Finite Group Databasebeta

← #44398705

Subgroups — #44398705

Order 29056.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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#28409613C211yesmaximal, characteristic
2#28409173C212yesmaximal, complemented
3#28408952C211yesmaximal, characteristic, complemented
4#28408767C212yesmaximal, complemented
5#324×C227C211yesmaximal, characteristic, complemented
6C2×#16813884—21no—
7C22×Dic454C2211yescharacteristic
8C2×#16813873—22no—
9#20300268—22no—
10#20300263—22no—
11#20300262—21no—
12#20300262C2212yes—
13C2×Dic227⋊C4—21no—
14#20300213—22no—
15C227⋊#95—21no—
16C227⋊#95C2211yescharacteristic
17C2×C4⋊C4×C227C2211yescharacteristic
18C22⋊C8×C227C2212yes—
19#16813884—42no—
20#16813884—42no—
21C2×Dic454—21no—
22C2×Dic454—22no—
23C2×Dic454—42no—
24C2×D908—21no—
25C2×D908—21no—
26C2×D908D412yes—
27C2×C22⋊D227—22no—
28C22×Dic227—21no—
29C22×Dic227—21no—
30C2×C4×D227—22no—
31#16813873—44no—
32C4⋊Dic227—21no—
33C4⋊Dic227D412yes—
34Dic227⋊C4—42no—
35(C2×D227)⋊C4—22no—
36(C2×D227)⋊C4—22no—
37C22×C4×C227D411yescharacteristic
38C22×C4×C227D411yescharacteristic
39C22×C4×C227C2311yescharacteristic
40C2×C8×C227—22no—
41C4⋊C4×C227—22no—
42C22×D227—22no—
43Dic454—24no—
44Dic454—42no—
45Dic454—42no—
46D908—42no—
47D908—42no—
48D908—81no—
49C22⋊D227—42no—
50C22⋊D227—42no—
51C2×Dic227—22no—
52C2×Dic227—22no—
53C2×Dic227—22no—
54C2×Dic227—41no—
55C2×Dic227—41no—
56C2×Dic227—41no—
57C2×Dic227—41no—
58C4×D227—44no—
59C23×C227C2×D411yescharacteristic
60C2×C4×C227—21no—
61C2×C4×C227—21no—
62C2×C4×C227—21no—
63C2×C4×C227—41no—
64C2×C4×C227—41no—
65C2×C4×C227SD1612yescomplemented
66C2×C4×C227C2×D412yes—
67C8×C227—42no—
68C22×C227—21no—
69C22×C227—21no—
70C22×C227#12011yescharacteristic
71C22×C227C2×SD1611yescharacteristic
72C22×C227AGL(1,ℤ/8)11yescharacteristic
73C2×D227—42no—
74C2×D227—42no—
75C2×D227—42no—
76C4×C227—22no—
77C4×C227—22no—
78C4×C227—41no—
79Dic227—42no—
80Dic227—42no—
81Dic227—42no—
82C2×C227—21no—
83C2×C227—21no—
84C2×C227#45011yescharacteristic
85C2×C227#45611yescharacteristic
86C2×C227B(4,𝔽2)11yescharacteristic
87D227—82no—
88C227#160411yescharacteristic, complemented, Sylow
89#1604—2271nomaximal, Sylow
90#537—2271no—
91#448—2272no—
92#395—2271no—
93#327—2272no—
94#324D22711yescharacteristic, complemented
95C2×#41—4541no—
96C22×Q8—2271no—
97C2×SD16—4542no—
98C22⋊Q8—4542no—
99#121—4542no—
100#121—4541no—
101#121—2272no—
102C2×C4⋊C4—4541no—
103C2×C4⋊C4C2×D22711yescharacteristic
104#103—4542no—
105C22⋊C8C2×D22712yes—
106#95—4541no—
107#95—2271no—
108#41—9082no—
109#41—9082no—
110C2×Q8—9082no—
111C2×Q8—4541no—
112C2×Q8—4542no—
113C2×D4—4541no—
114C2×D4—4541no—
115C2×D4—4542no—
116C2×D4—2272no—
117C22×C4—4542no—
118C22×C4—4541no—
119C22×C4—4541no—
120C22×C4D90811yescharacteristic
121C22×C4D90811yescharacteristic
122C22×C4C22×D22711yescharacteristic
123SD16—9084no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—9082no—
127C4⋊C4—4541no—
128C4⋊C4—2272no—
129C22⋊C4—4542no—
130C22⋊C4—4542no—
131C23—4542no—
132C23C2×D90811yescharacteristic
133Q8—9082no—
134Q8—9082no—
135Q8—4544no—
136D4—18161no—
137D4—9082no—
138D4—9082no—
139D4—9082no—
140D4—9082no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—9084no—
147C2×C4—9081no—
148C2×C4—9081no—
149C2×C4—9081no—
150C2×C4—9081no—
151C2×C4—4542no—
152C2×C4—4542no—
153C2×C4—4542no—
154C2×C4#1681387312yescomplemented
155C2×C4D4×D22712yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—9082no—
160C22—9082no—
161C22—9082no—
162C22#2030025811yescharacteristic
163C22C2×#1681387311yescharacteristic
164C22#2030031611yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—9082no—
169C4—9082no—
170C4—9082no—
171C2—21no—
172C2—21no—
173C2—18162no—
174C2#2840918511yescharacteristic
175C2#2840921711yescharacteristic
176C2#2840922011yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
Fitting#324×C227C2
socleC22×C227#120
prime radical#28409613C2
hypercenter#324D227
hypocenterC227#1604
LCSG ▷ C22×C4×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #28409173C2semidirect
3. #28408952C2semidirect
4. #28408767C2semidirect
5. #324×C227C2semidirect
65. C2×C4×C227SD16semidirect
88. C227#1604semidirect
94. #324D227semidirect
154. C2×C4#16813873semidirect

Maximal subgroups

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

subgroupindex
#284096132
#284091732
#284089522
#284087672
#324×C2272
#1604227