Finite Group Databasebeta

← #44509446

Subgroups — #44509446

Order 30848.

270 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 45 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#29507903C212yesmaximal, complemented
2#29507894C211yesmaximal, characteristic, complemented
3#29507512C211yesmaximal, characteristic, complemented
4#29507504C212yesmaximal, complemented
5H(3,𝔽4)×C241C211yesmaximal, characteristic, complemented
6#21440687C2211yescharacteristic
7#21440673—21no—
8#21440611—22no—
9#21440570—21no—
10#21440559C2211yescharacteristic
11#21440557C2212yes—
12#21440556—22no—
13#124×C241—21no—
14#124×C241—22no—
15#124×C241—21no—
16#124×C241C2211yescharacteristic
17#120×C241—22no—
18#120×C241C2212yes—
19#16838770—21no—
20#16838769—41no—
21C2×C22⋊D241—22no—
22C2×C22⋊D241—42no—
23D4×D241—41no—
24#16838760—41no—
25#16838757—41no—
26#16838754—21no—
27C22⋊Dic241—22no—
28C22⋊Dic241—42no—
29C24×C241D412yescomplemented
30C2×Q8×C241—21no—
31C2×Q8×C241D411yescharacteristic, complemented
32C2×D4×C241—21no—
33C2×D4×C241—21no—
34C2×D4×C241—22no—
35C2×D4×C241C2311yescharacteristic
36C22⋊C4×C241—22no—
37C22⋊C4×C241—22no—
38C22⋊C4×C241—22no—
39C22⋊C4×C241—22no—
40C22⋊C4×C241D412yescomplemented
41C42×C241—21no—
42C42×C241D411yescharacteristic, complemented
43C22×D241—22no—
44Dic482—21no—
45D964—21no—
46C22⋊D241—42no—
47C22⋊D241—41no—
48C22⋊D241—84no—
49C2×Dic241—22no—
50C2×Dic241—42no—
51C4×D241—41no—
52#14515349—41no—
53C23×C241—22no—
54C23×C241—42no—
55C23×C241—82no—
56C23×C241C2×D412yes—
57Q8×C241—41no—
58Q8×C241—81no—
59D4×C241—41no—
60D4×C241—82no—
61D4×C241—81no—
62D4×C241—81no—
63C2×C4×C241—21no—
64C2×C4×C241—22no—
65C2×C4×C241—21no—
66C2×C4×C241C2×D411yescharacteristic
67C22×C241—22no—
68C22×C241—42no—
69C22×C241—42no—
70C22×C241—42no—
71C22×C241—44no—
72C22×C241—42no—
73C22×C241—82no—
74C22×C241#12011yescharacteristic
75C2×D241—41no—
76C2×D241—82no—
77C4×C241—21no—
78C4×C241—42no—
79C4×C241—41no—
80C4×C241—41no—
81Dic241—41no—
82Dic241—82no—
83C2×C241—21no—
84C2×C241—42no—
85C2×C241—82no—
86C2×C241B(4,𝔽2)11yescharacteristic
87D241—81no—
88C241#219911yescharacteristic, complemented, Sylow
89#2199—2411nomaximal, Sylow
90H(3,𝔽4)D24111yescharacteristic, complemented
91B(4,𝔽2)—2412no—
92#455—2411no—
93#354—2411no—
94AΣL(1,𝔽16)—2412no—
95H(5,𝔽2)—2411no—
96#137—4821no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C2×D24111yescharacteristic
101#120—22no—
102#120—4822no—
103#120C2×D24112yes—
104#104—4821no—
105#100—2411no—
106#99—4822no—
107#99—2412no—
108C24C22⋊D24112yescomplemented
109#41—9641no—
110#41—4821no—
111C2×Q8—21no—
112C2×Q8C22⋊D24111yescharacteristic, complemented
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—9641no—
117C2×D4—9642no—
118C2×D4—4822no—
119C2×D4C22×D24111yescharacteristic
120Q16—9641no—
121SD16—9641no—
122M16—4821no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—9642no—
128C22⋊C4—4822no—
129C22⋊C4C22⋊D24112yescomplemented
130C42—21no—
131C42C22⋊D24111yescharacteristic, complemented
132C23—22no—
133C23—42no—
134C23—82no—
135C23—4822no—
136C23C2×C22⋊D24112yes—
137Q8—41no—
138Q8—81no—
139Q8—4821no—
140D4—41no—
141D4—82no—
142D4—81no—
143D4—81no—
144D4—19284no—
145D4—9642no—
146D4—9641no—
147D4—4821no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—9641no—
152C2×C4—9642no—
153C2×C4—4822no—
154C2×C4C2×C22⋊D24111yescharacteristic
155C8—9641no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—19282no—
164C22—9641no—
165C22#2144061111yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—19282no—
171C4—9641no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—19281no—
176C2#2950790411yescharacteristic
subgroupG/H
centerC2#29507904
commutator, cosocleC2×D4×C241C23
FrattiniC2×D4C22×D241
FittingH(3,𝔽4)×C241C2
socleC2×C241B(4,𝔽2)
hypercenterH(3,𝔽4)D241
hypocenterC241#2199
LCSG ▷ C2×D4×C241 ▷ C22×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #29507903C2semidirect
2. #29507894C2semidirect
3. #29507512C2semidirect
4. #29507504C2semidirect
5. H(3,𝔽4)×C241C2semidirect
29. C24×C241D4semidirect
31. C2×Q8×C241D4semidirect
40. C22⋊C4×C241D4semidirect
42. C42×C241D4semidirect
88. C241#2199semidirect
90. H(3,𝔽4)D241semidirect
108. C24C22⋊D241semidirect
112. C2×Q8C22⋊D241semidirect
129. C22⋊C4C22⋊D241semidirect
131. C42C22⋊D241semidirect

Maximal subgroups

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

subgroupindex
#295079032
#295078942
#295075122
#295075042
H(3,𝔽4)×C2412
#2199241