Finite Group Databasebeta

← #44559638

Subgroups — #44559638

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×#22548760C211yesmaximal, characteristic
2C2×#22548746C211yesmaximal, characteristic, complemented
3#30632261C211yesmaximal, characteristic, complemented
4#30631846C214yesmaximal, complemented
5#30631836C214yesmaximal, complemented
6#30631692C211yesmaximal, characteristic, complemented
7#30631684C211yesmaximal, characteristic, complemented
8#30631653C211yesmaximal, characteristic, complemented
9C22×SD16×C251C211yesmaximal, characteristic, complemented
10#22548760—24no—
11#22548760—24no—
12C2×#16860810C2211yescharacteristic
13C2×#16860810C2211yescharacteristic
14C2×#16860809—21no—
15C2×#16860809C2212yes—
16C22×C22⋊D251—22no—
17C2×D4×D251—21no—
18C2×D4×D251C2212yes—
19#22548746—24no—
20C2×#16860800C2212yescomplemented
21C2×#16860797C2212yescomplemented
22C2×#16860794C2211yescharacteristic, complemented
23#22548713—21no—
24#22548704—21no—
25#22548698—22no—
26#22548691C2214yescomplemented
27#22548674C2211yescharacteristic, complemented
28#22548640C2212yescomplemented
29#22548637C2212yescomplemented
30#22548636C2214yescomplemented
31(C2×D251)⋊C8C2214yescomplemented
32C22×Q8×C251C2211yescharacteristic, complemented
33C22×D4×C251C2211yescharacteristic
34C2×SD16×C251—24no—
35C2×SD16×C251C2214yescomplemented
36C22×C8×C251C2211yescharacteristic, complemented
37C23×D251—22no—
38#16860810—22no—
39#16860810—22no—
40#16860810—22no—
41#16860810D414yes—
42#16860809—28no—
43#16860809—24no—
44#16860809—24no—
45C2×Dic502D411yescharacteristic
46C2×Dic502C2311yescharacteristic
47C2×D1004D411yescharacteristic
48C2×D1004C2311yescharacteristic
49C2×C22⋊D251—216no—
50C2×C22⋊D251—22no—
51C2×C22⋊D251—22no—
52C2×C22⋊D251—21no—
53C2×C22⋊D251D412yes—
54D4×D251—28no—
55D4×D251—24no—
56D4×D251—24no—
57C22×Dic251—22no—
58C2×C4×D251—21no—
59C2×C4×D251C2312yes—
60#16860800—24no—
61#16860797—24no—
62#16860794—22no—
63C2×C251⋊C8C2312yes—
64C4⋊Dic251C2312yes—
65Dic251⋊C4—24no—
66C22⋊Dic251—21no—
67(C2×D251)⋊C4—24no—
68C4×Dic251—21no—
69C24×C251—21no—
70C2×Q8×C251—21no—
71C2×Q8×C251—22no—
72C2×Q8×C251—22no—
73C2×Q8×C251C2312yes—
74C2×D4×C251—21no—
75C2×D4×C251—22no—
76C2×D4×C251D414yes—
77C2×D4×C251C2312yes—
78C22×C4×C251—21no—
79C22×C4×C251C2311yescharacteristic
80SD16×C251—28no—
81C2×C8×C251—22no—
82C2×C8×C251C2312yes—
83C22×D251—21no—
84C22×D251—28no—
85C22×D251—84no—
86C22×D251C2×D412yes—
87Dic502—22no—
88Dic502—21no—
89Dic502C2×D412yes—
90D1004—22no—
91D1004—21no—
92D1004C2×D412yes—
93C22⋊D251—28no—
94C22⋊D251—28no—
95C22⋊D251—24no—
96C22⋊D251—24no—
97C2×Dic251—28no—
98C2×Dic251—21no—
99C2×Dic251—22no—
100C2×Dic251C2×D412yes—
101C4×D251—24no—
102C4×D251—22no—
103C4×D251—22no—
104C251⋊C8—22no—
105C23×C251—22no—
106C23×C251—24no—
107C23×C251—82no—
108C23×C251C2×D411yescharacteristic
109Q8×C251—22no—
110Q8×C251—22no—
111Q8×C251—42no—
112D4×C251—24no—
113D4×C251—22no—
114D4×C251C2×D414yes—
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—24no—
124C22×C251—44no—
125C22×C251—84no—
126C22×C251#12012yes—
127C22×C251C22×D411yescharacteristic
128C2×D251—22no—
129C2×D251—22no—
130C2×D251—24no—
131C2×D251—42no—
132C2×D251—88no—
133C4×C251—42no—
134C4×C251#12012yes—
135C4×C251C22×D411yescharacteristic
136C4×C251C22×D411yescharacteristic
137Dic251—22no—
138Dic251—22no—
139Dic251—42no—
140C2×C251—22no—
141C2×C251—44no—
142C2×C251C2×#12011yescharacteristic
143C2×C251#57712yes—
144D251—42no—
145D251—42no—
146C251#300911yescharacteristic, complemented, Sylow
147#3009—2511nomaximal, Sylow
148C2×H(5,𝔽2)—2511no—
149C2×#137—2511no—
150C22×SD16D25111yescharacteristic, complemented
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×#41—2511no—
163C22×Q8C2×D25111yescharacteristic, complemented
164C22×D4—5021no—
165C22×D4—5022no—
166C22×D4—2512no—
167C22×D4C2×D25111yescharacteristic
168#137—5024no—
169C2×Q16—2512no—
170C2×SD16—24no—
171C2×SD16—2512no—
172C2×SD16C2×D25114yescomplemented
173C2×M16—2511no—
174C22×C8C2×D25111yescharacteristic, complemented
175C4⋊Q8—5021no—
176#124—5021no—
177#123—5022no—
178C22⋊Q8—2514no—
179#117—2511no—
180#103—2512no—
181#102—2514no—
182#102—2512no—
183C22⋊C8—2514no—
184C24—21no—
185C24—5022no—
186#41—5028no—
187#41—5024no—
188#41—5024no—
189#41—5022no—
190#41—5022no—
191#41—5022no—
192#41—2514no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—2511no—
197C2×Q8—2511no—
198C2×Q8C22×D25112yes—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—5028no—
202C2×D4—5024no—
203C2×D4—50216no—
204C2×D4—5022no—
205C2×D4—5021no—
206C2×D4—5024no—
207C2×D4—5022no—
208C2×D4—2512no—
209C2×D4—2511no—
210C2×D4—2511no—
211C2×D4C22⋊D25114yes—
212C2×D4C22×D25112yes—
213C22×C4—21no—
214C22×C4—5022no—
215C22×C4—5021no—
216C22×C4—2512no—
217C22×C4C22×D25111yescharacteristic
218Q16—5024no—
219SD16—28no—
220SD16—5024no—
221M16—5022no—
222C2×C8—22no—
223C2×C8—2512no—
224C2×C8C22×D25112yes—
225C4⋊C4—5024no—
226C4⋊C4—2512no—
227C22⋊C4—5024no—
228C22⋊C4—5021no—
229C42—5021no—
230C23—22no—
231C23—24no—
232C23—82no—
233C23—20084no—
234C23—5028no—
235C23—5021no—
236C23—2512no—
237C23D4×D25111yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—42no—
241Q8—5021no—
242Q8—5022no—
243Q8—2512no—
244D4—22no—
245D4—24no—
246D4—5028no—
247D4—5028no—
248D4—5024no—
249D4—5024no—
250D4—5021no—
251D4—5022no—
252D4—2512no—
253D4C2×C22⋊D25114yes—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—5022no—
257C2×C4—5028no—
258C2×C4—5021no—
259C2×C4—5022no—
260C2×C4—5022no—
261C2×C4—5024no—
262C2×C4—2512no—
263C2×C4D4×D25112yes—
264C2×C4D4×D25111yescharacteristic
265C2×C4C2×C22⋊D25112yes—
266C2×C4C23×D25111yescharacteristic
267C8—22no—
268C8—5022no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—84no—
273C22—20088no—
274C22—10042no—
275C22—5024no—
276C22—5022no—
277C22—5022no—
278C22#2254868312yes—
279C22C2×D4×D25111yescharacteristic
280C4—42no—
281C4—10042no—
282C4—5022no—
283C4—5022no—
284C4#2254868312yes—
285C4C2×D4×D25111yescharacteristic
286C4C22×C22⋊D25111yescharacteristic
287C2—22no—
288C2—44no—
289C2—10042no—
290C2—10042no—
291C2C2×#2254868311yescharacteristic
292C2#3063249312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×SD16×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548760C2
hypercenterC22×SD16D251
hypocenterC251#3009
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#22548746C2semidirect
3. #30632261C2semidirect
4. #30631846C2semidirect
5. #30631836C2semidirect
6. #30631692C2semidirect
7. #30631684C2semidirect
8. #30631653C2semidirect
9. C22×SD16×C251C2semidirect
20. C2×#16860800C22semidirect
21. C2×#16860797C22semidirect
22. C2×#16860794C22semidirect
26. #22548691C22semidirect
27. #22548674C22semidirect
28. #22548640C22semidirect
29. #22548637C22semidirect
30. #22548636C22semidirect
31. (C2×D251)⋊C8C22semidirect
32. C22×Q8×C251C22semidirect
35. C2×SD16×C251C22semidirect
36. C22×C8×C251C22semidirect
146. C251#3009semidirect
150. C22×SD16D251semidirect
163. C22×Q8C2×D251semidirect
172. C2×SD16C2×D251semidirect
174. C22×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
C2×#225487462
#306322612
#306318462
#306318362
#306316922
#306316842
#306316532
C22×SD16×C2512
#3009251