Finite Group Databasebeta

← #5482437

Subgroups — #5482437

Order 512.

2619 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17229C211yesmaximal, characteristic, complemented
2#17096C214yesmaximal, complemented
3#16521C214yesmaximal, complemented
4#16333C214yesmaximal, complemented
5#15616C211yesmaximal, characteristic, complemented
6#15462C214yesmaximal, complemented
7#15309C214yesmaximal, complemented
8#15283C212yesmaximal
9#14726C214yesmaximal, complemented
10#14652C212yesmaximal, complemented
11#14134C211yesmaximal, characteristic
12C22×#120C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14C2×#399C2212yes—
15C2×#398C2214yescomplemented
16C2×#397C2212yes—
17C2×#397C2214yescomplemented
18C2×#396C2214yes—
19C2×#396C2214yes—
20C2×#396C2214yescomplemented
21C2×#394C2214yes—
22C2×#390C2212yes—
23C2×#389C2214yescomplemented
24C2×#386C2214yes—
25C2×#385C2214yes—
26C2×#385C2214yes—
27C2×#380C2214yes—
28C2×#379C2214yes—
29#1444C2214yescomplemented
30#1444C2214yescomplemented
31#1443C2214yescomplemented
32#1442C2214yescomplemented
33#1442C2214yescomplemented
34#1442C2214yescomplemented
35#1441C2214yescomplemented
36#1441C2214yescomplemented
37#1441C2214yescomplemented
38#1440C2211yescharacteristic
39#1440C2211yescharacteristic
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1438C2212yes—
50#1438C2214yes—
51#1438C2212yes—
52#1437C2212yes—
53#1437C2212yes—
54#1436C2214yescomplemented
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C26C2311yescharacteristic
58C23×D4C2312yes—
59C23×D4C2311yescharacteristic
60C24×C4C2314yes—
61C2×#120—24no—
62C2×#120—44no—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C22⋊C4—48no—
72C22×C22⋊C4—44no—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2311yescharacteristic
87C22×C42C2312yes—
88#399—44no—
89#398—48no—
90#397—28no—
91#397—48no—
92#396—48no—
93#396—48no—
94#396—48no—
95#394—48no—
96#390—44no—
97#389—48no—
98#386—48no—
99#385—48no—
100#385—48no—
101#380—48no—
102#379—48no—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2312yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2312yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C25—22no—
122C25—24no—
123C25—28no—
124C25—44no—
125C25—44no—
126C25C2411yescharacteristic
127C25C2411yescharacteristic
128C25C2412yes—
129C22×D4—24no—
130C22×D4—28no—
131C22×D4—22no—
132C22×D4—24no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—44no—
136C22×D4—44no—
137C23×C4—28no—
138C23×C4—28no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4C2414yes—
145C23×C4C2414yes—
146C23×C4C2414yes—
147C23×C4C2414yes—
148C23×C4C2411yescharacteristic
149C23×C4C2412yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152#120—416no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—432no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4#4114yes—
217C2×C42—44no—
218C2×C42—42no—
219#95—24no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—42no—
237C24—22no—
238C24—24no—
239C24—24no—
240C24—24no—
241C24—24no—
242C24—22no—
243C24—24no—
244C24—28no—
245C24—24no—
246C24—24no—
247C24—22no—
248C24—24no—
249C24—21no—
250C24—24no—
251C24—24no—
252C24—22no—
253C24—24no—
254C24—416no—
255C24—416no—
256C24—416no—
257C24—48no—
258C24—48no—
259C24—88no—
260C24—88no—
261C24—88no—
262C24—84no—
263C24—88no—
264C24C2×#4112yes—
265C24H(5,𝔽2)14yes—
266C242−1+418yes—
267C24C2511yescharacteristic
268C2×D4—48no—
269C2×D4—416no—
270C2×D4—44no—
271C2×D4—48no—
272C2×D4—84no—
273C2×D4—88no—
274C2×D4—84no—
275C2×D4—88no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
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—21no—
295C22×C4—24no—
296C22×C4—24no—
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—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—416no—
324C22×C4—416no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4—88no—
330C22×C4C2×#4114yes—
331C22×C4H(5,𝔽2)18yes—
332C22×C4H(5,𝔽2)18yes—
333C4⋊C4—416no—
334C4⋊C4—84no—
335C4⋊C4—88no—
336C4⋊C4—88no—
337C4⋊C4—88no—
338C4⋊C4—88no—
339C4⋊C4—88no—
340C4⋊C4—88no—
341C22⋊C4—44no—
342C22⋊C4—48no—
343C22⋊C4—416no—
344C22⋊C4—416no—
345C22⋊C4—416no—
346C22⋊C4—48no—
347C22⋊C4—48no—
348C22⋊C4—88no—
349C22⋊C4—88no—
350C22⋊C4—88no—
351C22⋊C4—88no—
352C22⋊C4—88no—
353C22⋊C4—88no—
354C22⋊C4—88no—
355C42—84no—
356C23—24no—
357C23—28no—
358C23—24no—
359C23—28no—
360C23—28no—
361C23—28no—
362C23—28no—
363C23—28no—
364C23—28no—
365C23—28no—
366C23—22no—
367C23—24no—
368C23—28no—
369C23—432no—
370C23—42no—
371C23—42no—
372C23—44no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—42no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—41no—
389C23—42no—
390C23—44no—
391C23—42no—
392C23—42no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—832no—
400C23—88no—
401C23—832no—
402C23—832no—
403C23#56318yes—
404C23C22×#4111yescharacteristic
405C23C2×H(5,𝔽2)12yes—
406C23C2×H(5,𝔽2)14yes—
407C23C2×H(5,𝔽2)14yes—
408C23C2×2−1+414yes—
409D4—816no—
410D4—88no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
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—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—832no—
462C22—28no—
463C22—216no—
464C22—48no—
465C22—48no—
466C22—48no—
467C22—48no—
468C22—44no—
469C22—44no—
470C22—48no—
471C22—48no—
472C22—42no—
473C22—44no—
474C22—832no—
475C22—84no—
476C22—84no—
477C22—84no—
478C22—84no—
479C22—81no—
480C22—82no—
481C22—84no—
482C22#349011yescharacteristic
483C22#350812yes—
484C22#352014yes—
485C22C2×#56312yes—
486C22#353212yes—
487C22#353614yes—
488C22#353614yes—
489C22#353614yes—
490C22#353714yes—
491C22#353714yes—
492C22#353814yes—
493C4—48no—
494C4—48no—
495C4—44no—
496C4—84no—
497C4—84no—
498C4—84no—
499C4—84no—
500C4—81no—
501C2—48no—
502C2—82no—
503C2—161no—
504C2#4572812yes—
505C2#4645411yescharacteristic
506C2#4822714yes—
507C2#5247314yes—
508C2#5361314yes—
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. #17096C2semidirect
3. #16521C2semidirect
4. #16333C2semidirect
5. #15616C2semidirect
6. #15462C2semidirect
7. #15309C2semidirect
9. #14726C2semidirect
10. #14652C2semidirect
15. C2×#398C22semidirect
17. C2×#397C22semidirect
20. C2×#396C22semidirect
23. C2×#389C22semidirect
29. #1444C22semidirect
30. #1444C22semidirect
31. #1443C22semidirect
32. #1442C22semidirect
33. #1442C22semidirect
34. #1442C22semidirect
35. #1441C22semidirect
36. #1441C22semidirect
37. #1441C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
54. #1436C22semidirect

Maximal subgroups

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

subgroupindex
#172292
#170962
#165212
#163332
#156162
#154622
#153092
#152832
#147262
#146522
#141342