Finite Group Databasebeta

← #45663407

Subgroups — #45663407

Order 32896.

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

normal subgroups only

This group has 256 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#30657391C211yesmaximal, characteristic, complemented
2#30657083C211yesmaximal, characteristic, complemented
3#30656929C211yesmaximal, characteristic, complemented
4#30656484C211yesmaximal, characteristic, complemented
5D4⋊#14523339C211yesmaximal, characteristic, complemented
6#30656464C211yesmaximal, characteristic, complemented
7#459×C257C211yesmaximal, characteristic, complemented
8C2×#16861617—21no—
9C2×#16861603—21no—
10#22551592C2211yescharacteristic
11#22551576—22no—
12#22551572—21no—
13#22551566—21no—
14C4×D1028C2211yescharacteristic
15C2×C4⋊Dic257—21no—
16C4⋊#14523339C2211yescharacteristic
17#22551513—22no—
18#22551513—21no—
19C4×#14523339C2211yescharacteristic
20C2×D8×C257—21no—
21#127×C257C2211yescharacteristic
22C4×D4×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic
24#102×C257—21no—
25#16861617—42no—
26#16861617—42no—
27C2×Dic514—21no—
28C2×D1028D411yescharacteristic
29C2×C22⋊D257—22no—
30C22×Dic257—22no—
31C2×C4×D257—21no—
32#16861603—42no—
33C2×#14523339—21no—
34C2×#14523339—21no—
35C4⋊Dic257—22no—
36C4⋊Dic257—21no—
37C4⋊Dic257D411yescharacteristic
38C4⋊Dic257D412yes—
39#16861595—22no—
40#16861595—21no—
41C2×D4×C257—21no—
42C2×D4×C257—21no—
43C2×D4×C257—21no—
44C2×D4×C257D411yescharacteristic
45C22×C4×C257—21no—
46D8×C257—42no—
47C2×C8×C257—21no—
48C4⋊C4×C257D411yescharacteristic
49C22⋊C4×C257—21no—
50C42×C257C2311yescharacteristic
51C22×D257—21no—
52Dic514—41no—
53Dic514—41no—
54D1028—21no—
55D1028—21no—
56C22⋊D257—44no—
57C2×Dic257—21no—
58C2×Dic257—22no—
59C2×Dic257—44no—
60C4×D257—41no—
61C4×D257—41no—
62#14523339—41no—
63#14523339—41no—
64C23×C257—21no—
65C23×C257—41no—
66D4×C257—21no—
67D4×C257—42no—
68D4×C257—41no—
69D4×C257—41no—
70D4×C257—41no—
71D4×C257—41no—
72D4×C257D812yescomplemented
73C2×C4×C257—21no—
74C2×C4×C257—41no—
75C2×C4×C257—41no—
76C2×C4×C257C2×D411yescharacteristic
77C2×C4×C257C2×D411yescharacteristic
78C2×C4×C257C2×D411yescharacteristic
79C8×C257—41no—
80C22×C257—22no—
81C22×C257—41no—
82C22×C257—41no—
83C22×C257—81no—
84C22×C257—81no—
85C22×C257—81no—
86C22×C257#12011yescharacteristic
87C2×D257—41no—
88C2×D257—41no—
89C2×D257—41no—
90C4×C257—21no—
91C4×C257—41no—
92C4×C257C2×D811yescharacteristic
93C4×C257#13511yescharacteristic
94C4×C257AGL(1,ℤ/8)11yescharacteristic
95C4×C257AGL(1,ℤ/8)11yescharacteristic
96Dic257—41no—
97Dic257—42no—
98C2×C257—42no—
99C2×C257—161no—
100C2×C257#44711yescharacteristic
101C2×C257#44911yescharacteristic
102C2×C257#45411yescharacteristic
103D257—81no—
104C257#162511yescharacteristic, complemented, Sylow
105#1625—2571nomaximal, Sylow
106#547—2571no—
107#486—2571no—
108#459—2571no—
109#459D25711yescharacteristic, complemented
110#332—2571no—
111D4⋊C8—2571no—
112D4⋊C8—2571no—
113C2×#41—5141no—
114C2×D8—21no—
115C2×D8—5141no—
116C4⋊Q8—2571no—
117#127C2×D25711yescharacteristic
118#123—5142no—
119C22⋊Q8—5141no—
120#121—5141no—
121C4×D4—2571no—
122C4×D4C2×D25711yescharacteristic
123C2×C4⋊C4—5141no—
124C4⋊C8—2571no—
125C4⋊C8C2×D25711yescharacteristic
126#102—21no—
127#102—5141no—
128#102—5142no—
129C4×C8—2571no—
130#41—10282no—
131#41—10282no—
132C2×Q8—5141no—
133C2×D4—2571no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—5142no—
138C2×D4C22⋊D25711yescharacteristic
139C22×C4—21no—
140C22×C4—5142no—
141C22×C4—5141no—
142D8—42no—
143D8—10282no—
144C2×C8—21no—
145C2×C8—5141no—
146C2×C8—5141no—
147C4⋊C4—2571no—
148C4⋊C4—2572no—
149C4⋊C4—5142no—
150C4⋊C4—5141no—
151C4⋊C4C22⋊D25711yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—5142no—
154C22⋊C4—5141no—
155C42C22×D25711yescharacteristic
156C23—21no—
157C23—5141no—
158C23—41no—
159Q8—10281no—
160Q8—10281no—
161D4—21no—
162D4—5141no—
163D4—5141no—
164D4—42no—
165D4—41no—
166D4—41no—
167D4—41no—
168D4—41no—
169D4—10284no—
170D4#1686160312yescomplemented
171C2×C4—21no—
172C2×C4—5141no—
173C2×C4—5142no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—10284no—
177C2×C4—10281no—
178C2×C4—10281no—
179C2×C4D4×D25711yescharacteristic
180C2×C4D4×D25711yescharacteristic
181C2×C4C2×C22⋊D25711yescharacteristic
182C8—41no—
183C8—10281no—
184C8—10281no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—10281no—
189C22—10281no—
190C22—10281no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22#2255156011yescharacteristic
195C4—21no—
196C4—41no—
197C4—10281no—
198C4—10282no—
199C4C2×#1686160311yescharacteristic
200C4#2255161111yescharacteristic
201C4#2255161611yescharacteristic
202C4#2255161711yescharacteristic
203C2—42no—
204C2—20561no—
205C2—161no—
206C2#3065687011yescharacteristic
207C2#3065687911yescharacteristic
208C2#3065690811yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#459×C257C2
socleC22×C257#120
hypercenter#459D257
hypocenterC257#1625
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #30657391C2semidirect
2. #30657083C2semidirect
3. #30656929C2semidirect
4. #30656484C2semidirect
5. D4⋊#14523339C2semidirect
6. #30656464C2semidirect
7. #459×C257C2semidirect
72. D4×C257D8semidirect
104. C257#1625semidirect
109. #459D257semidirect
170. D4#16861603semidirect

Maximal subgroups

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

subgroupindex
#306573912
#306570832
#306569292
#306564842
D4⋊#145233392
#306564642
#459×C2572
#1625257