—88no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—24no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—832no—
453C22—28no—
454C22—216no—
455C22—44no—
456C22—48no—
457C22—44no—
458C22—48no—
459C22—48no—
460C22—48no—
461C22—48no—
462C22—44no—
463C22—42no—
464C22—48no—
465C22—832no—
466C22—84no—
467C22—84no—
468C22—82no—
469C22—84no—
470C22—81no—
471C22—84no—
472C22—84no—
473C22C2×#55112yes—
474C22#347514yes—
475C22#348114yes—
476C22#349011yescharacteristic
477C22#349414yes—
478C22#350812yes—
479C22#351314yes—
480C22#351414yes—
481C22#353312yes—
482C22#353714yes—
483C22#356014yes—
484C4—44no—
485C4—48no—
486C4—48no—
487C4—48no—
488C4—84no—
489C4—81no—
490C4—84no—
491C4—84no—
492C2—48no—
493C2—82no—
494C2—161no—
495C2#4645411yescharacteristic
496C2#4650014yes—
497C2#5919312yes—
498C2#5946814yes—
499C2#5952014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17229C2semidirect
2. #16247C2semidirect
3. #16012C2semidirect
4. #15658C2semidirect
5. #15432C2semidirect
7. #14253C2semidirect
9. #14086C2semidirect
10. #14012C2semidirect
11. #13873C2semidirect
20. C2×#388C22semidirect
21. C2×#387C22semidirect
26. #1445C22semidirect
27. #1444C22semidirect
28. #1444C22semidirect
29. #1443C22semidirect
30. #1443C22semidirect
31. #1442C22semidirect
36. #1440C22semidirect
40. #1439C22semidirect
48. #1437C22semidirect
49. #1436C22semidirect
50. #1436C22semidirect
51. #1434C22semidirect
52. #1434C22semidirect
56. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
#172292
#162472
#160122
#156582
#154322
#152832
#142532
#141342
#140862
#140122
#138732