Finite Group Databasebeta

← #44549609

Subgroups — #44549609

Order 32128.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632415C211yesmaximal, characteristic
2#30632092C211yesmaximal, characteristic
3#30631973C211yesmaximal, characteristic
4#30631897C211yesmaximal, characteristic
5#30631432C211yesmaximal, characteristic
6#30631419C211yesmaximal, characteristic
7Q8⋊C8×C251C211yesmaximal, characteristic
8C2×#16860800—21no—
9#22548715C2211yescharacteristic
10#22548715C2211yescharacteristic
11#22548713—21no—
12#22548707—21no—
13#22548705—22no—
14C4×Dic502—21no—
15C4×Dic502C2211yescharacteristic
16#22548680—21no—
17#22548649—22no—
18#22548647—21no—
19C4⋊(C251⋊C8)C2211yescharacteristic
20#22548640—21no—
21#22548638—21no—
22C4×Q8×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4×C8×C251C2211yescharacteristic
25C2×Dic502—21no—
26C2×Dic502—21no—
27C2×Dic502D411yescharacteristic
28#16860800—42no—
29C2×C251⋊C8—21no—
30C4⋊Dic251—21no—
31C4⋊Dic251—21no—
32C4⋊Dic251—21no—
33C4⋊Dic251D411yescharacteristic
34C4⋊Dic251D412yes—
35Dic251⋊C4—22no—
36Dic251⋊C4—22no—
37Dic251⋊C4—22no—
38Dic251⋊C4—21no—
39C4×Dic251—22no—
40C4×Dic251—21no—
41C2×Q8×C251D411yescharacteristic
42C2×C8×C251—21no—
43C2×C8×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251D411yescharacteristic
46C42×C251—21no—
47C42×C251C2311yescharacteristic
48Dic502—21no—
49Dic502—41no—
50Dic502—41no—
51Dic502—41no—
52Dic502—41no—
53Dic502Q1612yescomplemented
54C2×Dic251—21no—
55C2×Dic251—21no—
56C2×Dic251—22no—
57C2×Dic251—41no—
58C251⋊C8—41no—
59Q8×C251—21no—
60Q8×C251—21no—
61C2×C4×C251—21no—
62C2×C4×C251—21no—
63C2×C4×C251C2×D411yescharacteristic
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251C2×D411yescharacteristic
66C8×C251—22no—
67C8×C251—41no—
68C22×C251#12011yescharacteristic
69C4×C251—21no—
70C4×C251—41no—
71C4×C251—41no—
72C4×C251C2×Q1611yescharacteristic
73C4×C251#13511yescharacteristic
74C4×C251AGL(1,ℤ/8)11yescharacteristic
75C4×C251#13711yescharacteristic
76Dic251—22no—
77Dic251—42no—
78Dic251—41no—
79Dic251—81no—
80C2×C251#44911yescharacteristic
81C2×C251#45111yescharacteristic
82C2×C251#45611yescharacteristic
83C251#164811yescharacteristic, complemented, Sylow
84#1648—2511nomaximal, Sylow
85#557—2511no—
86#500—2511no—
87#475—2511no—
88#462—2511no—
89#332—2511no—
90Q8⋊C8—2511no—
91Q8⋊C8D25111yescharacteristic
92C2×Q16—5021no—
93C4⋊Q8—5021no—
94C4⋊Q8—2511no—
95C4⋊Q8—2511no—
96#125—5021no—
97#125—5022no—
98C4×Q8—5021no—
99C4×Q8—5021no—
100C4×Q8—2511no—
101C4×Q8C2×D25111yescharacteristic
102#107—5022no—
103#106—5021no—
104C4⋊C8—2511no—
105C4⋊C8C2×D25111yescharacteristic
106#103—5021no—
107#103—5021no—
108C4×C8C2×D25111yescharacteristic
109C2×Q8—5021no—
110C2×Q8—5021no—
111C2×Q8—2511no—
112C2×Q8D100411yescharacteristic
113Q16—10042no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—5021no—
117C4⋊C4—21no—
118C4⋊C4—5022no—
119C4⋊C4—5022no—
120C4⋊C4—5022no—
121C4⋊C4—5021no—
122C4⋊C4—5021no—
123C4⋊C4—5021no—
124C4⋊C4—5021no—
125C4⋊C4—2511no—
126C4⋊C4—2512no—
127C4⋊C4D100411yescharacteristic
128C42—21no—
129C42—5022no—
130C42—5021no—
131C42C22×D25111yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—10041no—
135Q8—10041no—
136Q8—10041no—
137Q8—10041no—
138Q8—5021no—
139Q8—2512no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—10041no—
143C2×C4—5021no—
144C2×C4—5022no—
145C2×C4—5021no—
146C2×C4D4×D25111yescharacteristic
147C2×C4D4×D25111yescharacteristic
148C2×C4C2×D100411yescharacteristic
149C8—22no—
150C8—41no—
151C8—10041no—
152C22#2254868511yescharacteristic
153C4—21no—
154C4—41no—
155C4—41no—
156C4—20081no—
157C4—10042no—
158C4—10041no—
159C4—5022no—
160C4Q16×D25111yescharacteristic
161C4#2254873711yescharacteristic
162C4#2254874311yescharacteristic
163C4#2254874411yescharacteristic
164C2#3063183411yescharacteristic
165C2#3063184411yescharacteristic
166C2#3063187111yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
FittingQ8⋊C8×C251C2
socle, prime radicalC22×C251#120
hypercenterQ8⋊C8D251
hypocenterC251#1648
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
53. Dic502Q16semidirect
83. C251#1648semidirect

Maximal subgroups

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

subgroupindex
#306324152
#306320922
#306319732
#306318972
#306314322
#306314192
Q8⋊C8×C2512
#1648251