Finite Group Databasebeta

← #44399019

Subgroups — #44399019

Order 29056.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#28409685C211yesmaximal, characteristic
2#28409428C211yesmaximal, characteristic, complemented
3#28409313C211yesmaximal, characteristic, complemented
4#28409238C211yesmaximal, characteristic, complemented
5#28408759C211yesmaximal, characteristic, complemented
6#28408759C211yesmaximal, characteristic, complemented
7#332×C227C211yesmaximal, characteristic, complemented
8C2×#16813885—21no—
9C2×#16813873—21no—
10#20300288C2211yescharacteristic
11#20300273—22no—
12#20300264—21no—
13#20300263—21no—
14C4×D908C2211yescharacteristic
15C4×D908C2211yescharacteristic
16C4⋊C4×D227—21no—
17#20300222—21no—
18#20300219—22no—
19#20300213—21no—
20#20300210—21no—
21C4×C227⋊C8C2211yescharacteristic
22C4⋊Q8×C227C2211yescharacteristic
23C4⋊C8×C227C2211yescharacteristic
24C4⋊C8×C227C2211yescharacteristic
25#16813885—42no—
26#16813885—42no—
27C2×Dic454—21no—
28C2×D908—22no—
29C2×D908D411yescharacteristic
30C2×D908D411yescharacteristic
31C2×C4×D227—22no—
32C2×C4×D227—21no—
33C2×C4×D227—21no—
34#16813873—42no—
35C2×C227⋊C8—21no—
36C4⋊Dic227—21no—
37C4⋊Dic227—21no—
38C4⋊Dic227D411yescharacteristic
39C4⋊Dic227D411yescharacteristic
40Dic227⋊C4—22no—
41(C2×D227)⋊C4—22no—
42(C2×D227)⋊C4—21no—
43(C2×D227)⋊C4—21no—
44C2×Q8×C227—21no—
45C2×C8×C227—21no—
46C2×C8×C227—21no—
47C4⋊C4×C227—21no—
48C4⋊C4×C227D412yes—
49C42×C227C2311yescharacteristic
50C22×D227—21no—
51C22×D227—21no—
52Dic454—41no—
53Dic454—41no—
54D908—21no—
55D908—21no—
56D908—21no—
57D908—44no—
58D908SD1612yescomplemented
59C2×Dic227—21no—
60C2×Dic227—21no—
61C2×Dic227—41no—
62C4×D227—44no—
63C4×D227—41no—
64C4×D227—41no—
65C4×D227—41no—
66C4×D227—41no—
67C227⋊C8—22no—
68Q8×C227—41no—
69Q8×C227—41no—
70C2×C4×C227—22no—
71C2×C4×C227C2×D411yescharacteristic
72C2×C4×C227C2×D411yescharacteristic
73C2×C4×C227C2×D411yescharacteristic
74C8×C227—41no—
75C8×C227—41no—
76C22×C227#12011yescharacteristic
77C2×D227—22no—
78C2×D227—41no—
79C2×D227—41no—
80C2×D227—41no—
81C2×D227—41no—
82C2×D227—41no—
83C4×C227—21no—
84C4×C227—42no—
85C4×C227C2×SD1611yescharacteristic
86C4×C227#13511yescharacteristic
87C4×C227AGL(1,ℤ/8)11yescharacteristic
88C4×C227AGL(1,ℤ/8)11yescharacteristic
89Dic227—41no—
90Dic227—41no—
91Dic227—81no—
92C2×C227#44911yescharacteristic
93C2×C227#45011yescharacteristic
94C2×C227#45611yescharacteristic
95D227—42no—
96D227—81no—
97C227#165411yescharacteristic, complemented, Sylow
98#1654—2271nomaximal, Sylow
99#547—2271no—
100#498—2271no—
101#474—2271no—
102#461—2271no—
103#332D22711yescharacteristic, complemented
104D4⋊C8—2271no—
105D4⋊C8—2271no—
106C2×#41—4541no—
107C2×SD16—4541no—
108C4⋊Q8—2271no—
109C4⋊Q8C2×D22711yescharacteristic
110#123—4542no—
111C22⋊Q8—4541no—
112#121—4541no—
113C4×D4—2271no—
114C4×D4—2271no—
115C2×C4⋊C4—4541no—
116#107—4541no—
117#106—4542no—
118C4⋊C8C2×D22711yescharacteristic
119C4⋊C8C2×D22711yescharacteristic
120#103—4541no—
121#102—4541no—
122C4×C8—2271no—
123#41—9082no—
124#41—9082no—
125C2×Q8—21no—
126C2×Q8—4541no—
127C2×D4—4542no—
128C2×D4—2271no—
129C2×D4—2271no—
130C22×C4—4542no—
131C22×C4—4541no—
132C22×C4—4541no—
133SD16—9082no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—4541no—
137C4⋊C4—21no—
138C4⋊C4—4542no—
139C4⋊C4—4541no—
140C4⋊C4—4541no—
141C4⋊C4—2271no—
142C4⋊C4—2271no—
143C4⋊C4D90812yes—
144C22⋊C4—4542no—
145C22⋊C4—4541no—
146C22⋊C4—4541no—
147C42C22×D22711yescharacteristic
148C23—4541no—
149C23—4541no—
150Q8—41no—
151Q8—41no—
152Q8—9081no—
153Q8—9081no—
154D4—9084no—
155D4—4541no—
156D4—4541no—
157D4—4541no—
158D4—2272no—
159C2×C4—22no—
160C2×C4—9084no—
161C2×C4—9081no—
162C2×C4—9081no—
163C2×C4—9081no—
164C2×C4—9081no—
165C2×C4—9081no—
166C2×C4—4541no—
167C2×C4—4541no—
168C2×C4D4×D22711yescharacteristic
169C2×C4D4×D22711yescharacteristic
170C2×C4C2×D90811yescharacteristic
171C8—41no—
172C8—41no—
173C8—4542no—
174C22—9081no—
175C22—9081no—
176C22—9081no—
177C22—9081no—
178C22—9081no—
179C22—4542no—
180C22#2030025811yescharacteristic
181C4—21no—
182C4—42no—
183C4—18161no—
184C4—9081no—
185C4—9081no—
186C4SD16×D22711yescharacteristic
187C4#2030031111yescharacteristic
188C4#2030031211yescharacteristic
189C4#2030031511yescharacteristic
190C2—18161no—
191C2—9082no—
192C2#2840918011yescharacteristic
193C2#2840918411yescharacteristic
194C2#2840921711yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#332×C227C2
socleC22×C227#120
prime radical#28409685C2
hypercenter#332D227
hypocenterC227#1654
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #28409428C2semidirect
3. #28409313C2semidirect
4. #28409238C2semidirect
5. #28408759C2semidirect
6. #28408759C2semidirect
7. #332×C227C2semidirect
58. D908SD16semidirect
97. C227#1654semidirect
103. #332D227semidirect

Maximal subgroups

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

subgroupindex
#284096852
#284094282
#284093132
#284092382
#284087592
#332×C2272
#1654227