Finite Group Databasebeta

← #45663452

Subgroups — #45663452

Order 32896.

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

normal subgroups only

This group has 214 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#30657444C211yesmaximal, characteristic
2#30657116C211yesmaximal, characteristic, complemented
3#30657029C211yesmaximal, characteristic, complemented
4#30656485C211yesmaximal, characteristic, complemented
5#30656467C211yesmaximal, characteristic, complemented
6D4⋊#14523339C211yesmaximal, characteristic
7#476×C257C211yesmaximal, characteristic
8C2×#16861608—22no—
9#22551592C2211yescharacteristic
10#22551583—21no—
11#22551571—21no—
12#22551569—22no—
13C4×Dic514C2211yescharacteristic
14D4×Dic257—21no—
15C2×C4⋊Dic257—21no—
16#22551525—21no—
17C4⋊#14523339C2211yescharacteristic
18#22551515—21no—
19C4×#14523339C2211yescharacteristic
20C4⋊Q8×C257C2211yescharacteristic
21C4×D4×C257C2211yescharacteristic
22#106×C257—21no—
23C4⋊C8×C257C2211yescharacteristic
24#102×C257—21no—
25C2×Dic514D411yescharacteristic
26C2×Dic514D412yes—
27C22×Dic257—22no—
28#16861608—44no—
29C2×#14523339—21no—
30C2×#14523339—21no—
31C4⋊Dic257—22no—
32C4⋊Dic257—21no—
33C4⋊Dic257—21no—
34C4⋊Dic257D411yescharacteristic
35#16861597—22no—
36#16861597—21no—
37C22⋊Dic257—22no—
38C4×Dic257—21no—
39C2×Q8×C257—21no—
40C2×D4×C257D411yescharacteristic
41C22×C4×C257—21no—
42C2×C8×C257—21no—
43C4⋊C4×C257—21no—
44C4⋊C4×C257—21no—
45C4⋊C4×C257D411yescharacteristic
46C22⋊C4×C257—21no—
47C42×C257C2311yescharacteristic
48Dic514—21no—
49Dic514—21no—
50Dic514—42no—
51C2×Dic257—21no—
52C2×Dic257—21no—
53C2×Dic257—22no—
54C2×Dic257—82no—
55#14523339—22no—
56#14523339—41no—
57C23×C257—21no—
58Q8×C257—42no—
59D4×C257—21no—
60D4×C257Q1612yescomplemented
61C2×C4×C257—21no—
62C2×C4×C257—41no—
63C2×C4×C257—41no—
64C2×C4×C257—41no—
65C2×C4×C257C2×D411yescharacteristic
66C2×C4×C257C2×D411yescharacteristic
67C2×C4×C257C2×D411yescharacteristic
68C8×C257—41no—
69C22×C257—22no—
70C22×C257—41no—
71C22×C257—41no—
72C22×C257#12011yescharacteristic
73C4×C257—21no—
74C4×C257—41no—
75C4×C257—81no—
76C4×C257C2×Q1611yescharacteristic
77C4×C257#13511yescharacteristic
78C4×C257#13711yescharacteristic
79C4×C257#13711yescharacteristic
80Dic257—41no—
81Dic257—41no—
82Dic257—42no—
83C2×C257—42no—
84C2×C257#45111yescharacteristic
85C2×C257#45211yescharacteristic
86C2×C257#45311yescharacteristic
87C257#163211yescharacteristic, complemented, Sylow
88#1632—2571nomaximal, Sylow
89#554—2571no—
90#494—2571no—
91#477—2571no—
92#476D25711yescharacteristic
93#333—2571no—
94Q8⋊C8—2571no—
95D4⋊C8—2571no—
96C2×Q16—5142no—
97C4⋊Q8—2571no—
98C4⋊Q8C2×D25711yescharacteristic
99#125—5141no—
100C22⋊Q8—5142no—
101C22⋊Q8—5141no—
102C4×Q8—2571no—
103C4×D4—5141no—
104C4×D4C2×D25711yescharacteristic
105C2×C4⋊C4—5141no—
106#107—5141no—
107#106—21no—
108C4⋊C8—2571no—
109C4⋊C8C2×D25711yescharacteristic
110#103—5141no—
111#102—21no—
112C4×C8—2571no—
113C2×Q8—2571no—
114C2×Q8—2572no—
115C2×Q8—21no—
116C2×D4C22⋊D25711yescharacteristic
117C22×C4—21no—
118C22×C4—5142no—
119Q16—10284no—
120C2×C8—21no—
121C2×C8—5141no—
122C2×C8—5141no—
123C4⋊C4—2571no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—5142no—
127C4⋊C4—5142no—
128C4⋊C4—5141no—
129C4⋊C4—5141no—
130C4⋊C4—5141no—
131C4⋊C4C22⋊D25711yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—5142no—
134C42—5141no—
135C42C22×D25711yescharacteristic
136C23—21no—
137Q8—5141no—
138Q8—5141no—
139Q8—42no—
140Q8—10282no—
141D4—21no—
142D4#1686160812yescomplemented
143C2×C4—21no—
144C2×C4—5141no—
145C2×C4—5141no—
146C2×C4—5142no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—20562no—
151C2×C4D4×D25711yescharacteristic
152C2×C4D4×D25711yescharacteristic
153C2×C4C2×C22⋊D25711yescharacteristic
154C8—5142no—
155C8—41no—
156C8—10281no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#2255156011yescharacteristic
161C4—21no—
162C4—41no—
163C4—10281no—
164C4—10281no—
165C4—10282no—
166C4—81no—
167C4C2×#1686160811yescharacteristic
168C4#2255161311yescharacteristic
169C4#2255162111yescharacteristic
170C4#2255162211yescharacteristic
171C2—42no—
172C2#3065689211yescharacteristic
173C2#3065689611yescharacteristic
174C2#3065690311yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#476×C257C2
socleC22×C257#120
prime radicalC2×D4×C257D4
hypercenter#476D257
hypocenterC257#1632
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #30657116C2semidirect
3. #30657029C2semidirect
4. #30656485C2semidirect
5. #30656467C2semidirect
60. D4×C257Q16semidirect
87. C257#1632semidirect
142. D4#16861608semidirect

Maximal subgroups

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

subgroupindex
#306574442
#306571162
#306570292
#306564852
#306564672
D4⋊#145233392
#476×C2572
#1632257