Finite Group Databasebeta

← #44559643

Subgroups — #44559643

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×#22548748C211yesmaximal, characteristic, complemented
3C22×#16860798C211yesmaximal, characteristic, complemented
4#30632263C211yesmaximal, characteristic, complemented
5#30631850C214yesmaximal, complemented
6#30631839C214yesmaximal, complemented
7#30631681C211yesmaximal, characteristic, complemented
8#30631655C211yesmaximal, characteristic, complemented
9#418×C251C211yesmaximal, characteristic, complemented
10#22548760—24no—
11#22548760—24no—
12C2×#16860810C2211yescharacteristic
13C2×#16860810C2211yescharacteristic
14C2×#16860809—21no—
15C2×#16860809C2212yes—
16C22×Dic502C2211yescharacteristic, complemented
17C22×C22⋊D251—22no—
18C2×D4×D251—21no—
19C2×D4×D251C2212yes—
20#22548748—24no—
21C2×Dic1004C2212yescomplemented
22C2×#16860799C2212yescomplemented
23C2×#16860798—24no—
24C2×#16860798C2214yescomplemented
25C22×C251⋊C8C2211yescharacteristic, complemented
26#22548715—21no—
27#22548703—21no—
28#22548699—22no—
29#22548695C2214yescomplemented
30#22548638C2212yescomplemented
31#22548635C2212yescomplemented
32(C2×D251)⋊C8C2214yescomplemented
33C22×D4×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic, complemented
35#117×C251C2211yescharacteristic, complemented
36#102×C251C2214yescomplemented
37C23×D251—22no—
38#16860810—22no—
39#16860810—22no—
40#16860810—22no—
41#16860810D414yes—
42#16860809—28no—
43#16860809—24no—
44#16860809—24no—
45C2×Dic502—21no—
46C2×Dic502—22no—
47C2×Dic502—22no—
48C2×Dic502D411yescharacteristic
49C2×Dic502C2312yes—
50C2×Dic502C2311yescharacteristic
51C2×D1004D411yescharacteristic
52C2×D1004C2311yescharacteristic
53C2×C22⋊D251—216no—
54C2×C22⋊D251—22no—
55C2×C22⋊D251—22no—
56C2×C22⋊D251—21no—
57C2×C22⋊D251D412yes—
58D4×D251—28no—
59D4×D251—24no—
60D4×D251—24no—
61C22×Dic251—22no—
62C22×Dic251—21no—
63C2×C4×D251—21no—
64C2×C4×D251C2312yes—
65Dic1004—24no—
66#16860799—24no—
67#16860798—28no—
68C2×C251⋊C8—22no—
69C2×C251⋊C8C2312yes—
70C4⋊Dic251—24no—
71(C2×D251)⋊C4—24no—
72C24×C251—21no—
73C2×D4×C251—21no—
74C2×D4×C251—22no—
75C2×D4×C251D414yes—
76C2×D4×C251C2312yes—
77C22×C4×C251C2311yescharacteristic
78M16×C251—22no—
79C2×C8×C251C2312yes—
80C4⋊C4×C251C2312yes—
81C22⋊C4×C251—21no—
82C42×C251—21no—
83C22×D251—21no—
84C22×D251—28no—
85C22×D251—84no—
86C22×D251C2×D412yes—
87Dic502—22no—
88Dic502—21no—
89Dic502—22no—
90Dic502—22no—
91Dic502—42no—
92Dic502C2×D412yes—
93D1004—22no—
94D1004—21no—
95D1004C2×D412yes—
96C22⋊D251—28no—
97C22⋊D251—28no—
98C22⋊D251—24no—
99C22⋊D251—24no—
100C2×Dic251—28no—
101C2×Dic251—21no—
102C2×Dic251—22no—
103C2×Dic251—42no—
104C2×Dic251C2×D412yes—
105C4×D251—24no—
106C4×D251—22no—
107C4×D251—22no—
108C251⋊C8—22no—
109C23×C251—22no—
110C23×C251—24no—
111C23×C251—82no—
112C23×C251C2×D411yescharacteristic
113D4×C251—24no—
114D4×C251—22no—
115D4×C251C2×D414yes—
116C2×C4×C251—22no—
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×SD16—2511no—
151#536—2511no—
152#452—2514no—
153#450—2514no—
154#418D25111yescharacteristic, complemented
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×Q8—2511no—
164C22×D4—5021no—
165C22×D4—5022no—
166C22×D4—2512no—
167C22×D4C2×D25111yescharacteristic
168#137—5024no—
169C2×Q16—2512no—
170C2×SD16—5024no—
171C2×SD16—2514no—
172C2×SD16—2512no—
173C2×M16C2×D25111yescharacteristic, complemented
174C22×C8—2511no—
175C4⋊Q8—5021no—
176#124—5021no—
177#123—5022no—
178C22⋊Q8—2514no—
179#117C2×D25111yescharacteristic, complemented
180#103—2512no—
181#102—2512no—
182#102C2×D25114yescomplemented
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—5022no—
194C2×Q8—5022no—
195C2×Q8—5021no—
196C2×Q8—2512no—
197C2×Q8—2511no—
198C2×Q8—2511no—
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×D4D100414yes—
212C2×D4C22×D25112yes—
213C22×C4—5021no—
214C22×C4—5022no—
215C22×C4—5021no—
216C22×C4—2512no—
217C22×C4C22×D25111yescharacteristic
218Q16—5024no—
219SD16—5024no—
220SD16—5028no—
221M16—22no—
222C2×C8—5022no—
223C2×C8—2512no—
224C2×C8C22×D25112yes—
225C4⋊C4—5024no—
226C4⋊C4C22×D25112yes—
227C22⋊C4—21no—
228C22⋊C4—5024no—
229C42—21no—
230C23—22no—
231C23—24no—
232C23—82no—
233C23—20084no—
234C23—5028no—
235C23—5021no—
236C23—2512no—
237C23D4×D25111yescharacteristic
238Q8—10042no—
239Q8—5022no—
240Q8—5022no—
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×D100414yes—
254C2×C4—22no—
255C2×C4—10042no—
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×D100412yes—
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#2254868512yes—
279C22C2×D4×D25111yescharacteristic
280C4—42no—
281C4—10042no—
282C4—5022no—
283C4—5022no—
284C4#2254868512yes—
285C4C2×D4×D25111yescharacteristic
286C4C22×D100411yescharacteristic
287C2—22no—
288C2—44no—
289C2—10042no—
290C2—10042no—
291C2C2×#2254868511yescharacteristic
292C2#3063249712yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548760C2
hypercenter#418D251
hypocenterC251#3009
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
2. C2×#22548748C2semidirect
3. C22×#16860798C2semidirect
4. #30632263C2semidirect
5. #30631850C2semidirect
6. #30631839C2semidirect
7. #30631681C2semidirect
8. #30631655C2semidirect
9. #418×C251C2semidirect
16. C22×Dic502C22semidirect
21. C2×Dic1004C22semidirect
22. C2×#16860799C22semidirect
24. C2×#16860798C22semidirect
25. C22×C251⋊C8C22semidirect
29. #22548695C22semidirect
30. #22548638C22semidirect
31. #22548635C22semidirect
32. (C2×D251)⋊C8C22semidirect
34. C2×M16×C251C22semidirect
35. #117×C251C22semidirect
36. #102×C251C22semidirect
146. C251#3009semidirect
154. #418D251semidirect
173. C2×M16C2×D251semidirect
179. #117C2×D251semidirect
182. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
C2×#225487482
C22×#168607982
#306322632
#306318502
#306318392
#306316812
#306316552
#418×C2512
#3009251