Finite Group Databasebeta

← #44549485

Subgroups — #44549485

Order 32128.

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

normal subgroups only

This group has 218 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#30632061C211yesmaximal, characteristic
3#30631979C211yesmaximal, characteristic, complemented
4#30631430C211yesmaximal, characteristic
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631417C211yesmaximal, characteristic, complemented
7#477×C251C211yesmaximal, characteristic, complemented
8C2×#16860797—22no—
9#22548712C2211yescharacteristic
10#22548704—21no—
11#22548703—21no—
12#22548700—22no—
13C4×Dic502C2211yescharacteristic
14C4×D1004—21no—
15#22548677—21no—
16#22548648—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548638—21no—
19C4×C251⋊C8C2211yescharacteristic
20C4⋊Q8×C251C2211yescharacteristic
21C4×Q8×C251C2211yescharacteristic
22#107×C251—21no—
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25C2×Dic502D411yescharacteristic
26C2×D1004—21no—
27C2×D1004—21no—
28C2×D1004D412yes—
29C2×C4×D251—22no—
30#16860797—44no—
31C2×C251⋊C8—21no—
32C2×C251⋊C8—21no—
33C4⋊Dic251D411yescharacteristic
34Dic251⋊C4—21no—
35(C2×D251)⋊C4—22no—
36(C2×D251)⋊C4—22no—
37(C2×D251)⋊C4—22no—
38C4×Dic251—21no—
39C2×Q8×C251—21no—
40C2×Q8×C251D411yescharacteristic
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—22no—
60C251⋊C8—41no—
61Q8×C251—21no—
62Q8×C251—21no—
63Q8×C251—42no—
64C2×C4×C251—21no—
65C2×C4×C251—21no—
66C2×C4×C251—41no—
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C2×C4×C251C2×D411yescharacteristic
70C8×C251—41no—
71C22×C251#12011yescharacteristic
72C2×D251—42no—
73C2×D251—42no—
74C2×D251—42no—
75C4×C251—21no—
76C4×C251—41no—
77C4×C251—41no—
78C4×C251—81no—
79C4×C251#13511yescharacteristic
80C4×C251#13511yescharacteristic
81C4×C251#13711yescharacteristic
82C4×C251#13711yescharacteristic
83Dic251—41no—
84Dic251—41no—
85C2×C251#45211yescharacteristic
86C2×C251#45211yescharacteristic
87C2×C251#45311yescharacteristic
88D251—82no—
89C251#163011yescharacteristic, complemented, Sylow
90#1630—2511nomaximal, Sylow
91#559—2511no—
92#492—2511no—
93#477—2511no—
94#477D25111yescharacteristic, complemented
95#331—2511no—
96Q8⋊C8—2511no—
97Q8⋊C8—2511no—
98C2×SD16—5022no—
99C4⋊Q8C2×D25111yescharacteristic
100#127—2511no—
101#124—5021no—
102#124—5021no—
103#123—5022no—
104C4×Q8—2511no—
105C4×Q8C2×D25111yescharacteristic
106C4×D4—5021no—
107C4×D4—5021no—
108#107—21no—
109#107—5021no—
110C4⋊C8—2511no—
111C4⋊C8C2×D25111yescharacteristic
112#103—21no—
113#103—5021no—
114C4×C8—2511no—
115C2×Q8—21no—
116C2×Q8—2511no—
117C2×Q8C22⋊D25111yescharacteristic
118C2×D4—5021no—
119C2×D4—5021no—
120C2×D4—2512no—
121C22×C4—5022no—
122SD16—10044no—
123C2×C8—21no—
124C2×C8—5021no—
125C2×C8—5021no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—5021no—
130C4⋊C4—2511no—
131C4⋊C4C22⋊D25111yescharacteristic
132C22⋊C4—5022no—
133C22⋊C4—5022no—
134C22⋊C4—5022no—
135C42—21no—
136C42—5021no—
137C42C22×D25111yescharacteristic
138C23—5022no—
139Q8—21no—
140Q8—21no—
141Q8—42no—
142Q8—5021no—
143Q8—5021no—
144D4—10042no—
145D4—10041no—
146D4—10041no—
147D4—10041no—
148D4—10041no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—20082no—
153C2×C4—5021no—
154C2×C4—5021no—
155C2×C4D4×D25111yescharacteristic
156C2×C4D4×D25111yescharacteristic
157C2×C4C2×C22⋊D25111yescharacteristic
158C8—41no—
159C8—10041no—
160C8—5022no—
161C22—10042no—
162C22—10042no—
163C22—10042no—
164C22#2254868311yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—81no—
169C4—10041no—
170C4—10041no—
171C4#2254873511yescharacteristic
172C4#2254873811yescharacteristic
173C4#2254874611yescharacteristic
174C4#2254874711yescharacteristic
175C2—20082no—
176C2#3063184711yescharacteristic
177C2#3063184811yescharacteristic
178C2#3063185311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#477×C251C2
socleC22×C251#120
prime radical#22548712C22
hypercenter#477D251
hypocenterC251#1630
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #30631979C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
6. #30631417C2semidirect
7. #477×C251C2semidirect
89. C251#1630semidirect
94. #477D251semidirect

Maximal subgroups

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

subgroupindex
#306324222
#306320612
#306319792
#306314302
Q8⋊(C251⋊C8)2
#306314172
#477×C2512
#1630251