Finite Group Databasebeta

← #45663604

Subgroups — #45663604

Order 32896.

240 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 61 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#30657133C211yesmaximal, characteristic, complemented
2#30657017C211yesmaximal, characteristic, complemented
3#30656942C211yesmaximal, characteristic, complemented
4#30656483C211yesmaximal, characteristic, complemented
5D4⋊#14523339C211yesmaximal, characteristic, complemented
6D4⋊#14523339C211yesmaximal, characteristic, complemented
7#547×C257C211yesmaximal, characteristic
8C2×#16861606—21no—
9#22551592C2211yescharacteristic
10#22551525—21no—
11#22551523—22no—
12C4⋊#14523339C2211yescharacteristic
13C4⋊#14523339C2211yescharacteristic
14#22551515—21no—
15#22551513—21no—
16C4×#14523339C2211yescharacteristic
17C2×#41×C257—21no—
18C4⋊Q8×C257C2211yescharacteristic
19#123×C257—22no—
20C22⋊Q8×C257—21no—
21#121×C257—21no—
22C4×D4×C257C2211yescharacteristic
23C4×D4×C257C2211yescharacteristic
24C2×C4⋊C4×C257—21no—
25C2×Dic514—21no—
26#16861606—42no—
27C2×#14523339—21no—
28C2×#14523339—21no—
29C2×#14523339—21no—
30C4⋊Dic257—21no—
31C4⋊Dic257—21no—
32#41×C257—42no—
33#41×C257—42no—
34C2×Q8×C257—21no—
35C2×D4×C257—22no—
36C2×D4×C257D411yescharacteristic
37C2×D4×C257D411yescharacteristic
38C22×C4×C257—22no—
39C22×C4×C257—21no—
40C22×C4×C257—21no—
41C4⋊C4×C257—22no—
42C4⋊C4×C257—21no—
43C4⋊C4×C257D411yescharacteristic
44C4⋊C4×C257D411yescharacteristic
45C4⋊C4×C257D412yes—
46C22⋊C4×C257—22no—
47C22⋊C4×C257—21no—
48C22⋊C4×C257—21no—
49C42×C257C2311yescharacteristic
50Dic514—41no—
51Dic514—41no—
52C2×Dic257—41no—
53#14523339—22no—
54#14523339—41no—
55#14523339—41no—
56C23×C257—21no—
57C23×C257—21no—
58Q8×C257—41no—
59Q8×C257—41no—
60D4×C257—21no—
61D4×C257—21no—
62D4×C257—21no—
63D4×C257—44no—
64D4×C257SD1612yescomplemented
65C2×C4×C257—21no—
66C2×C4×C257—21no—
67C2×C4×C257—22no—
68C2×C4×C257—44no—
69C2×C4×C257—41no—
70C2×C4×C257—41no—
71C2×C4×C257—41no—
72C2×C4×C257—41no—
73C2×C4×C257C2×D411yescharacteristic
74C2×C4×C257C2×D411yescharacteristic
75C2×C4×C257C2×D411yescharacteristic
76C22×C257—22no—
77C22×C257—41no—
78C22×C257—41no—
79C22×C257—41no—
80C22×C257—41no—
81C22×C257—41no—
82C22×C257#12011yescharacteristic
83C4×C257—21no—
84C4×C257—41no—
85C4×C257—41no—
86C4×C257—42no—
87C4×C257C2×SD1611yescharacteristic
88C4×C257#13511yescharacteristic
89C4×C257AGL(1,ℤ/8)11yescharacteristic
90C4×C257AGL(1,ℤ/8)11yescharacteristic
91Dic257—81no—
92C2×C257—42no—
93C2×C257—81no—
94C2×C257#44911yescharacteristic
95C2×C257#45011yescharacteristic
96C2×C257#45611yescharacteristic
97C257#165411yescharacteristic, complemented, Sylow
98#1654—2571nomaximal, Sylow
99#547D25711yescharacteristic
100#498—2571no—
101#474—2571no—
102#461—2571no—
103#332—2571no—
104D4⋊C8—2571no—
105D4⋊C8—2571no—
106C2×#41—21no—
107C2×SD16—5141no—
108C4⋊Q8—2571no—
109C4⋊Q8C2×D25711yescharacteristic
110#123—22no—
111C22⋊Q8—21no—
112#121—21no—
113C4×D4C2×D25711yescharacteristic
114C4×D4C2×D25711yescharacteristic
115C2×C4⋊C4—21no—
116#107—5141no—
117#106—5142no—
118C4⋊C8—2571no—
119C4⋊C8—2571no—
120#103—5141no—
121#102—5141no—
122C4×C8—2571no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—5141no—
127C2×D4—22no—
128C2×D4C22⋊D25711yescharacteristic
129C2×D4C22⋊D25711yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—10282no—
134C2×C8—5141no—
135C2×C8—5141no—
136C2×C8—5141no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—5141no—
140C4⋊C4—5141no—
141C4⋊C4C22⋊D25711yescharacteristic
142C4⋊C4C22⋊D25712yes—
143C4⋊C4C22⋊D25711yescharacteristic
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C22×D25711yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—10281no—
153Q8—10281no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4#1686160612yescomplemented
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—10281no—
168C2×C4C2×C22⋊D25711yescharacteristic
169C2×C4C2×C22⋊D25711yescharacteristic
170C2×C4C2×C22⋊D25711yescharacteristic
171C8—5142no—
172C8—10281no—
173C8—10281no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#2255156111yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—20561no—
186C4C2×#1686160611yescharacteristic
187C4#2255161211yescharacteristic
188C4#2255161711yescharacteristic
189C4#2255161811yescharacteristic
190C2—42no—
191C2—81no—
192C2#3065688211yescharacteristic
193C2#3065688811yescharacteristic
194C2#3065692011yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting, prime radical#547×C257C2
socleC22×C257#120
hypercenter#547D257
hypocenterC257#1654
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #30657133C2semidirect
2. #30657017C2semidirect
3. #30656942C2semidirect
4. #30656483C2semidirect
5. D4⋊#14523339C2semidirect
6. D4⋊#14523339C2semidirect
64. D4×C257SD16semidirect
97. C257#1654semidirect
158. D4#16861606semidirect

Maximal subgroups

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

subgroupindex
#306571332
#306570172
#306569422
#306564832
D4⋊#145233392
#547×C2572
#1654257