Finite Group Databasebeta

← #44559635

Subgroups — #44559635

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×#22548743C211yesmaximal, characteristic, complemented
3C22×D2008C211yesmaximal, characteristic, complemented
4#30632251C211yesmaximal, characteristic, complemented
5#30631834C214yesmaximal, complemented
6#30631823C214yesmaximal, complemented
7#30631693C211yesmaximal, characteristic, complemented
8#30631684C211yesmaximal, characteristic, complemented
9#408×C251C211yesmaximal, characteristic, complemented
10#22548761—24no—
11#22548761—24no—
12C2×#16860811—21no—
13C2×#16860810C2211yescharacteristic
14C2×#16860810C2212yescomplemented
15C22×D1004C2211yescharacteristic
16C22×D1004C2211yescharacteristic, complemented
17C22×D1004C2212yescomplemented
18C2×D4×D251—22no—
19C2×D4×D251—21no—
20#22548743—24no—
21C2×#16860799C2212yescomplemented
22C2×D2008—24no—
23C2×D2008C2214yescomplemented
24C2×D2008C2212yescomplemented
25#22548711—21no—
26#22548704—21no—
27#22548689C2214yescomplemented
28#22548683—22no—
29#22548674C2211yescharacteristic, complemented
30#22548640C2212yescomplemented
31#22548637C2212yescomplemented
32#22548637C2214yescomplemented
33C2×#41×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic, complemented
35C22×C8×C251C2211yescharacteristic, complemented
36C22⋊C8×C251C2214yescomplemented
37C23×D251—21no—
38C23×D251—22no—
39C23×D251—21no—
40#16860811—28no—
41#16860810—24no—
42#16860810—24no—
43#16860810—22no—
44#16860810D414yescomplemented
45C2×Dic502C2311yescharacteristic
46C2×D1004—24no—
47C2×D1004—24no—
48C2×D1004—21no—
49C2×D1004—22no—
50C2×D1004—22no—
51C2×D1004—22no—
52C2×D1004—22no—
53C2×D1004—21no—
54C2×D1004D414yescomplemented
55C2×D1004C2312yes—
56C2×D1004C2312yes—
57C2×D1004C2311yescharacteristic
58C2×C22⋊D251—21no—
59C2×C22⋊D251—22no—
60C2×C22⋊D251—24no—
61D4×D251—216no—
62D4×D251—28no—
63C2×C4×D251—21no—
64C2×C4×D251—22no—
65#16860799—24no—
66D2008—24no—
67D2008—28no—
68C4⋊Dic251C2312yes—
69C22⋊Dic251—21no—
70(C2×D251)⋊C4—24no—
71C4×Dic251—21no—
72#41×C251—22no—
73#41×C251—22no—
74C2×Q8×C251D411yescharacteristic
75C2×D4×C251D412yes—
76C2×D4×C251D411yescharacteristic
77C22×C4×C251C2312yes—
78C22×C4×C251C2311yescharacteristic
79M16×C251—22no—
80C2×C8×C251—22no—
81C2×C8×C251C2312yes—
82C2×C8×C251C2312yes—
83C22×D251—21no—
84C22×D251—22no—
85C22×D251—28no—
86C22×D251—24no—
87C22×D251—22no—
88C22×D251—42no—
89C22×D251—82no—
90C22×D251—84no—
91C22×D251—82no—
92Dic502—21no—
93Dic502C2×D412yes—
94D1004—28no—
95D1004—24no—
96D1004—22no—
97D1004—21no—
98D1004—22no—
99D1004—22no—
100D1004—42no—
101D1004C2×D412yes—
102D1004C2×D414yes—
103C22⋊D251—28no—
104C22⋊D251—24no—
105C22⋊D251—84no—
106C2×Dic251—21no—
107C2×Dic251—22no—
108C4×D251—28no—
109C4×D251—24no—
110C23×C251C2×D412yes—
111C23×C251C2×D411yescharacteristic
112Q8×C251—22no—
113D4×C251—24no—
114D4×C251—22no—
115C2×C4×C251—22no—
116C2×C4×C251—22no—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2411yescharacteristic
122C8×C251—22no—
123C8×C251—22no—
124C22×C251—22no—
125C22×C251—22no—
126C22×C251—24no—
127C22×C251#12012yes—
128C22×C251C22×D411yescharacteristic
129C2×D251—24no—
130C2×D251—22no—
131C2×D251—44no—
132C2×D251—42no—
133C2×D251—42no—
134C2×D251—44no—
135C2×D251—84no—
136C2×D251—88no—
137C2×D251—84no—
138C4×C251—22no—
139C4×C251#12012yes—
140C4×C251C22×D411yescharacteristic
141C4×C251C22×D411yescharacteristic
142Dic251—22no—
143Dic251—42no—
144C2×C251—22no—
145C2×C251—42no—
146C2×C251C2×#12011yescharacteristic
147C2×C251#57612yes—
148D251—44no—
149D251—42no—
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#416—2511no—
161#408D25111yescharacteristic, complemented
162H(5,𝔽2)—5024no—
163H(5,𝔽2)—5024no—
164C2×#41—5021no—
165C2×#41—2512no—
166C2×#41—2511no—
167C2×#41C2×D25111yescharacteristic
168C22×D4—5022no—
169C22×D4—5021no—
170C22×D4—2511no—
171C22×D4—2512no—
172C22×D4—2511no—
173AGL(1,ℤ/8)—5024no—
174C2×SD16—2512no—
175C2×D8—5024no—
176C2×D8—2514no—
177C2×D8—2512no—
178C2×M16C2×D25111yescharacteristic, complemented
179C22×C8C2×D25111yescharacteristic, complemented
180#127—5021no—
181#124—5021no—
182#121—2514no—
183#120—5022no—
184#117—2511no—
185#103—2512no—
186#102—2512no—
187#102—2514no—
188C22⋊C8C2×D25114yescomplemented
189C24—5021no—
190C24—5022no—
191C24—5021no—
192#41—22no—
193#41—22no—
194#41—5024no—
195#41—5024no—
196#41—5028no—
197#41—5022no—
198#41—2514no—
199C2×Q8—2511no—
200C2×Q8D100411yescharacteristic
201C2×D4—5024no—
202C2×D4—5024no—
203C2×D4—50216no—
204C2×D4—5028no—
205C2×D4—5024no—
206C2×D4—5022no—
207C2×D4—5021no—
208C2×D4—5022no—
209C2×D4—5022no—
210C2×D4—5022no—
211C2×D4—5021no—
212C2×D4—5022no—
213C2×D4—5021no—
214C2×D4—2514no—
215C2×D4—2512no—
216C2×D4—2511no—
217C2×D4—2512no—
218C2×D4D100412yes—
219C2×D4D100411yescharacteristic
220C22×C4—5022no—
221C22×C4—5021no—
222C22×C4C22×D25112yes—
223C22×C4C22×D25111yescharacteristic
224SD16—5024no—
225D8—5028no—
226D8—5024no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D25112yes—
230C2×C8C22×D25112yes—
231C4⋊C4—2512no—
232C22⋊C4—5021no—
233C22⋊C4—5024no—
234C42—5021no—
235C23—20082no—
236C23—20082no—
237C23—20084no—
238C23—10042no—
239C23—5028no—
240C23—5024no—
241C23—5022no—
242C23—5022no—
243C23—5021no—
244C23C2×D100412yes—
245C23C2×D100411yescharacteristic
246Q8—22no—
247Q8—5021no—
248Q8—2512no—
249D4—24no—
250D4—22no—
251D4—20084no—
252D4—10042no—
253D4—5028no—
254D4—5028no—
255D4—5024no—
256D4—5022no—
257D4—5022no—
258D4—5022no—
259D4—5021no—
260D4—5024no—
261D4—2512no—
262D4—2514no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—5028no—
266C2×C4—5022no—
267C2×C4—5021no—
268C2×C4—5024no—
269C2×C4D4×D25112yes—
270C2×C4D4×D25112yes—
271C2×C4C2×D100412yes—
272C2×C4C2×D100411yescharacteristic
273C2×C4C23×D25111yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—20084no—
280C22—20084no—
281C22—20088no—
282C22—10044no—
283C22—10042no—
284C22—10044no—
285C22—10042no—
286C22—5024no—
287C22—5022no—
288C22#2254868512yes—
289C22C22×D100411yescharacteristic
290C4—22no—
291C4—10042no—
292C4—5022no—
293C4#2254868512yes—
294C4C2×D4×D25111yescharacteristic
295C4C2×D4×D25111yescharacteristic
296C2—22no—
297C2—42no—
298C2—20082no—
299C2—10044no—
300C2—10042no—
301C2C2×#2254868511yescharacteristic
302C2#3063249212yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#408×C251C2
socleC22×C251C22×D4
hypercenter#408D251
hypocenterC251#3008
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2semidirect
2. C2×#22548743C2semidirect
3. C22×D2008C2semidirect
4. #30632251C2semidirect
5. #30631834C2semidirect
6. #30631823C2semidirect
7. #30631693C2semidirect
8. #30631684C2semidirect
9. #408×C251C2semidirect
14. C2×#16860810C22semidirect
16. C22×D1004C22semidirect
17. C22×D1004C22semidirect
21. C2×#16860799C22semidirect
23. C2×D2008C22semidirect
24. C2×D2008C22semidirect
27. #22548689C22semidirect
29. #22548674C22semidirect
30. #22548640C22semidirect
31. #22548637C22semidirect
32. #22548637C22semidirect
34. C2×M16×C251C22semidirect
35. C22×C8×C251C22semidirect
36. C22⋊C8×C251C22semidirect
44. #16860810D4semidirect
54. C2×D1004D4semidirect
151. C251#3008semidirect
161. #408D251semidirect
178. C2×M16C2×D251semidirect
179. C22×C8C2×D251semidirect
188. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
C2×#225487432
C22×D20082
#306322512
#306318342
#306318232
#306316932
#306316842
#408×C2512
#3008251