Finite Group Databasebeta

← #44549633

Subgroups — #44549633

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#30632410C211yesmaximal, characteristic
2#30632083C211yesmaximal, characteristic, complemented
3#30631964C211yesmaximal, characteristic
4#30631432C211yesmaximal, characteristic, complemented
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic
7#462×C251C211yesmaximal, characteristic, complemented
8#22548715C2211yescharacteristic
9#22548710—22no—
10#22548703—21no—
11#22548691—21no—
12C4×Dic502—21no—
13C4×D1004C2211yescharacteristic
14#22548673—21no—
15#22548648—21no—
16#22548646—22no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548635—21no—
19C4×C251⋊C8C2211yescharacteristic
20C2×Q16×C251—21no—
21C4⋊Q8×C251C2211yescharacteristic
22C4×Q8×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25C2×Dic502—21no—
26C2×D1004D411yescharacteristic
27C2×C4×D251—21no—
28C2×C251⋊C8—21no—
29C2×C251⋊C8—21no—
30C4⋊Dic251—21no—
31C4⋊Dic251D412yes—
32C4⋊Dic251D411yescharacteristic
33Dic251⋊C4—22no—
34(C2×D251)⋊C4—22no—
35(C2×D251)⋊C4—22no—
36(C2×D251)⋊C4—21no—
37C4×Dic251—22no—
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—21no—
49Dic502—41no—
50Dic502—41no—
51D1004—21no—
52D1004—21no—
53C2×Dic251—21no—
54C2×Dic251—22no—
55C4×D251—41no—
56C4×D251—41no—
57C251⋊C8—22no—
58C251⋊C8—41no—
59Q8×C251—21no—
60Q8×C251—21no—
61Q8×C251—41no—
62Q8×C251—41no—
63C2×C4×C251—21no—
64C2×C4×C251—21no—
65C2×C4×C251—41no—
66C2×C4×C251C2×D411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C8×C251—41no—
70C22×C251#12011yescharacteristic
71C2×D251—41no—
72C2×D251—41no—
73C2×D251—41no—
74C4×C251—21no—
75C4×C251—41no—
76C4×C251—41no—
77C4×C251—81no—
78C4×C251#13511yescharacteristic
79C4×C251#13511yescharacteristic
80C4×C251AGL(1,ℤ/8)11yescharacteristic
81C4×C251#13711yescharacteristic
82Dic251—41no—
83Dic251—42no—
84C2×C251#44911yescharacteristic
85C2×C251#45211yescharacteristic
86C2×C251#45611yescharacteristic
87D251—81no—
88C251#165211yescharacteristic, complemented, Sylow
89#1652—2511nomaximal, Sylow
90#556—2511no—
91#498—2511no—
92#474—2511no—
93#462D25111yescharacteristic, complemented
94#332—2511no—
95Q8⋊C8—2511no—
96D4⋊C8—2511no—
97C2×Q16—21no—
98C4⋊Q8—2511no—
99C4⋊Q8C2×D25111yescharacteristic
100#126—5022no—
101#124—5021no—
102C22⋊Q8—5021no—
103C4×Q8—5021no—
104C4×Q8C2×D25111yescharacteristic
105C4×D4—2511no—
106#117—5021no—
107#107—5021no—
108#106—5022no—
109C4⋊C8—2511no—
110C4⋊C8C2×D25111yescharacteristic
111#103—21no—
112#102—5021no—
113C4×C8—2511no—
114C2×Q8—21no—
115C2×Q8—5021no—
116C2×Q8C22⋊D25111yescharacteristic
117C2×D4—2511no—
118C22×C4—5021no—
119Q16—42no—
120C2×C8—21no—
121C2×C8—5021no—
122C2×C8—5021no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—5022no—
127C4⋊C4—5021no—
128C4⋊C4—2512no—
129C4⋊C4—2511no—
130C4⋊C4C22⋊D25111yescharacteristic
131C22⋊C4—5022no—
132C22⋊C4—5022no—
133C22⋊C4—5021no—
134C42—21no—
135C42—5022no—
136C42C22×D25111yescharacteristic
137C23—5021no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—10041no—
143Q8—10041no—
144D4—5021no—
145D4—5021no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—10041no—
150C2×C4—10041no—
151C2×C4—5021no—
152C2×C4—5022no—
153C2×C4D4×D25111yescharacteristic
154C2×C4D4×D25111yescharacteristic
155C2×C4C2×C22⋊D25111yescharacteristic
156C8—41no—
157C8—10041no—
158C8—5022no—
159C22—10041no—
160C22—10041no—
161C22—10041no—
162C22#2254868311yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—10041no—
168C4—10042no—
169C4#2254873511yescharacteristic
170C4#2254873811yescharacteristic
171C4#2254874111yescharacteristic
172C4#2254874711yescharacteristic
173C2—20081no—
174C2#3063183011yescharacteristic
175C2#3063184811yescharacteristic
176C2#3063186811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#462×C251C2
socleC22×C251#120
prime radicalC2×D1004D4
hypercenter#462D251
hypocenterC251#1652
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #30632083C2semidirect
4. #30631432C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
7. #462×C251C2semidirect
88. C251#1652semidirect
93. #462D251semidirect

Maximal subgroups

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

subgroupindex
#306324102
#306320832
#306319642
#306314322
Q8⋊(C251⋊C8)2
#306314142
#462×C2512
#1652251