Finite Group Databasebeta

← #45663012

Subgroups — #45663012

Order 32896.

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

normal subgroups only

This group has 252 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#30657285C211yesmaximal, characteristic
2#30656793C211yesmaximal, characteristic, complemented
3C4⋊C8×D257C211yesmaximal, characteristic, complemented
4#30656485C211yesmaximal, characteristic, complemented
5#30656485C211yesmaximal, characteristic, complemented
6#30656481C211yesmaximal, characteristic, complemented
7#331×C257C211yesmaximal, characteristic, complemented
8C2×#16861617C412yes—
9C2×C8×D257—21no—
10#22551592C2211yescharacteristic
11#22551579C412yes—
12#22551563—21no—
13#22551563—21no—
14D4×Dic257—21no—
15D4×Dic257—21no—
16C42×D257C2211yescharacteristic
17C4⋊#14523339C2211yescharacteristic
18C4⋊#14523339C2211yescharacteristic
19#22551521—21no—
20#22551504—21no—
21#22551502—21no—
22#127×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24C4⋊C8×C257C2211yescharacteristic
25#16861617—44no—
26C2×Dic514C2×C412yes—
27C2×C22⋊D257—22no—
28C22×Dic257—22no—
29C2×C4×D257C2×C411yescharacteristic
30C2×C4×D257D411yescharacteristic
31C2×C4×D257D411yescharacteristic
32C2×#14523339—21no—
33C2×#14523339—21no—
34C8×D257—42no—
35C4⋊Dic257—21no—
36C4⋊Dic257—21no—
37C22⋊Dic257—22no—
38C22⋊Dic257—22no—
39C4×Dic257C2×C411yescharacteristic
40C4×Dic257D411yescharacteristic
41C4×Dic257D411yescharacteristic
42C2×D4×C257—21no—
43C2×D4×C257—21no—
44C2×D4×C257C2×C412yes—
45C2×C8×C257—21no—
46C2×C8×C257—21no—
47C42×C257C2311yescharacteristic
48C22×D257C22⋊C411yescharacteristic
49Dic514—42no—
50C22⋊D257—44no—
51C2×Dic257—22no—
52C2×Dic257—44no—
53C2×Dic257C22⋊C411yescharacteristic
54C2×Dic257C22⋊C411yescharacteristic
55C2×Dic257C22⋊C411yescharacteristic
56C4×D257—22no—
57C4×D257—21no—
58C4×D257—21no—
59C4×D257—21no—
60C4×D257—21no—
61#14523339—41no—
62#14523339—41no—
63C23×C257—22no—
64D4×C257—42no—
65D4×C257—41no—
66D4×C257—41no—
67D4×C257—41no—
68D4×C257—41no—
69C2×C4×C257C22×C411yescharacteristic
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D411yescharacteristic
72C8×C257—41no—
73C8×C257—41no—
74C22×C257—42no—
75C22×C257—42no—
76C22×C257—42no—
77C22×C257C2×C22⋊C411yescharacteristic
78C2×D257—21no—
79C2×D257—21no—
80C2×D257#10012yes—
81C4×C257—21no—
82C4×C257#13511yescharacteristic
83C4×C257#13511yescharacteristic
84C4×C257AGL(1,ℤ/8)11yescharacteristic
85C4×C257#13711yescharacteristic
86Dic257—21no—
87Dic257—21no—
88Dic257—21no—
89Dic257—42no—
90C2×C257—82no—
91C2×C257C2×#10011yescharacteristic
92C2×C257#41611yescharacteristic
93C2×C257#41711yescharacteristic
94D257—22no—
95C257#155111yescharacteristic, complemented, Sylow
96#1551—2571nomaximal, Sylow
97#532—2571no—
98#432—2571no—
99C2×C4⋊C8—2571no—
100#333—2571no—
101#333—2571no—
102#331—2571no—
103#331D25711yescharacteristic, complemented
104C2×#41—2572no—
105C22×C8—5141no—
106C4⋊Q8—2571no—
107#127C2×D25711yescharacteristic
108#124—2572no—
109#121—5141no—
110#121—5141no—
111C4×D4—5141no—
112C4×D4—5141no—
113C2×C42—2571no—
114C4⋊C8—2571no—
115C4⋊C8—2571no—
116C4⋊C8—5141no—
117C4⋊C8C2×D25711yescharacteristic
118C4⋊C8C2×D25711yescharacteristic
119C22⋊C8—5141no—
120#97—5141no—
121#41—10284no—
122C2×Q8—2572no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—5142no—
126C2×D4C4×D25712yes—
127C22×C4—2571no—
128C22×C4—2571no—
129C22×C4—2571no—
130C22×C4—5142no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—5141no—
134C2×C8—5141no—
135C2×C8—10282no—
136C4⋊C4—5141no—
137C4⋊C4—5141no—
138C22⋊C4—5142no—
139C22⋊C4—5142no—
140C42—2571no—
141C42—2571no—
142C42—2571no—
143C42C22×D25711yescharacteristic
144C23—2571no—
145C23—22no—
146Q8—10282no—
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152D4—10284no—
153C2×C4—2571no—
154C2×C4—2571no—
155C2×C4—2571no—
156C2×C4—5142no—
157C2×C4—5141no—
158C2×C4—5141no—
159C2×C4—5141no—
160C2×C4—5141no—
161C2×C4—5142no—
162C2×C4—10284no—
163C2×C4C2×C4×D25711yescharacteristic
164C2×C4D4×D25711yescharacteristic
165C2×C4D4×D25711yescharacteristic
166C8—41no—
167C8—41no—
168C8—10281no—
169C8—10281no—
170C22—2572no—
171C22—5141no—
172C22—5141no—
173C22—42no—
174C22—42no—
175C22—42no—
176C22C22⋊C4×D25711yescharacteristic
177C4—21no—
178C4—5141no—
179C4—5141no—
180C4—5141no—
181C4—10282no—
182C4#2255161111yescharacteristic
183C4#2255161311yescharacteristic
184C4#2255161611yescharacteristic
185C4#2255162111yescharacteristic
186C2—5142no—
187C2—82no—
188C2#100×D25711yescharacteristic
189C2#3065673011yescharacteristic
190C2#3065673511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D257
commutatorC2×C4×C257C22×C4
FrattiniC42C22×D257
Fitting#331×C257C2
socleC22×C257C2×C22⋊C4
prime radical#30657285C2
cosocleC42×C257C23
hypercenter#331D257
hypocenterC257#1551
LCSG ▷ C2×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #30656793C2semidirect
3. C4⋊C8×D257C2semidirect
4. #30656485C2semidirect
5. #30656485C2semidirect
6. #30656481C2semidirect
7. #331×C257C2semidirect
95. C257#1551semidirect
103. #331D257semidirect

Maximal subgroups

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

subgroupindex
#306572852
#306567932
C4⋊C8×D2572
#306564852
#306564812
#331×C2572
#1551257