Finite Group Databasebeta

← #44549570

Subgroups — #44549570

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#30632039C211yesmaximal, characteristic, complemented
3#30631969C211yesmaximal, characteristic, complemented
4#30631435C211yesmaximal, characteristic, complemented
5#30631417C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic
7#461×C251C211yesmaximal, characteristic
8#22548715C2211yescharacteristic
9#22548706—21no—
10#22548694—21no—
11#22548692—22no—
12C4×Dic502C2211yescharacteristic
13D4×Dic251—21no—
14C2×C4⋊Dic251—21no—
15#22548646—21no—
16C4⋊(C251⋊C8)C2211yescharacteristic
17#22548638—21no—
18#22548638—22no—
19C4×C251⋊C8C2211yescharacteristic
20C2×SD16×C251—21no—
21C4⋊Q8×C251C2211yescharacteristic
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25C2×Dic502D411yescharacteristic
26C2×Dic502D412yes—
27C22×Dic251—22no—
28C2×C251⋊C8—21no—
29C2×C251⋊C8—21no—
30C4⋊Dic251—22no—
31C4⋊Dic251—21no—
32C4⋊Dic251—21no—
33C4⋊Dic251D411yescharacteristic
34Dic251⋊C4—22no—
35Dic251⋊C4—21no—
36C22⋊Dic251—22no—
37C4×Dic251—21no—
38C2×Q8×C251—21no—
39C2×D4×C251D411yescharacteristic
40C22×C4×C251—21no—
41SD16×C251—42no—
42C2×C8×C251—21no—
43C4⋊C4×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251D411yescharacteristic
46C22⋊C4×C251—21no—
47C42×C251C2311yescharacteristic
48Dic502—21no—
49Dic502—21no—
50Dic502—42no—
51C2×Dic251—21no—
52C2×Dic251—21no—
53C2×Dic251—22no—
54C2×Dic251—82no—
55C251⋊C8—41no—
56C251⋊C8—41no—
57C23×C251—21no—
58Q8×C251—41no—
59Q8×C251—41no—
60D4×C251—21no—
61D4×C251SD1612yes—
62C2×C4×C251—21no—
63C2×C4×C251—41no—
64C2×C4×C251—41no—
65C2×C4×C251—41no—
66C2×C4×C251C2×D411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C8×C251—41no—
70C22×C251—22no—
71C22×C251—41no—
72C22×C251—41no—
73C22×C251#12011yescharacteristic
74C4×C251—21no—
75C4×C251—41no—
76C4×C251—81no—
77C4×C251C2×SD1611yescharacteristic
78C4×C251#13511yescharacteristic
79C4×C251#13711yescharacteristic
80C4×C251#13711yescharacteristic
81Dic251—41no—
82Dic251—41no—
83Dic251—42no—
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#461D25111yescharacteristic
94#333—2511no—
95Q8⋊C8—2511no—
96D4⋊C8—2511no—
97C2×SD16—21no—
98C4⋊Q8—2511no—
99C4⋊Q8C2×D25111yescharacteristic
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—21no—
111#103—5021no—
112#103—5022no—
113C4×C8—2511no—
114C2×Q8—21no—
115C2×Q8—2512no—
116C2×Q8—2511no—
117C2×D4C22⋊D25111yescharacteristic
118C22×C4—21no—
119C22×C4—5022no—
120SD16—42no—
121C2×C8—21no—
122C2×C8—5021no—
123C2×C8—5021no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—5022no—
127C4⋊C4—5022no—
128C4⋊C4—5021no—
129C4⋊C4—5021no—
130C4⋊C4—5021no—
131C4⋊C4—2511no—
132C4⋊C4C22⋊D25111yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—5022no—
135C42—5021no—
136C42C22×D25111yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—10042no—
141Q8—5021no—
142Q8—5021no—
143D4—21no—
144D4#1686079812yes—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—20082no—
150C2×C4—5021no—
151C2×C4—5021no—
152C2×C4—5022no—
153C2×C4D4×D25111yescharacteristic
154C2×C4D4×D25111yescharacteristic
155C2×C4C2×C22⋊D25111yescharacteristic
156C8—41no—
157C8—10041no—
158C8—10041no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#2254868311yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—10041no—
167C4—10041no—
168C4—10042no—
169C4C2×#1686079811yescharacteristic
170C4#2254873611yescharacteristic
171C4#2254874411yescharacteristic
172C4#2254874511yescharacteristic
173C2—42no—
174C2#3063182511yescharacteristic
175C2#3063184611yescharacteristic
176C2#3063186311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#461×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenter#461D251
hypocenterC251#1643
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #30632039C2semidirect
3. #30631969C2semidirect
4. #30631435C2semidirect
5. #30631417C2semidirect
88. C251#1643semidirect

Maximal subgroups

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

subgroupindex
#306323942
#306320392
#306319692
#306314352
#306314172
D4⋊(C251⋊C8)2
#461×C2512
#1643251