Finite Group Databasebeta

← #44549563

Subgroups — #44549563

Order 32128.

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

normal subgroups only

This group has 250 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#30632358C211yesmaximal, characteristic
2#30632035C211yesmaximal, characteristic, complemented
3#30631969C211yesmaximal, characteristic, complemented
4#30631430C211yesmaximal, characteristic, complemented
5#30631417C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
7#461×C251C211yesmaximal, characteristic, complemented
8C2×#16860810—21no—
9#22548712C2211yescharacteristic
10#22548711—21no—
11#22548689—21no—
12#22548688—22no—
13C4×Dic502C2211yescharacteristic
14#22548677—21no—
15#22548646—21no—
16C4⋊(C251⋊C8)C2211yescharacteristic
17#22548638—21no—
18#22548635—22no—
19C4×C251⋊C8C2211yescharacteristic
20C2×SD16×C251—21no—
21C4⋊Q8×C251C2211yescharacteristic
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25#16860810—42no—
26#16860810—42no—
27C2×Dic502D411yescharacteristic
28C2×D1004—21no—
29C2×D1004—21no—
30C2×D1004D412yes—
31C2×C22⋊D251—22no—
32C2×C22⋊D251—22no—
33C2×C4×D251—22no—
34C2×C251⋊C8—21no—
35C2×C251⋊C8—21no—
36C4⋊Dic251D411yescharacteristic
37Dic251⋊C4—21no—
38(C2×D251)⋊C4—22no—
39C4×Dic251—21no—
40C2×Q8×C251—21no—
41C2×D4×C251D411yescharacteristic
42C22×C4×C251—21no—
43SD16×C251—42no—
44C2×C8×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C22⋊C4×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—22no—
51Dic502—21no—
52Dic502SD1612yescomplemented
53D1004—42no—
54D1004—41no—
55D1004—41no—
56D1004—41no—
57D1004—41no—
58C22⋊D251—44no—
59C22⋊D251—82no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C4×D251—44no—
63C251⋊C8—41no—
64C251⋊C8—41no—
65C23×C251—21no—
66Q8×C251—41no—
67Q8×C251—41no—
68D4×C251—21no—
69D4×C251—21no—
70C2×C4×C251—21no—
71C2×C4×C251—41no—
72C2×C4×C251—41no—
73C2×C4×C251—41no—
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251C2×D411yescharacteristic
76C2×C4×C251C2×D411yescharacteristic
77C8×C251—41no—
78C22×C251—41no—
79C22×C251—41no—
80C22×C251—41no—
81C22×C251#12011yescharacteristic
82C2×D251—42no—
83C2×D251—42no—
84C2×D251—42no—
85C4×C251—21no—
86C4×C251—41no—
87C4×C251—81no—
88C4×C251C2×SD1611yescharacteristic
89C4×C251#13511yescharacteristic
90C4×C251AGL(1,ℤ/8)11yescharacteristic
91C4×C251AGL(1,ℤ/8)11yescharacteristic
92Dic251—22no—
93Dic251—41no—
94C2×C251—81no—
95C2×C251#44911yescharacteristic
96C2×C251#45011yescharacteristic
97C2×C251#45511yescharacteristic
98D251—82no—
99C251#164211yescharacteristic, complemented, Sylow
100#1642—2511nomaximal, Sylow
101#550—2511no—
102#486—2511no—
103#475—2511no—
104#461D25111yescharacteristic, complemented
105#331—2511no—
106Q8⋊C8—2511no—
107D4⋊C8—2511no—
108C2×#41—5021no—
109C2×SD16—21no—
110C4⋊Q8C2×D25111yescharacteristic
111#127—5021no—
112#127—2511no—
113#121—5022no—
114#121—5021no—
115C4×Q8—2511no—
116C4×D4—5021no—
117C4×D4C2×D25111yescharacteristic
118#106—5021no—
119C4⋊C8—2511no—
120C4⋊C8C2×D25111yescharacteristic
121#103—21no—
122#103—5021no—
123#102—5022no—
124C4×C8—2511no—
125#41—10042no—
126#41—10042no—
127C2×Q8—21no—
128C2×Q8—2511no—
129C2×D4—5022no—
130C2×D4—5022no—
131C2×D4—5021no—
132C2×D4—5021no—
133C2×D4—2512no—
134C2×D4C22⋊D25111yescharacteristic
135C22×C4—21no—
136C22×C4—5022no—
137SD16—42no—
138C2×C8—21no—
139C2×C8—5021no—
140C2×C8—5021no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—5021no—
144C4⋊C4—2511no—
145C4⋊C4C22⋊D25111yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—5022no—
148C42—5021no—
149C42C22×D25111yescharacteristic
150C23—21no—
151C23—5022no—
152Q8—41no—
153Q8—41no—
154Q8—5021no—
155Q8—2512no—
156D4—21no—
157D4—21no—
158D4—20082no—
159D4—10042no—
160D4—10044no—
161D4—10041no—
162D4—10041no—
163D4—10041no—
164D4—10041no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—10044no—
170C2×C4—5021no—
171C2×C4—5021no—
172C2×C4D4×D25111yescharacteristic
173C2×C4D4×D25111yescharacteristic
174C2×C4C2×C22⋊D25111yescharacteristic
175C8—41no—
176C8—10041no—
177C8—10041no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—10042no—
182C22—10042no—
183C22—10042no—
184C22#2254868311yescharacteristic
185C4—21no—
186C4—41no—
187C4—81no—
188C4—10041no—
189C4—5022no—
190C4SD16×D25111yescharacteristic
191C4#2254873511yescharacteristic
192C4#2254874111yescharacteristic
193C4#2254874211yescharacteristic
194C2—81no—
195C2—20082no—
196C2#3063183011yescharacteristic
197C2#3063183611yescharacteristic
198C2#3063186311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#461×C251C2
socleC22×C251#120
prime radical#30632358C2
hypercenter#461D251
hypocenterC251#1642
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #30632035C2semidirect
3. #30631969C2semidirect
4. #30631430C2semidirect
5. #30631417C2semidirect
6. D4⋊(C251⋊C8)C2semidirect
7. #461×C251C2semidirect
52. Dic502SD16semidirect
99. C251#1642semidirect
104. #461D251semidirect

Maximal subgroups

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

subgroupindex
#306323582
#306320352
#306319692
#306314302
#306314172
D4⋊(C251⋊C8)2
#461×C2512
#1642251