Finite Group Databasebeta

← #44559644

Subgroups — #44559644

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
2C2×#22548748C211yesmaximal, characteristic, complemented
3C22×#16860799C211yesmaximal, characteristic, complemented
4#30632258C211yesmaximal, characteristic, complemented
5#30631851C214yesmaximal, complemented
6#30631840C214yesmaximal, complemented
7#30631693C211yesmaximal, characteristic, complemented
8#30631684C211yesmaximal, characteristic, complemented
9#408×C251C211yesmaximal, characteristic, complemented
10#22548761—24no—
11#22548761—24no—
12C2×#16860811—21no—
13C2×#16860810C2212yes—
14C2×#16860810C2211yescharacteristic
15C22×Dic502C2211yescharacteristic, complemented
16C22×D1004C2212yes—
17C22×D1004C2211yescharacteristic
18C2×D4×D251—22no—
19C2×D4×D251—21no—
20#22548748—24no—
21C2×Dic1004C2212yescomplemented
22C2×#16860799—24no—
23C2×#16860799C2214yescomplemented
24C2×#16860799C2212yescomplemented
25#22548713—21no—
26#22548702—21no—
27#22548696—22no—
28#22548694C2214yescomplemented
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—22no—
38C23×D251—21no—
39#16860811—28no—
40#16860810—24no—
41#16860810—24no—
42#16860810—22no—
43#16860810D414yes—
44C2×Dic502—21no—
45C2×Dic502—22no—
46C2×Dic502—22no—
47C2×Dic502C2312yes—
48C2×Dic502C2311yescharacteristic
49C2×D1004—24no—
50C2×D1004—24no—
51C2×D1004—22no—
52C2×D1004—21no—
53C2×D1004—22no—
54C2×D1004D414yes—
55C2×D1004C2311yescharacteristic
56C2×D1004C2312yes—
57C2×C22⋊D251—22no—
58C2×C22⋊D251—21no—
59D4×D251—216no—
60D4×D251—28no—
61C22×Dic251—21no—
62C2×C4×D251—22no—
63C2×C4×D251—21no—
64Dic1004—24no—
65#16860799—24no—
66#16860799—28no—
67C4⋊Dic251C2312yes—
68Dic251⋊C4—24no—
69C22⋊Dic251—21no—
70C22⋊Dic251—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—22no—
84C22×D251—21no—
85C22×D251—28no—
86C22×D251—24no—
87C22×D251—82no—
88C22×D251—84no—
89Dic502—21no—
90Dic502—22no—
91Dic502—22no—
92Dic502—42no—
93Dic502C2×D412yes—
94D1004—28no—
95D1004—24no—
96D1004—22no—
97D1004—21no—
98D1004C2×D412yes—
99D1004C2×D414yes—
100C22⋊D251—28no—
101C22⋊D251—24no—
102C2×Dic251—21no—
103C2×Dic251—22no—
104C2×Dic251—22no—
105C2×Dic251—42no—
106C4×D251—28no—
107C4×D251—24no—
108C23×C251C2×D412yes—
109C23×C251C2×D411yescharacteristic
110Q8×C251—22no—
111D4×C251—24no—
112D4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D412yes—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2411yescharacteristic
120C8×C251—22no—
121C8×C251—22no—
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—24no—
125C22×C251#12012yes—
126C22×C251C22×D411yescharacteristic
127C2×D251—24no—
128C2×D251—22no—
129C2×D251—44no—
130C2×D251—42no—
131C2×D251—84no—
132C2×D251—88no—
133C4×C251—22no—
134C4×C251#12012yes—
135C4×C251C22×D411yescharacteristic
136C4×C251C22×D411yescharacteristic
137Dic251—22no—
138Dic251—42no—
139Dic251—42no—
140C2×C251—22no—
141C2×C251—42no—
142C2×C251C2×#12011yescharacteristic
143C2×C251#57712yes—
144D251—44no—
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#418—2511no—
155#416—2511no—
156#408D25111yescharacteristic, complemented
157H(5,𝔽2)—5024no—
158H(5,𝔽2)—5024no—
159C2×#41—5021no—
160C2×#41—2512no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C22×Q8—2511no—
164C22×D4—5022no—
165C22×D4—5021no—
166C22×D4—2512no—
167C22×D4—2511no—
168#137—5024no—
169C2×Q16—2512no—
170C2×SD16—5024no—
171C2×SD16—2514no—
172C2×SD16—2512no—
173C2×M16C2×D25111yescharacteristic, complemented
174C22×C8C2×D25111yescharacteristic, complemented
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⋊C8C2×D25114yescomplemented
184C24—5022no—
185C24—5021no—
186#41—22no—
187#41—22no—
188#41—5024no—
189#41—5024no—
190#41—5028no—
191#41—5022no—
192#41—2514no—
193C2×Q8—5022no—
194C2×Q8—5022no—
195C2×Q8—5021no—
196C2×Q8—2512no—
197C2×Q8—2511no—
198C2×Q8D100411yescharacteristic
199C2×D4—5024no—
200C2×D4—5024no—
201C2×D4—50216no—
202C2×D4—5028no—
203C2×D4—5022no—
204C2×D4—5021no—
205C2×D4—5022no—
206C2×D4—5022no—
207C2×D4—5021no—
208C2×D4—2514no—
209C2×D4—2511no—
210C2×D4—2512no—
211C2×D4D100412yes—
212C2×D4D100411yescharacteristic
213C22×C4—5021no—
214C22×C4—5022no—
215C22×C4—5021no—
216C22×C4C22×D25112yes—
217C22×C4C22×D25111yescharacteristic
218Q16—5024no—
219SD16—5028no—
220SD16—5024no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D25112yes—
224C2×C8C22×D25112yes—
225C4⋊C4—5024no—
226C4⋊C4—2512no—
227C22⋊C4—5021no—
228C22⋊C4—5024no—
229C42—5021no—
230C23—20082no—
231C23—20084no—
232C23—5028no—
233C23—5024no—
234C23—5022no—
235C23—5021no—
236C23C2×D100412yes—
237C23C2×D100411yescharacteristic
238Q8—22no—
239Q8—10042no—
240Q8—5022no—
241Q8—5022no—
242Q8—5021no—
243Q8—2512no—
244D4—24no—
245D4—22no—
246D4—5028no—
247D4—5028no—
248D4—5024no—
249D4—5022no—
250D4—5021no—
251D4—5024no—
252D4—2512no—
253D4—2514no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—10042no—
257C2×C4—5028no—
258C2×C4—5022no—
259C2×C4—5022no—
260C2×C4—5021no—
261C2×C4—5024no—
262C2×C4D4×D25112yes—
263C2×C4D4×D25112yes—
264C2×C4C2×D100412yes—
265C2×C4C2×D100411yescharacteristic
266C2×C4C23×D25111yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—20084no—
273C22—20088no—
274C22—10044no—
275C22—10042no—
276C22—5024no—
277C22—5022no—
278C22#2254868512yes—
279C22C22×D100411yescharacteristic
280C4—22no—
281C4—10042no—
282C4—10042no—
283C4—5022no—
284C4#2254868512yes—
285C4C2×D4×D25111yescharacteristic
286C4C2×D4×D25111yescharacteristic
287C2—22no—
288C2—42no—
289C2—10044no—
290C2—10042no—
291C2C2×#2254868511yescharacteristic
292C2#3063249912yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#408×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548761C2
hypercenter#408D251
hypocenterC251#3009
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
2. C2×#22548748C2semidirect
3. C22×#16860799C2semidirect
4. #30632258C2semidirect
5. #30631851C2semidirect
6. #30631840C2semidirect
7. #30631693C2semidirect
8. #30631684C2semidirect
9. #408×C251C2semidirect
15. C22×Dic502C22semidirect
21. C2×Dic1004C22semidirect
23. C2×#16860799C22semidirect
24. C2×#16860799C22semidirect
28. #22548694C22semidirect
29. #22548674C22semidirect
30. #22548640C22semidirect
31. #22548637C22semidirect
32. #22548637C22semidirect
34. C2×M16×C251C22semidirect
35. C22×C8×C251C22semidirect
36. C22⋊C8×C251C22semidirect
146. C251#3009semidirect
156. #408D251semidirect
173. C2×M16C2×D251semidirect
174. C22×C8C2×D251semidirect
183. 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×#225487482
C22×#168607992
#306322582
#306318512
#306318402
#306316932
#306316842
#408×C2512
#3009251