Finite Group Databasebeta

← #44550291

Subgroups — #44550291

Order 32128.

584 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#16860810C211yesmaximal, characteristic
2C2×#22548633C212yesmaximal, complemented
3C2×#22548631C212yesmaximal, complemented
4#30631651C211yesmaximal, characteristic, complemented
5#407×C251C211yesmaximal, characteristic, complemented
6C2×#16860810—24no—
7C2×#16860810—24no—
8C2×#16860810—22no—
9C2×#16860810—22no—
10C22×Dic502C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12C22×C22⋊D251C412yes—
13C22×C4×D251C412yes—
14C2×#16860794C2212yes—
15#22548633—24no—
16#22548631—24no—
17C22⋊(C251⋊C8)—22no—
18C23×C4×C251C2211yescharacteristic
19C2×M16×C251C2212yes—
20C22⋊C8×C251—22no—
21C23×D251C2×C412yes—
22#16860810—216no—
23#16860810—216no—
24C2×Dic502—22no—
25C2×Dic502—21no—
26C2×Dic502—21no—
27C2×Dic502D414yes—
28C2×D1004—22no—
29C2×D1004—21no—
30C2×D1004—21no—
31C2×D1004D414yes—
32C2×C22⋊D251—24no—
33C2×C22⋊D251—24no—
34C2×C22⋊D251—24no—
35C22×Dic251C2×C412yes—
36C2×C4×D251—24no—
37C2×C4×D251—24no—
38C2×C4×D251—24no—
39#16860794—22no—
40#16860794—22no—
41C2×C251⋊C8—22no—
42C24×C251C2×C411yescharacteristic
43C22×C4×C251—22no—
44C22×C4×C251—22no—
45C22×C4×C251C2×C411yescharacteristic
46C22×C4×C251D412yes—
47C22×C4×C251D412yes—
48C22×C4×C251C2311yescharacteristic
49M16×C251—22no—
50M16×C251—22no—
51C2×C8×C251—22no—
52C22×D251—22no—
53C22×D251—22no—
54C22×D251—44no—
55C22×D251C22⋊C414yes—
56Dic502—24no—
57Dic502—24no—
58D1004—24no—
59D1004—24no—
60C22⋊D251—216no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251C22⋊C414yes—
64C4×D251—216no—
65C251⋊C8—42no—
66C23×C251—21no—
67C23×C251—21no—
68C23×C251—42no—
69C23×C251C22⋊C412yes—
70C23×C251C22×C411yescharacteristic
71C2×C4×C251—21no—
72C2×C4×C251—21no—
73C2×C4×C251—24no—
74C2×C4×C251—24no—
75C2×C4×C251C22⋊C412yes—
76C2×C4×C251C2×D412yes—
77C2×C4×C251C2×D412yes—
78C2×C4×C251C2×D411yescharacteristic
79C2×C4×C251C2×D411yescharacteristic
80C8×C251—42no—
81C22×C251—22no—
82C22×C251—22no—
83C22×C251—24no—
84C22×C251—42no—
85C22×C251—42no—
86C22×C251C2×C22⋊C411yescharacteristic
87C22×C251C2×C22⋊C411yescharacteristic
88C22×C251C2×C22⋊C411yescharacteristic
89C2×D251—24no—
90C2×D251—44no—
91C2×D251—44no—
92C2×D251—44no—
93C4×C251—22no—
94C4×C251#12012yes—
95C4×C251#12012yes—
96Dic251—24no—
97C2×C251—21no—
98C2×C251—21no—
99C2×C251—42no—
100C2×C251#37911yescharacteristic
101C2×C251#41312yes—
102D251—44no—
103C251#178511yescharacteristic, complemented, Sylow
104#1785—2511nomaximal, Sylow
105C22×#41—2511no—
106C2×#101—2512no—
107C2×#100—2512no—
108#407—2511no—
109#407D25111yescharacteristic, complemented
110C2×#41—5024no—
111C2×#41—5024no—
112C2×#41—5022no—
113C2×#41—5022no—
114C22×Q8—2511no—
115C22×D4—2512no—
116C22×D4—2511no—
117C23×C4—2512no—
118C23×C4C2×D25111yescharacteristic
119C2×M16—2512no—
120C2×M16C2×D25112yes—
121#101—5024no—
122#100—5024no—
123C22⋊C8—22no—
124C22⋊C8—5022no—
125C24—2512no—
126C24C4×D25111yescharacteristic
127#41—50216no—
128#41—50216no—
129C2×Q8—5022no—
130C2×Q8—5021no—
131C2×Q8—5021no—
132C2×Q8—2514no—
133C2×D4—5024no—
134C2×D4—5024no—
135C2×D4—5024no—
136C2×D4—5022no—
137C2×D4—5021no—
138C2×D4—5021no—
139C2×D4—2514no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—5024no—
143C22×C4—5024no—
144C22×C4—5024no—
145C22×C4—2512no—
146C22×C4C4×D25111yescharacteristic
147C22×C4C22⋊D25112yes—
148C22×C4D100412yes—
149C22×C4C22×D25111yescharacteristic
150M16—22no—
151M16—22no—
152M16—5022no—
153M16—5022no—
154C2×C8—22no—
155C2×C8—5022no—
156C23—21no—
157C23—21no—
158C23—42no—
159C23—10044no—
160C23—5022no—
161C23—5022no—
162C23—2514no—
163C23(C2×D251)⋊C412yes—
164C23C2×C4×D25111yescharacteristic
165Q8—5024no—
166Q8—5024no—
167D4—50216no—
168D4—5024no—
169D4—5024no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—50216no—
175C2×C4—5022no—
176C2×C4—5022no—
177C2×C4—2514no—
178C2×C4(C2×D251)⋊C412yes—
179C2×C4D4×D25112yes—
180C2×C4D4×D25112yes—
181C2×C4C2×C22⋊D25111yescharacteristic
182C2×C4C2×D100411yescharacteristic
183C8—42no—
184C8—10042no—
185C22—22no—
186C22—22no—
187C22—24no—
188C22—42no—
189C22—42no—
190C22—10044no—
191C22—10044no—
192C22—10044no—
193C22—5024no—
194C22C22⋊C4×D25111yescharacteristic
195C22C22⋊C4×D25111yescharacteristic
196C22C2×(C2×D251)⋊C411yescharacteristic
197C4—22no—
198C4—5024no—
199C4#2254868312yes—
200C4#2254868512yes—
201C2—21no—
202C2—21no—
203C2—42no—
204C2—10044no—
205C2#3063154211yescharacteristic
206C2#3063166912yes—
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#407×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×#16860810C2
cosocleC22×C4×C251C23
hypercenter#407D251
hypocenterC251#1785
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#22548633C2semidirect
3. C2×#22548631C2semidirect
4. #30631651C2semidirect
5. #407×C251C2semidirect
103. C251#1785semidirect
109. #407D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168608102
C2×#225486332
C2×#225486312
#306316512
#407×C2512
#1785251