Finite Group Databasebeta

← #44559885

Subgroups — #44559885

Order 32128.

490 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548748C211yesmaximal, characteristic, complemented
2C22×#16860798C211yesmaximal, characteristic, complemented
3#30632254C211yesmaximal, characteristic
4#30632164C211yesmaximal, characteristic, complemented
5#30631908C214yesmaximal, complemented
6#30631886C214yesmaximal, complemented
7#30631715C211yesmaximal, characteristic, complemented
8#30631681C211yesmaximal, characteristic, complemented
9#418×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×Dic502C2211yescharacteristic
12#22548748—24no—
13C2×Dic1004C2212yescomplemented
14C2×#16860799C2212yescomplemented
15C2×#16860798—24no—
16C2×#16860798C2214yes—
17C22×C251⋊C8C2211yescharacteristic
18#22548711C2212yes—
19#22548702C2212yes—
20#22548686—22no—
21#22548684—22no—
22C4×Dic502—22no—
23#22548680C2214yescomplemented
24#22548674C2211yescharacteristic
25#22548671—22no—
26Dic251⋊C8C2214yescomplemented
27#22548638C2212yescomplemented
28#22548635C2212yescomplemented
29C22×D4×C251C2211yescharacteristic
30C2×M16×C251C2211yescharacteristic
31#117×C251C2211yescharacteristic
32#102×C251C2214yes—
33#16860810—22no—
34#16860810—22no—
35C2×Dic502—21no—
36C2×Dic502—22no—
37C2×Dic502D414yescomplemented
38C2×Dic502C2311yescharacteristic
39C2×Dic502C2312yes—
40C2×D1004C2311yescharacteristic
41C2×C22⋊D251—24no—
42C2×C22⋊D251—21no—
43C22×Dic251—21no—
44C2×C4×D251—21no—
45Dic1004—24no—
46#16860799—24no—
47#16860798—28no—
48C2×C251⋊C8—22no—
49C2×C251⋊C8C2312yes—
50C4⋊Dic251D412yes—
51Dic251⋊C4—24no—
52C22⋊Dic251—24no—
53C22⋊Dic251D412yes—
54C4×Dic251—24no—
55C4×Dic251C2312yes—
56C24×C251—21no—
57C2×D4×C251—22no—
58C2×D4×C251—21no—
59C2×D4×C251—22no—
60C2×D4×C251C2312yes—
61C22×C4×C251C2311yescharacteristic
62M16×C251—22no—
63C2×C8×C251C2312yes—
64C4⋊C4×C251C2312yes—
65C22⋊C4×C251—21no—
66C42×C251—21no—
67C22×D251—21no—
68Dic502—24no—
69Dic502—22no—
70Dic502—21no—
71Dic502—21no—
72Dic502C2×D414yes—
73D1004—21no—
74D1004—21no—
75C22⋊D251—42no—
76C22⋊D251—84no—
77C2×Dic251—24no—
78C2×Dic251—22no—
79C2×Dic251—21no—
80C2×Dic251C2×D414yes—
81C4×D251—42no—
82C251⋊C8—22no—
83C23×C251—22no—
84C23×C251—42no—
85C23×C251—82no—
86C23×C251C2×D411yescharacteristic
87D4×C251—22no—
88D4×C251—22no—
89D4×C251—42no—
90C2×C4×C251—22no—
91C2×C4×C251C2×D411yescharacteristic
92C2×C4×C251C2×D412yes—
93C2×C4×C251#4112yes—
94C2×C4×C251C2411yescharacteristic
95C8×C251—22no—
96C22×C251—22no—
97C22×C251—42no—
98C22×C251—44no—
99C22×C251—84no—
100C22×C251#12112yes—
101C22×C251C22×D411yescharacteristic
102C2×D251—41no—
103C2×D251—42no—
104C4×C251—42no—
105C4×C251#12112yes—
106C4×C251C22×D411yescharacteristic
107C4×C251C2×#4111yescharacteristic
108Dic251—24no—
109Dic251—24no—
110Dic251—41no—
111C2×C251—22no—
112C2×C251—82no—
113C2×C251C2×#12111yescharacteristic
114C2×C251#57712yes—
115D251—81no—
116C251#304311yescharacteristic, complemented, Sylow
117#3043—2511nomaximal, Sylow
118C2×#137—2511no—
119C22×SD16—2511no—
120#535—2511no—
121#519—2511no—
122#464—2514no—
123#460—2514no—
124#424—2511no—
125#418D25111yescharacteristic, complemented
126#416—2511no—
127C2×#41—2511no—
128C22×Q8—2511no—
129C22×D4C2×D25111yescharacteristic
130#137—5024no—
131C2×Q16—2512no—
132C2×SD16—5024no—
133C2×SD16—2514no—
134C2×SD16—2512no—
135C2×M16C2×D25111yescharacteristic
136C22×C8—2511no—
137#127—2512no—
138#124—2512no—
139#121—5022no—
140#120—5022no—
141C4×Q8—5022no—
142C4×Q8—2514no—
143#117—5022no—
144#117—2511no—
145#117C2×D25111yescharacteristic
146C4⋊C8—2514no—
147#103—2512no—
148#102—2512no—
149#102C2×D25114yes—
150C24—21no—
151#41—5022no—
152#41—5022no—
153C2×Q8—5021no—
154C2×Q8—5022no—
155C2×Q8—2514no—
156C2×Q8—2511no—
157C2×Q8—2512no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—5021no—
162C2×D4—5024no—
163C2×D4—2511no—
164C2×D4C22×D25112yes—
165C22×C4—5021no—
166C22×C4—5021no—
167C22×C4C22×D25111yescharacteristic
168Q16—5024no—
169SD16—5024no—
170SD16—5028no—
171M16—22no—
172C2×C8—5022no—
173C2×C8—2512no—
174C2×C8C22×D25112yes—
175C4⋊C4—5024no—
176C4⋊C4—2512no—
177C4⋊C4C22×D25112yes—
178C22⋊C4—21no—
179C22⋊C4—5024no—
180C22⋊C4—2512no—
181C42—21no—
182C42—5024no—
183C42—2512no—
184C23—22no—
185C23—42no—
186C23—82no—
187C23—5021no—
188C23D4×D25111yescharacteristic
189Q8—5021no—
190Q8—5021no—
191Q8—5022no—
192Q8—5024no—
193Q8—2514no—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—20084no—
198D4—10042no—
199D4—5021no—
200D4—5021no—
201C2×C4—22no—
202C2×C4—10042no—
203C2×C4—5021no—
204C2×C4—5022no—
205C2×C4—5024no—
206C2×C4—2514no—
207C2×C4D4×D25112yes—
208C2×C4D4×D25111yescharacteristic
209C2×C4#1686081012yes—
210C2×C4C23×D25111yescharacteristic
211C8—22no—
212C8—5022no—
213C22—22no—
214C22—44no—
215C22—42no—
216C22—84no—
217C22—10042no—
218C22—10041no—
219C22#2254868812yes—
220C22C2×D4×D25111yescharacteristic
221C4—42no—
222C4—10041no—
223C4—5024no—
224C4—5024no—
225C4#2254868812yes—
226C4C2×D4×D25111yescharacteristic
227C4C2×#1686081011yescharacteristic
228C2—22no—
229C2—82no—
230C2—20081no—
231C2C2×#2254868811yescharacteristic
232C2#3063249712yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
prime radical#30632254C2
hypercenter#418D251
hypocenterC251#3043
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
1. C2×#22548748C2semidirect
2. C22×#16860798C2semidirect
4. #30632164C2semidirect
5. #30631908C2semidirect
6. #30631886C2semidirect
7. #30631715C2semidirect
8. #30631681C2semidirect
9. #418×C251C2semidirect
13. C2×Dic1004C22semidirect
14. C2×#16860799C22semidirect
23. #22548680C22semidirect
26. Dic251⋊C8C22semidirect
27. #22548638C22semidirect
28. #22548635C22semidirect
37. C2×Dic502D4semidirect
116. C251#3043semidirect
125. #418D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487482
C22×#168607982
#306322542
#306321642
#306319082
#306318862
#306317152
#306316812
#418×C2512
#3043251