Finite Group Databasebeta

← #44399024

Subgroups — #44399024

Order 29056.

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

normal subgroups only

This group has 262 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#28409774C211yesmaximal, characteristic, complemented
2#28409422C211yesmaximal, characteristic, complemented
3#28409421C212yesmaximal, complemented
4#28408771C212yesmaximal, complemented
5#329×C227C211yesmaximal, characteristic, complemented
6C2×#16813873—22no—
7C2×#16813871—21no—
8C2×D1816—22no—
9C2×#16813869—21no—
10#20300285C2211yescharacteristic, complemented
11#20300277—21no—
12#20300276C2212yes—
13#20300274—22no—
14#20300258—22no—
15#20300256—21no—
16#20300199C2211yescharacteristic, complemented
17#124×C227C2211yescharacteristic
18#97×C227C2212yescomplemented
19C23×D227—21no—
20C2×Dic454D412yes—
21C2×D908—21no—
22C2×D908—21no—
23C2×D908—22no—
24C2×D908D412yes—
25C2×C22⋊D227—22no—
26#16813873—44no—
27#16813871—42no—
28D1816—44no—
29#16813869—42no—
30C2×C227⋊C8—21no—
31C22⋊Dic227—21no—
32(C2×D227)⋊C4—22no—
33(C2×D227)⋊C4—22no—
34(C2×D227)⋊C4—22no—
35C4×Dic227—21no—
36C2×Q8×C227D411yescharacteristic
37C2×D4×C227D411yescharacteristic
38C2×C8×C227—22no—
39C22⋊C4×C227—22no—
40C42×C227C2311yescharacteristic
41C22×D227—22no—
42C22×D227—41no—
43C22×D227—82no—
44C22×D227—81no—
45Dic454—42no—
46D908—42no—
47D908—41no—
48D908—42no—
49D908—41no—
50D908—42no—
51D908—81no—
52C22⋊D227—82no—
53C2×Dic227—22no—
54C227⋊C8—22no—
55C23×C227—21no—
56Q8×C227—41no—
57D4×C227—41no—
58C2×C4×C227—21no—
59C2×C4×C227C2×D412yes—
60C2×C4×C227C2×D411yescharacteristic
61C8×C227—24no—
62C22×C227—41no—
63C22×C227—42no—
64C22×C227#12011yescharacteristic
65C2×D227—42no—
66C2×D227—42no—
67C2×D227—42no—
68C2×D227—84no—
69C2×D227—82no—
70C2×D227—81no—
71C4×C227—22no—
72C4×C227—21no—
73C4×C227—41no—
74Dic227—42no—
75C2×C227—81no—
76C2×C227#45312yes—
77C2×C227#45311yescharacteristic
78D227—82no—
79D227—161no—
80C227#165511yescharacteristic, complemented, Sylow
81#1655—2271nomaximal, Sylow
82H(3,𝔽4)—2271no—
83#496—2271no—
84#496—2272no—
85#329—2272no—
86#329D22711yescharacteristic, complemented
87C2×SD16—4542no—
88C2×SD16—4541no—
89C2×D8—4542no—
90C2×D8—4541no—
91#127—2271no—
92#124—4542no—
93#124—4541no—
94#124—2272no—
95#124C2×D22711yescharacteristic
96#120—4542no—
97#120—4541no—
98#97—2271no—
99#97C2×D22712yescomplemented
100C24—4541no—
101C2×Q8—2272no—
102C2×Q8D90811yescharacteristic
103C2×D4—4541no—
104C2×D4—4542no—
105C2×D4—4542no—
106C2×D4—4541no—
107C2×D4—2272no—
108C2×D4D90811yescharacteristic
109SD16—9084no—
110SD16—9082no—
111D8—9084no—
112D8—9082no—
113C2×C8—22no—
114C2×C8—4541no—
115C22⋊C4—22no—
116C22⋊C4—4542no—
117C22⋊C4—4541no—
118C22⋊C4—4542no—
119C22⋊C4—4542no—
120C42—4541no—
121C42C22×D22711yescharacteristic
122C23—21no—
123C23—18162no—
124C23—18161no—
125C23—9081no—
126C23—4542no—
127Q8—41no—
128Q8—9082no—
129D4—41no—
130D4—18161no—
131D4—18162no—
132D4—9082no—
133D4—9081no—
134D4—9082no—
135D4—9081no—
136D4—9082no—
137C2×C4—21no—
138C2×C4—4542no—
139C2×C4D4×D22712yes—
140C2×C4C2×D90811yescharacteristic
141C8—24no—
142C8—4542no—
143C22—42no—
144C22—41no—
145C22—18164no—
146C22—18162no—
147C22—18161no—
148C22—9082no—
149C22—9082no—
150C22—9082no—
151C22#2030025811yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—9082no—
156C2—81no—
157C2—18162no—
158C2—36321no—
159C2#2840920112yes—
160C2#2840920211yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#329×C227C2
socleC22×C227#120
hypercenter#329D227
hypocenterC227#1655
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #28409774C2semidirect
2. #28409422C2semidirect
3. #28409421C2semidirect
4. #28408771C2semidirect
5. #329×C227C2semidirect
10. #20300285C22semidirect
16. #20300199C22semidirect
18. #97×C227C22semidirect
80. C227#1655semidirect
86. #329D227semidirect
99. #97C2×D227semidirect

Maximal subgroups

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

subgroupindex
#284097742
#284094222
#284094212
#284087712
#329×C2272
#1655227