Finite Group Databasebeta

← #44559619

Subgroups — #44559619

Order 32128.

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

normal subgroups only

This group has 806 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×#16860810C211yesmaximal, characteristic, complemented
2C2×#22548743C212yesmaximal, complemented
3C2×#22548689C211yesmaximal, characteristic, complemented
4#30631834C214yesmaximal, complemented
5#30631828C212yesmaximal, complemented
6#30631823C212yesmaximal, complemented
7#30631689C212yesmaximal, complemented
8#407×C251C211yesmaximal, characteristic, complemented
9C2×#16860810—28no—
10C2×#16860810—21no—
11C2×#16860810—21no—
12C2×#16860810C2212yes—
13C2×#16860810C2212yescomplemented
14C22×Dic502C2211yescharacteristic, complemented
15C22×D1004C2211yescharacteristic, complemented
16C22×D1004C2211yescharacteristic, complemented
17C22×C22⋊D251—21no—
18C22×C22⋊D251—21no—
19C22×C4×D251—21no—
20#22548743—28no—
21C2×#16860799C2214yescomplemented
22C2×D2008C2214yescomplemented
23#22548689—21no—
24#22548689—21no—
25#22548689C2212yes—
26#22548689C2212yescomplemented
27C2×C4⋊Dic251C2211yescharacteristic, complemented
28C2×(C2×D251)⋊C4—21no—
29#22548640C2214yescomplemented
30#22548637C2214yescomplemented
31C23×C4×C251C2211yescharacteristic
32C2×M16×C251C2212yescomplemented
33C22⋊C8×C251C2212yescomplemented
34C22⋊C8×C251C2212yescomplemented
35C23×D251—21no—
36C23×D251—21no—
37#16860810—28no—
38#16860810—28no—
39#16860810—24no—
40#16860810—24no—
41#16860810—24no—
42#16860810D418yescomplemented
43C2×Dic502—24no—
44C2×Dic502—21no—
45C2×Dic502C2312yes—
46C2×D1004—24no—
47C2×D1004—21no—
48C2×D1004—24no—
49C2×D1004—21no—
50C2×D1004C2312yes—
51C2×D1004C2312yes—
52C2×C22⋊D251—28no—
53C2×C22⋊D251—22no—
54C2×C22⋊D251—22no—
55C2×C22⋊D251—22no—
56C2×C22⋊D251—22no—
57C2×C22⋊D251—44no—
58C22×Dic251—21no—
59C22×Dic251—21no—
60C2×C4×D251—28no—
61C2×C4×D251—22no—
62C2×C4×D251—22no—
63#16860799—28no—
64D2008—28no—
65C4⋊Dic251—21no—
66C4⋊Dic251C2312yes—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—22no—
69C24×C251D411yescharacteristic
70C22×C4×C251—24no—
71C22×C4×C251D411yescharacteristic
72C22×C4×C251D411yescharacteristic
73C22×C4×C251D411yescharacteristic
74C22×C4×C251C2311yescharacteristic
75C22×C4×C251C2311yescharacteristic
76C22×C4×C251C2311yescharacteristic
77M16×C251—24no—
78C2×C8×C251C2314yes—
79C22×D251—24no—
80C22×D251—22no—
81C22×D251—22no—
82C22×D251—42no—
83C22×D251—42no—
84C22×D251—42no—
85C22×D251—42no—
86C22×D251—42no—
87Dic502—24no—
88Dic502—22no—
89Dic502C2×D414yescomplemented
90D1004—24no—
91D1004—22no—
92D1004—24no—
93D1004—42no—
94D1004C2×D414yescomplemented
95C22⋊D251—28no—
96C22⋊D251—28no—
97C22⋊D251—44no—
98C22⋊D251—44no—
99C2×Dic251—24no—
100C2×Dic251—22no—
101C2×Dic251—22no—
102C2×Dic251—42no—
103C4×D251—28no—
104C4×D251—28no—
105C23×C251—22no—
106C23×C251—22no—
107C23×C251—41no—
108C23×C251C2×D411yescharacteristic
109C23×C251C2×D411yescharacteristic
110C23×C251C2×D411yescharacteristic
111C2×C4×C251—22no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251—22no—
116C2×C4×C251C2×D414yes—
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251C2411yescharacteristic
121C8×C251—24no—
122C22×C251—21no—
123C22×C251—21no—
124C22×C251—21no—
125C22×C251—21no—
126C22×C251—21no—
127C22×C251—21no—
128C22×C251—21no—
129C22×C251—44no—
130C22×C251#12012yes—
131C22×C251AGL(1,ℤ/8)12yescomplemented
132C22×C251C22×D411yescharacteristic
133C2×D251—24no—
134C2×D251—48no—
135C2×D251—42no—
136C2×D251—42no—
137C2×D251—42no—
138C2×D251—42no—
139C2×D251—42no—
140C2×D251—84no—
141C4×C251—21no—
142C4×C251—21no—
143C4×C251#12012yes—
144C4×C251C22×D412yes—
145Dic251—24no—
146Dic251—42no—
147C2×C251—22no—
148C2×C251—22no—
149C2×C251—41no—
150C2×C251C2×#12011yescharacteristic
151C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C251#57511yescharacteristic
153D251—44no—
154D251—82no—
155C251#300611yescharacteristic, complemented, Sylow
156#3006—2511nomaximal, Sylow
157C22×#41—2511no—
158C2×AGL(1,ℤ/8)—2512no—
159C2×#121—2511no—
160#449—2514no—
161#448—2512no—
162#447—2512no—
163#417—2512no—
164#407D25111yescharacteristic, complemented
165C2×#41—5028no—
166C2×#41—5021no—
167C2×#41—5021no—
168C2×#41—2512no—
169C2×#41—2512no—
170C22×Q8—2511no—
171C22×D4—5021no—
172C22×D4—5021no—
173C22×D4—2511no—
174C22×D4—2511no—
175C23×C4—5021no—
176C23×C4C2×D25111yescharacteristic
177AGL(1,ℤ/8)—5028no—
178C2×SD16—2514no—
179C2×D8—2514no—
180C2×M16C2×D25112yescomplemented
181#121—5021no—
182#121—5021no—
183#121—2512no—
184#121—2512no—
185C2×C4⋊C4—2511no—
186C2×C22⋊C4—5021no—
187#103—2514no—
188#102—2514no—
189C22⋊C8C2×D25112yescomplemented
190C22⋊C8C2×D25112yescomplemented
191C24—5021no—
192C24—5021no—
193C24D100411yescharacteristic
194#41—5028no—
195#41—5028no—
196#41—5024no—
197#41—5024no—
198#41—5024no—
199#41—2518no—
200C2×Q8—5024no—
201C2×Q8—5021no—
202C2×Q8—2512no—
203C2×D4—10044no—
204C2×D4—5024no—
205C2×D4—5022no—
206C2×D4—5022no—
207C2×D4—5028no—
208C2×D4—5022no—
209C2×D4—5022no—
210C2×D4—5024no—
211C2×D4—5021no—
212C2×D4—5021no—
213C2×D4—2512no—
214C2×D4—2512no—
215C22×C4—24no—
216C22×C4—5028no—
217C22×C4—5021no—
218C22×C4—5021no—
219C22×C4—5022no—
220C22×C4—5022no—
221C22×C4D100411yescharacteristic
222C22×C4D100411yescharacteristic
223C22×C4D100411yescharacteristic
224C22×C4C22×D25111yescharacteristic
225C22×C4C22×D25111yescharacteristic
226C22×C4C22×D25111yescharacteristic
227SD16—5028no—
228D8—5028no—
229M16—24no—
230C2×C8C22×D25114yes—
231C4⋊C4—5021no—
232C4⋊C4—2512no—
233C22⋊C4—5022no—
234C22⋊C4—5022no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—10042no—
239C23—10042no—
240C23—10042no—
241C23—10042no—
242C23—10042no—
243C23—5024no—
244C23—5022no—
245C23—5022no—
246C23C2×D100411yescharacteristic
247C23C2×D100411yescharacteristic
248C23C2×D100411yescharacteristic
249Q8—5024no—
250Q8—5022no—
251Q8—2514no—
252D4—10044no—
253D4—10044no—
254D4—10042no—
255D4—5024no—
256D4—5028no—
257D4—5028no—
258D4—5024no—
259D4—5022no—
260D4—2514no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—10042no—
267C2×C4—5028no—
268C2×C4—5022no—
269C2×C4—5024no—
270C2×C4—5022no—
271C2×C4—5028no—
272C2×C4D4×D25114yes—
273C2×C4C2×D100411yescharacteristic
274C2×C4C2×D100411yescharacteristic
275C2×C4C2×D100411yescharacteristic
276C2×C4C23×D25111yescharacteristic
277C8—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—20084no—
287C22—10048no—
288C22—10042no—
289C22—10042no—
290C22—10042no—
291C22—10042no—
292C22—10042no—
293C22—5024no—
294C22#2254868512yes—
295C22#2254874312yescomplemented
296C22C22×D100411yescharacteristic
297C4—21no—
298C4—21no—
299C4—10042no—
300C4—5024no—
301C4#2254868512yes—
302C4C2×D4×D25112yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—20082no—
307C2—10044no—
308C2C2×#2254868511yescharacteristic
309C2C2×#2254874311yescharacteristic
310C2#3063248711yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#407×C251C2
socleC22×C251C22×D4
hypercenter#407D251
hypocenterC251#3006
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#16860810C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548689C2semidirect
4. #30631834C2semidirect
5. #30631828C2semidirect
6. #30631823C2semidirect
7. #30631689C2semidirect
8. #407×C251C2semidirect
13. C2×#16860810C22semidirect
14. C22×Dic502C22semidirect
15. C22×D1004C22semidirect
16. C22×D1004C22semidirect
21. C2×#16860799C22semidirect
22. C2×D2008C22semidirect
26. #22548689C22semidirect
27. C2×C4⋊Dic251C22semidirect
29. #22548640C22semidirect
30. #22548637C22semidirect
32. C2×M16×C251C22semidirect
33. C22⋊C8×C251C22semidirect
34. C22⋊C8×C251C22semidirect
42. #16860810D4semidirect
89. Dic502C2×D4semidirect
94. D1004C2×D4semidirect
131. C22×C251AGL(1,ℤ/8)semidirect
155. C251#3006semidirect
164. #407D251semidirect
180. C2×M16C2×D251semidirect
189. C22⋊C8C2×D251semidirect
190. C22⋊C8C2×D251semidirect
295. C22#22548743semidirect

Maximal subgroups

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

subgroupindex
C22×#168608102
C2×#225487432
C2×#225486892
#306318342
#306318282
#306318232
#306316892
#407×C2512
#3006251