Finite Group Databasebeta

← #45663602

Subgroups — #45663602

Order 32896.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#30657392C211yesmaximal, characteristic
2#30657133C211yesmaximal, characteristic, complemented
3#30657014C211yesmaximal, characteristic, complemented
4#30656482C211yesmaximal, characteristic, complemented
5D4⋊#14523339C211yesmaximal, characteristic, complemented
6#30656464C211yesmaximal, characteristic, complemented
7#461×C257C211yesmaximal, characteristic, complemented
8C2×#16861618—21no—
9#22551592C2211yescharacteristic
10#22551574—22no—
11#22551571—21no—
12#22551563—21no—
13C4×D1028C2211yescharacteristic
14C4⋊C4×D257—21no—
15#22551525—21no—
16#22551523—22no—
17C4⋊#14523339C2211yescharacteristic
18#22551512—21no—
19C4×#14523339C2211yescharacteristic
20C2×SD16×C257—21no—
21C4⋊Q8×C257C2211yescharacteristic
22C4×D4×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24#103×C257—21no—
25#16861618—42no—
26#16861618—42no—
27C2×Dic514—21no—
28C2×D1028D411yescharacteristic
29C2×C22⋊D257—22no—
30C2×C4×D257—22no—
31C2×C4×D257—21no—
32C2×#14523339—21no—
33C2×#14523339—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257D411yescharacteristic
36C4⋊Dic257D412yes—
37#16861597—22no—
38C22⋊Dic257—22no—
39#16861595—21no—
40C2×Q8×C257—21no—
41C2×D4×C257D411yescharacteristic
42C22×C4×C257—21no—
43SD16×C257—42no—
44C2×C8×C257—21no—
45C4⋊C4×C257—21no—
46C4⋊C4×C257—21no—
47C4⋊C4×C257D411yescharacteristic
48C22⋊C4×C257—21no—
49C42×C257C2311yescharacteristic
50C22×D257—21no—
51Dic514—41no—
52Dic514—41no—
53D1028—21no—
54D1028SD1612yescomplemented
55C22⋊D257—44no—
56C2×Dic257—21no—
57C2×Dic257—22no—
58C4×D257—44no—
59C4×D257—41no—
60C4×D257—41no—
61#14523339—22no—
62#14523339—41no—
63C23×C257—21no—
64Q8×C257—41no—
65Q8×C257—41no—
66D4×C257—21no—
67D4×C257—21no—
68C2×C4×C257—21no—
69C2×C4×C257—41no—
70C2×C4×C257—41no—
71C2×C4×C257—41no—
72C2×C4×C257C2×D411yescharacteristic
73C2×C4×C257C2×D411yescharacteristic
74C2×C4×C257C2×D411yescharacteristic
75C8×C257—41no—
76C22×C257—41no—
77C22×C257—41no—
78C22×C257—41no—
79C22×C257#12011yescharacteristic
80C2×D257—22no—
81C2×D257—41no—
82C2×D257—41no—
83C4×C257—21no—
84C4×C257—41no—
85C4×C257—81no—
86C4×C257C2×SD1611yescharacteristic
87C4×C257#13511yescharacteristic
88C4×C257AGL(1,ℤ/8)11yescharacteristic
89C4×C257AGL(1,ℤ/8)11yescharacteristic
90Dic257—41no—
91Dic257—42no—
92C2×C257—81no—
93C2×C257#44911yescharacteristic
94C2×C257#45011yescharacteristic
95C2×C257#45611yescharacteristic
96D257—42no—
97C257#165411yescharacteristic, complemented, Sylow
98#1654—2571nomaximal, Sylow
99#547—2571no—
100#498—2571no—
101#474—2571no—
102#461D25711yescharacteristic, complemented
103#332—2571no—
104D4⋊C8—2571no—
105D4⋊C8—2571no—
106C2×#41—5141no—
107C2×SD16—21no—
108C4⋊Q8—2571no—
109C4⋊Q8C2×D25711yescharacteristic
110#123—5142no—
111C22⋊Q8—5141no—
112#121—5141no—
113C4×D4—2571no—
114C4×D4C2×D25711yescharacteristic
115C2×C4⋊C4—5141no—
116#107—5141no—
117#106—5142no—
118C4⋊C8—2571no—
119C4⋊C8C2×D25711yescharacteristic
120#103—21no—
121#102—5141no—
122C4×C8—2571no—
123#41—10282no—
124#41—10282no—
125C2×Q8—21no—
126C2×Q8—5141no—
127C2×D4—2571no—
128C2×D4—5142no—
129C2×D4C22⋊D25711yescharacteristic
130C22×C4—21no—
131C22×C4—5142no—
132C22×C4—5141no—
133SD16—42no—
134C2×C8—21no—
135C2×C8—5141no—
136C2×C8—5141no—
137C4⋊C4—2572no—
138C4⋊C4—2571no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—5142no—
142C4⋊C4—5141no—
143C4⋊C4C22⋊D25711yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—5142no—
146C22⋊C4—5141no—
147C42C22×D25711yescharacteristic
148C23—21no—
149C23—5141no—
150Q8—41no—
151Q8—41no—
152Q8—10281no—
153Q8—10281no—
154D4—2572no—
155D4—21no—
156D4—21no—
157D4—5141no—
158D4—10284no—
159C2×C4—21no—
160C2×C4—5141no—
161C2×C4—5142no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—10284no—
166C2×C4—10281no—
167C2×C4—10281no—
168C2×C4D4×D25711yescharacteristic
169C2×C4D4×D25711yescharacteristic
170C2×C4C2×C22⋊D25711yescharacteristic
171C8—5142no—
172C8—41no—
173C8—10281no—
174C22—5142no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—10281no—
179C22—10281no—
180C22#2255156011yescharacteristic
181C4—21no—
182C4—41no—
183C4—10281no—
184C4—10282no—
185C4—81no—
186C4SD16×D25711yescharacteristic
187C4#2255161211yescharacteristic
188C4#2255161811yescharacteristic
189C4#2255161911yescharacteristic
190C2—10282no—
191C2—81no—
192C2#3065688011yescharacteristic
193C2#3065688611yescharacteristic
194C2#3065691811yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#461×C257C2
socleC22×C257#120
prime radical#30657392C2
hypercenter#461D257
hypocenterC257#1654
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #30657133C2semidirect
3. #30657014C2semidirect
4. #30656482C2semidirect
5. D4⋊#14523339C2semidirect
6. #30656464C2semidirect
7. #461×C257C2semidirect
54. D1028SD16semidirect
97. C257#1654semidirect
102. #461D257semidirect

Maximal subgroups

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

subgroupindex
#306573922
#306571332
#306570142
#306564822
D4⋊#145233392
#306564642
#461×C2572
#1654257