Finite Group Databasebeta

← #45665451

Subgroups — #45665451

Order 32896.

430 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 73 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#30657299C211yesmaximal, characteristic
2C2×#22551575C212yesmaximal, complemented
3C2×#22551506C212yesmaximal, complemented
4#30656504C211yesmaximal, characteristic, complemented
5#381×C257C211yesmaximal, characteristic, complemented
6C22×C22⋊D257C2212yes—
7#22551576—22no—
8#22551575—24no—
9#22551575—24no—
10#22551566—22no—
11#22551562—22no—
12C2×#16861597—22no—
13C2×C22⋊Dic257C2212yes—
14C2×#16861595—22no—
15#22551506—24no—
16C23×C4×C257C2211yescharacteristic
17C2×C22⋊C4×C257C2211yescharacteristic
18C2×C22⋊C4×C257C2211yescharacteristic
19#95×C257—22no—
20C23×D257D412yes—
21C2×D1028—22no—
22C2×C22⋊D257—22no—
23C2×C22⋊D257—22no—
24C2×C22⋊D257—22no—
25C2×C22⋊D257—22no—
26C2×C22⋊D257—22no—
27C2×C22⋊D257—22no—
28C22×Dic257—22no—
29C22×Dic257D412yes—
30C4⋊Dic257—22no—
31#16861597—28no—
32C22⋊Dic257—22no—
33C22⋊Dic257D414yescomplemented
34#16861595—24no—
35#16861595—28no—
36C24×C257C2311yescharacteristic
37C22×C4×C257—22no—
38C22×C4×C257—21no—
39C22×C4×C257—22no—
40C22×C4×C257—21no—
41C22×C4×C257—21no—
42C22×C4×C257D412yes—
43C22×C4×C257D412yescomplemented
44C22⋊C4×C257—22no—
45C22⋊C4×C257—22no—
46C22×D257—22no—
47C22×D257—22no—
48C22×D257—22no—
49C22×D257—44no—
50D1028—82no—
51C22⋊D257—42no—
52C22⋊D257—42no—
53C22⋊D257—42no—
54C22⋊D257—42no—
55C2×Dic257—24no—
56C2×Dic257—22no—
57C2×Dic257—22no—
58C2×Dic257—22no—
59C2×Dic257—44no—
60C23×C257—42no—
61C23×C257C2×D412yes—
62C23×C257C2×D412yes—
63C23×C257C2×D411yescharacteristic
64C23×C257C2×D411yescharacteristic
65C23×C257#4111yescharacteristic
66C2×C4×C257—22no—
67C2×C4×C257—24no—
68C2×C4×C257—22no—
69C2×C4×C257—41no—
70C2×C4×C257—42no—
71C2×C4×C257—41no—
72C2×C4×C257—42no—
73C2×C4×C257—42no—
74C2×C4×C257—41no—
75C22×C257—22no—
76C22×C257—24no—
77C22×C257—42no—
78C22×C257—41no—
79C22×C257—41no—
80C22×C257#12011yescharacteristic
81C22×C257#12012yes—
82C22×C257#12112yes—
83C22×C257#12111yescharacteristic
84C22×C257#12711yescharacteristic
85C2×D257—44no—
86C2×D257—42no—
87C2×D257—44no—
88C2×D257—82no—
89C4×C257—42no—
90C4×C257—42no—
91C4×C257—81no—
92Dic257—44no—
93Dic257—42no—
94C2×C257—22no—
95C2×C257—21no—
96C2×C257—22no—
97C2×C257—21no—
98C2×C257#39211yescharacteristic
99C2×C257#45812yes—
100D257—82no—
101C257#202211yescharacteristic, complemented, Sylow
102#2022—2571nomaximal, Sylow
103#534—2571no—
104C2×#123—2572no—
105C2×#99—2572no—
106#381D25711yescharacteristic, complemented
107#342—2571no—
108C22×D4—2572no—
109C23×C4C2×D25711yescharacteristic
110#123—5144no—
111#123—5142no—
112#123—5144no—
113#121—5142no—
114#120—5142no—
115C2×C4⋊C4—5142no—
116C2×C22⋊C4—2572no—
117C2×C22⋊C4—5142no—
118C2×C22⋊C4C2×D25711yescharacteristic
119C2×C22⋊C4C2×D25711yescharacteristic
120#99—5144no—
121#95—22no—
122C24—2572no—
123C24C22×D25711yescharacteristic
124C2×D4—5142no—
125C2×D4—5142no—
126C2×D4—5142no—
127C2×D4—5142no—
128C2×D4—5142no—
129C2×D4—5142no—
130C2×D4—5142no—
131C22×C4—2572no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—5142no—
138C22×C4C22⋊D25712yes—
139C22×C4D102812yescomplemented
140C4⋊C4—5148no—
141C4⋊C4—5142no—
142C22⋊C4—2574no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—5148no—
146C22⋊C4—5144no—
147C22⋊C4—5142no—
148C23—5142no—
149C23—5142no—
150C23—5142no—
151C23—42no—
152C23—10284no—
153C23D4×D25712yes—
154C23D4×D25712yes—
155C23C2×C22⋊D25711yescharacteristic
156C23C2×D102811yescharacteristic
157C23#1686161811yescharacteristic
158D4—10282no—
159D4—10282no—
160D4—10282no—
161D4—10282no—
162D4—20562no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—5142no—
167C2×C4—5144no—
168C2×C4—5142no—
169C2×C4—5142no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—10284no—
177C22—22no—
178C22—24no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—10284no—
183C22—10282no—
184C22—10284no—
185C22—20562no—
186C22#2255156011yescharacteristic
187C22#2255156212yes—
188C22#2255156512yes—
189C22#2255156611yescharacteristic
190C22#2255158811yescharacteristic
191C4—42no—
192C4—42no—
193C4—10284no—
194C4—10282no—
195C4—81no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—20562no—
201C2#3065664611yescharacteristic
202C2#3065692612yes—
subgroupG/H
centerC22#22551560
commutator, cosocleC24×C257C23
FrattiniC24C22×D257
Fitting#381×C257C2
socleC22×C257#120
prime radical#30657299C2
hypercenter#381D257
hypocenterC257#2022
LCSG ▷ C24×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#22551575C2semidirect
3. C2×#22551506C2semidirect
4. #30656504C2semidirect
5. #381×C257C2semidirect
33. C22⋊Dic257D4semidirect
43. C22×C4×C257D4semidirect
101. C257#2022semidirect
106. #381D257semidirect
139. C22×C4D1028semidirect

Maximal subgroups

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

subgroupindex
#306572992
C2×#225515752
C2×#225515062
#306565042
#381×C2572
#2022257