Finite Group Databasebeta

← #44549619

Subgroups — #44549619

Order 32128.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#30632407C211yesmaximal, characteristic
2#30632090C211yesmaximal, characteristic, complemented
3#30631969C211yesmaximal, characteristic, complemented
4#30631434C211yesmaximal, characteristic, complemented
5#30631417C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic
7#461×C251C211yesmaximal, characteristic
8#22548715C2211yescharacteristic
9#22548708—22no—
10#22548702—21no—
11#22548694—21no—
12C4×Dic502C2211yescharacteristic
13#22548680—21no—
14#22548674—21no—
15#22548648—22no—
16#22548646—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548638—21no—
19C4×C251⋊C8C2211yescharacteristic
20C2×SD16×C251—21no—
21C4⋊Q8×C251C2211yescharacteristic
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25C2×Dic502—21no—
26C2×Dic502D411yescharacteristic
27C2×C251⋊C8—21no—
28C2×C251⋊C8—21no—
29C4⋊Dic251—21no—
30C4⋊Dic251D411yescharacteristic
31C4⋊Dic251D412yes—
32Dic251⋊C4—22no—
33Dic251⋊C4—21no—
34C22⋊Dic251—22no—
35C22⋊Dic251—22no—
36C4×Dic251—22no—
37C4×Dic251—21no—
38C2×Q8×C251—21no—
39C2×D4×C251D411yescharacteristic
40C22×C4×C251—21no—
41SD16×C251—42no—
42C2×C8×C251—21no—
43C4⋊C4×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251D411yescharacteristic
46C22⋊C4×C251—21no—
47C42×C251C2311yescharacteristic
48Dic502—21no—
49Dic502—21no—
50Dic502—41no—
51Dic502—41no—
52C2×Dic251—21no—
53C2×Dic251—21no—
54C2×Dic251—22no—
55C251⋊C8—22no—
56C251⋊C8—41no—
57C23×C251—21no—
58Q8×C251—41no—
59Q8×C251—41no—
60D4×C251—21no—
61D4×C251—21no—
62C2×C4×C251—21no—
63C2×C4×C251—41no—
64C2×C4×C251—41no—
65C2×C4×C251—41no—
66C2×C4×C251C2×D411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C8×C251—41no—
70C22×C251—41no—
71C22×C251—41no—
72C22×C251—41no—
73C22×C251#12011yescharacteristic
74C4×C251—21no—
75C4×C251—41no—
76C4×C251—81no—
77C4×C251#13511yescharacteristic
78C4×C251#13511yescharacteristic
79C4×C251AGL(1,ℤ/8)11yescharacteristic
80C4×C251#13711yescharacteristic
81Dic251—42no—
82Dic251—41no—
83Dic251—41no—
84C2×C251—81no—
85C2×C251#44911yescharacteristic
86C2×C251#45211yescharacteristic
87C2×C251#45611yescharacteristic
88C251#165011yescharacteristic, complemented, Sylow
89#1650—2511nomaximal, Sylow
90#556—2511no—
91#500—2511no—
92#475—2511no—
93#461D25111yescharacteristic
94#332—2511no—
95Q8⋊C8—2511no—
96D4⋊C8—2511no—
97C2×SD16—21no—
98C4⋊Q8—2511no—
99C4⋊Q8C2×D25111yescharacteristic
100#126—5022no—
101#124—5021no—
102C22⋊Q8—5021no—
103C4×Q8—5021no—
104C4×Q8—2511no—
105C4×D4C2×D25111yescharacteristic
106#117—5021no—
107#107—5022no—
108#106—5021no—
109C4⋊C8—2511no—
110C4⋊C8C2×D25111yescharacteristic
111#103—21no—
112#103—5021no—
113C4×C8—2511no—
114C2×Q8—21no—
115C2×Q8—5021no—
116C2×Q8—2511no—
117C2×D4C22⋊D25111yescharacteristic
118C22×C4—21no—
119SD16—42no—
120C2×C8—21no—
121C2×C8—5021no—
122C2×C8—5021no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—5022no—
126C4⋊C4—5021no—
127C4⋊C4—5021no—
128C4⋊C4—2511no—
129C4⋊C4—2512no—
130C4⋊C4C22⋊D25111yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—5022no—
133C22⋊C4—5022no—
134C42—5022no—
135C42—5021no—
136C42C22×D25111yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—10041no—
141Q8—10041no—
142Q8—5021no—
143Q8—5021no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—5021no—
151C2×C4—5021no—
152C2×C4—5022no—
153C2×C4D4×D25111yescharacteristic
154C2×C4D4×D25111yescharacteristic
155C2×C4C2×C22⋊D25111yescharacteristic
156C8—41no—
157C8—10041no—
158C8—5022no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2254868311yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—10042no—
167C4—10041no—
168C4—10041no—
169C4#2254873511yescharacteristic
170C4#2254873611yescharacteristic
171C4#2254874111yescharacteristic
172C4#2254874411yescharacteristic
173C2—81no—
174C2#3063183011yescharacteristic
175C2#3063184611yescharacteristic
176C2#3063186811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#461×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenter#461D251
hypocenterC251#1650
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #30632090C2semidirect
3. #30631969C2semidirect
4. #30631434C2semidirect
5. #30631417C2semidirect
88. C251#1650semidirect

Maximal subgroups

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

subgroupindex
#306324072
#306320902
#306319692
#306314342
#306314172
D4⋊(C251⋊C8)2
#461×C2512
#1650251