Finite Group Databasebeta

← #44401752

Subgroups — #44401752

Order 29056.

248 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 41 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#28409774C211yesmaximal, characteristic, complemented
2#28409207C212yesmaximal, complemented
3#28409204C211yesmaximal, characteristic, complemented
4#28408828C212yesmaximal, complemented
5#355×C227C211yesmaximal, characteristic, complemented
6#20300336C2211yescharacteristic, complemented
7#20300316—22no—
8#20300314—21no—
9#20300277C2212yes—
10#20300276—21no—
11#20300274—22no—
12#20300258—22no—
13#20300256—21no—
14#20300216—21no—
15#20300214—22no—
16#20300207C2211yescharacteristic, complemented
17#124×C227C2211yescharacteristic
18#101×C227C2212yescomplemented
19C23×D227—21no—
20#16813885—41no—
21#16813884—21no—
22#16813884—22no—
23C2×Dic454—21no—
24Q8×D227—41no—
25C2×D908—21no—
26C2×D908D412yes—
27C2×C22⋊D227—22no—
28#16813873—42no—
29#16813872—41no—
30D1816—42no—
31#16813869—41no—
32#16813868—21no—
33C22⋊Dic227—21no—
34(C2×D227)⋊C4—22no—
35(C2×D227)⋊C4—22no—
36(C2×D227)⋊C4—22no—
37C4×Dic227D412yes—
38C2×Q8×C227C2311yescharacteristic
39C2×D4×C227D411yescharacteristic
40M16×C227—22no—
41C22⋊C4×C227—22no—
42C42×C227D411yescharacteristic
43C22×D227—22no—
44C22×D227—82no—
45C22×D227—81no—
46Dic454—21no—
47Dic454—22no—
48Dic454—81no—
49D908—21no—
50D908—22no—
51D908—42no—
52D908—81no—
53C22⋊D227—41no—
54C22⋊D227—82no—
55C2×Dic227—22no—
56C4×D227—42no—
57C4×D227—41no—
58C227⋊C8—41no—
59C23×C227—21no—
60Q8×C227—41no—
61D4×C227—41no—
62C2×C4×C227—21no—
63C2×C4×C227C2×D412yes—
64C2×C4×C227C2×D411yescharacteristic
65C8×C227—42no—
66C22×C227—42no—
67C22×C227—41no—
68C22×C227#12011yescharacteristic
69C2×D227—42no—
70C2×D227—42no—
71C2×D227—41no—
72C2×D227—42no—
73C2×D227—84no—
74C4×C227—22no—
75C4×C227—21no—
76C4×C227—41no—
77Dic227—42no—
78Dic227—41no—
79C2×C227—21no—
80C2×C227—81no—
81C2×C227B(4,𝔽2)11yescharacteristic
82D227—81no—
83D227—82no—
84C227#220211yescharacteristic, complemented, Sylow
85#2202—2271nomaximal, Sylow
86H(3,𝔽4)—2271no—
87#454—2271no—
88#454—2272no—
89#355—2272no—
90#355D22711yescharacteristic, complemented
912−1+4—2271no—
92AGL(1,ℤ/8)—4542no—
93AGL(1,ℤ/8)—4541no—
94#124—4542no—
95#124—4541no—
96#124—2272no—
97#124C2×D22711yescharacteristic
98#120—4542no—
99#120—4541no—
100#104—4542no—
101#104—4541no—
102#101—2271no—
103#101C2×D22712yescomplemented
104C24—4541no—
105#41—9081no—
106#41—4542no—
107#41—4541no—
108C2×Q8—9081no—
109C2×Q8—4541no—
110C2×Q8C22×D22711yescharacteristic
111C2×D4—4541no—
112C2×D4—4542no—
113C2×D4—2272no—
114C2×D4D90811yescharacteristic
115SD16—9082no—
116SD16—9081no—
117D8—9082no—
118D8—9081no—
119M16—22no—
120M16—4541no—
121C22⋊C4—22no—
122C22⋊C4—4542no—
123C22⋊C4—4541no—
124C22⋊C4—4542no—
125C22⋊C4—4542no—
126C42—2272no—
127C42D90811yescharacteristic
128C23—21no—
129C23—18162no—
130C23—18161no—
131C23—4542no—
132Q8—41no—
133Q8—18161no—
134Q8—4542no—
135Q8—4541no—
136D4—41no—
137D4—18161no—
138D4—18162no—
139D4—9082no—
140D4—9081no—
141D4—4542no—
142D4—4541no—
143C2×C4—21no—
144C2×C4—9082no—
145C2×C4—9081no—
146C2×C4—4542no—
147C2×C4D4×D22712yes—
148C2×C4C2×D90811yescharacteristic
149C8—42no—
150C8—9081no—
151C22—42no—
152C22—41no—
153C22—18164no—
154C22—9082no—
155C22—9082no—
156C22—9081no—
157C22—9082no—
158C22#2030025811yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—9082no—
163C4—9081no—
164C2—21no—
165C2—81no—
166C2—18162no—
167C2—18161no—
168C2#2840922011yescharacteristic
subgroupG/H
centerC2#28409220
commutator, cosocleC2×Q8×C227C23
FrattiniC2×Q8C22×D227
Fitting#355×C227C2
socleC2×C227B(4,𝔽2)
hypercenter#355D227
hypocenterC227#2202
LCSG ▷ C2×Q8×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #28409774C2semidirect
2. #28409207C2semidirect
3. #28409204C2semidirect
4. #28408828C2semidirect
5. #355×C227C2semidirect
6. #20300336C22semidirect
16. #20300207C22semidirect
18. #101×C227C22semidirect
84. C227#2202semidirect
90. #355D227semidirect
103. #101C2×D227semidirect

Maximal subgroups

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

subgroupindex
#284097742
#284092072
#284092042
#284088282
#355×C2272
#2202227