Finite Group Databasebeta

← #44550685

Subgroups — #44550685

Order 32128.

412 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#22548689C211yesmaximal, characteristic
2C2×#22548674C211yesmaximal, characteristic, complemented
3C2×#22548637C212yesmaximal, complemented
4#30631450C212yesmaximal, complemented
5C2×C22⋊C8×C251C211yesmaximal, characteristic, complemented
6C22×D1004C2211yescharacteristic
7C22×C22⋊D251—21no—
8#22548689—22no—
9#22548689C414yescomplemented
10#22548674—24no—
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2212yes—
13C2×C22⋊Dic251—21no—
14C2×(C2×D251)⋊C4—21no—
15C2×C4×Dic251—21no—
16#22548637—24no—
17C23×C4×C251C2211yescharacteristic
18C22×C8×C251C2212yes—
19C22⋊C8×C251—22no—
20C23×D251—21no—
21C2×D1004—22no—
22C2×D1004—21no—
23C2×D1004—22no—
24C2×D1004C2×C412yes—
25C2×C22⋊D251—24no—
26C2×C22⋊D251—44no—
27C22×Dic251—21no—
28C22×Dic251—22no—
29C4⋊Dic251—22no—
30C4⋊Dic251—21no—
31C4⋊Dic251C2×C412yes—
32C4⋊Dic251D414yescomplemented
33C22⋊Dic251—24no—
34(C2×D251)⋊C4—24no—
35C4×Dic251—24no—
36C24×C251D411yescharacteristic
37C22×C4×C251—22no—
38C22×C4×C251—22no—
39C22×C4×C251C2×C412yes—
40C22×C4×C251D412yes—
41C22×C4×C251D411yescharacteristic
42C22×C4×C251C2311yescharacteristic
43C2×C8×C251—22no—
44C2×C8×C251—22no—
45C2×C8×C251—22no—
46C22×D251—22no—
47C22×D251—41no—
48C22×D251—41no—
49C22×D251—42no—
50C22×D251—42no—
51D1004—24no—
52D1004—42no—
53C22⋊D251—48no—
54C2×Dic251—22no—
55C2×Dic251—24no—
56C2×Dic251—42no—
57C2×Dic251—42no—
58C2×Dic251—42no—
59C23×C251—21no—
60C23×C251—21no—
61C23×C251—22no—
62C23×C251—22no—
63C23×C251C22⋊C412yes—
64C23×C251C2×D411yescharacteristic
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—24no—
68C2×C4×C251—24no—
69C2×C4×C251C22⋊C412yes—
70C2×C4×C251C22×C411yescharacteristic
71C2×C4×C251C2×D412yes—
72C2×C4×C251C2×D411yescharacteristic
73C2×C4×C251#4112yes—
74C8×C251—42no—
75C22×C251—22no—
76C22×C251—22no—
77C22×C251—22no—
78C22×C251—42no—
79C22×C251—42no—
80C22×C251C2×C22⋊C411yescharacteristic
81C22×C251#12011yescharacteristic
82C22×C251#12311yescharacteristic
83C22×C251#13512yes—
84C22×C251AGL(1,ℤ/8)12yes—
85C2×D251—42no—
86C2×D251—42no—
87C2×D251—42no—
88C2×D251—82no—
89C2×D251—82no—
90C4×C251—22no—
91C4×C251C4×D412yes—
92C4×C251#12112yes—
93Dic251—42no—
94Dic251—44no—
95C2×C251—42no—
96C2×C251#38611yescharacteristic
97C2×C251#41611yescharacteristic
98C2×C251#41811yescharacteristic
99C2×C251#44912yes—
100C2×C251#48212yes—
101D251—82no—
102C251#187511yescharacteristic, complemented, Sylow
103#1875—2511nomaximal, Sylow
104C2×#121—2511no—
105C2×#117—2511no—
106C2×#102—2512no—
107C2×C22⋊C8D25111yescharacteristic, complemented
108#340—2512no—
109C22×D4—5021no—
110C22×D4—2511no—
111C23×C4C2×D25111yescharacteristic
112C22×C8C2×D25112yes—
113#121—5022no—
114#121—2514no—
115#117—5024no—
116C2×C4⋊C4—2512no—
117C2×C4⋊C4—2511no—
118C2×C22⋊C4—5021no—
119C2×C22⋊C4—5021no—
120C2×C42—5021no—
121#102—5024no—
122C22⋊C8—22no—
123C24—5021no—
124C24D100411yescharacteristic
125C2×D4—10044no—
126C2×D4—5024no—
127C2×D4—5022no—
128C2×D4—5021no—
129C2×D4—5022no—
130C2×D4—2512no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—5022no—
134C22×C4—5021no—
135C22×C4C4×D25112yes—
136C22×C4C22⋊D25112yes—
137C22×C4D100411yescharacteristic
138C22×C4C22×D25111yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C4⋊C4—5022no—
143C4⋊C4—5021no—
144C4⋊C4—2512no—
145C4⋊C4—2514no—
146C22⋊C4—5024no—
147C22⋊C4—5024no—
148C42—5024no—
149C23—22no—
150C23—22no—
151C23—21no—
152C23—21no—
153C23—10042no—
154C23—10042no—
155C23—10041no—
156C23—10041no—
157C23—5022no—
158C23(C2×D251)⋊C412yes—
159C23C2×D100411yescharacteristic
160D4—10048no—
161D4—10042no—
162D4—5024no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—10042no—
168C2×C4—10042no—
169C2×C4—10042no—
170C2×C4—5022no—
171C2×C4—5024no—
172C2×C4(C2×D251)⋊C412yes—
173C2×C4C2×C4×D25111yescharacteristic
174C2×C4D4×D25112yes—
175C2×C4C2×C22⋊D25111yescharacteristic
176C2×C4#1686080912yes—
177C8—42no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—20082no—
184C22—20082no—
185C22—10042no—
186C22—10042no—
187C22—10042no—
188C22C2×(C2×D251)⋊C411yescharacteristic
189C22#2254868511yescharacteristic
190C22#2254869911yescharacteristic
191C22#2254873712yes—
192C22#2254874312yes—
193C4—22no—
194C4—10042no—
195C4—10044no—
196C4(C22⋊D251)⋊C412yes—
197C4#2254868712yes—
198C2—42no—
199C2—20082no—
200C2#3063156411yescharacteristic
201C2#3063168411yescharacteristic
202C2#3063169311yescharacteristic
203C2#3063183412yes—
204C2#3063201212yes—
subgroupG/H
centerC23C2×D1004
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC2×C22⋊C8×C251C2
socleC23×C251C2×D4
prime radicalC2×#22548689C2
cosocleC22×C4×C251C23
hypercenterC2×C22⋊C8D251
hypocenterC251#1875
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#22548674C2semidirect
3. C2×#22548637C2semidirect
4. #30631450C2semidirect
5. C2×C22⋊C8×C251C2semidirect
9. #22548689C4semidirect
32. C4⋊Dic251D4semidirect
102. C251#1875semidirect
107. C2×C22⋊C8D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486892
C2×#225486742
C2×#225486372
#306314502
C2×C22⋊C8×C2512
#1875251