Finite Group Databasebeta

← #45664510

Subgroups — #45664510

Order 32896.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C22×C4⋊Dic257C211yesmaximal, characteristic
2C2×#22551526C212yesmaximal, complemented
3#30656597C211yesmaximal, characteristic
4#30656500C212yesmaximal, complemented
5C2×C22⋊C8×C257C211yesmaximal, characteristic
6C23×Dic257—21no—
7C2×C4⋊Dic257—24no—
8C2×C4⋊Dic257—21no—
9C2×C4⋊Dic257C2211yescharacteristic
10C2×C4⋊Dic257C2211yescharacteristic
11C2×C4⋊Dic257C2212yes—
12C2×C22⋊Dic257—21no—
13#22551526—24no—
14#22551499—21no—
15C23×C4×C257C2211yescharacteristic
16C22×C8×C257C2212yes—
17C22⋊C8×C257C414yescomplemented
18C22×Dic257—24no—
19C22×Dic257—22no—
20C22×Dic257—21no—
21C22×Dic257—21no—
22C22×Dic257—42no—
23C22×Dic257—42no—
24C4⋊Dic257—24no—
25C4⋊Dic257—22no—
26C4⋊Dic257—22no—
27C4⋊Dic257—22no—
28C4⋊Dic257D414yes—
29C22⋊Dic257—42no—
30C24×C257D411yescharacteristic
31C22×C4×C257—22no—
32C22×C4×C257—22no—
33C22×C4×C257C2×C412yes—
34C22×C4×C257D411yescharacteristic
35C22×C4×C257Q812yes—
36C22×C4×C257C2311yescharacteristic
37C2×C8×C257—22no—
38C2×C8×C257—22no—
39C2×C8×C257C2×C414yescomplemented
40C2×Dic257—24no—
41C2×Dic257—48no—
42C2×Dic257—42no—
43C2×Dic257—42no—
44C2×Dic257—41no—
45C2×Dic257—41no—
46C2×Dic257—41no—
47C2×Dic257—41no—
48C2×Dic257—41no—
49C2×Dic257—41no—
50C23×C257—21no—
51C23×C257—21no—
52C23×C257—21no—
53C23×C257—21no—
54C23×C257C4⋊C412yes—
55C23×C257D812yes—
56C23×C257Q1612yes—
57C23×C257C2×D411yescharacteristic
58C2×C4×C257—21no—
59C2×C4×C257—21no—
60C2×C4×C257—24no—
61C2×C4×C257—24no—
62C2×C4×C257C4⋊C412yes—
63C2×C4×C257C22×C411yescharacteristic
64C2×C4×C257C2×D412yes—
65C2×C4×C257C2×Q811yescharacteristic
66C2×C4×C257#4112yes—
67C8×C257—24no—
68C22×C257—22no—
69C22×C257—22no—
70C22×C257—22no—
71C22×C257—22no—
72C22×C257—22no—
73C22×C257—22no—
74C22×C257#10714yescomplemented
75C22×C257C2×C4⋊C411yescharacteristic
76C22×C257#12011yescharacteristic
77C22×C257#12311yescharacteristic
78C22×C257C2×D811yescharacteristic
79C22×C257C2×Q1611yescharacteristic
80C22×C257AGL(1,ℤ/8)11yescharacteristic
81C22×C257#13711yescharacteristic
82C4×C257—22no—
83C4×C257C4×D412yes—
84C4×C257C22⋊Q812yes—
85Dic257—44no—
86Dic257—81no—
87Dic257—81no—
88C2×C257—24no—
89C2×C257#38511yescharacteristic
90C2×C257C2×#10711yescharacteristic
91C2×C257#42811yescharacteristic
92C2×C257#44711yescharacteristic
93C2×C257#45111yescharacteristic
94C2×C257#48011yescharacteristic
95C2×C257#48411yescharacteristic
96C257#185211yescharacteristic, complemented, Sylow
97#1852—2571nomaximal, Sylow
98C22×C4⋊C4—2571no—
99C2×#107—2572no—
100C2×C22⋊C8D25711yescharacteristic
101#380—2571no—
102#340—2572no—
103C23×C4—5141no—
104C23×C4C2×D25711yescharacteristic
105C22×C8C2×D25712yes—
106C2×C4⋊C4—2572no—
107C2×C4⋊C4—2571no—
108C2×C4⋊C4—2571no—
109C2×C4⋊C4—5144no—
110C2×C4⋊C4—5141no—
111C2×C22⋊C4—5141no—
112#107—5144no—
113C22⋊C8Dic25714yescomplemented
114#95—5141no—
115C24D102811yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—5144no—
119C22×C4—5142no—
120C22×C4—5141no—
121C22×C4—5141no—
122C22×C4—10282no—
123C22×C4—10282no—
124C22×C4C2×Dic25712yes—
125C22×C4D102811yescharacteristic
126C22×C4Dic51412yes—
127C22×C4C22×D25711yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×Dic25714yescomplemented
131C4⋊C4—2574no—
132C4⋊C4—5144no—
133C4⋊C4—5142no—
134C4⋊C4—5142no—
135C4⋊C4—5142no—
136C22⋊C4—10282no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊Dic25712yes—
142C23D205612yes—
143C23Dic102812yes—
144C23C2×D102811yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—5144no—
150C2×C4—10288no—
151C2×C4—10282no—
152C2×C4—10282no—
153C2×C4—10281no—
154C2×C4—10281no—
155C2×C4—10281no—
156C2×C4—10281no—
157C2×C4—10281no—
158C2×C4—10281no—
159C2×C4C4⋊Dic25712yes—
160C2×C4C22×Dic25711yescharacteristic
161C2×C4D4×D25712yes—
162C2×C4C2×Dic51411yescharacteristic
163C2×C4#1686161712yes—
164C8—24no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#2255152614yescomplemented
172C22C2×C4⋊Dic25711yescharacteristic
173C22#2255156211yescharacteristic
174C22#2255157611yescharacteristic
175C22C2×D205611yescharacteristic
176C22C2×Dic102811yescharacteristic
177C22#2255162011yescharacteristic
178C22#2255162511yescharacteristic
179C4—22no—
180C4—10284no—
181C4—20561no—
182C4—20561no—
183C4D4×Dic25712yes—
184C4#2255156912yes—
185C2—24no—
186C2#3065661211yescharacteristic
187C2C2×#2255152611yescharacteristic
188C2#3065677911yescharacteristic
189C2#3065687311yescharacteristic
190C2#3065689511yescharacteristic
191C2#3065705111yescharacteristic
192C2#3065707311yescharacteristic
subgroupG/H
centerC23C2×D1028
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
FittingC2×C22⋊C8×C257C2
socleC23×C257C2×D4
prime radicalC24×C257D4
cosocleC22×C4×C257C23
hypercenterC2×C22⋊C8D257
hypocenterC257#1852
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#22551526C2semidirect
4. #30656500C2semidirect
17. C22⋊C8×C257C4semidirect
39. C2×C8×C257C2×C4semidirect
74. C22×C257#107semidirect
96. C257#1852semidirect
113. C22⋊C8Dic257semidirect
130. C2×C8C2×Dic257semidirect
171. C22#22551526semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic2572
C2×#225515262
#306565972
#306565002
C2×C22⋊C8×C2572
#1852257