Finite Group Databasebeta

← #44549549

Subgroups — #44549549

Order 32128.

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

normal subgroups only

This group has 212 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#30632422C211yesmaximal, characteristic
2#30632035C211yesmaximal, characteristic
3#30631969C211yesmaximal, characteristic, complemented
4#30631430C211yesmaximal, characteristic
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631417C211yesmaximal, characteristic, complemented
7#462×C251C211yesmaximal, characteristic, complemented
8#22548712C2211yescharacteristic
9#22548704—21no—
10#22548703—21no—
11#22548700—22no—
12C4×Dic502C2211yescharacteristic
13C4×D1004—21no—
14#22548677—21no—
15#22548646—21no—
16C4⋊(C251⋊C8)C2211yescharacteristic
17#22548638—21no—
18#22548635—22no—
19C4×C251⋊C8C2211yescharacteristic
20C2×Q16×C251—21no—
21C4⋊Q8×C251C2211yescharacteristic
22C4×Q8×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25C2×Dic502D411yescharacteristic
26C2×D1004—21no—
27C2×D1004—21no—
28C2×D1004D412yes—
29C2×C4×D251—22no—
30C2×C251⋊C8—21no—
31C2×C251⋊C8—21no—
32C4⋊Dic251D411yescharacteristic
33Dic251⋊C4—21no—
34(C2×D251)⋊C4—22no—
35(C2×D251)⋊C4—22no—
36(C2×D251)⋊C4—22no—
37C4×Dic251—21no—
38C2×Q8×C251—21no—
39C2×Q8×C251D411yescharacteristic
40Q16×C251—42no—
41C2×C8×C251—21no—
42C4⋊C4×C251—21no—
43C4⋊C4×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251D411yescharacteristic
46C42×C251—21no—
47C42×C251C2311yescharacteristic
48C22×D251—22no—
49Dic502—21no—
50Dic502—21no—
51D1004—42no—
52D1004—41no—
53D1004—41no—
54D1004—41no—
55D1004—41no—
56C2×Dic251—21no—
57C2×Dic251—21no—
58C4×D251—82no—
59C251⋊C8—41no—
60C251⋊C8—41no—
61Q8×C251—21no—
62Q8×C251—21no—
63Q8×C251—41no—
64Q8×C251—41no—
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—41no—
68C2×C4×C251C2×D411yescharacteristic
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251C2×D411yescharacteristic
71C8×C251—41no—
72C22×C251#12011yescharacteristic
73C2×D251—42no—
74C2×D251—42no—
75C2×D251—42no—
76C4×C251—21no—
77C4×C251—41no—
78C4×C251—41no—
79C4×C251—81no—
80C4×C251#13511yescharacteristic
81C4×C251#13511yescharacteristic
82C4×C251AGL(1,ℤ/8)11yescharacteristic
83C4×C251#13711yescharacteristic
84Dic251—41no—
85Dic251—41no—
86C2×C251#44911yescharacteristic
87C2×C251#45211yescharacteristic
88C2×C251#45511yescharacteristic
89D251—82no—
90C251#164011yescharacteristic, complemented, Sylow
91#1640—2511nomaximal, Sylow
92#559—2511no—
93#486—2511no—
94#475—2511no—
95#462D25111yescharacteristic, complemented
96#331—2511no—
97Q8⋊C8—2511no—
98Q8⋊C8—2511no—
99C2×Q16—21no—
100C4⋊Q8C2×D25111yescharacteristic
101#127—2511no—
102#124—5021no—
103#124—5021no—
104#123—5022no—
105C4×Q8—2511no—
106C4×Q8C2×D25111yescharacteristic
107C4×D4—5021no—
108C4×D4—5021no—
109#106—5021no—
110C4⋊C8—2511no—
111C4⋊C8C2×D25111yescharacteristic
112#103—21no—
113#103—5021no—
114#102—5022no—
115C4×C8—2511no—
116C2×Q8—21no—
117C2×Q8—2511no—
118C2×Q8C22⋊D25111yescharacteristic
119C2×D4—5021no—
120C2×D4—5021no—
121C2×D4—2512no—
122C22×C4—5022no—
123Q16—42no—
124C2×C8—21no—
125C2×C8—5021no—
126C2×C8—5021no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—5021no—
131C4⋊C4—2511no—
132C4⋊C4C22⋊D25111yescharacteristic
133C22⋊C4—5022no—
134C22⋊C4—5022no—
135C22⋊C4—5022no—
136C42—21no—
137C42—5021no—
138C42C22×D25111yescharacteristic
139C23—5022no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—5021no—
145Q8—5021no—
146D4—10042no—
147D4—10041no—
148D4—10041no—
149D4—10041no—
150D4—10041no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—20082no—
155C2×C4—5021no—
156C2×C4—5021no—
157C2×C4D4×D25111yescharacteristic
158C2×C4D4×D25111yescharacteristic
159C2×C4C2×C22⋊D25111yescharacteristic
160C8—41no—
161C8—10041no—
162C8—10041no—
163C22—10042no—
164C22—10042no—
165C22—10042no—
166C22#2254868311yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—10041no—
172C4—10041no—
173C4#2254873511yescharacteristic
174C4#2254873811yescharacteristic
175C4#2254874111yescharacteristic
176C4#2254874711yescharacteristic
177C2—20082no—
178C2#3063183011yescharacteristic
179C2#3063184811yescharacteristic
180C2#3063186311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#462×C251C2
socleC22×C251#120
prime radical#22548712C22
hypercenter#462D251
hypocenterC251#1640
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #30631969C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
6. #30631417C2semidirect
7. #462×C251C2semidirect
90. C251#1640semidirect
95. #462D251semidirect

Maximal subgroups

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

subgroupindex
#306324222
#306320352
#306319692
#306314302
Q8⋊(C251⋊C8)2
#306314172
#462×C2512
#1640251