Finite Group Databasebeta

← #44549700

Subgroups — #44549700

Order 32128.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#30632441C211yesmaximal, characteristic
2#30632096C211yesmaximal, characteristic, complemented
3#30632057C211yesmaximal, characteristic, complemented
4#30632047C211yesmaximal, characteristic
5#30631428C211yesmaximal, characteristic, complemented
6#30631426C211yesmaximal, characteristic
7#330×C251C211yesmaximal, characteristic, complemented
8#22548715C2211yescharacteristic
9#22548714—21no—
10#22548710—21no—
11#22548710—21no—
12#22548707C2211yescharacteristic
13#22548703C2211yescharacteristic
14#22548700—21no—
15#22548695—21no—
16#22548693—21no—
17#22548649—21no—
18#22548647—21no—
19#22548640—21no—
20#22548638—21no—
21#22548638—21no—
22#22548637—21no—
23#22548626C2211yescharacteristic
24#125×C251C2211yescharacteristic
25#97×C251C2211yescharacteristic
26#97×C251C2211yescharacteristic
27C2×Dic502—21no—
28C2×Dic502—21no—
29C2×Dic502D411yescharacteristic
30C2×D1004D411yescharacteristic
31C2×C4×D251—21no—
32C2×C251⋊C8—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251D411yescharacteristic
37C4⋊Dic251D411yescharacteristic
38Dic251⋊C4—21no—
39Dic251⋊C4—21no—
40Dic251⋊C4—21no—
41(C2×D251)⋊C4—21no—
42(C2×D251)⋊C4—21no—
43(C2×D251)⋊C4—21no—
44(C2×D251)⋊C4—21no—
45(C2×D251)⋊C4—21no—
46C4×Dic251—21no—
47C2×C8×C251—21no—
48C2×C8×C251—21no—
49C4⋊C4×C251—21no—
50C4⋊C4×C251—21no—
51C4⋊C4×C251D411yescharacteristic
52C4⋊C4×C251D411yescharacteristic
53C42×C251C2311yescharacteristic
54C22×D251—21no—
55Dic502—41no—
56Dic502—42no—
57Dic502—81no—
58D1004—41no—
59C2×Dic251—21no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—41no—
63C4×D251—81no—
64C251⋊C8—41no—
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C2×C4×C251C2×D411yescharacteristic
70C8×C251—22no—
71C8×C251—41no—
72C22×C251#12011yescharacteristic
73C2×D251—41no—
74C2×D251—41no—
75C2×D251—41no—
76C4×C251—21no—
77C4×C251—21no—
78C4×C251—21no—
79C4×C251—41no—
80C4×C251—41no—
81Dic251—41no—
82Dic251—41no—
83Dic251—41no—
84Dic251—81no—
85C2×C251#45411yescharacteristic
86C2×C251#45511yescharacteristic
87C2×C251#45611yescharacteristic
88D251—81no—
89C251#166411yescharacteristic, complemented, Sylow
90#1664—2511nomaximal, Sylow
91#563—2511no—
92#501—2511no—
93#491—2511no—
94#489—2511no—
95#330—2511no—
96#330D25111yescharacteristic, complemented
97#329—2511no—
98C4⋊Q8—5021no—
99C4⋊Q8—2511no—
100#126—5021no—
101#126—5021no—
102#125—2511no—
103#125C2×D25111yescharacteristic
104#124—2511no—
105#123—5021no—
106C22⋊Q8—5021no—
107C22⋊Q8—5021no—
108#107—5021no—
109#106—5021no—
110#103—5021no—
111#103—5021no—
112#103—5021no—
113#102—5021no—
114#97—2511no—
115#97C2×D25111yescharacteristic
116#97C2×D25111yescharacteristic
117C2×Q8—5021no—
118C2×Q8—5021no—
119C2×Q8—2511no—
120C2×D4—2511no—
121C22×C4—5021no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—5021no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—5021no—
128C4⋊C4—5021no—
129C4⋊C4—5021no—
130C4⋊C4—5021no—
131C4⋊C4—5021no—
132C4⋊C4—5021no—
133C4⋊C4—2511no—
134C4⋊C4—2511no—
135C4⋊C4D100411yescharacteristic
136C4⋊C4D100411yescharacteristic
137C22⋊C4—5021no—
138C22⋊C4—5021no—
139C22⋊C4—5021no—
140C22⋊C4—5021no—
141C22⋊C4—5021no—
142C42—5021no—
143C42C22×D25111yescharacteristic
144C23—5021no—
145Q8—20081no—
146Q8—10042no—
147Q8—10041no—
148D4—10041no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—20081no—
152C2×C4—10041no—
153C2×C4—5021no—
154C2×C4—5021no—
155C2×C4—5021no—
156C2×C4D4×D25111yescharacteristic
157C2×C4D4×D25111yescharacteristic
158C2×C4C2×D100411yescharacteristic
159C8—22no—
160C8—41no—
161C8—10041no—
162C22—10041no—
163C22—10041no—
164C22—10041no—
165C22#2254868511yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—20081no—
172C4—10041no—
173C4—10041no—
174C4—10041no—
175C2—20081no—
176C2#3063186211yescharacteristic
177C2#3063186611yescharacteristic
178C2#3063187111yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#330×C251C2
socleC22×C251#120
prime radicalC2×D1004D4
hypercenter#330D251
hypocenterC251#1664
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #30632096C2semidirect
3. #30632057C2semidirect
5. #30631428C2semidirect
7. #330×C251C2semidirect
89. C251#1664semidirect
96. #330D251semidirect

Maximal subgroups

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

subgroupindex
#306324412
#306320962
#306320572
#306320472
#306314282
#306314262
#330×C2512
#1664251