Finite Group Databasebeta

← #45663444

Subgroups — #45663444

Order 32896.

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#30657407C211yesmaximal, characteristic
2#30657029C211yesmaximal, characteristic, complemented
3#30657024C211yesmaximal, characteristic, complemented
4#30656481C211yesmaximal, characteristic, complemented
5#30656469C211yesmaximal, characteristic, complemented
6#30656466C211yesmaximal, characteristic, complemented
7#492×C257C211yesmaximal, characteristic, complemented
8C2×#16861617—21no—
9#22551588—21no—
10#22551564—22no—
11#22551563—21no—
12C4×Dic514C2211yescharacteristic
13C4×D1028C2211yescharacteristic
14D4×Dic257—21no—
15#22551525—21no—
16#22551523—21no—
17C4⋊#14523339C2211yescharacteristic
18C4⋊#14523339C2211yescharacteristic
19#22551515—21no—
20#22551512—21no—
21C2×SD16×C257—22no—
22C4⋊Q8×C257C2211yescharacteristic
23#127×C257C2211yescharacteristic
24C4×C8×C257C2211yescharacteristic
25#16861617—42no—
26#16861617—42no—
27C2×Dic514D411yescharacteristic
28C2×D1028D411yescharacteristic
29C2×C22⋊D257—22no—
30C2×C22⋊D257—22no—
31C22×Dic257—22no—
32C2×C4×D257—21no—
33C2×#14523339—21no—
34C2×#14523339—21no—
35C4⋊Dic257D411yescharacteristic
36C4⋊Dic257D411yescharacteristic
37#16861597—21no—
38C22⋊Dic257—22no—
39#16861595—21no—
40C4×Dic257—21no—
41C2×Q8×C257—21no—
42C2×D4×C257—21no—
43C2×D4×C257—21no—
44C2×D4×C257D412yes—
45SD16×C257—44no—
46C2×C8×C257—21no—
47C4⋊C4×C257—21no—
48C4⋊C4×C257—21no—
49C42×C257C2311yescharacteristic
50C22×D257—21no—
51Dic514—21no—
52Dic514SD1612yescomplemented
53D1028—21no—
54D1028—21no—
55C22⋊D257—44no—
56C22⋊D257—82no—
57C2×Dic257—21no—
58C2×Dic257—21no—
59C2×Dic257—21no—
60C2×Dic257—44no—
61C4×D257—41no—
62C4×D257—41no—
63#14523339—41no—
64#14523339—41no—
65C23×C257—22no—
66Q8×C257—42no—
67D4×C257—42no—
68D4×C257—41no—
69D4×C257—41no—
70D4×C257—41no—
71D4×C257—41no—
72C2×C4×C257—41no—
73C2×C4×C257C2×D411yescharacteristic
74C2×C4×C257C2×D411yescharacteristic
75C2×C4×C257C2×D411yescharacteristic
76C8×C257—22no—
77C22×C257—42no—
78C22×C257—42no—
79C22×C257—42no—
80C22×C257#12011yescharacteristic
81C2×D257—41no—
82C2×D257—41no—
83C2×D257—41no—
84C4×C257—21no—
85C4×C257—81no—
86C4×C257C2×SD1611yescharacteristic
87C4×C257#13511yescharacteristic
88C4×C257AGL(1,ℤ/8)11yescharacteristic
89C4×C257#13711yescharacteristic
90Dic257—22no—
91Dic257—41no—
92Dic257—41no—
93C2×C257—82no—
94C2×C257#45011yescharacteristic
95C2×C257#45211yescharacteristic
96C2×C257#45311yescharacteristic
97D257—81no—
98C257#163111yescharacteristic, complemented, Sylow
99#1631—2571nomaximal, Sylow
100#550—2571no—
101#492D25711yescharacteristic, complemented
102#477—2571no—
103#476—2571no—
104#331—2571no—
105Q8⋊C8—2571no—
106D4⋊C8—2571no—
107C2×#41—5141no—
108C2×SD16—22no—
109C4⋊Q8C2×D25711yescharacteristic
110#127—5141no—
111#127C2×D25711yescharacteristic
112#121—5141no—
113#121—5142no—
114C4×Q8—2571no—
115C4×D4—2571no—
116C4×D4—5141no—
117#107—5141no—
118#106—5141no—
119C4⋊C8—2571no—
120C4⋊C8—2571no—
121#103—5141no—
122#102—5141no—
123C4×C8C2×D25711yescharacteristic
124#41—10282no—
125#41—10282no—
126C2×Q8—2571no—
127C2×Q8—21no—
128C2×D4—2571no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—5142no—
132C2×D4—5142no—
133C2×D4C22⋊D25712yes—
134C22×C4—5142no—
135C22×C4—5141no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—5141no—
139C2×C8—5141no—
140C4⋊C4—2571no—
141C4⋊C4—2571no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—5141no—
145C22⋊C4—5142no—
146C22⋊C4—5141no—
147C42—5141no—
148C42C22×D25711yescharacteristic
149C23—22no—
150C23—5141no—
151Q8—2572no—
152Q8—5141no—
153Q8—42no—
154D4—5141no—
155D4—5141no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—10284no—
162D4—20562no—
163C2×C4—5141no—
164C2×C4—5141no—
165C2×C4—5141no—
166C2×C4—41no—
167C2×C4—10284no—
168C2×C4—10281no—
169C2×C4—10281no—
170C2×C4D4×D25711yescharacteristic
171C2×C4D4×D25711yescharacteristic
172C2×C4C2×C22⋊D25711yescharacteristic
173C8—22no—
174C8—10281no—
175C8—10281no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—10281no—
180C22—10281no—
181C22—10281no—
182C22#2255156011yescharacteristic
183C4—21no—
184C4—5142no—
185C4—10281no—
186C4—10281no—
187C4—81no—
188C4SD16×D25711yescharacteristic
189C4#2255161311yescharacteristic
190C4#2255161911yescharacteristic
191C4#2255162111yescharacteristic
192C2—82no—
193C2—20561no—
194C2#3065688611yescharacteristic
195C2#3065689611yescharacteristic
196C2#3065690411yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#492×C257C2
socleC22×C257#120
prime radical#30657407C2
hypercenter#492D257
hypocenterC257#1631
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #30657029C2semidirect
3. #30657024C2semidirect
4. #30656481C2semidirect
5. #30656469C2semidirect
6. #30656466C2semidirect
7. #492×C257C2semidirect
52. Dic514SD16semidirect
98. C257#1631semidirect
101. #492D257semidirect

Maximal subgroups

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

subgroupindex
#306574072
#306570292
#306570242
#306564812
#306564692
#306564662
#492×C2572
#1631257