Finite Group Databasebeta

← #45665452

Subgroups — #45665452

Order 32896.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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×#22551576C212yesmaximal, complemented
3C2×#22551506C212yesmaximal, complemented
4#30656598C211yesmaximal, characteristic, complemented
5#342×C257C211yesmaximal, characteristic, complemented
6C22×C22⋊D257C2212yes—
7C23×Dic257C2211yescharacteristic
8#22551576—22no—
9#22551576—24no—
10#22551576—24no—
11#22551566—22no—
12#22551562—22no—
13C2×C4⋊Dic257—22no—
14C2×C22⋊Dic257C2211yescharacteristic
15C2×#16861595—22no—
16#22551506—24no—
17#22551499—22no—
18C2×C22⋊C4×C257C2211yescharacteristic
19C2×C22⋊C4×C257C2212yes—
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×Dic257—21no—
30C22×Dic257—22no—
31C22×Dic257—21no—
32C22×Dic257—21no—
33C22×Dic257D412yes—
34C22×Dic257D412yescomplemented
35C4⋊Dic257—22no—
36C4⋊Dic257—28no—
37C22⋊Dic257—22no—
38#16861595—24no—
39#16861595—28no—
40C24×C257C2311yescharacteristic
41C22×C4×C257—22no—
42C22×C4×C257D412yes—
43C22⋊C4×C257—22no—
44C22⋊C4×C257—22no—
45C22⋊C4×C257D414yescomplemented
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—22no—
60C2×Dic257—41no—
61C2×Dic257—42no—
62C2×Dic257—41no—
63C2×Dic257—42no—
64C2×Dic257—41no—
65C23×C257—42no—
66C23×C257C2×D412yes—
67C23×C257C2×D411yescharacteristic
68C23×C257C2×D411yescharacteristic
69C23×C257C2×D412yes—
70C23×C257#4111yescharacteristic
71C2×C4×C257—22no—
72C2×C4×C257—24no—
73C2×C4×C257—44no—
74C2×C4×C257—42no—
75C22×C257—22no—
76C22×C257—24no—
77C22×C257—41no—
78C22×C257—42no—
79C22×C257—41no—
80C22×C257#12012yes—
81C22×C257#12011yescharacteristic
82C22×C257#12111yescharacteristic
83C22×C257#12112yes—
84C22×C257#12711yescharacteristic
85C2×D257—44no—
86C2×D257—42no—
87C2×D257—44no—
88C2×D257—82no—
89C4×C257—44no—
90C4×C257—42no—
91Dic257—42no—
92Dic257—42no—
93Dic257—81no—
94C2×C257—21no—
95C2×C257—22no—
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#381—2571no—
107#342D25711yescharacteristic, complemented
108C22×D4—2572no—
109C23×C4—2571no—
110#123—5144no—
111#123—5142no—
112#123—5144no—
113#121—5142no—
114#120—5142no—
115C2×C4⋊C4—5142no—
116C2×C22⋊C4—2571no—
117C2×C22⋊C4—5142no—
118C2×C22⋊C4C2×D25711yescharacteristic
119C2×C22⋊C4C2×D25712yes—
120#99—5144no—
121#95—5142no—
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—2572no—
133C22×C4—22no—
134C22×C4—5142no—
135C22×C4—5141no—
136C22×C4—5142no—
137C22×C4—5141no—
138C22×C4—5141no—
139C22×C4D102812yes—
140C4⋊C4—5148no—
141C4⋊C4—5142no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—5148no—
145C22⋊C4—5144no—
146C22⋊C4—5142no—
147C22⋊C4D102814yescomplemented
148C23—5142no—
149C23—5142no—
150C23—5142no—
151C23—42no—
152C23—10284no—
153C23D4×D25712yes—
154C23D4×D25711yescharacteristic
155C23C2×D102811yescharacteristic
156C23C2×D102812yes—
157C23#1686161911yescharacteristic
158D4—10282no—
159D4—10282no—
160D4—10282no—
161D4—10282no—
162D4—20562no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—5144no—
166C2×C4—5142no—
167C2×C4—5142no—
168C2×C4—5142no—
169C2×C4—5142no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—10281no—
173C2×C4—10282no—
174C2×C4—10281no—
175C2×C4—10282no—
176C2×C4—10281no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—10284no—
183C22—10282no—
184C22—10284no—
185C22—20562no—
186C22#2255156212yes—
187C22#2255156211yescharacteristic
188C22#2255156711yescharacteristic
189C22#2255156712yes—
190C22#2255158911yescharacteristic
191C4—44no—
192C4—42no—
193C4—10282no—
194C4—10282no—
195C4—20561no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—20562no—
201C2#3065664711yescharacteristic
202C2#3065692812yes—
subgroupG/H
centerC22#22551562
commutator, cosocleC24×C257C23
FrattiniC24C22×D257
Fitting#342×C257C2
socleC22×C257#120
prime radical#30657299C2
hypercenter#342D257
hypocenterC257#2022
LCSG ▷ C24×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#22551576C2semidirect
3. C2×#22551506C2semidirect
4. #30656598C2semidirect
5. #342×C257C2semidirect
34. C22×Dic257D4semidirect
45. C22⋊C4×C257D4semidirect
101. C257#2022semidirect
107. #342D257semidirect
147. C22⋊C4D1028semidirect

Maximal subgroups

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

subgroupindex
#306572992
C2×#225515762
C2×#225515062
#306565982
#342×C2572
#2022257