Finite Group Databasebeta

← #44549489

Subgroups — #44549489

Order 32128.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632357C211yesmaximal, characteristic
2#30631979C211yesmaximal, characteristic, complemented
3#30631974C211yesmaximal, characteristic, complemented
4#30631431C211yesmaximal, characteristic, complemented
5#30631419C211yesmaximal, characteristic, complemented
6#30631416C211yesmaximal, characteristic, complemented
7#492×C251C211yesmaximal, characteristic, complemented
8C2×#16860809—21no—
9#22548711—21no—
10#22548687—22no—
11#22548686—21no—
12C4×Dic502C2211yescharacteristic
13C4×D1004C2211yescharacteristic
14D4×Dic251—21no—
15#22548648—21no—
16#22548646—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18C4⋊(C251⋊C8)C2211yescharacteristic
19#22548638—21no—
20#22548635—21no—
21C2×SD16×C251—22no—
22C4⋊Q8×C251C2211yescharacteristic
23#127×C251C2211yescharacteristic
24C4×C8×C251C2211yescharacteristic
25#16860809—42no—
26#16860809—42no—
27C2×Dic502D411yescharacteristic
28C2×D1004D411yescharacteristic
29C2×C22⋊D251—22no—
30C2×C22⋊D251—22no—
31C22×Dic251—22no—
32C2×C4×D251—21no—
33C2×C251⋊C8—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251D411yescharacteristic
36C4⋊Dic251D411yescharacteristic
37Dic251⋊C4—21no—
38C22⋊Dic251—22no—
39(C2×D251)⋊C4—21no—
40C4×Dic251—21no—
41C2×Q8×C251—21no—
42C2×D4×C251—21no—
43C2×D4×C251—21no—
44C2×D4×C251D412yes—
45SD16×C251—44no—
46C2×C8×C251—21no—
47C4⋊C4×C251—21no—
48C4⋊C4×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51Dic502—21no—
52Dic502SD1612yescomplemented
53D1004—21no—
54D1004—21no—
55C22⋊D251—44no—
56C22⋊D251—82no—
57C2×Dic251—21no—
58C2×Dic251—21no—
59C2×Dic251—21no—
60C2×Dic251—44no—
61C4×D251—41no—
62C4×D251—41no—
63C251⋊C8—41no—
64C251⋊C8—41no—
65C23×C251—22no—
66Q8×C251—42no—
67D4×C251—42no—
68D4×C251—41no—
69D4×C251—41no—
70D4×C251—41no—
71D4×C251—41no—
72C2×C4×C251—41no—
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251C2×D411yescharacteristic
76C8×C251—22no—
77C22×C251—42no—
78C22×C251—42no—
79C22×C251—42no—
80C22×C251#12011yescharacteristic
81C2×D251—41no—
82C2×D251—41no—
83C2×D251—41no—
84C4×C251—21no—
85C4×C251—81no—
86C4×C251C2×SD1611yescharacteristic
87C4×C251#13511yescharacteristic
88C4×C251AGL(1,ℤ/8)11yescharacteristic
89C4×C251#13711yescharacteristic
90Dic251—22no—
91Dic251—41no—
92Dic251—41no—
93C2×C251—82no—
94C2×C251#45011yescharacteristic
95C2×C251#45211yescharacteristic
96C2×C251#45311yescharacteristic
97D251—81no—
98C251#163111yescharacteristic, complemented, Sylow
99#1631—2511nomaximal, Sylow
100#550—2511no—
101#492D25111yescharacteristic, complemented
102#477—2511no—
103#476—2511no—
104#331—2511no—
105Q8⋊C8—2511no—
106D4⋊C8—2511no—
107C2×#41—5021no—
108C2×SD16—22no—
109C4⋊Q8C2×D25111yescharacteristic
110#127—5021no—
111#127C2×D25111yescharacteristic
112#121—5022no—
113#121—5021no—
114C4×Q8—2511no—
115C4×D4—5021no—
116C4×D4—2511no—
117#107—5021no—
118#106—5021no—
119C4⋊C8—2511no—
120C4⋊C8—2511no—
121#103—5021no—
122#102—5021no—
123C4×C8C2×D25111yescharacteristic
124#41—10042no—
125#41—10042no—
126C2×Q8—21no—
127C2×Q8—2511no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—5022no—
131C2×D4—5022no—
132C2×D4—2511no—
133C2×D4C22⋊D25112yes—
134C22×C4—5022no—
135C22×C4—5021no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—5021no—
139C2×C8—5021no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—5021no—
143C4⋊C4—2511no—
144C4⋊C4—2511no—
145C22⋊C4—5022no—
146C22⋊C4—5021no—
147C42—5021no—
148C42C22×D25111yescharacteristic
149C23—22no—
150C23—5021no—
151Q8—42no—
152Q8—5021no—
153Q8—2512no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—20082no—
160D4—10044no—
161D4—5021no—
162D4—5021no—
163C2×C4—41no—
164C2×C4—10044no—
165C2×C4—10041no—
166C2×C4—10041no—
167C2×C4—5021no—
168C2×C4—5021no—
169C2×C4—5021no—
170C2×C4D4×D25111yescharacteristic
171C2×C4D4×D25111yescharacteristic
172C2×C4C2×C22⋊D25111yescharacteristic
173C8—22no—
174C8—10041no—
175C8—10041no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—10041no—
180C22—10041no—
181C22—10041no—
182C22#2254868311yescharacteristic
183C4—21no—
184C4—81no—
185C4—10041no—
186C4—10041no—
187C4—5022no—
188C4SD16×D25111yescharacteristic
189C4#2254873611yescharacteristic
190C4#2254874211yescharacteristic
191C4#2254874411yescharacteristic
192C2—82no—
193C2—20081no—
194C2#3063183611yescharacteristic
195C2#3063184611yescharacteristic
196C2#3063185411yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#492×C251C2
socleC22×C251#120
prime radical#30632357C2
hypercenter#492D251
hypocenterC251#1631
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #30631979C2semidirect
3. #30631974C2semidirect
4. #30631431C2semidirect
5. #30631419C2semidirect
6. #30631416C2semidirect
7. #492×C251C2semidirect
52. Dic502SD16semidirect
98. C251#1631semidirect
101. #492D251semidirect

Maximal subgroups

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

subgroupindex
#306323572
#306319792
#306319742
#306314312
#306314192
#306314162
#492×C2512
#1631251