Finite Group Databasebeta

← #450×D251 #44559544

Subgroups — #450×D251 #44559544

Order 32128.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D251C211yesmaximal, characteristic
2C2×SD16×D251C212yesmaximal, complemented
3C22⋊Q8×D251C211yesmaximal, characteristic, complemented
4#30631840C211yesmaximal, characteristic, complemented
5#30631839C212yesmaximal, complemented
6#30631838C211yesmaximal, characteristic, complemented
7#30631837C211yesmaximal, characteristic, complemented
8#30631836C212yesmaximal, complemented
9#102×D251C212yesmaximal, complemented
10C22⋊C8×D251C211yesmaximal, characteristic, complemented
11#450×C251C211yesmaximal, characteristic, complemented
12C24×D251—21no—
13C2×Q8×D251C2211yescharacteristic, complemented
14C22×D1004C2211yescharacteristic
15C22×C22⋊D251—21no—
16C2×D4×D251—28no—
17C2×D4×D251—21no—
18C2×D4×D251—21no—
19C2×D4×D251C2212yes—
20C2×D4×D251C2212yes—
21C22×C4×D251C2211yescharacteristic
22C2×#16860799C2212yescomplemented
23C2×#16860798C2212yescomplemented
24C2×#16860797C2212yescomplemented
25SD16×D251—28no—
26C2×C8×D251C2212yescomplemented
27#22548695C2211yescharacteristic, complemented
28#22548694C2211yescharacteristic, complemented
29#22548693—21no—
30#22548692—21no—
31#22548691C2211yescharacteristic, complemented
32C4⋊C4×D251—21no—
33C4⋊C4×D251C2211yescharacteristic, complemented
34C22⋊C4×D251—21no—
35#22548637C2212yescomplemented
36#22548636C2212yescomplemented
37#22548635C2212yescomplemented
38C22⋊(C251⋊C8)C2211yescharacteristic, complemented
39(C2×D251)⋊C8C2212yescomplemented
40C22×D4×C251C2211yescharacteristic
41C2×SD16×C251C2212yescomplemented
42C22⋊Q8×C251C2211yescharacteristic, complemented
43#102×C251C2212yescomplemented
44C22⋊C8×C251C2211yescharacteristic, complemented
45C23×D251—28no—
46C23×D251—22no—
47C23×D251—22no—
48C23×D251—21no—
49C23×D251—44no—
50C23×D251—44no—
51C23×D251D411yescharacteristic
52C2×Dic502—21no—
53C2×Dic502C2311yescharacteristic, complemented
54Q8×D251—24no—
55C2×D1004—24no—
56C2×D1004—21no—
57C2×D1004C2312yes—
58C2×C22⋊D251—28no—
59C2×C22⋊D251—22no—
60C2×C22⋊D251—22no—
61D4×D251—28no—
62D4×D251—28no—
63D4×D251—24no—
64D4×D251—24no—
65D4×D251—24no—
66D4×D251D418yes—
67C22×Dic251D411yescharacteristic
68C2×C4×D251—24no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251D411yescharacteristic
72C2×C4×D251D411yescharacteristic
73C2×C4×D251C2311yescharacteristic
74C2×C4×D251C2311yescharacteristic
75#16860799—24no—
76#16860798—24no—
77#16860797—24no—
78C2×C251⋊C8C2312yescomplemented
79C8×D251—24no—
80C4⋊Dic251—21no—
81C4⋊Dic251C2311yescharacteristic, complemented
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85C22⋊Dic251—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88C24×C251—21no—
89C2×Q8×C251C2311yescharacteristic, complemented
90C2×D4×C251—24no—
91C2×D4×C251—21no—
92C2×D4×C251C2312yes—
93C22×C4×C251C2311yescharacteristic
94SD16×C251—24no—
95C2×C8×C251C2312yescomplemented
96C4⋊C4×C251—21no—
97C4⋊C4×C251C2311yescharacteristic, complemented
98C22⋊C4×C251—21no—
99C22×D251—22no—
100C22×D251—22no—
101C22×D251—28no—
102C22×D251—24no—
103C22×D251—22no—
104C22×D251—28no—
105C22×D251—44no—
106C22×D251—432no—
107C22×D251—42no—
108C22×D251—42no—
109C22×D251—44no—
110C22×D251—44no—
111C22×D251—44no—
112C22×D251SD1614yescomplemented
113C22×D251C2×D411yescharacteristic
114C22×D251C2×D411yescharacteristic
115Dic502—22no—
116Dic502—42no—
117D1004—24no—
118D1004—22no—
119D1004C2×D414yes—
120C22⋊D251—28no—
121C22⋊D251—28no—
122C2×Dic251—22no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C4×D251—22no—
128C4×D251—22no—
129C4×D251—22no—
130C4×D251—42no—
131C4×D251—42no—
132C4×D251C2×D414yes—
133C251⋊C8—22no—
134C23×C251—24no—
135C23×C251—22no—
136C23×C251—42no—
137C23×C251—42no—
138C23×C251C2×D411yescharacteristic
139Q8×C251—22no—
140D4×C251—24no—
141D4×C251—22no—
142D4×C251C2×D414yes—
143C2×C4×C251—22no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C8×C251—22no—
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—24no—
152C22×C251—48no—
153C22×C251—42no—
154C22×C251—42no—
155C22×C251C2×SD1612yescomplemented
156C22×C251C22×D411yescharacteristic
157C2×D251—21no—
158C2×D251—21no—
159C2×D251—28no—
160C2×D251—24no—
161C2×D251—416no—
162C2×D251—48no—
163C2×D251—42no—
164C2×D251—42no—
165C2×D251—416no—
166C2×D251#12012yes—
167C2×D251C2×SD1612yes—
168C2×D251C2×SD1612yescomplemented
169C2×D251AGL(1,ℤ/8)12yes—
170C4×C251—21no—
171C4×C251—41no—
172C4×C251—41no—
173C4×C251C22×D412yes—
174Dic251—21no—
175Dic251—41no—
176Dic251—41no—
177Dic251#12012yes—
178C2×C251—22no—
179C2×C251—44no—
180C2×C251C2×#12011yescharacteristic
181C2×C251C22×SD1611yescharacteristic
182C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
183D251—22no—
184D251—44no—
185D251#45014yesdirect product factor
186C251C2×#45011yescharacteristic, complemented, Sylow
187C2×#450—2511nomaximal, Sylow
188C23×D4—2511no—
189C22×SD16—2512no—
190C2×C22⋊Q8—2511no—
191#450—2511no—
192#450—2512no—
193#450—2511no—
194#450—2512no—
195#450—2511no—
196#450D25111yescharacteristic, direct product factor
197C2×#102—2512no—
198C2×C22⋊C8—2511no—
199C25—5021no—
200C22×Q8—2511no—
201C22×D4—5028no—
202C22×D4—5021no—
203C22×D4—5021no—
204C22×D4—5021no—
205C22×D4—2512no—
206C22×D4—2512no—
207C22×D4—2511no—
208C22×D4C2×D25111yescharacteristic
209C23×C4—2511no—
210C2×SD16—5028no—
211C2×SD16—2512no—
212C2×SD16—2512no—
213C2×SD16—2512no—
214C2×SD16C2×D25112yescomplemented
215C22×C8—2512no—
216C22⋊Q8—5021no—
217C22⋊Q8—5021no—
218C22⋊Q8—2511no—
219C22⋊Q8—2511no—
220C22⋊Q8—2511no—
221C22⋊Q8C2×D25111yescharacteristic, complemented
222C2×C4⋊C4—5021no—
223C2×C4⋊C4—2511no—
224C2×C22⋊C4—5021no—
225#102—2512no—
226#102—2512no—
227#102—2512no—
228#102C2×D25112yescomplemented
229C22⋊C8—2511no—
230C22⋊C8—2512no—
231C22⋊C8C2×D25111yescharacteristic, complemented
232C24—21no—
233C24—10044no—
234C24—10044no—
235C24—5028no—
236C24—5022no—
237C24—5022no—
238C24—5021no—
239C24—2511no—
240C2×Q8—5021no—
241C2×Q8—5024no—
242C2×Q8—2511no—
243C2×Q8C22×D25111yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—5024no—
247C2×D4—5028no—
248C2×D4—5028no—
249C2×D4—5024no—
250C2×D4—5028no—
251C2×D4—5024no—
252C2×D4—5024no—
253C2×D4—5022no—
254C2×D4—5022no—
255C2×D4—5021no—
256C2×D4—2518no—
257C2×D4—2512no—
258C2×D4C22×D25112yes—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—5024no—
262C22×C4—2511no—
263C22×C4—2511no—
264C22×C4—2511no—
265C22×C4—2511no—
266C22×C4—2511no—
267C22×C4C22×D25111yescharacteristic
268SD16—24no—
269SD16—5024no—
270SD16—5024no—
271SD16—5024no—
272C2×C8—5024no—
273C2×C8—2512no—
274C2×C8C22×D25112yescomplemented
275C4⋊C4—21no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—5021no—
279C4⋊C4—5021no—
280C4⋊C4—2511no—
281C4⋊C4C22×D25111yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—5021no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—10042no—
291C23—100432no—
292C23—10044no—
293C23—10044no—
294C23—10044no—
295C23—10044no—
296C23—10042no—
297C23—5024no—
298C23—5028no—
299C23—5028no—
300C23—5022no—
301C23—5022no—
302C23—5022no—
303C23—2511no—
304C23—2514no—
305C23—2511no—
306C23D4×D25111yescharacteristic
307Q8—22no—
308Q8—10042no—
309Q8—5022no—
310D4—24no—
311D4—22no—
312D4—5024no—
313D4—5028no—
314D4—5028no—
315D4—5022no—
316D4—2514no—
317D4D4×D25114yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—10042no—
322C2×C4—10042no—
323C2×C4—5022no—
324C2×C4—5022no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5022no—
328C2×C4—5022no—
329C2×C4—2511no—
330C2×C4—2511no—
331C2×C4—2514no—
332C2×C4D4×D25111yescharacteristic
333C2×C4C23×D25111yescharacteristic
334C8—22no—
335C8—5022no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—10048no—
343C22—100416no—
344C22—100416no—
345C22—10042no—
346C22—10042no—
347C22—5028no—
348C22—5024no—
349C22—5021no—
350C22—5021no—
351C22—2512no—
352C22—2512no—
353C22—2512no—
354C22—2512no—
355C22SD16×D25112yescomplemented
356C22C2×D4×D25111yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—10041no—
361C4—10041no—
362C4—5021no—
363C4—2512no—
364C4C2×D4×D25112yes—
365C2—22no—
366C2—44no—
367C2—10044no—
368C2—5022no—
369C2—2514no—
370C2#120×D25111yescharacteristic
371C2C2×SD16×D25111yescharacteristic
372C2AGL(1,ℤ/8)×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#450×C251C2
socleC22×C251C22×D4
prime radicalC22×D4×D251C2
hypercenter#450D251
hypocenterC251C2×#450
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D251C2semidirect
3. C22⋊Q8×D251C2semidirect
4. #30631840C2semidirect
5. #30631839C2semidirect
6. #30631838C2semidirect
7. #30631837C2semidirect
8. #30631836C2semidirect
9. #102×D251C2semidirect
10. C22⋊C8×D251C2semidirect
11. #450×C251C2semidirect
13. C2×Q8×D251C22semidirect
22. C2×#16860799C22semidirect
23. C2×#16860798C22semidirect
24. C2×#16860797C22semidirect
26. C2×C8×D251C22semidirect
27. #22548695C22semidirect
28. #22548694C22semidirect
31. #22548691C22semidirect
33. C4⋊C4×D251C22semidirect
35. #22548637C22semidirect
36. #22548636C22semidirect
37. #22548635C22semidirect
38. C22⋊(C251⋊C8)C22semidirect
39. (C2×D251)⋊C8C22semidirect
41. C2×SD16×C251C22semidirect
42. C22⋊Q8×C251C22semidirect
43. #102×C251C22semidirect
44. C22⋊C8×C251C22semidirect
53. C2×Dic502C23semidirect
78. C2×C251⋊C8C23semidirect
81. C4⋊Dic251C23semidirect
89. C2×Q8×C251C23semidirect
95. C2×C8×C251C23semidirect
97. C4⋊C4×C251C23semidirect
112. C22×D251SD16semidirect
155. C22×C251C2×SD16semidirect
168. C2×D251C2×SD16semidirect
185. D251#450direct (N × H)
186. C251C2×#450semidirect
196. #450D251direct (N × H)
214. C2×SD16C2×D251semidirect
221. C22⋊Q8C2×D251semidirect
228. #102C2×D251semidirect
231. C22⋊C8C2×D251semidirect
243. C2×Q8C22×D251semidirect
274. C2×C8C22×D251semidirect
281. C4⋊C4C22×D251semidirect
355. C22SD16×D251semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2512
C2×SD16×D2512
C22⋊Q8×D2512
#306318402
#306318392
#306318382
#306318372
#306318362
#102×D2512
C22⋊C8×D2512
#450×C2512
C2×#450251