Finite Group Databasebeta

← #45663506

Subgroups — #45663506

Order 32896.

212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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#30657083C211yesmaximal, characteristic
2#30657022C211yesmaximal, characteristic, complemented
3#30656947C211yesmaximal, characteristic, complemented
4#30656481C211yesmaximal, characteristic
5Q8⋊#14523339C211yesmaximal, characteristic, complemented
6Q8⋊#14523339C211yesmaximal, characteristic, complemented
7#559×C257C211yesmaximal, characteristic
8C2×#16861608—21no—
9#22551592C2211yescharacteristic
10#22551523—21no—
11C4⋊#14523339C2211yescharacteristic
12C4⋊#14523339C2211yescharacteristic
13#22551516—21no—
14#22551515—21no—
15#22551513—22no—
16C4×#14523339C2211yescharacteristic
17#127×C257C2211yescharacteristic
18#124×C257—21no—
19#124×C257—21no—
20#123×C257—22no—
21C4×Q8×C257C2211yescharacteristic
22C4×Q8×C257C2211yescharacteristic
23C4×D4×C257—21no—
24C4×D4×C257—21no—
25C2×Dic514—21no—
26#16861608—42no—
27C2×#14523339—21no—
28C2×#14523339—21no—
29C2×#14523339—21no—
30C4⋊Dic257—21no—
31C4⋊Dic257—21no—
32C2×Q8×C257D411yescharacteristic
33C2×Q8×C257D411yescharacteristic
34C2×D4×C257—21no—
35C2×D4×C257—21no—
36C2×D4×C257D412yes—
37C22×C4×C257—22no—
38C4⋊C4×C257—21no—
39C4⋊C4×C257—21no—
40C4⋊C4×C257D411yescharacteristic
41C4⋊C4×C257D411yescharacteristic
42C22⋊C4×C257—22no—
43C22⋊C4×C257—22no—
44C22⋊C4×C257—22no—
45C42×C257—21no—
46C42×C257—21no—
47C42×C257C2311yescharacteristic
48Dic514—41no—
49Dic514—41no—
50C2×Dic257—41no—
51#14523339—41no—
52#14523339—41no—
53#14523339—41no—
54C23×C257—22no—
55Q8×C257—21no—
56Q8×C257—21no—
57Q8×C257—21no—
58Q8×C257—21no—
59D4×C257—42no—
60D4×C257—41no—
61D4×C257—41no—
62D4×C257—41no—
63D4×C257—41no—
64C2×C4×C257—21no—
65C2×C4×C257—21no—
66C2×C4×C257—21no—
67C2×C4×C257—21no—
68C2×C4×C257—82no—
69C2×C4×C257C2×D411yescharacteristic
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D411yescharacteristic
72C22×C257—42no—
73C22×C257—42no—
74C22×C257—42no—
75C22×C257#12011yescharacteristic
76C4×C257—21no—
77C4×C257—41no—
78C4×C257—41no—
79C4×C257—41no—
80C4×C257—41no—
81C4×C257#13511yescharacteristic
82C4×C257#13511yescharacteristic
83C4×C257AGL(1,ℤ/8)11yescharacteristic
84C4×C257#13711yescharacteristic
85Dic257—81no—
86C2×C257—82no—
87C2×C257#44911yescharacteristic
88C2×C257#45211yescharacteristic
89C2×C257#45511yescharacteristic
90C257#164011yescharacteristic, complemented, Sylow
91#1640—2571nomaximal, Sylow
92#559D25711yescharacteristic
93#486—2571no—
94#475—2571no—
95#462—2571no—
96#331—2571no—
97Q8⋊C8—2571no—
98Q8⋊C8—2571no—
99C2×Q16—5141no—
100C4⋊Q8—2571no—
101#127C2×D25711yescharacteristic
102#124—21no—
103#124—21no—
104#123—22no—
105C4×Q8C2×D25711yescharacteristic
106C4×Q8C2×D25711yescharacteristic
107C4×D4—21no—
108C4×D4—21no—
109#106—5141no—
110C4⋊C8—2571no—
111C4⋊C8—2571no—
112#103—5141no—
113#103—5141no—
114#102—5142no—
115C4×C8—2571no—
116C2×Q8—5141no—
117C2×Q8C22⋊D25711yescharacteristic
118C2×Q8C22⋊D25711yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D25712yes—
122C22×C4—22no—
123Q16—10282no—
124C2×C8—5141no—
125C2×C8—5141no—
126C2×C8—5141no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—5141no—
130C4⋊C4—5141no—
131C4⋊C4C22⋊D25711yescharacteristic
132C4⋊C4C22⋊D25711yescharacteristic
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C42—21no—
137C42—21no—
138C42C22×D25711yescharacteristic
139C23—22no—
140Q8—21no—
141Q8—21no—
142Q8—21no—
143Q8—21no—
144Q8—10281no—
145Q8—10281no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—10281no—
156C2×C4—82no—
157C2×C4C2×C22⋊D25711yescharacteristic
158C2×C4C2×C22⋊D25711yescharacteristic
159C2×C4C2×C22⋊D25711yescharacteristic
160C8—10281no—
161C8—10281no—
162C8—10281no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#2255156111yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—20561no—
173C4#2255161211yescharacteristic
174C4#2255161211yescharacteristic
175C4#2255161711yescharacteristic
176C4#2255162211yescharacteristic
177C2—82no—
178C2#3065688211yescharacteristic
179C2#3065689911yescharacteristic
180C2#3065691511yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#559×C257C2
socleC22×C257#120
prime radical#127×C257C22
hypercenter#559D257
hypocenterC257#1640
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #30657022C2semidirect
3. #30656947C2semidirect
5. Q8⋊#14523339C2semidirect
6. Q8⋊#14523339C2semidirect
90. C257#1640semidirect

Maximal subgroups

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

subgroupindex
#306570832
#306570222
#306569472
#306564812
Q8⋊#145233392
#559×C2572
#1640257