Finite Group Databasebeta

← #45663243

Subgroups — #45663243

Order 32896.

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

normal subgroups only

This group has 316 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#30656887C212yesmaximal, complemented
2#30656646C211yesmaximal, characteristic
3#30656474C212yesmaximal, complemented
4#30656461C211yesmaximal, characteristic, complemented
5#543×C257C211yesmaximal, characteristic, complemented
6C22×D1028C2211yescharacteristic
7C22×C22⋊D257—21no—
8C2×#16861605—22no—
9C2×#16861595—21no—
10#22551512—22no—
11C22⋊#14523339C2212yes—
12#22551499C2211yescharacteristic
13C22⋊Q8×C257—22no—
14C22⋊Q8×C257—21no—
15C22⋊Q8×C257—22no—
16C22⋊Q8×C257C2212yes—
17C2×C22⋊C4×C257—21no—
18C2×C22⋊C4×C257C2211yescharacteristic
19C23×D257—21no—
20C2×D1028—22no—
21C2×D1028—42no—
22C2×C22⋊D257—42no—
23C2×C22⋊D257—42no—
24C2×C22⋊D257—42no—
25C22×Dic257—21no—
26#16861605—44no—
27C2×#14523339—22no—
28#16861595—42no—
29C24×C257D411yescharacteristic
30C2×Q8×C257—21no—
31C2×Q8×C257D412yescomplemented
32C22×C4×C257—21no—
33C22×C4×C257—21no—
34C22×C4×C257D411yescharacteristic
35C22×C4×C257C2311yescharacteristic
36C4⋊C4×C257—21no—
37C4⋊C4×C257—22no—
38C4⋊C4×C257—22no—
39C4⋊C4×C257D412yescomplemented
40C22⋊C4×C257—22no—
41C22⋊C4×C257—22no—
42C22⋊C4×C257—22no—
43C22×D257—22no—
44C22×D257—41no—
45C22×D257—41no—
46C22×D257—41no—
47C22×D257—41no—
48C22×D257—41no—
49C22×D257—41no—
50D1028—24no—
51D1028—42no—
52C22⋊D257—48no—
53C2×Dic257—22no—
54C2×Dic257—41no—
55C2×Dic257—41no—
56#14523339—42no—
57C23×C257—21no—
58C23×C257—21no—
59C23×C257—41no—
60C23×C257—41no—
61C23×C257SD1612yescomplemented
62C23×C257C2×D411yescharacteristic
63Q8×C257—42no—
64Q8×C257—81no—
65C2×C4×C257—22no—
66C2×C4×C257—21no—
67C2×C4×C257—22no—
68C2×C4×C257—41no—
69C2×C4×C257—41no—
70C2×C4×C257—81no—
71C2×C4×C257—81no—
72C2×C4×C257C2×D412yes—
73C22×C257—21no—
74C22×C257—21no—
75C22×C257—22no—
76C22×C257—42no—
77C22×C257—41no—
78C22×C257—41no—
79C22×C257—81no—
80C22×C257#12011yescharacteristic
81C22×C257C2×SD1611yescharacteristic
82C22×C257#13711yescharacteristic
83C2×D257—42no—
84C2×D257—42no—
85C2×D257—42no—
86C2×D257—81no—
87C2×D257—81no—
88C2×D257—81no—
89C2×D257—81no—
90C4×C257—22no—
91C4×C257—41no—
92C4×C257—42no—
93C4×C257—42no—
94Dic257—42no—
95C2×C257—21no—
96C2×C257—21no—
97C2×C257—42no—
98C2×C257#44811yescharacteristic
99C2×C257#45311yescharacteristic
100C2×C257B(4,𝔽2)11yescharacteristic
101D257—82no—
102C257#159611yescharacteristic, complemented, Sylow
103#1596—2571nomaximal, Sylow
104#543D25711yescharacteristic, complemented
105#450—2572no—
106#392—2571no—
107#328—2572no—
108#323—2571no—
109C22×D4—2571no—
110C22×D4—5141no—
111C2×SD16—5142no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C2×D25712yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—5141no—
118C2×C22⋊C4C2×D25711yescharacteristic
119#102—5142no—
120C22⋊C8—2572no—
121#95—2571no—
122C24—5141no—
123C24C22⋊D25711yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D25712yescomplemented
126C2×D4—5142no—
127C2×D4—10282no—
128C2×D4—10282no—
129C2×D4—10282no—
130C2×D4—10282no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—5141no—
134C22×C4C22⋊D25711yescharacteristic
135C22×C4C22×D25711yescharacteristic
136SD16—10284no—
137C2×C8—5142no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D25712yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—10282no—
146C23—21no—
147C23—21no—
148C23—5142no—
149C23—41no—
150C23—41no—
151C23—10281no—
152C23—10281no—
153C23—10281no—
154C23—10281no—
155C23—10281no—
156C23—10281no—
157C23#1686160512yescomplemented
158C23C2×C22⋊D25711yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—5144no—
162D4—10282no—
163D4—10288no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—5142no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—10281no—
171C2×C4—10281no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4C2×C22⋊D25712yes—
175C8—10282no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—10282no—
183C22—10282no—
184C22—10282no—
185C22—81no—
186C22—20561no—
187C22—20561no—
188C22—20561no—
189C22—20561no—
190C22#2255156111yescharacteristic
191C22C2×#1686160511yescharacteristic
192C22#2255162311yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—10282no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—20562no—
202C2#3065687611yescharacteristic
203C2#3065690511yescharacteristic
204C2#3065692411yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#543×C257C2
socleC22×C257#120
prime radical#30656646C2
hypercenter#543D257
hypocenterC257#1596
LCSG ▷ C22×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #30656887C2semidirect
3. #30656474C2semidirect
4. #30656461C2semidirect
5. #543×C257C2semidirect
31. C2×Q8×C257D4semidirect
39. C4⋊C4×C257D4semidirect
61. C23×C257SD16semidirect
102. C257#1596semidirect
104. #543D257semidirect
125. C2×Q8C22⋊D257semidirect
141. C4⋊C4C22⋊D257semidirect
157. C23#16861605semidirect

Maximal subgroups

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

subgroupindex
#306568872
#306566462
#306564742
#306564612
#543×C2572
#1596257