Finite Group Databasebeta

← #44559970

Subgroups — #44559970

Order 32128.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16860793C211yesmaximal, characteristic, complemented
2#30632194C212yesmaximal, complemented
3#30631942C212yesmaximal, complemented
4#30631937C212yesmaximal, complemented
5#30631930C214yesmaximal, complemented
6#30631725C211yesmaximal, characteristic, complemented
7#30631683C212yesmaximal, complemented
8C2×#135×C251C211yesmaximal, characteristic, complemented
9C22×C4×D251C2211yescharacteristic
10C2×#16860793—24no—
11C2×#16860793C2212yes—
12C2×#16860793C2212yes—
13C22×C251⋊C8C2211yescharacteristic
14#22548696—22no—
15#22548691C2214yes—
16#22548686C2214yes—
17#22548683—22no—
18C4×C22⋊D251—24no—
19#22548674C2212yes—
20#22548649C2212yescomplemented
21#22548647C2212yescomplemented
22#22548639C2214yescomplemented
23#22548636C2214yescomplemented
24C2×#41×C251C2212yes—
25#135×C251—24no—
26C2×Q16×C251C2211yescharacteristic, complemented
27C2×SD16×C251C2212yescomplemented
28C2×D8×C251C2211yescharacteristic, complemented
29C22×C8×C251C2211yescharacteristic
30C23×D251D411yescharacteristic
31C2×C22⋊D251—24no—
32C22×Dic251D411yescharacteristic
33C2×C4×D251—22no—
34C2×C4×D251—22no—
35C2×C4×D251D412yes—
36C2×C4×D251C2312yes—
37#16860793—24no—
38#16860793—24no—
39C2×C251⋊C8—22no—
40C2×C251⋊C8C2312yes—
41C4⋊Dic251C2314yes—
42Dic251⋊C4—24no—
43C22⋊Dic251—24no—
44C22⋊Dic251—22no—
45(C2×D251)⋊C4—24no—
46C4×Dic251—22no—
47#41×C251—24no—
48#41×C251—24no—
49C2×Q8×C251C2312yes—
50C2×D4×C251—22no—
51C2×D4×C251C2312yes—
52C22×C4×C251—22no—
53C22×C4×C251C2311yescharacteristic
54Q16×C251—22no—
55SD16×C251—24no—
56D8×C251—22no—
57C2×C8×C251D414yescomplemented
58C2×C8×C251C2311yescharacteristic
59C2×C8×C251C2311yescharacteristic
60C22×D251—22no—
61C22×D251—42no—
62C22×D251C2×D412yes—
63C22⋊D251—48no—
64C2×Dic251—22no—
65C2×Dic251—24no—
66C2×Dic251C2×D412yes—
67C4×D251—24no—
68C4×D251—22no—
69C4×D251—22no—
70C251⋊C8—22no—
71C23×C251—22no—
72C23×C251C2×D411yescharacteristic
73Q8×C251—22no—
74Q8×C251—22no—
75D4×C251—22no—
76D4×C251—22no—
77D4×C251—44no—
78C2×C4×C251—22no—
79C2×C4×C251—44no—
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D412yes—
82C2×C4×C251#4112yes—
83C2×C4×C251C2411yescharacteristic
84C8×C251C2×D412yes—
85C8×C251C2×D412yes—
86C22×C251—22no—
87C22×C251—42no—
88C22×C251—44no—
89C22×C251#12112yes—
90C22×C251C22×D411yescharacteristic
91C2×D251—22no—
92C2×D251—24no—
93C2×D251—44no—
94C4×C251—42no—
95C4×C251#12112yes—
96C4×C251C22×D411yescharacteristic
97C4×C251C2×#4111yescharacteristic
98Dic251—22no—
99Dic251—44no—
100C2×C251—22no—
101C2×C251—82no—
102C2×C251C2×#12111yescharacteristic
103C2×C251#57512yes—
104D251—42no—
105C251#305411yescharacteristic, complemented, Sylow
106#3054—2511nomaximal, Sylow
107C2×#135D25111yescharacteristic, complemented
108C22×M16—2511no—
109#525—2512no—
110#470—2512no—
111#469—2512no—
112#468—2514no—
113#427—2511no—
114#416—2512no—
115C2×#41C2×D25112yes—
116C23×C4—2511no—
117#135—24no—
118C2×Q16C2×D25111yescharacteristic, complemented
119C2×SD16C2×D25112yescomplemented
120C2×D8C2×D25111yescharacteristic, complemented
121C2×M16—5024no—
122C2×M16—2512no—
123C2×M16—2512no—
124C22×C8—2511no—
125C22×C8C2×D25111yescharacteristic
126#123—5022no—
127C22⋊Q8—2514no—
128#121—2514no—
129#120—5022no—
130C4×D4—5024no—
131#117—2512no—
132#107—2512no—
133#106—2512no—
134#103—2514no—
135#102—2514no—
136C24—2511no—
137#41—24no—
138#41—24no—
139C2×Q8C22×D25112yes—
140C2×D4—22no—
141C2×D4—5024no—
142C2×D4C22×D25112yes—
143C22×C4—22no—
144C22×C4—5022no—
145C22×C4—5022no—
146C22×C4—2511no—
147C22×C4—2512no—
148C22×C4—2512no—
149C22×C4C22×D25111yescharacteristic
150Q16—22no—
151SD16—24no—
152D8—22no—
153M16—5024no—
154M16—5024no—
155C2×C8—5022no—
156C2×C8—2512no—
157C2×C8C22⋊D25114yescomplemented
158C2×C8C22×D25111yescharacteristic
159C2×C8C22×D25111yescharacteristic
160C4⋊C4—5024no—
161C4⋊C4—2514no—
162C22⋊C4—5024no—
163C22⋊C4—5022no—
164C22⋊C4—5024no—
165C42—5022no—
166C23—22no—
167C23—10042no—
168C23—5022no—
169C23—2512no—
170C23D4×D25111yescharacteristic
171Q8—22no—
172Q8—22no—
173D4—22no—
174D4—22no—
175D4—44no—
176D4—10048no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—5024no—
180C2×C4—5022no—
181C2×C4—5022no—
182C2×C4—5022no—
183C2×C4—5024no—
184C2×C4—2512no—
185C2×C4D4×D25111yescharacteristic
186C2×C4C2×C22⋊D25112yes—
187C2×C4#1686080912yes—
188C2×C4C23×D25111yescharacteristic
189C8—5022no—
190C8C2×C22⋊D25112yes—
191C8C2×C22⋊D25112yes—
192C22—22no—
193C22—44no—
194C22—42no—
195C22—10044no—
196C22—5024no—
197C22—5022no—
198C22#2254868612yes—
199C22C2×D4×D25111yescharacteristic
200C4—42no—
201C4—10044no—
202C4—5022no—
203C4#2254868612yes—
204C4C22×C22⋊D25111yescharacteristic
205C4C2×#1686080911yescharacteristic
206C2—22no—
207C2—82no—
208C2—10042no—
209C2C2×#2254868611yescharacteristic
210C2#3063248412yes—
subgroupG/H
centerC2×C4D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#135×C251C2
socleC22×C251C22×D4
hypercenterC2×#135D251
hypocenterC251#3054
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×#16860793C2semidirect
2. #30632194C2semidirect
3. #30631942C2semidirect
4. #30631937C2semidirect
5. #30631930C2semidirect
6. #30631725C2semidirect
7. #30631683C2semidirect
8. C2×#135×C251C2semidirect
20. #22548649C22semidirect
21. #22548647C22semidirect
22. #22548639C22semidirect
23. #22548636C22semidirect
26. C2×Q16×C251C22semidirect
27. C2×SD16×C251C22semidirect
28. C2×D8×C251C22semidirect
57. C2×C8×C251D4semidirect
105. C251#3054semidirect
107. C2×#135D251semidirect
118. C2×Q16C2×D251semidirect
119. C2×SD16C2×D251semidirect
120. C2×D8C2×D251semidirect
157. C2×C8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168607932
#306321942
#306319422
#306319372
#306319302
#306317252
#306316832
C2×#135×C2512
#3054251