Finite Group Databasebeta

← #44549646

Subgroups — #44549646

Order 32128.

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

normal subgroups only

This group has 240 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#30632340C211yesmaximal, characteristic
2#30631964C211yesmaximal, characteristic, complemented
3#30631889C211yesmaximal, characteristic, complemented
4#30631433C211yesmaximal, characteristic, complemented
5#30631416C211yesmaximal, characteristic, complemented
6#30631416C211yesmaximal, characteristic, complemented
7#498×C251C211yesmaximal, characteristic, complemented
8C2×#16860811—21no—
9C2×#16860797—21no—
10#22548700—22no—
11#22548691—21no—
12#22548690—21no—
13C4×D1004C2211yescharacteristic
14C4×D1004C2211yescharacteristic
15C4⋊C4×D251—21no—
16#22548648—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18C4⋊(C251⋊C8)C2211yescharacteristic
19#22548639—21no—
20#22548635—21no—
21C4⋊Q8×C251C2211yescharacteristic
22C4⋊Q8×C251C2211yescharacteristic
23#106×C251—22no—
24C4×C8×C251C2211yescharacteristic
25#16860811—42no—
26#16860811—42no—
27C2×D1004—22no—
28C2×D1004D411yescharacteristic
29C2×D1004D411yescharacteristic
30C2×C4×D251—22no—
31C2×C4×D251—21no—
32C2×C4×D251—21no—
33#16860797—42no—
34C2×C251⋊C8—21no—
35C2×C251⋊C8—21no—
36C4⋊Dic251D411yescharacteristic
37C4⋊Dic251D411yescharacteristic
38Dic251⋊C4—22no—
39(C2×D251)⋊C4—22no—
40(C2×D251)⋊C4—21no—
41(C2×D251)⋊C4—21no—
42C2×Q8×C251—21no—
43C2×Q8×C251—21no—
44C2×C8×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251—21no—
48C4⋊C4×C251D412yes—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51C22×D251—21no—
52D1004—21no—
53D1004—21no—
54D1004—21no—
55D1004—44no—
56D1004SD1612yescomplemented
57C2×Dic251—21no—
58C2×Dic251—21no—
59C4×D251—44no—
60C4×D251—41no—
61C4×D251—41no—
62C4×D251—41no—
63C4×D251—41no—
64C251⋊C8—41no—
65C251⋊C8—41no—
66Q8×C251—41no—
67Q8×C251—41no—
68Q8×C251—41no—
69Q8×C251—41no—
70C2×C4×C251—22no—
71C2×C4×C251—41no—
72C2×C4×C251C2×D411yescharacteristic
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251C2×D411yescharacteristic
75C8×C251—22no—
76C22×C251#12011yescharacteristic
77C2×D251—22no—
78C2×D251—41no—
79C2×D251—41no—
80C2×D251—41no—
81C2×D251—41no—
82C2×D251—41no—
83C4×C251—21no—
84C4×C251—42no—
85C4×C251—81no—
86C4×C251C2×SD1611yescharacteristic
87C4×C251#13511yescharacteristic
88C4×C251AGL(1,ℤ/8)11yescharacteristic
89C4×C251AGL(1,ℤ/8)11yescharacteristic
90Dic251—41no—
91Dic251—41no—
92C2×C251#44911yescharacteristic
93C2×C251#45011yescharacteristic
94C2×C251#45611yescharacteristic
95D251—42no—
96D251—81no—
97C251#165411yescharacteristic, complemented, Sylow
98#1654—2511nomaximal, Sylow
99#547—2511no—
100#498D25111yescharacteristic, complemented
101#474—2511no—
102#461—2511no—
103#332—2511no—
104D4⋊C8—2511no—
105D4⋊C8—2511no—
106C2×#41—5021no—
107C2×SD16—5021no—
108C4⋊Q8C2×D25111yescharacteristic
109C4⋊Q8C2×D25111yescharacteristic
110#123—5022no—
111C22⋊Q8—5021no—
112#121—5021no—
113C4×D4—2511no—
114C4×D4—2511no—
115C2×C4⋊C4—5021no—
116#107—5021no—
117#106—22no—
118C4⋊C8—2511no—
119C4⋊C8—2511no—
120#103—5021no—
121#102—5021no—
122C4×C8C2×D25111yescharacteristic
123#41—10042no—
124#41—10042no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—5022no—
128C2×D4—2511no—
129C2×D4—2511no—
130C22×C4—5022no—
131C22×C4—5021no—
132C22×C4—5021no—
133SD16—10042no—
134C2×C8—21no—
135C2×C8—5021no—
136C2×C8—5021no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—5022no—
141C4⋊C4—2511no—
142C4⋊C4—2511no—
143C4⋊C4C22⋊D25112yes—
144C22⋊C4—5022no—
145C22⋊C4—5021no—
146C22⋊C4—5021no—
147C42C22×D25111yescharacteristic
148C23—5021no—
149C23—5021no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—10044no—
155D4—5021no—
156D4—5021no—
157D4—5021no—
158D4—2512no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—10044no—
162C2×C4—10041no—
163C2×C4—10041no—
164C2×C4—10041no—
165C2×C4—10041no—
166C2×C4—5021no—
167C2×C4—5021no—
168C2×C4D4×D25111yescharacteristic
169C2×C4D4×D25111yescharacteristic
170C2×C4C2×C22⋊D25111yescharacteristic
171C8—22no—
172C8—10041no—
173C8—10041no—
174C22—10041no—
175C22—10041no—
176C22—10041no—
177C22—10041no—
178C22—10041no—
179C22—5022no—
180C22#2254868311yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—10041no—
185C4—10041no—
186C4SD16×D25111yescharacteristic
187C4#2254873611yescharacteristic
188C4#2254874211yescharacteristic
189C4#2254874211yescharacteristic
190C2—20081no—
191C2—10042no—
192C2#3063183111yescharacteristic
193C2#3063183611yescharacteristic
194C2#3063186911yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#498×C251C2
socleC22×C251#120
prime radical#30632340C2
hypercenter#498D251
hypocenterC251#1654
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #30631964C2semidirect
3. #30631889C2semidirect
4. #30631433C2semidirect
5. #30631416C2semidirect
6. #30631416C2semidirect
7. #498×C251C2semidirect
56. D1004SD16semidirect
97. C251#1654semidirect
100. #498D251semidirect

Maximal subgroups

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

subgroupindex
#306323402
#306319642
#306318892
#306314332
#306314162
#498×C2512
#1654251