Finite Group Databasebeta

← #45665488

Subgroups — #45665488

Order 32896.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#30657363C211yesmaximal, characteristic
2#30656598C212yesmaximal
3#30656598C211yesmaximal, characteristic
4#30656503C212yesmaximal
5#342×C257C211yesmaximal, characteristic
6C23×Dic257C2211yescharacteristic
7#22551571—22no—
8#22551569—24no—
9C2×C22⋊Dic257C2211yescharacteristic
10C2×C22⋊Dic257C2212yes—
11#22551499—24no—
12#22551499—22no—
13C2×C22⋊C4×C257C2212yes—
14C2×C22⋊C4×C257C2211yescharacteristic
15C2×Dic514—22no—
16C22×Dic257—22no—
17C22×Dic257—21no—
18C22×Dic257—21no—
19C22×Dic257—21no—
20C22×Dic257—22no—
21C22×Dic257D414yes—
22C22×Dic257Q812yescomplemented
23C4⋊Dic257—22no—
24#16861597—24no—
25C22⋊Dic257—22no—
26C22⋊Dic257—22no—
27C22⋊Dic257—22no—
28C24×C257C2311yescharacteristic
29C22×C4×C257—22no—
30C22×C4×C257D412yes—
31C22⋊C4×C257—22no—
32C22⋊C4×C257—22no—
33C22⋊C4×C257—22no—
34Dic514—82no—
35C2×Dic257—24no—
36C2×Dic257—24no—
37C2×Dic257—22no—
38C2×Dic257—41no—
39C2×Dic257—42no—
40C2×Dic257—41no—
41C2×Dic257—44no—
42C2×Dic257—42no—
43C2×Dic257—41no—
44C23×C257—42no—
45C23×C257C2×D412yes—
46C23×C257C2×D411yescharacteristic
47C23×C257C2×Q811yescharacteristic
48C23×C257#4111yescharacteristic
49C23×C257#4112yes—
50C2×C4×C257—22no—
51C2×C4×C257—42no—
52C2×C4×C257—44no—
53C2×C4×C257—42no—
54C22×C257—24no—
55C22×C257—22no—
56C22×C257—41no—
57C22×C257—41no—
58C22×C257—42no—
59C22×C257#12011yescharacteristic
60C22×C257C22⋊Q812yes—
61C22×C257C22⋊Q811yescharacteristic
62C22×C257#12412yes—
63C22×C257#12411yescharacteristic
64C4×C257—42no—
65C4×C257—82no—
66Dic257—42no—
67Dic257—44no—
68Dic257—81no—
69C2×C257—21no—
70C2×C257—22no—
71C2×C257—22no—
72C2×C257—21no—
73C2×C257#39311yescharacteristic
74C2×C257#45812yes—
75C257#203011yescharacteristic, complemented, Sylow
76#2030—2571nomaximal, Sylow
77#543—2571no—
78#381—2572no—
79#381—2571no—
80#342—2572no—
81#342D25711yescharacteristic
82C23×C4—2571no—
83C22⋊Q8—5144no—
84C22⋊Q8—5142no—
85C2×C22⋊C4—2572no—
86C2×C22⋊C4—2571no—
87C2×C22⋊C4C2×D25712yes—
88C2×C22⋊C4C2×D25711yescharacteristic
89#95—5144no—
90#95—5142no—
91C24C22×D25711yescharacteristic
92C2×Q8—5142no—
93C22×C4—2572no—
94C22×C4—2574no—
95C22×C4—22no—
96C22×C4—5142no—
97C22×C4—5141no—
98C22×C4—5141no—
99C22×C4—5142no—
100C22×C4—5141no—
101C22×C4D102812yes—
102C4⋊C4—5144no—
103C4⋊C4—5142no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—5142no—
108C22⋊C4—5142no—
109C22⋊C4—5142no—
110C23—42no—
111C23D4×D25712yes—
112C23C2×D102811yescharacteristic
113C23Q8×D25711yescharacteristic
114C23#1686161711yescharacteristic
115C23#1686161812yes—
116Q8—20562no—
117C2×C4—22no—
118C2×C4—5144no—
119C2×C4—5144no—
120C2×C4—5142no—
121C2×C4—42no—
122C2×C4—44no—
123C2×C4—42no—
124C2×C4—10281no—
125C2×C4—10282no—
126C2×C4—10281no—
127C2×C4—10284no—
128C2×C4—10282no—
129C2×C4—10281no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#2255156211yescharacteristic
136C22#2255157012yes—
137C22#2255157211yescharacteristic
138C22#2255157812yes—
139C22#2255158011yescharacteristic
140C4—42no—
141C4—10282no—
142C4—10284no—
143C4—82no—
144C4—20561no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#3065665011yescharacteristic
150C2#3065692812yes—
subgroupG/H
centerC22#22551562
commutator, prime radical, cosocleC24×C257C23
FrattiniC24C22×D257
Fitting#342×C257C2
socleC22×C257#120
hypercenter#342D257
hypocenterC257#2030
LCSG ▷ C24×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
22. C22×Dic257Q8semidirect
75. C257#2030semidirect

Maximal subgroups

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

subgroupindex
#306573632
#306565982
#306565032
#342×C2572
#2030257