Finite Group Databasebeta

← #49120478

Subgroups — #49120478

Order 36224.

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#31827172C211yesmaximal, characteristic
2#31826816C211yesmaximal, characteristic, complemented
3#31826745C211yesmaximal, characteristic
4#31826676C211yesmaximal, characteristic, complemented
5#31826213C211yesmaximal, characteristic, complemented
6#31826197C211yesmaximal, characteristic, complemented
7D4⋊C8×C283C211yesmaximal, characteristic
8C2×Dic1132—21no—
9#22583610C2211yescharacteristic
10#22583610C2211yescharacteristic
11#22583601—21no—
12#22583589—21no—
13#22583587—22no—
14C4×Dic566C2211yescharacteristic
15D4×Dic283—21no—
16C2×C4⋊Dic283—21no—
17#22583543—21no—
18C4⋊(C283⋊C8)C2211yescharacteristic
19#22583535—22no—
20#22583535—21no—
21#22583531—21no—
22C4×D4×C283C2211yescharacteristic
23C4⋊C8×C283C2211yescharacteristic
24C4×C8×C283C2211yescharacteristic
25C2×Dic566—21no—
26C2×Dic566D411yescharacteristic
27C2×Dic566D412yes—
28C22×Dic283—22no—
29Dic1132—42no—
30C2×C283⋊C8—21no—
31C4⋊Dic283—22no—
32C4⋊Dic283—21no—
33C4⋊Dic283—21no—
34C4⋊Dic283—21no—
35C4⋊Dic283—21no—
36C4⋊Dic283D411yescharacteristic
37Dic283⋊C4—22no—
38Dic283⋊C4—21no—
39C22⋊Dic283—22no—
40C4×Dic283—21no—
41C2×D4×C283D411yescharacteristic
42C22×C4×C283—21no—
43C2×C8×C283—21no—
44C2×C8×C283—21no—
45C4⋊C4×C283D411yescharacteristic
46C22⋊C4×C283—21no—
47C42×C283C2311yescharacteristic
48Dic566—21no—
49Dic566—21no—
50Dic566—42no—
51Dic566—41no—
52Dic566—41no—
53C2×Dic283—21no—
54C2×Dic283—21no—
55C2×Dic283—22no—
56C2×Dic283—41no—
57C2×Dic283—82no—
58C283⋊C8—41no—
59C23×C283—21no—
60D4×C283—21no—
61D4×C283Q1612yescomplemented
62C2×C4×C283—21no—
63C2×C4×C283—41no—
64C2×C4×C283—41no—
65C2×C4×C283C2×D411yescharacteristic
66C2×C4×C283C2×D411yescharacteristic
67C2×C4×C283C2×D411yescharacteristic
68C8×C283—41no—
69C8×C283—41no—
70C22×C283—22no—
71C22×C283—41no—
72C22×C283—41no—
73C22×C283#12011yescharacteristic
74C4×C283—21no—
75C4×C283—41no—
76C4×C283C2×Q1611yescharacteristic
77C4×C283#13511yescharacteristic
78C4×C283AGL(1,ℤ/8)11yescharacteristic
79C4×C283#13711yescharacteristic
80Dic283—41no—
81Dic283—41no—
82Dic283—42no—
83Dic283—81no—
84C2×C283—42no—
85C2×C283#44911yescharacteristic
86C2×C283#45111yescharacteristic
87C2×C283#45511yescharacteristic
88C283#164411yescharacteristic, complemented, Sylow
89#1644—2831nomaximal, Sylow
90#554—2831no—
91#487—2831no—
92#474—2831no—
93#462—2831no—
94#333—2831no—
95Q8⋊C8—2831no—
96D4⋊C8D28311yescharacteristic
97C2×Q16—5661no—
98C4⋊Q8—2831no—
99C4⋊Q8—2831no—
100#125—5661no—
101C22⋊Q8—5662no—
102C22⋊Q8—5661no—
103C4×Q8—2831no—
104C4×D4—5661no—
105C4×D4C2×D28311yescharacteristic
106C2×C4⋊C4—5661no—
107#107—5661no—
108C4⋊C8—2831no—
109C4⋊C8C2×D28311yescharacteristic
110#103—5661no—
111#103—5662no—
112#102—5661no—
113C4×C8C2×D28311yescharacteristic
114C2×Q8—2831no—
115C2×Q8—2832no—
116C2×Q8—5661no—
117C2×D4D113211yescharacteristic
118C22×C4—21no—
119C22×C4—5662no—
120Q16—11322no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—5661no—
124C4⋊C4—2831no—
125C4⋊C4—5662no—
126C4⋊C4—5662no—
127C4⋊C4—5661no—
128C4⋊C4—5661no—
129C4⋊C4—5661no—
130C4⋊C4—5661no—
131C4⋊C4—5661no—
132C4⋊C4D113211yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—5662no—
135C42—5661no—
136C42C22×D28311yescharacteristic
137C23—21no—
138Q8—5661no—
139Q8—5661no—
140Q8—11322no—
141Q8—11321no—
142Q8—11321no—
143D4—21no—
144D4Dic113212yescomplemented
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—5661no—
149C2×C4—5661no—
150C2×C4—5662no—
151C2×C4—11321no—
152C2×C4—22642no—
153C2×C4D4×D28311yescharacteristic
154C2×C4D4×D28311yescharacteristic
155C2×C4C2×D113211yescharacteristic
156C8—41no—
157C8—41no—
158C8—11321no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#2258358011yescharacteristic
163C4—21no—
164C4—41no—
165C4—11321no—
166C4—11321no—
167C4—11322no—
168C4—22641no—
169C4C2×Dic113211yescharacteristic
170C4#2258363111yescharacteristic
171C4#2258363411yescharacteristic
172C4#2258364311yescharacteristic
173C2—42no—
174C2#3182661111yescharacteristic
175C2#3182662311yescharacteristic
176C2#3182664411yescharacteristic
subgroupG/H
centerC22#22583580
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
FittingD4⋊C8×C283C2
socleC22×C283#120
prime radicalC2×D4×C283D4
hypercenterD4⋊C8D283
hypocenterC283#1644
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #31826816C2semidirect
4. #31826676C2semidirect
5. #31826213C2semidirect
6. #31826197C2semidirect
61. D4×C283Q16semidirect
88. C283#1644semidirect
144. D4Dic1132semidirect

Maximal subgroups

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

subgroupindex
#318271722
#318268162
#318267452
#318266762
#318262132
#318261972
D4⋊C8×C2832
#1644283