Finite Group Databasebeta

← #44559636

Subgroups — #44559636

Order 32128.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#22548761C211yesmaximal, characteristic, complemented
2C2×#22548740C211yesmaximal, characteristic, complemented
3C22×D2008C211yesmaximal, characteristic, complemented
4#30632256C211yesmaximal, characteristic, complemented
5#30631835C214yesmaximal, complemented
6#30631822C214yesmaximal, complemented
7#30631691C211yesmaximal, characteristic, complemented
8#30631653C211yesmaximal, characteristic, complemented
9#416×C251C211yesmaximal, characteristic, complemented
10#22548761—24no—
11#22548761—24no—
12C2×#16860811C2212yescomplemented
13C2×#16860810—21no—
14C2×#16860810C2211yescharacteristic
15C22×D1004—21no—
16C22×D1004C2211yescharacteristic
17C22×D1004C2211yescharacteristic, complemented
18C2×D4×D251—22no—
19C2×D4×D251C2212yescomplemented
20#22548740—24no—
21C2×#16860797C2212yescomplemented
22C2×D2008—24no—
23C2×D2008C2214yescomplemented
24C2×#16860795C2212yescomplemented
25C2×#16860794C2211yescharacteristic, complemented
26#22548712—21no—
27#22548703—21no—
28#22548690C2214yescomplemented
29#22548685—22no—
30#22548635C2214yescomplemented
31(C2×D251)⋊C8C2214yescomplemented
32C2×#41×C251C2211yescharacteristic
33C22×C8×C251C2211yescharacteristic, complemented
34#117×C251C2211yescharacteristic, complemented
35#103×C251C2212yescomplemented
36#102×C251C2212yescomplemented
37C23×D251—22no—
38C23×D251—21no—
39C23×D251—21no—
40#16860811—24no—
41#16860811—24no—
42#16860810—28no—
43#16860810—22no—
44#16860810—22no—
45C2×Dic502D411yescharacteristic
46C2×D1004—28no—
47C2×D1004—22no—
48C2×D1004—24no—
49C2×D1004—21no—
50C2×D1004—22no—
51C2×D1004—22no—
52C2×D1004—22no—
53C2×D1004—21no—
54C2×D1004D411yescharacteristic
55C2×D1004D414yescomplemented
56C2×D1004C2312yes—
57C2×D1004C2312yes—
58C2×C22⋊D251—22no—
59C2×C22⋊D251D412yes—
60D4×D251—216no—
61D4×D251—24no—
62D4×D251—24no—
63C2×C4×D251—22no—
64C2×C4×D251C2312yes—
65#16860797—24no—
66D2008—28no—
67#16860795—24no—
68#16860794—22no—
69C2×C251⋊C8C2312yes—
70(C2×D251)⋊C4—24no—
71#41×C251—22no—
72#41×C251D414yescomplemented
73C2×Q8×C251C2311yescharacteristic
74C2×D4×C251—21no—
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C22×C4×C251C2311yescharacteristic
78C2×C8×C251—22no—
79C2×C8×C251C2312yes—
80C4⋊C4×C251C2312yes—
81C22⋊C4×C251—21no—
82C42×C251—21no—
83C22×D251—22no—
84C22×D251—28no—
85C22×D251—24no—
86C22×D251—22no—
87C22×D251—42no—
88C22×D251—84no—
89C22×D251—82no—
90C22×D251—82no—
91C22×D251C2×D412yes—
92Dic502—22no—
93D1004—28no—
94D1004—22no—
95D1004—24no—
96D1004—22no—
97D1004—22no—
98D1004—22no—
99D1004—42no—
100D1004—84no—
101D1004C2×D414yes—
102C22⋊D251—28no—
103C22⋊D251—24no—
104C2×Dic251C2×D412yes—
105C4×D251—28no—
106C4×D251—22no—
107C4×D251—22no—
108C251⋊C8—22no—
109C23×C251—21no—
110C23×C251C2×D411yescharacteristic
111Q8×C251—21no—
112Q8×C251C2×D412yes—
113D4×C251—24no—
114D4×C251—21no—
115D4×C251C2×D412yes—
116C2×C4×C251—21no—
117C2×C4×C251—24no—
118C2×C4×C251—22no—
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251C2×D412yes—
121C2×C4×C251C2×D412yes—
122C2×C4×C251C2411yescharacteristic
123C8×C251—22no—
124C22×C251—22no—
125C22×C251—22no—
126C22×C251—42no—
127C22×C251#12012yes—
128C22×C251C22×D411yescharacteristic
129C2×D251—22no—
130C2×D251—24no—
131C2×D251—24no—
132C2×D251—44no—
133C2×D251—42no—
134C2×D251—44no—
135C2×D251—84no—
136C2×D251—88no—
137C2×D251—84no—
138C4×C251—22no—
139C4×C251—42no—
140C4×C251#12012yes—
141C4×C251C22×D411yescharacteristic
142C4×C251C22×D411yescharacteristic
143Dic251—22no—
144C2×C251—22no—
145C2×C251—42no—
146C2×C251C2×#12011yescharacteristic
147C2×C251#57612yes—
148D251—42no—
149D251—44no—
150D251—82no—
151C251#300811yescharacteristic, complemented, Sylow
152#3008—2511nomaximal, Sylow
153C2×H(5,𝔽2)—2511no—
154C2×AGL(1,ℤ/8)—2511no—
155C22×D8—2511no—
156#535—2511no—
157#449—2514no—
158#447—2514no—
159#418—2511no—
160#416D25111yescharacteristic, complemented
161#408—2511no—
162H(5,𝔽2)—5024no—
163H(5,𝔽2)—5024no—
164C2×#41—5021no—
165C2×#41—2512no—
166C2×#41—2511no—
167C2×#41C2×D25111yescharacteristic
168C22×D4—5021no—
169C22×D4—5022no—
170C22×D4—2512no—
171C22×D4—2511no—
172C22×D4—2511no—
173AGL(1,ℤ/8)—5024no—
174C2×SD16—2512no—
175C2×D8—5024no—
176C2×D8—2512no—
177C2×D8—2514no—
178C2×M16—2511no—
179C22×C8C2×D25111yescharacteristic, complemented
180#127—5021no—
181#124—5021no—
182#121—2514no—
183#120—5022no—
184#117C2×D25111yescharacteristic, complemented
185#103C2×D25112yescomplemented
186#102—2514no—
187#102C2×D25112yescomplemented
188C22⋊C8—2514no—
189C24—5022no—
190C24—5021no—
191C24—5021no—
192#41—22no—
193#41—5028no—
194#41—5024no—
195#41—5024no—
196#41—5022no—
197#41—5022no—
198#41D100414yescomplemented
199C2×Q8—2511no—
200C2×Q8C22×D25111yescharacteristic
201C2×D4—21no—
202C2×D4—5024no—
203C2×D4—5028no—
204C2×D4—5022no—
205C2×D4—50216no—
206C2×D4—5024no—
207C2×D4—5024no—
208C2×D4—5022no—
209C2×D4—5021no—
210C2×D4—5022no—
211C2×D4—5021no—
212C2×D4—5022no—
213C2×D4—5022no—
214C2×D4—2512no—
215C2×D4—2511no—
216C2×D4—2514no—
217C2×D4—2512no—
218C2×D4—2512no—
219C2×D4C22×D25111yescharacteristic
220C22×C4—21no—
221C22×C4—5022no—
222C22×C4—2512no—
223C22×C4C22×D25111yescharacteristic
224SD16—5024no—
225D8—5028no—
226D8—5024no—
227M16—5022no—
228C2×C8—22no—
229C2×C8—2512no—
230C2×C8C22×D25112yes—
231C4⋊C4C22×D25112yes—
232C22⋊C4—21no—
233C22⋊C4—5024no—
234C42—21no—
235C23—21no—
236C23—20082no—
237C23—20082no—
238C23—20084no—
239C23—10042no—
240C23—5028no—
241C23—5024no—
242C23—5022no—
243C23—5022no—
244C23—2512no—
245C23D4×D25111yescharacteristic
246Q8—21no—
247Q8—5022no—
248Q8C2×D100412yes—
249D4—24no—
250D4—21no—
251D4—20084no—
252D4—10042no—
253D4—5028no—
254D4—5028no—
255D4—5024no—
256D4—5022no—
257D4—5022no—
258D4—5022no—
259D4—5022no—
260D4—5024no—
261D4—2514no—
262D4C2×D100412yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—5028no—
267C2×C4—5022no—
268C2×C4—5022no—
269C2×C4—2512no—
270C2×C4D4×D25112yes—
271C2×C4D4×D25111yescharacteristic
272C2×C4C2×D100412yes—
273C2×C4C23×D25111yescharacteristic
274C8—22no—
275C8—5022no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—20084no—
280C22—20084no—
281C22—20088no—
282C22—10044no—
283C22—10042no—
284C22—10044no—
285C22—5024no—
286C22—5022no—
287C22—5024no—
288C22#2254868512yes—
289C22C2×D4×D25111yescharacteristic
290C4—22no—
291C4—42no—
292C4—5022no—
293C4#2254868512yes—
294C4C2×D4×D25111yescharacteristic
295C4C22×D100411yescharacteristic
296C2—22no—
297C2—42no—
298C2—20082no—
299C2—10044no—
300C2—10042no—
301C2C2×#2254868511yescharacteristic
302C2#3063248912yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
hypercenter#416D251
hypocenterC251#3008
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2semidirect
2. C2×#22548740C2semidirect
3. C22×D2008C2semidirect
4. #30632256C2semidirect
5. #30631835C2semidirect
6. #30631822C2semidirect
7. #30631691C2semidirect
8. #30631653C2semidirect
9. #416×C251C2semidirect
12. C2×#16860811C22semidirect
17. C22×D1004C22semidirect
19. C2×D4×D251C22semidirect
21. C2×#16860797C22semidirect
23. C2×D2008C22semidirect
24. C2×#16860795C22semidirect
25. C2×#16860794C22semidirect
28. #22548690C22semidirect
30. #22548635C22semidirect
31. (C2×D251)⋊C8C22semidirect
33. C22×C8×C251C22semidirect
34. #117×C251C22semidirect
35. #103×C251C22semidirect
36. #102×C251C22semidirect
55. C2×D1004D4semidirect
72. #41×C251D4semidirect
151. C251#3008semidirect
160. #416D251semidirect
179. C22×C8C2×D251semidirect
184. #117C2×D251semidirect
185. #103C2×D251semidirect
187. #102C2×D251semidirect
198. #41D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
C2×#225487402
C22×D20082
#306322562
#306318352
#306318222
#306316912
#306316532
#416×C2512
#3008251