Finite Group Databasebeta

← #44559641

Subgroups — #44559641

Order 32128.

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

normal subgroups only

This group has 752 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
2C22×#16860797C211yesmaximal, characteristic, complemented
3#30632261C211yesmaximal, characteristic, complemented
4#30631848C214yesmaximal, complemented
5#30631836C214yesmaximal, complemented
6#30631693C211yesmaximal, characteristic, complemented
7#30631683C211yesmaximal, characteristic, complemented
8#30631655C211yesmaximal, characteristic, complemented
9C2×#137×C251C211yesmaximal, characteristic, complemented
10#22548761—24no—
11#22548761—24no—
12C2×#16860811C2212yes—
13C2×#16860810—21no—
14C2×#16860810C2211yescharacteristic
15C22×D1004—21no—
16C22×D1004C2211yescharacteristic
17C2×D4×D251—22no—
18C2×D4×D251C2212yes—
19C2×#16860797—24no—
20C2×#16860797C2214yescomplemented
21C22×C251⋊C8C2211yescharacteristic, complemented
22#22548713—21no—
23#22548704—21no—
24#22548698—22no—
25#22548691C2214yescomplemented
26#22548674C2211yescharacteristic, complemented
27#22548639C2212yescomplemented
28#22548637C2214yescomplemented
29#22548636C2212yescomplemented
30(C2×D251)⋊C8C2214yescomplemented
31C2×#41×C251C2211yescharacteristic
32C22×Q8×C251C2211yescharacteristic, complemented
33#137×C251—24no—
34C2×Q16×C251C2212yescomplemented
35C2×SD16×C251C2212yescomplemented
36C2×M16×C251C2211yescharacteristic, complemented
37C23×D251—22no—
38C23×D251—21no—
39#16860811—24no—
40#16860811—24no—
41#16860810—28no—
42#16860810—22no—
43#16860810—22no—
44C2×Dic502D411yescharacteristic
45C2×D1004—28no—
46C2×D1004—22no—
47C2×D1004—21no—
48C2×D1004—22no—
49C2×D1004D411yescharacteristic
50C2×D1004D414yes—
51C2×D1004C2312yes—
52C2×C22⋊D251—22no—
53C2×C22⋊D251D412yes—
54D4×D251—216no—
55D4×D251—24no—
56D4×D251—24no—
57C2×C4×D251—22no—
58C2×C4×D251C2312yes—
59#16860797—28no—
60C2×C251⋊C8—22no—
61C2×C251⋊C8C2312yes—
62C4⋊Dic251C2312yes—
63Dic251⋊C4—24no—
64C22⋊Dic251—21no—
65(C2×D251)⋊C4—24no—
66C4×Dic251—21no—
67#41×C251—22no—
68#41×C251D414yes—
69C2×Q8×C251—21no—
70C2×Q8×C251—22no—
71C2×Q8×C251—22no—
72C2×Q8×C251C2312yes—
73C2×Q8×C251C2311yescharacteristic
74C2×D4×C251—21no—
75C2×D4×C251C2311yescharacteristic
76C22×C4×C251—21no—
77C22×C4×C251—21no—
78C22×C4×C251C2311yescharacteristic
79Q16×C251—24no—
80SD16×C251—24no—
81M16×C251—22no—
82C2×C8×C251C2312yes—
83C22×D251—22no—
84C22×D251—28no—
85C22×D251—24no—
86C22×D251—84no—
87C22×D251—82no—
88C22×D251C2×D412yes—
89Dic502—22no—
90D1004—28no—
91D1004—22no—
92D1004—24no—
93D1004—22no—
94D1004C2×D414yes—
95C22⋊D251—28no—
96C22⋊D251—24no—
97C2×Dic251—22no—
98C2×Dic251C2×D412yes—
99C4×D251—28no—
100C4×D251—22no—
101C4×D251—22no—
102C251⋊C8—22no—
103C23×C251—21no—
104C23×C251C2×D411yescharacteristic
105Q8×C251—21no—
106Q8×C251—22no—
107Q8×C251—22no—
108Q8×C251—42no—
109Q8×C251C2×D412yes—
110D4×C251—24no—
111D4×C251—21no—
112D4×C251C2×D412yes—
113C2×C4×C251—21no—
114C2×C4×C251—24no—
115C2×C4×C251—22no—
116C2×C4×C251—42no—
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2411yescharacteristic
121C8×C251—22no—
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—42no—
125C22×C251#12012yes—
126C22×C251C22×D411yescharacteristic
127C2×D251—22no—
128C2×D251—24no—
129C2×D251—24no—
130C2×D251—44no—
131C2×D251—84no—
132C2×D251—88no—
133C4×C251—22no—
134C4×C251—42no—
135C4×C251#12012yes—
136C4×C251C22×D411yescharacteristic
137C4×C251C22×D411yescharacteristic
138Dic251—22no—
139Dic251—42no—
140C2×C251—22no—
141C2×C251—42no—
142C2×C251C2×#12011yescharacteristic
143C2×C251#57712yes—
144D251—42no—
145D251—44no—
146C251#300911yescharacteristic, complemented, Sylow
147#3009—2511nomaximal, Sylow
148C2×H(5,𝔽2)—2511no—
149C2×#137D25111yescharacteristic, complemented
150C22×SD16—2511no—
151#536—2511no—
152#452—2514no—
153#450—2514no—
154#418—2511no—
155#416—2511no—
156#408—2511no—
157H(5,𝔽2)—5024no—
158H(5,𝔽2)—5024no—
159C2×#41—5021no—
160C2×#41—2512no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C22×Q8C2×D25111yescharacteristic, complemented
164C22×D4—5021no—
165C22×D4—5022no—
166C22×D4—2512no—
167C22×D4—2511no—
168#137—24no—
169C2×Q16C2×D25112yescomplemented
170C2×SD16—5024no—
171C2×SD16—2514no—
172C2×SD16C2×D25112yescomplemented
173C2×M16C2×D25111yescharacteristic, complemented
174C22×C8—2511no—
175C4⋊Q8—5021no—
176#124—5021no—
177#123—5022no—
178C22⋊Q8—2514no—
179#117—2511no—
180#103—2512no—
181#102—2512no—
182#102—2514no—
183C22⋊C8—2514no—
184C24—5022no—
185C24—5021no—
186#41—22no—
187#41—5028no—
188#41—5024no—
189#41—5024no—
190#41—5022no—
191#41—5022no—
192#41C22⋊D25114yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—2511no—
197C2×Q8C22×D25112yes—
198C2×Q8C22×D25111yescharacteristic
199C2×D4—21no—
200C2×D4—5028no—
201C2×D4—5022no—
202C2×D4—50216no—
203C2×D4—5024no—
204C2×D4—5024no—
205C2×D4—5022no—
206C2×D4—5021no—
207C2×D4—5022no—
208C2×D4—2512no—
209C2×D4—2511no—
210C2×D4—2514no—
211C2×D4—2512no—
212C2×D4C22×D25111yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—5022no—
216C22×C4—2512no—
217C22×C4C22×D25111yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—5028no—
221M16—22no—
222C2×C8—5022no—
223C2×C8—2512no—
224C2×C8C22×D25112yes—
225C4⋊C4—5024no—
226C4⋊C4—2512no—
227C22⋊C4—5024no—
228C22⋊C4—5021no—
229C42—5021no—
230C23—21no—
231C23—20082no—
232C23—20084no—
233C23—5028no—
234C23—5024no—
235C23—5022no—
236C23—2512no—
237C23D4×D25111yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—5022no—
243Q8C2×C22⋊D25112yes—
244D4—24no—
245D4—21no—
246D4—5028no—
247D4—5028no—
248D4—5024no—
249D4—5022no—
250D4—5022no—
251D4—5024no—
252D4—2514no—
253D4C2×C22⋊D25112yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—5028no—
259C2×C4—5022no—
260C2×C4—5022no—
261C2×C4—5022no—
262C2×C4—2512no—
263C2×C4D4×D25112yes—
264C2×C4D4×D25111yescharacteristic
265C2×C4C2×C22⋊D25112yes—
266C2×C4C23×D25111yescharacteristic
267C8—22no—
268C8—5022no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—20084no—
273C22—20088no—
274C22—10044no—
275C22—5024no—
276C22—5022no—
277C22—5024no—
278C22#2254868312yes—
279C22C2×D4×D25111yescharacteristic
280C4—22no—
281C4—42no—
282C4—10042no—
283C4—5022no—
284C4#2254868312yes—
285C4C2×D4×D25111yescharacteristic
286C4C22×C22⋊D25111yescharacteristic
287C2—22no—
288C2—42no—
289C2—10044no—
290C2—10042no—
291C2C2×#2254868311yescharacteristic
292C2#3063249612yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#137×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548761C2
hypercenterC2×#137D251
hypocenterC251#3009
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#16860797C2semidirect
3. #30632261C2semidirect
4. #30631848C2semidirect
5. #30631836C2semidirect
6. #30631693C2semidirect
7. #30631683C2semidirect
8. #30631655C2semidirect
9. C2×#137×C251C2semidirect
20. C2×#16860797C22semidirect
21. C22×C251⋊C8C22semidirect
25. #22548691C22semidirect
26. #22548674C22semidirect
27. #22548639C22semidirect
28. #22548637C22semidirect
29. #22548636C22semidirect
30. (C2×D251)⋊C8C22semidirect
32. C22×Q8×C251C22semidirect
34. C2×Q16×C251C22semidirect
35. C2×SD16×C251C22semidirect
36. C2×M16×C251C22semidirect
146. C251#3009semidirect
149. C2×#137D251semidirect
163. C22×Q8C2×D251semidirect
169. C2×Q16C2×D251semidirect
172. C2×SD16C2×D251semidirect
173. C2×M16C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
C22×#168607972
#306322612
#306318482
#306318362
#306316932
#306316832
#306316552
C2×#137×C2512
#3009251