Finite Group Databasebeta

← #45663427

Subgroups — #45663427

Order 32896.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#30657461C211yesmaximal, characteristic, complemented
2#30656934C211yesmaximal, characteristic, complemented
3#30656929C211yesmaximal, characteristic
4#30656483C211yesmaximal, characteristic, complemented
5#30656469C211yesmaximal, characteristic, complemented
6#30656466C211yesmaximal, characteristic, complemented
7#486×C257C211yesmaximal, characteristic, complemented
8C2×#16861606—21no—
9C2×#16861603—21no—
10#22551587—22no—
11#22551581—21no—
12#22551572—21no—
13Q8×Dic257—21no—
14C4×Dic514C2211yescharacteristic
15C4×D1028C2211yescharacteristic
16#22551550—21no—
17C4⋊#14523339C2211yescharacteristic
18C4⋊#14523339C2211yescharacteristic
19#22551513—21no—
20#22551513—21no—
21C4⋊Q8×C257C2211yescharacteristic
22#127×C257C2211yescharacteristic
23#102×C257—22no—
24C4×C8×C257C2211yescharacteristic
25C2×Dic514D411yescharacteristic
26C2×D1028D411yescharacteristic
27C2×C4×D257—21no—
28#16861606—42no—
29#16861603—42no—
30C2×#14523339—21no—
31C2×#14523339—21no—
32C4⋊Dic257—22no—
33C4⋊Dic257D411yescharacteristic
34C4⋊Dic257D411yescharacteristic
35#16861597—21no—
36#16861595—22no—
37#16861595—22no—
38#16861595—21no—
39C4×Dic257—22no—
40C4×Dic257—21no—
41C2×Q8×C257—21no—
42C2×D4×C257—21no—
43C2×D4×C257—21no—
44C2×D4×C257—21no—
45C2×C8×C257—21no—
46C4⋊C4×C257—21no—
47C4⋊C4×C257D412yes—
48C42×C257C2311yescharacteristic
49C22×D257—21no—
50Dic514—21no—
51Dic514—21no—
52D1028—21no—
53D1028—21no—
54C2×Dic257—21no—
55C2×Dic257—21no—
56C2×Dic257—21no—
57C4×D257—41no—
58C4×D257—41no—
59#14523339—41no—
60#14523339—41no—
61C23×C257—41no—
62Q8×C257—41no—
63Q8×C257—41no—
64D4×C257—42no—
65D4×C257—41no—
66D4×C257—41no—
67D4×C257—41no—
68D4×C257—41no—
69C2×C4×C257—22no—
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D411yescharacteristic
72C2×C4×C257C2×D411yescharacteristic
73C8×C257—41no—
74C22×C257—81no—
75C22×C257—81no—
76C22×C257—81no—
77C22×C257#12011yescharacteristic
78C2×D257—41no—
79C2×D257—41no—
80C2×D257—41no—
81C4×C257—21no—
82C4×C257—42no—
83C4×C257#13511yescharacteristic
84C4×C257#13511yescharacteristic
85C4×C257AGL(1,ℤ/8)11yescharacteristic
86C4×C257AGL(1,ℤ/8)11yescharacteristic
87Dic257—41no—
88Dic257—41no—
89Dic257—41no—
90C2×C257—161no—
91C2×C257#44911yescharacteristic
92C2×C257#44911yescharacteristic
93C2×C257#45411yescharacteristic
94D257—81no—
95C257#162811yescharacteristic, complemented, Sylow
96#1628—2571nomaximal, Sylow
97#556—2571no—
98#486D25711yescharacteristic, complemented
99#460—2571no—
100#459—2571no—
101#332—2571no—
102Q8⋊C8—2571no—
103D4⋊C8—2571no—
104C2×SD16—5141no—
105C2×D8—5141no—
106C4⋊Q8C2×D25711yescharacteristic
107#127C2×D25711yescharacteristic
108#126—5142no—
109#124—5141no—
110C22⋊Q8—5141no—
111C4×Q8—2571no—
112C4×Q8—5141no—
113C4×D4—2571no—
114#117—5141no—
115C4⋊C8—2571no—
116C4⋊C8—2571no—
117#102—22no—
118#102—5141no—
119#102—5141no—
120C4×C8C2×D25711yescharacteristic
121C2×Q8—2571no—
122C2×Q8—21no—
123C2×D4—2571no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C22×C4—5141no—
128SD16—10282no—
129D8—10282no—
130C2×C8—21no—
131C2×C8—5141no—
132C2×C8—5141no—
133C4⋊C4—2571no—
134C4⋊C4—2571no—
135C4⋊C4—21no—
136C4⋊C4—5142no—
137C4⋊C4—5141no—
138C4⋊C4C22⋊D25712yes—
139C22⋊C4—5142no—
140C22⋊C4—5142no—
141C22⋊C4—5141no—
142C42—5142no—
143C42—5141no—
144C42C22×D25711yescharacteristic
145C23—5141no—
146C23—41no—
147Q8—5141no—
148Q8—5141no—
149Q8—41no—
150Q8—41no—
151D4—5141no—
152D4—5141no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158C2×C4—22no—
159C2×C4—5141no—
160C2×C4—5141no—
161C2×C4—5141no—
162C2×C4—10281no—
163C2×C4—10281no—
164C2×C4D4×D25711yescharacteristic
165C2×C4D4×D25711yescharacteristic
166C2×C4C2×C22⋊D25711yescharacteristic
167C8—41no—
168C8—10281no—
169C8—10281no—
170C22—10281no—
171C22—10281no—
172C22—10281no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22#2255156011yescharacteristic
177C4—21no—
178C4—42no—
179C4—10281no—
180C4—10281no—
181C4—10281no—
182C4#2255161111yescharacteristic
183C4#2255161311yescharacteristic
184C4#2255161611yescharacteristic
185C4#2255161911yescharacteristic
186C2—20561no—
187C2—161no—
188C2#3065687911yescharacteristic
189C2#3065688111yescharacteristic
190C2#3065690911yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#486×C257C2
socleC22×C257#120
prime radical#30656929C2
hypercenter#486D257
hypocenterC257#1628
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #30657461C2semidirect
2. #30656934C2semidirect
4. #30656483C2semidirect
5. #30656469C2semidirect
6. #30656466C2semidirect
7. #486×C257C2semidirect
95. C257#1628semidirect
98. #486D257semidirect

Maximal subgroups

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

subgroupindex
#306574612
#306569342
#306569292
#306564832
#306564692
#306564662
#486×C2572
#1628257