Finite Group Databasebeta

← #44559889

Subgroups — #44559889

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×#22548745C211yesmaximal, characteristic, complemented
2C22×#16860799C211yesmaximal, characteristic, complemented
3#30632251C211yesmaximal, characteristic
4#30632164C211yesmaximal, characteristic, complemented
5#30631912C214yesmaximal, complemented
6#30631886C214yesmaximal, complemented
7#30631713C211yesmaximal, characteristic, complemented
8#30631691C211yesmaximal, characteristic, complemented
9#416×C251C211yesmaximal, characteristic, complemented
10C22×Dic502C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12#22548745—24no—
13C2×#16860800C2212yescomplemented
14C2×#16860799—24no—
15C2×#16860799C2214yes—
16C2×#16860798C2212yescomplemented
17C2×#16860794C2211yescharacteristic
18#22548711C2212yes—
19#22548704C2212yes—
20#22548689—22no—
21#22548683—22no—
22C4×Dic502—22no—
23#22548680C2214yescomplemented
24#22548674C2211yescharacteristic
25#22548671—22no—
26Dic251⋊C8C2214yescomplemented
27#22548635C2214yes—
28C2×#41×C251C2211yescharacteristic
29C22×C8×C251C2211yescharacteristic
30#117×C251C2211yescharacteristic
31#103×C251C2212yescomplemented
32#102×C251C2212yescomplemented
33C23×D251—21no—
34C2×Dic502—21no—
35C2×Dic502—22no—
36C2×Dic502D414yescomplemented
37C2×Dic502C2312yes—
38C2×D1004—22no—
39C2×D1004—21no—
40C2×D1004—22no—
41C2×D1004C2312yes—
42C2×C22⋊D251—24no—
43C22×Dic251—21no—
44#16860800—24no—
45#16860799—28no—
46#16860798—24no—
47#16860794—22no—
48C2×C251⋊C8C2312yes—
49C4⋊Dic251D412yes—
50Dic251⋊C4—24no—
51C22⋊Dic251D412yes—
52(C2×D251)⋊C4—24no—
53C4×Dic251—24no—
54C4×Dic251C2312yes—
55#41×C251—22no—
56#41×C251—22no—
57C2×Q8×C251C2311yescharacteristic
58C2×D4×C251—21no—
59C2×D4×C251C2311yescharacteristic
60C22×C4×C251—21no—
61C22×C4×C251C2311yescharacteristic
62C2×C8×C251—22no—
63C2×C8×C251C2312yes—
64C4⋊C4×C251C2312yes—
65C22⋊C4×C251—21no—
66C42×C251—21no—
67C22×D251—22no—
68C22×D251—42no—
69C22×D251—82no—
70Dic502—24no—
71Dic502—22no—
72Dic502C2×D414yes—
73D1004—22no—
74D1004—22no—
75D1004—42no—
76C22⋊D251—84no—
77C2×Dic251—24no—
78C2×Dic251—22no—
79C2×Dic251C2×D414yes—
80C251⋊C8—22no—
81C23×C251—21no—
82C23×C251C2×D411yescharacteristic
83Q8×C251—21no—
84Q8×C251—21no—
85D4×C251—21no—
86D4×C251—21no—
87D4×C251—42no—
88C2×C4×C251—22no—
89C2×C4×C251—21no—
90C2×C4×C251—42no—
91C2×C4×C251C2×D411yescharacteristic
92C2×C4×C251C2×D412yes—
93C2×C4×C251#4112yes—
94C2×C4×C251C2411yescharacteristic
95C8×C251—22no—
96C22×C251—22no—
97C22×C251—41no—
98C22×C251—42no—
99C22×C251#12112yes—
100C22×C251C22×D411yescharacteristic
101C2×D251—42no—
102C2×D251—44no—
103C2×D251—84no—
104C4×C251—41no—
105C4×C251—42no—
106C4×C251#12112yes—
107C4×C251C22×D411yescharacteristic
108C4×C251C2×#4111yescharacteristic
109Dic251—24no—
110Dic251—24no—
111C2×C251—22no—
112C2×C251—81no—
113C2×C251C2×#12111yescharacteristic
114C2×C251#57712yes—
115D251—82no—
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#418—2511no—
126#416D25111yescharacteristic, complemented
127C2×#41C2×D25111yescharacteristic
128C22×Q8—2511no—
129C22×D4—2511no—
130#137—5024no—
131C2×Q16—2512no—
132C2×SD16—5024no—
133C2×SD16—2512no—
134C2×SD16—2514no—
135C2×M16—2511no—
136C22×C8C2×D25111yescharacteristic
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#103C2×D25112yescomplemented
148#102—2514no—
149#102C2×D25112yescomplemented
150C24—5021no—
151#41—22no—
152#41—22no—
153C2×Q8—5021no—
154C2×Q8—5022no—
155C2×Q8—2514no—
156C2×Q8—2512no—
157C2×Q8C22×D25111yescharacteristic
158C2×D4—21no—
159C2×D4—5024no—
160C2×D4—5021no—
161C2×D4—5022no—
162C2×D4—5022no—
163C2×D4—2512no—
164C2×D4C22×D25111yescharacteristic
165C22×C4—21no—
166C22×C4—5021no—
167C22×C4C22×D25111yescharacteristic
168Q16—5024no—
169SD16—5028no—
170SD16—5024no—
171M16—5022no—
172C2×C8—22no—
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—21no—
185C23—20082no—
186C23—10042no—
187C23—5022no—
188C23D4×D25111yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—5022no—
192Q8—5024no—
193Q8—2514no—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—20084no—
198D4—10042no—
199D4—5022no—
200D4—5022no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—42no—
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—42no—
215C22—41no—
216C22—20084no—
217C22—10044no—
218C22—10042no—
219C22#2254868812yes—
220C22C2×D4×D25111yescharacteristic
221C4—42no—
222C4—41no—
223C4—5024no—
224C4—5024no—
225C4#2254868812yes—
226C4C2×D4×D25111yescharacteristic
227C4C2×#1686081011yescharacteristic
228C2—22no—
229C2—81no—
230C2—20082no—
231C2C2×#2254868811yescharacteristic
232C2#3063249412yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
prime radical#30632251C2
hypercenter#416D251
hypocenterC251#3043
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#22548745C2semidirect
2. C22×#16860799C2semidirect
4. #30632164C2semidirect
5. #30631912C2semidirect
6. #30631886C2semidirect
7. #30631713C2semidirect
8. #30631691C2semidirect
9. #416×C251C2semidirect
13. C2×#16860800C22semidirect
16. C2×#16860798C22semidirect
23. #22548680C22semidirect
26. Dic251⋊C8C22semidirect
31. #103×C251C22semidirect
32. #102×C251C22semidirect
36. C2×Dic502D4semidirect
116. C251#3043semidirect
126. #416D251semidirect
147. #103C2×D251semidirect
149. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487452
C22×#168607992
#306322512
#306321642
#306319122
#306318862
#306317132
#306316912
#416×C2512
#3043251