Finite Group Databasebeta

← #44549574

Subgroups — #44549574

Order 32128.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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#30632394C211yesmaximal, characteristic
2#30632038C211yesmaximal, characteristic, complemented
3#30631973C211yesmaximal, characteristic, complemented
4#30631892C211yesmaximal, characteristic
5#30631435C211yesmaximal, characteristic, complemented
6#30631419C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic
8C2×#16860798—21no—
9#22548715C2211yescharacteristic
10#22548715C2211yescharacteristic
11#22548706—21no—
12#22548694—21no—
13#22548692—22no—
14C4×Dic502C2211yescharacteristic
15D4×Dic251—21no—
16C2×C4⋊Dic251—21no—
17#22548647—21no—
18C4⋊(C251⋊C8)C2211yescharacteristic
19#22548640—21no—
20#22548640—22no—
21#22548638—21no—
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4×C8×C251C2211yescharacteristic
25C2×Dic502—21no—
26C2×Dic502D411yescharacteristic
27C2×Dic502D412yes—
28C22×Dic251—22no—
29#16860798—42no—
30C2×C251⋊C8—21no—
31C4⋊Dic251—22no—
32C4⋊Dic251—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251D411yescharacteristic
37Dic251⋊C4—22no—
38Dic251⋊C4—21no—
39C22⋊Dic251—22no—
40C4×Dic251—21no—
41C2×D4×C251D411yescharacteristic
42C22×C4×C251—21no—
43C2×C8×C251—21no—
44C2×C8×C251—21no—
45C4⋊C4×C251D411yescharacteristic
46C22⋊C4×C251—21no—
47C42×C251C2311yescharacteristic
48Dic502—21no—
49Dic502—21no—
50Dic502—42no—
51Dic502—41no—
52Dic502—41no—
53C2×Dic251—21no—
54C2×Dic251—21no—
55C2×Dic251—22no—
56C2×Dic251—41no—
57C2×Dic251—82no—
58C251⋊C8—41no—
59C23×C251—21no—
60D4×C251—21no—
61D4×C251SD1612yes—
62C2×C4×C251—21no—
63C2×C4×C251—41no—
64C2×C4×C251—41no—
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251C2×D411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C8×C251—41no—
69C8×C251—41no—
70C22×C251—22no—
71C22×C251—41no—
72C22×C251—41no—
73C22×C251#12011yescharacteristic
74C4×C251—21no—
75C4×C251—41no—
76C4×C251C2×SD1611yescharacteristic
77C4×C251#13511yescharacteristic
78C4×C251#13711yescharacteristic
79C4×C251#13711yescharacteristic
80Dic251—41no—
81Dic251—41no—
82Dic251—42no—
83Dic251—81no—
84C2×C251—42no—
85C2×C251#44811yescharacteristic
86C2×C251#45211yescharacteristic
87C2×C251#45511yescharacteristic
88C251#164311yescharacteristic, complemented, Sylow
89#1643—2511nomaximal, Sylow
90#554—2511no—
91#487—2511no—
92#475—2511no—
93#461—2511no—
94#333—2511no—
95Q8⋊C8—2511no—
96D4⋊C8D25111yescharacteristic
97C2×SD16—5021no—
98C4⋊Q8—2511no—
99C4⋊Q8—2511no—
100#125—5021no—
101C22⋊Q8—5022no—
102C22⋊Q8—5021no—
103C4×Q8—2511no—
104C4×D4—5021no—
105C4×D4C2×D25111yescharacteristic
106C2×C4⋊C4—5021no—
107#106—5021no—
108C4⋊C8—2511no—
109C4⋊C8C2×D25111yescharacteristic
110#103—5021no—
111#103—5022no—
112#103—5021no—
113C4×C8C2×D25111yescharacteristic
114C2×Q8—5021no—
115C2×Q8—2512no—
116C2×Q8—2511no—
117C2×D4D100411yescharacteristic
118C22×C4—21no—
119C22×C4—5022no—
120SD16—10042no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—5021no—
124C4⋊C4—5022no—
125C4⋊C4—5022no—
126C4⋊C4—5021no—
127C4⋊C4—5021no—
128C4⋊C4—5021no—
129C4⋊C4—5021no—
130C4⋊C4—5021no—
131C4⋊C4—2511no—
132C4⋊C4D100411yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—5022no—
135C42—5021no—
136C42C22×D25111yescharacteristic
137C23—21no—
138Q8—10042no—
139Q8—10041no—
140Q8—10041no—
141Q8—5021no—
142Q8—5021no—
143D4—21no—
144D4#1686079912yes—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—20082no—
149C2×C4—10041no—
150C2×C4—5021no—
151C2×C4—5021no—
152C2×C4—5022no—
153C2×C4D4×D25111yescharacteristic
154C2×C4D4×D25111yescharacteristic
155C2×C4C2×D100411yescharacteristic
156C8—41no—
157C8—41no—
158C8—10041no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#2254868511yescharacteristic
163C4—21no—
164C4—41no—
165C4—20081no—
166C4—10041no—
167C4—10041no—
168C4—10042no—
169C4C2×#1686079911yescharacteristic
170C4#2254873411yescharacteristic
171C4#2254874411yescharacteristic
172C4#2254874811yescharacteristic
173C2—42no—
174C2#3063182811yescharacteristic
175C2#3063185011yescharacteristic
176C2#3063186611yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
FittingD4⋊C8×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenterD4⋊C8D251
hypocenterC251#1643
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #30632038C2semidirect
3. #30631973C2semidirect
5. #30631435C2semidirect
6. #30631419C2semidirect
88. C251#1643semidirect

Maximal subgroups

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

subgroupindex
#306323942
#306320382
#306319732
#306318922
#306314352
#306314192
D4⋊C8×C2512
#1643251