Finite Group Databasebeta

← #45663601

Subgroups — #45663601

Order 32896.

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#30657390C211yesmaximal, characteristic
2#30657014C211yesmaximal, characteristic, complemented
3#30656939C211yesmaximal, characteristic, complemented
4#30656483C211yesmaximal, characteristic, complemented
5#30656466C211yesmaximal, characteristic, complemented
6#30656466C211yesmaximal, characteristic, complemented
7#498×C257C211yesmaximal, characteristic, complemented
8C2×#16861619—21no—
9C2×#16861605—21no—
10#22551577—22no—
11#22551568—21no—
12#22551567—21no—
13C4×D1028C2211yescharacteristic
14C4×D1028C2211yescharacteristic
15C4⋊C4×D257—21no—
16#22551525—21no—
17C4⋊#14523339C2211yescharacteristic
18C4⋊#14523339C2211yescharacteristic
19#22551516—21no—
20#22551512—21no—
21C4⋊Q8×C257C2211yescharacteristic
22C4⋊Q8×C257C2211yescharacteristic
23#106×C257—22no—
24C4×C8×C257C2211yescharacteristic
25#16861619—42no—
26#16861619—42no—
27C2×D1028—22no—
28C2×D1028D411yescharacteristic
29C2×D1028D411yescharacteristic
30C2×C4×D257—22no—
31C2×C4×D257—21no—
32C2×C4×D257—21no—
33#16861605—42no—
34C2×#14523339—21no—
35C2×#14523339—21no—
36C4⋊Dic257D411yescharacteristic
37C4⋊Dic257D411yescharacteristic
38#16861597—22no—
39#16861595—22no—
40#16861595—21no—
41#16861595—21no—
42C2×Q8×C257—21no—
43C2×Q8×C257—21no—
44C2×C8×C257—21no—
45C4⋊C4×C257—21no—
46C4⋊C4×C257—21no—
47C4⋊C4×C257—21no—
48C4⋊C4×C257D412yes—
49C42×C257C2311yescharacteristic
50C22×D257—21no—
51C22×D257—21no—
52D1028—21no—
53D1028—21no—
54D1028—21no—
55D1028—44no—
56D1028SD1612yescomplemented
57C2×Dic257—21no—
58C2×Dic257—21no—
59C4×D257—44no—
60C4×D257—41no—
61C4×D257—41no—
62C4×D257—41no—
63C4×D257—41no—
64#14523339—41no—
65#14523339—41no—
66Q8×C257—41no—
67Q8×C257—41no—
68Q8×C257—41no—
69Q8×C257—41no—
70C2×C4×C257—22no—
71C2×C4×C257—41no—
72C2×C4×C257C2×D411yescharacteristic
73C2×C4×C257C2×D411yescharacteristic
74C2×C4×C257C2×D411yescharacteristic
75C8×C257—22no—
76C22×C257#12011yescharacteristic
77C2×D257—22no—
78C2×D257—41no—
79C2×D257—41no—
80C2×D257—41no—
81C2×D257—41no—
82C2×D257—41no—
83C4×C257—21no—
84C4×C257—42no—
85C4×C257—81no—
86C4×C257C2×SD1611yescharacteristic
87C4×C257#13511yescharacteristic
88C4×C257AGL(1,ℤ/8)11yescharacteristic
89C4×C257AGL(1,ℤ/8)11yescharacteristic
90Dic257—41no—
91Dic257—41no—
92C2×C257#44911yescharacteristic
93C2×C257#45011yescharacteristic
94C2×C257#45611yescharacteristic
95D257—42no—
96D257—81no—
97C257#165411yescharacteristic, complemented, Sylow
98#1654—2571nomaximal, Sylow
99#547—2571no—
100#498D25711yescharacteristic, complemented
101#474—2571no—
102#461—2571no—
103#332—2571no—
104D4⋊C8—2571no—
105D4⋊C8—2571no—
106C2×#41—5141no—
107C2×SD16—5141no—
108C4⋊Q8C2×D25711yescharacteristic
109C4⋊Q8C2×D25711yescharacteristic
110#123—5142no—
111C22⋊Q8—5141no—
112#121—5141no—
113C4×D4—2571no—
114C4×D4—2571no—
115C2×C4⋊C4—5141no—
116#107—5141no—
117#106—22no—
118C4⋊C8—2571no—
119C4⋊C8—2571no—
120#103—5141no—
121#102—5141no—
122C4×C8C2×D25711yescharacteristic
123#41—10282no—
124#41—10282no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—2571no—
128C2×D4—2571no—
129C2×D4—5142no—
130C22×C4—5142no—
131C22×C4—5141no—
132C22×C4—5141no—
133SD16—10282no—
134C2×C8—21no—
135C2×C8—5141no—
136C2×C8—5141no—
137C4⋊C4—2571no—
138C4⋊C4—2571no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—5142no—
143C4⋊C4C22⋊D25712yes—
144C22⋊C4—5142no—
145C22⋊C4—5141no—
146C22⋊C4—5141no—
147C42C22×D25711yescharacteristic
148C23—5141no—
149C23—5141no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—2572no—
155D4—5141no—
156D4—5141no—
157D4—5141no—
158D4—10284no—
159C2×C4—22no—
160C2×C4—5141no—
161C2×C4—5141no—
162C2×C4—41no—
163C2×C4—10284no—
164C2×C4—10281no—
165C2×C4—10281no—
166C2×C4—10281no—
167C2×C4—10281no—
168C2×C4D4×D25711yescharacteristic
169C2×C4D4×D25711yescharacteristic
170C2×C4C2×C22⋊D25711yescharacteristic
171C8—22no—
172C8—10281no—
173C8—10281no—
174C22—5142no—
175C22—10281no—
176C22—10281no—
177C22—10281no—
178C22—10281no—
179C22—10281no—
180C22#2255156011yescharacteristic
181C4—21no—
182C4—42no—
183C4—10281no—
184C4—10281no—
185C4—81no—
186C4SD16×D25711yescharacteristic
187C4#2255161311yescharacteristic
188C4#2255161911yescharacteristic
189C4#2255161911yescharacteristic
190C2—10282no—
191C2—20561no—
192C2#3065688111yescharacteristic
193C2#3065688611yescharacteristic
194C2#3065691911yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#498×C257C2
socleC22×C257#120
prime radical#30657390C2
hypercenter#498D257
hypocenterC257#1654
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #30657014C2semidirect
3. #30656939C2semidirect
4. #30656483C2semidirect
5. #30656466C2semidirect
6. #30656466C2semidirect
7. #498×C257C2semidirect
56. D1028SD16semidirect
97. C257#1654semidirect
100. #498D257semidirect

Maximal subgroups

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

subgroupindex
#306573902
#306570142
#306569392
#306564832
#306564662
#498×C2572
#1654257