Finite Group Databasebeta

← #44559594

Subgroups — #44559594

Order 32128.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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, complemented
2C2×#22548743C211yesmaximal, characteristic, complemented
3C2×#22548741C211yesmaximal, characteristic, complemented
4#30632199C211yesmaximal, characteristic, complemented
5#30631839C214yesmaximal, complemented
6#30631822C214yesmaximal, complemented
7#30631691C211yesmaximal, characteristic, complemented
8#30631650C211yesmaximal, characteristic, complemented
9#418×C251C211yesmaximal, characteristic, complemented
10C24×D251—21no—
11C2×#16860810C2211yescharacteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C22⋊D251—21no—
14C2×D4×D251—24no—
15C2×D4×D251—22no—
16C2×D4×D251—24no—
17C2×D4×D251C2214yes—
18C22×C4×D251C2211yescharacteristic
19#22548743—24no—
20#22548741—24no—
21C2×#16860799C2212yescomplemented
22C2×#16860798C2212yescomplemented
23C2×D2008C2212yescomplemented
24C2×#16860795C2212yescomplemented
25C2×#16860794C2211yescharacteristic, complemented
26#22548699—21no—
27#22548695C2212yescomplemented
28#22548690C2212yescomplemented
29#22548685—21no—
30C4×D1004—22no—
31#22548635C2214yescomplemented
32(C2×D251)⋊C8C2214yescomplemented
33C22×D4×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic, complemented
35#117×C251C2211yescharacteristic, complemented
36#102×C251C2214yescomplemented
37C23×D251—28no—
38C23×D251—24no—
39C23×D251—21no—
40C23×D251—48no—
41C23×D251D411yescharacteristic
42#16860810—22no—
43#16860810—22no—
44C2×Dic502C2311yescharacteristic, complemented
45C2×D1004—22no—
46C2×D1004—21no—
47C2×D1004—22no—
48C2×D1004D414yescomplemented
49C2×D1004C2312yes—
50C2×D1004C2311yescharacteristic, complemented
51C2×C22⋊D251—28no—
52C2×C22⋊D251—24no—
53C2×C22⋊D251—21no—
54D4×D251—216no—
55D4×D251—28no—
56D4×D251—28no—
57C22×Dic251D411yescharacteristic
58C2×C4×D251—22no—
59C2×C4×D251—22no—
60C2×C4×D251—21no—
61C2×C4×D251D412yes—
62C2×C4×D251C2312yes—
63#16860799—24no—
64#16860798—24no—
65D2008—24no—
66#16860795—24no—
67#16860794—22no—
68C2×C251⋊C8C2312yescomplemented
69C4⋊Dic251—22no—
70(C2×D251)⋊C4—22no—
71(C2×D251)⋊C4—22no—
72C24×C251—21no—
73C2×D4×C251—21no—
74C2×D4×C251—22no—
75C2×D4×C251D414yescomplemented
76C2×D4×C251C2312yes—
77C22×C4×C251C2311yescharacteristic
78M16×C251—22no—
79C2×C8×C251C2312yescomplemented
80C4⋊C4×C251C2312yescomplemented
81C22⋊C4×C251—21no—
82C42×C251—21no—
83C22×D251—22no—
84C22×D251—24no—
85C22×D251—216no—
86C22×D251—24no—
87C22×D251—22no—
88C22×D251—21no—
89C22×D251—432no—
90C22×D251—48no—
91C22×D251—44no—
92C22×D251—48no—
93C22×D251C2×D412yes—
94Dic502—21no—
95Dic502—21no—
96D1004—24no—
97D1004—22no—
98D1004—21no—
99D1004—21no—
100D1004—44no—
101D1004C2×D414yes—
102C22⋊D251—216no—
103C22⋊D251—42no—
104C2×Dic251—22no—
105C2×Dic251—21no—
106C2×Dic251C2×D412yes—
107C4×D251—24no—
108C4×D251—22no—
109C4×D251—22no—
110C4×D251—42no—
111C251⋊C8—22no—
112C23×C251—24no—
113C23×C251—22no—
114C23×C251—44no—
115C23×C251C2×D411yescharacteristic
116D4×C251—24no—
117D4×C251—22no—
118D4×C251C2×D414yes—
119C2×C4×C251—22no—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2×D412yes—
122C2×C4×C251C2×D412yes—
123C2×C4×C251C2411yescharacteristic
124C8×C251—22no—
125C22×C251—22no—
126C22×C251—24no—
127C22×C251—48no—
128C22×C251—44no—
129C22×C251#12012yes—
130C22×C251C22×D411yescharacteristic
131C2×D251—28no—
132C2×D251—22no—
133C2×D251—22no—
134C2×D251—24no—
135C2×D251—432no—
136C2×D251—48no—
137C2×D251—44no—
138C2×D251—41no—
139C2×D251—42no—
140C2×D251AGL(1,ℤ/8)14yescomplemented
141C4×C251—42no—
142C4×C251#12012yes—
143C4×C251C22×D411yescharacteristic
144C4×C251C22×D411yescharacteristic
145Dic251—22no—
146Dic251—41no—
147C2×C251—22no—
148C2×C251—44no—
149C2×C251C2×#12011yescharacteristic
150C2×C251C2×AGL(1,ℤ/8)12yes—
151D251—24no—
152D251—44no—
153D251—81no—
154C251#300311yescharacteristic, complemented, Sylow
155#3003—2511nomaximal, Sylow
156C23×D4—2511no—
157C2×AGL(1,ℤ/8)—2511no—
158C2×AGL(1,ℤ/8)—2511no—
159#525—2511no—
160#450—2514no—
161#447—2514no—
162#418—2511no—
163#418D25111yescharacteristic, complemented
164#407—2511no—
165C25—5021no—
166C2×#41—2511no—
167C22×D4—5024no—
168C22×D4—5024no—
169C22×D4—5022no—
170C22×D4—5021no—
171C22×D4—2514no—
172C22×D4—2511no—
173C22×D4C2×D25111yescharacteristic
174C23×C4—2511no—
175AGL(1,ℤ/8)—5024no—
176AGL(1,ℤ/8)—5024no—
177C2×SD16—2512no—
178C2×SD16—2512no—
179C2×D8—2512no—
180C2×D8—2512no—
181C2×M16—2511no—
182C2×M16C2×D25111yescharacteristic, complemented
183#123—5021no—
184C22⋊Q8—2512no—
185#121—2512no—
186#120—5021no—
187C4×D4—5022no—
188#117C2×D25111yescharacteristic, complemented
189#102—2514no—
190#102C2×D25114yescomplemented
191C22⋊C8—2514no—
192C24—21no—
193C24—10048no—
194C24—5028no—
195C24—5024no—
196C24—5021no—
197C24—2511no—
198#41—5022no—
199#41—5022no—
200C2×Q8—2511no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—5022no—
204C2×D4—50216no—
205C2×D4—5028no—
206C2×D4—5021no—
207C2×D4—5028no—
208C2×D4—5028no—
209C2×D4—5024no—
210C2×D4—5021no—
211C2×D4—5022no—
212C2×D4—2514no—
213C2×D4—2511no—
214C2×D4—2512no—
215C2×D4D100414yescomplemented
216C2×D4C22×D25112yes—
217C22×C4—5021no—
218C22×C4—5022no—
219C22×C4—5022no—
220C22×C4—2511no—
221C22×C4—2512no—
222C22×C4—2512no—
223C22×C4C22×D25111yescharacteristic
224SD16—5024no—
225SD16—5024no—
226D8—5024no—
227D8—5024no—
228M16—22no—
229M16—5022no—
230C2×C8—2512no—
231C2×C8C22×D25112yescomplemented
232C4⋊C4—5022no—
233C4⋊C4C22×D25112yescomplemented
234C22⋊C4—21no—
235C22⋊C4—5022no—
236C22⋊C4—5022no—
237C42—21no—
238C23—22no—
239C23—24no—
240C23—44no—
241C23—10044no—
242C23—100432no—
243C23—10048no—
244C23—10048no—
245C23—50216no—
246C23—5024no—
247C23—5021no—
248C23—5022no—
249C23—5024no—
250C23—5022no—
251C23—2512no—
252C23D4×D25111yescharacteristic
253Q8—5021no—
254Q8—5021no—
255D4—22no—
256D4—24no—
257D4—10044no—
258D4—10042no—
259D4—50216no—
260D4—5021no—
261D4—5021no—
262D4—5022no—
263D4—5024no—
264D4—2514no—
265D4C2×D100414yes—
266C2×C4—22no—
267C2×C4—10042no—
268C2×C4—5021no—
269C2×C4—5022no—
270C2×C4—5022no—
271C2×C4—5022no—
272C2×C4—5024no—
273C2×C4—2512no—
274C2×C4D4×D25112yes—
275C2×C4D4×D25111yescharacteristic
276C2×C4C2×D100412yes—
277C2×C4C23×D25111yescharacteristic
278C8—22no—
279C8—5022no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—10048no—
285C22—100432no—
286C22—10042no—
287C22—10041no—
288C22—10044no—
289C22—5028no—
290C22—5024no—
291C22—5022no—
292C22—5022no—
293C22—2514no—
294C22#2254868512yes—
295C22C2×D4×D25111yescharacteristic
296C4—42no—
297C4—10041no—
298C4—5022no—
299C4#2254868512yes—
300C4C2×D4×D25111yescharacteristic
301C4C22×D100411yescharacteristic
302C2—22no—
303C2—44no—
304C2—20081no—
305C2—10044no—
306C2—5024no—
307C2C2×#2254868511yescharacteristic
308C2AGL(1,ℤ/8)×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
hypercenter#418D251
hypocenterC251#3003
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D251C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548741C2semidirect
4. #30632199C2semidirect
5. #30631839C2semidirect
6. #30631822C2semidirect
7. #30631691C2semidirect
8. #30631650C2semidirect
9. #418×C251C2semidirect
11. C2×#16860810C22semidirect
21. C2×#16860799C22semidirect
22. C2×#16860798C22semidirect
23. C2×D2008C22semidirect
24. C2×#16860795C22semidirect
25. C2×#16860794C22semidirect
27. #22548695C22semidirect
28. #22548690C22semidirect
31. #22548635C22semidirect
32. (C2×D251)⋊C8C22semidirect
34. C2×M16×C251C22semidirect
35. #117×C251C22semidirect
36. #102×C251C22semidirect
44. C2×Dic502C23semidirect
48. C2×D1004D4semidirect
50. C2×D1004C23semidirect
68. C2×C251⋊C8C23semidirect
75. C2×D4×C251D4semidirect
79. C2×C8×C251C23semidirect
80. C4⋊C4×C251C23semidirect
140. C2×D251AGL(1,ℤ/8)semidirect
154. C251#3003semidirect
163. #418D251semidirect
182. C2×M16C2×D251semidirect
188. #117C2×D251semidirect
190. #102C2×D251semidirect
215. C2×D4D1004semidirect
231. C2×C8C22×D251semidirect
233. C4⋊C4C22×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×#225487432
C2×#225487412
#306321992
#306318392
#306318222
#306316912
#306316502
#418×C2512
#3003251