Finite Group Databasebeta

← #4474536

Subgroups — #4474536

Order 512.

2338 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17282C214yesmaximal, complemented
2#16903C211yesmaximal, characteristic, complemented
3#16753C214yesmaximal, complemented
4#16656C214yesmaximal, complemented
5#16254C211yesmaximal, characteristic, complemented
6#16133C214yesmaximal, complemented
7#15986C214yesmaximal
8#15910C214yesmaximal, complemented
9#14717C212yesmaximal
10#14449C212yesmaximal, complemented
11#14341C211yesmaximal, characteristic
12C22×#126C2214yes—
13C22×C22⋊Q8C2212yes—
14C23×C22⋊C4C2211yescharacteristic
15C2×#395C2214yescomplemented
16C2×#393C2212yes—
17C2×#393C2212yes—
18C2×#391C2214yescomplemented
19C2×#379C2212yes—
20C2×#379C2214yes—
21#1445C2214yescomplemented
22#1445C2214yescomplemented
23#1443C2214yescomplemented
24#1443C2214yescomplemented
25#1442C2214yescomplemented
26#1442C2214yescomplemented
27#1442C2214yescomplemented
28#1442C2214yescomplemented
29#1441C2214yescomplemented
30#1441C2214yescomplemented
31#1441C2214yescomplemented
32#1440C2214yes—
33#1440C2214yes—
34#1440C2214yes—
35#1440C2214yes—
36#1440C2214yes—
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2212yes—
45#1439C2214yes—
46#1439C2212yes—
47#1437C2214yes—
48#1437C2214yes—
49#1436C2214yescomplemented
50#1436C2214yescomplemented
51#1434C2214yescomplemented
52C22⋊#95C2212yes—
53C22⋊#95C2211yescharacteristic
54C22⋊#95C2211yescharacteristic
55C22⋊#95C2214yes—
56C22⋊#95C2212yes—
57C26C2311yescharacteristic
58C23×Q8C2312yes—
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C2×#126—48no—
62C2×#126—48no—
63C2×#126—48no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—48no—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C22⋊C4—24no—
72C22×C22⋊C4—44no—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C42C2314yes—
86#395—48no—
87#393—44no—
88#393—44no—
89#391—48no—
90#379—48no—
91#379—44no—
92C2×#95C2314yes—
93C2×#95C2312yes—
94C2×#95C2314yes—
95C2×#95C2312yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C25—22no—
116C25—24no—
117C25—24no—
118C25—41no—
119C25—41no—
120C25—44no—
121C25—44no—
122C25C2412yes—
123C25C2411yescharacteristic
124C22×Q8—24no—
125C22×Q8—28no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—44no—
129C23×C4—22no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4#4114yes—
135C23×C4C2414yes—
136C23×C4C2414yes—
137C23×C4C2414yes—
138C23×C4C2414yes—
139C23×C4C2414yes—
140C23×C4C2411yescharacteristic
141C23×C4C2411yescharacteristic
142C23×C4C2412yes—
143C23×C4C2414yes—
144#126—464no—
145C22⋊Q8—432no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—416no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
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—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C42—28no—
198C2×C42—44no—
199C2×C42—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—42no—
218#95—42no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223C24—24no—
224C24—22no—
225C24—21no—
226C24—21no—
227C24—24no—
228C24—28no—
229C24—22no—
230C24—24no—
231C24—24no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—24no—
238C24—24no—
239C24—48no—
240C24—48no—
241C24—44no—
242C24—416no—
243C24—44no—
244C24—84no—
245C24—84no—
246C24—88no—
247C24—88no—
248C24—88no—
249C24—88no—
250C24—88no—
251C24C2×#4112yes—
252C24H(5,𝔽2)14yes—
253C242−1+414yes—
254C24C2511yescharacteristic
255C2×Q8—216no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—88no—
259C2×Q8—88no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
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—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—44no—
312C22×C4—48no—
313C22×C4—44no—
314C22×C4—84no—
315C22×C4—88no—
316C22×C4—88no—
317C22×C4C2×#4114yes—
318C22×C4H(5,𝔽2)18yes—
319C4⋊C4—416no—
320C4⋊C4—416no—
321C4⋊C4—48no—
322C4⋊C4—416no—
323C4⋊C4—416no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—416no—
327C22⋊C4—416no—
328C22⋊C4—416no—
329C22⋊C4—416no—
330C22⋊C4—84no—
331C22⋊C4—84no—
332C22⋊C4—88no—
333C22⋊C4—88no—
334C22⋊C4—88no—
335C22⋊C4—88no—
336C42—416no—
337C23—24no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—28no—
344C23—28no—
345C23—24no—
346C23—28no—
347C23—24no—
348C23—24no—
349C23—28no—
350C23—416no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—44no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—816no—
374C23—88no—
375C23—88no—
376C23—832no—
377C23—832no—
378C23—816no—
379C23#54114yes—
380C23C22×#4111yescharacteristic
381C23C2×H(5,𝔽2)12yes—
382C23C2×H(5,𝔽2)14yes—
383C23C2×2−1+412yes—
384C23#58511yescharacteristic
385C23#58511yescharacteristic
386C23#58514yes—
387Q8—816no—
388C2×C4—24no—
389C2×C4—28no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
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—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—88no—
442C2×C4—88no—
443C22—28no—
444C22—24no—
445C22—48no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22—44no—
450C22—48no—
451C22—48no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—832no—
456C22—84no—
457C22—84no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22C2×#54111yescharacteristic
463C22#349012yes—
464C22#349011yescharacteristic
465C22#349311yescharacteristic
466C22#351012yes—
467C22#353214yes—
468C22#353512yes—
469C22#353512yes—
470C22#354114yes—
471C22#355114yes—
472C22#356014yes—
473C22#356114yes—
474C22#356214yes—
475C4—44no—
476C4—48no—
477C4—84no—
478C4—84no—
479C4—84no—
480C4—84no—
481C4—81no—
482C4—81no—
483C4—84no—
484C2—44no—
485C2—84no—
486C2#3673211yescharacteristic
487C2#4054112yes—
488C2#4645711yescharacteristic
489C2#4699611yescharacteristic
490C2#4959612yes—
491C2#5506314yes—
492C2#5613114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17282C2semidirect
2. #16903C2semidirect
3. #16753C2semidirect
4. #16656C2semidirect
5. #16254C2semidirect
6. #16133C2semidirect
8. #15910C2semidirect
10. #14449C2semidirect
15. C2×#395C22semidirect
18. C2×#391C22semidirect
21. #1445C22semidirect
22. #1445C22semidirect
23. #1443C22semidirect
24. #1443C22semidirect
25. #1442C22semidirect
26. #1442C22semidirect
27. #1442C22semidirect
28. #1442C22semidirect
29. #1441C22semidirect
30. #1441C22semidirect
31. #1441C22semidirect
49. #1436C22semidirect
50. #1436C22semidirect
51. #1434C22semidirect

Maximal subgroups

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

subgroupindex
#172822
#169032
#167532
#166562
#162542
#161332
#159862
#159102
#147172
#144492
#143412