Finite Group Databasebeta

← #45665461

Subgroups — #45665461

Order 32896.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#22551592C211yesmaximal, characteristic
2C2×#22551516C212yesmaximal
3#30656499C212yesmaximal
4#30656492C211yesmaximal, characteristic
5#391×C257C211yesmaximal, characteristic
6C22×Dic514—21no—
7C22×#14523339C2212yes—
8#22551592—24no—
9C2×C4⋊Dic257—21no—
10C2×C4⋊Dic257C2212yes—
11#22551516—24no—
12C22×Q8×C257C2211yescharacteristic
13C2×C4⋊C4×C257C2211yescharacteristic
14C2×C42×C257C2211yescharacteristic
15#95×C257—21no—
16#95×C257—21no—
17C2×Dic514—24no—
18C2×Dic514—42no—
19C2×Dic514—42no—
20C22×Dic257—22no—
21C2×#14523339—22no—
22C2×#14523339—22no—
23C2×#14523339—22no—
24C4⋊Dic257—24no—
25C4⋊Dic257—22no—
26C4⋊Dic257D414yes—
27C2×Q8×C257—21no—
28C2×Q8×C257—22no—
29C2×Q8×C257—22no—
30C2×Q8×C257Q812yescomplemented
31C22×C4×C257—21no—
32C22×C4×C257—21no—
33C22×C4×C257D411yescharacteristic
34C22×C4×C257D411yescharacteristic
35C22×C4×C257C2311yescharacteristic
36C4⋊C4×C257—22no—
37C42×C257—22no—
38Dic514—48no—
39C2×Dic257—24no—
40C2×Dic257—42no—
41C2×Dic257—42no—
42#14523339—42no—
43C23×C257C2×D411yescharacteristic
44Q8×C257—24no—
45Q8×C257—42no—
46C2×C4×C257—21no—
47C2×C4×C257—21no—
48C2×C4×C257—21no—
49C2×C4×C257—21no—
50C2×C4×C257—22no—
51C2×C4×C257—41no—
52C2×C4×C257—41no—
53C2×C4×C257—41no—
54C2×C4×C257—41no—
55C2×C4×C257—41no—
56C2×C4×C257SD1612yes—
57C2×C4×C257Q1612yes—
58C2×C4×C257C2×D412yes—
59C2×C4×C257C2×Q811yescharacteristic
60C2×C4×C257#4112yes—
61C2×C4×C257#4111yescharacteristic
62C22×C257#12011yescharacteristic
63C22×C257C22⋊Q811yescharacteristic
64C22×C257#12411yescharacteristic
65C22×C257C2×SD1611yescharacteristic
66C22×C257C2×Q1611yescharacteristic
67C22×C257AGL(1,ℤ/8)11yescharacteristic
68C22×C257#13711yescharacteristic
69C4×C257—22no—
70C4×C257—22no—
71C4×C257—42no—
72C4×C257—81no—
73C4×C257C22⋊Q812yes—
74C4×C257#12412yes—
75Dic257—44no—
76C2×C257#39311yescharacteristic
77C2×C257#45011yescharacteristic
78C2×C257#45111yescharacteristic
79C2×C257#47511yescharacteristic
80C2×C257#47711yescharacteristic
81C2×C257#48711yescharacteristic
82C2×C257#48911yescharacteristic
83C257#202411yescharacteristic, complemented, Sylow
84#2024—2571nomaximal, Sylow
85C2×C4⋊Q8—2571no—
86C2×#103—2572no—
87#391D25711yescharacteristic
88#340—2572no—
89#336—2571no—
90C22×Q8—5141no—
91C22×Q8C2×D25711yescharacteristic
92C22×C8—2572no—
93C4⋊Q8—5144no—
94C2×C4⋊C4—2572no—
95C2×C4⋊C4—5141no—
96C2×C4⋊C4C2×D25711yescharacteristic
97C2×C42C2×D25711yescharacteristic
98#103—5144no—
99#95—21no—
100#95—21no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—5144no—
105C2×Q8—10282no—
106C2×Q8—10282no—
107C2×Q8Dic51412yescomplemented
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—5142no—
111C22×C4C22⋊D25711yescharacteristic
112C22×C4C22⋊D25711yescharacteristic
113C22×C4C22×D25711yescharacteristic
114C2×C8—5142no—
115C2×C8—5142no—
116C2×C8—5142no—
117C4⋊C4—2574no—
118C4⋊C4—22no—
119C4⋊C4—5144no—
120C4⋊C4—5142no—
121C42—22no—
122C23C2×C22⋊D25711yescharacteristic
123Q8—24no—
124Q8—42no—
125Q8—10288no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—5144no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—10282no—
138C2×C4—10282no—
139C2×C4#1686160512yes—
140C2×C4#1686160812yes—
141C2×C4D4×D25712yes—
142C2×C4C2×Dic51411yescharacteristic
143C2×C4#1686161712yes—
144C2×C4#1686161811yescharacteristic
145C8—10282no—
146C22#2255156011yescharacteristic
147C22#2255157111yescharacteristic
148C22#2255157911yescharacteristic
149C22C2×#1686160511yescharacteristic
150C22C2×#1686160811yescharacteristic
151C22#2255161711yescharacteristic
152C22#2255162211yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—10284no—
157C4—81no—
158C4#2255156912yes—
159C4#2255157812yes—
160C2#3065664911yescharacteristic
161C2#3065688711yescharacteristic
162C2#3065689211yescharacteristic
163C2#3065702211yescharacteristic
164C2#3065703211yescharacteristic
165C2#3065708711yescharacteristic
166C2#3065709611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D257
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#391×C257C2
socle, prime radicalC23×C257C2×D4
hypercenter#391D257
hypocenterC257#2024
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C257Q8semidirect
83. C257#2024semidirect
107. C2×Q8Dic514semidirect

Maximal subgroups

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

subgroupindex
C2×#225515922
C2×#225515162
#306564992
#306564922
#391×C2572
#2024257