Finite Group Databasebeta

← #44550799

Subgroups — #44550799

Order 32128.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16860794C211yesmaximal, characteristic
2C2×#22548639C212yesmaximal, complemented
3#30631547C211yesmaximal, characteristic
4#30631449C212yesmaximal, complemented
5C2×C22⋊Q8×C251C211yesmaximal, characteristic
6C2×#16860794—24no—
7C2×#16860794—22no—
8C22×C251⋊C8C2212yes—
9C2×C4⋊Dic251C2212yes—
10C2×C22⋊Dic251—21no—
11#22548639—24no—
12C251⋊#95—21no—
13C22×Q8×C251C2211yescharacteristic
14C23×C4×C251C2211yescharacteristic
15C22⋊Q8×C251—22no—
16C22⋊Q8×C251C414yescomplemented
17C2×C4⋊C4×C251—21no—
18C2×C4⋊C4×C251C2211yescharacteristic
19C2×C22⋊C4×C251—21no—
20C22×Dic251—22no—
21#16860794—28no—
22C2×C251⋊C8—22no—
23C2×C251⋊C8—22no—
24C2×C251⋊C8D414yes—
25C4⋊Dic251—22no—
26C4⋊Dic251—22no—
27C22⋊Dic251—42no—
28C24×C251D411yescharacteristic
29C2×Q8×C251—21no—
30C2×Q8×C251—22no—
31C2×Q8×C251—22no—
32C2×Q8×C251C2×C412yes—
33C22×C4×C251—22no—
34C22×C4×C251—22no—
35C22×C4×C251—21no—
36C22×C4×C251—21no—
37C22×C4×C251C2×C412yes—
38C22×C4×C251D411yescharacteristic
39C22×C4×C251D412yes—
40C22×C4×C251C2311yescharacteristic
41C4⋊C4×C251—24no—
42C4⋊C4×C251—21no—
43C4⋊C4×C251C2×C412yes—
44C22⋊C4×C251—24no—
45C2×Dic251—42no—
46C2×Dic251—42no—
47C2×Dic251—42no—
48C251⋊C8—24no—
49C23×C251—22no—
50C23×C251—21no—
51C23×C251—21no—
52C23×C251—22no—
53C23×C251C22⋊C412yes—
54C23×C251C2×D411yescharacteristic
55Q8×C251—24no—
56Q8×C251—41no—
57Q8×C251—41no—
58C2×C4×C251—22no—
59C2×C4×C251—24no—
60C2×C4×C251—22no—
61C2×C4×C251—22no—
62C2×C4×C251—22no—
63C2×C4×C251—22no—
64C2×C4×C251—42no—
65C2×C4×C251—41no—
66C2×C4×C251—41no—
67C2×C4×C251C22⋊C412yes—
68C2×C4×C251C22×C411yescharacteristic
69C2×C4×C251C2×D412yes—
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251#4112yes—
72C22×C251—22no—
73C22×C251—22no—
74C22×C251—22no—
75C22×C251—42no—
76C22×C251—42no—
77C22×C251C2×C22⋊C411yescharacteristic
78C22×C251#12112yes—
79C22×C251AGL(1,ℤ/8)11yescharacteristic
80C22×C251#13711yescharacteristic
81C22×C251#13712yes—
82C4×C251—22no—
83C4×C251—42no—
84C4×C251—42no—
85C4×C251C4×D412yes—
86C4×C251#12011yescharacteristic
87C4×C251#12311yescharacteristic
88Dic251—82no—
89C2×C251—42no—
90C2×C251#38611yescharacteristic
91C2×C251#41711yescharacteristic
92C2×C251#41911yescharacteristic
93C2×C251#46812yes—
94C2×C251#47012yes—
95C251#189411yescharacteristic, complemented, Sylow
96#1894—2511nomaximal, Sylow
97C22×M16—2511no—
98C2×C22⋊Q8D25111yescharacteristic
99C2×#103—2512no—
100#380—2511no—
101#340—2512no—
102C22×Q8C2×D25111yescharacteristic
103C23×C4C2×D25111yescharacteristic
104C2×M16—5022no—
105C2×M16—5024no—
106C22×C8—2512no—
107C22⋊Q8—22no—
108C22⋊Q8Dic25114yescomplemented
109C2×C4⋊C4—21no—
110C2×C4⋊C4—2512no—
111C2×C4⋊C4C2×D25111yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—5021no—
114#103—5024no—
115#95—5021no—
116C24C22⋊D25111yescharacteristic
117C2×Q8—21no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8C2×Dic25112yes—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—5022no—
126C22×C4C2×Dic25112yes—
127C22×C4C22⋊D25111yescharacteristic
128C22×C4C22⋊D25112yes—
129C22×C4C22×D25111yescharacteristic
130M16—5028no—
131C2×C8—5022no—
132C2×C8—5022no—
133C2×C8—2514no—
134C4⋊C4—24no—
135C4⋊C4—21no—
136C4⋊C4—5022no—
137C4⋊C4—5022no—
138C4⋊C4C2×Dic25112yes—
139C22⋊C4—24no—
140C22⋊C4—10042no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—22no—
145C23C22⋊Dic25112yes—
146C23C2×C22⋊D25111yescharacteristic
147Q8—24no—
148Q8—41no—
149Q8—41no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—10042no—
160C2×C4—10042no—
161C2×C4—10042no—
162C2×C4C22⋊Dic25112yes—
163C2×C4C22×Dic25111yescharacteristic
164C2×C4D4×D25112yes—
165C2×C4C2×C22⋊D25111yescharacteristic
166C2×C4#1686080912yes—
167C8—5024no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22C2×C22⋊Dic25111yescharacteristic
174C22#2254868712yes—
175C22#2254874011yescharacteristic
176C22#2254874511yescharacteristic
177C22#2254874612yes—
178C4—22no—
179C4—42no—
180C4—42no—
181C4—20082no—
182C4D4×Dic25112yes—
183C4#2254868311yescharacteristic
184C4#2254869611yescharacteristic
185C2—42no—
186C2#3063156711yescharacteristic
187C2#3063168811yescharacteristic
188C2#3063169611yescharacteristic
189C2#3063193512yes—
190C2#3063194312yes—
subgroupG/H
centerC23C2×C22⋊D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC2×C22⋊Q8×C251C2
socleC23×C251C2×D4
prime radicalC24×C251D4
cosocleC22×C4×C251C23
hypercenterC2×C22⋊Q8D251
hypocenterC251#1894
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#22548639C2semidirect
4. #30631449C2semidirect
16. C22⋊Q8×C251C4semidirect
95. C251#1894semidirect
108. C22⋊Q8Dic251semidirect

Maximal subgroups

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

subgroupindex
C22×#168607942
C2×#225486392
#306315472
#306314492
C2×C22⋊Q8×C2512
#1894251