Finite Group Databasebeta

← #44399017

Subgroups — #44399017

Order 29056.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#28409686C211yesmaximal, characteristic
2#28409428C211yesmaximal, characteristic, complemented
3#28409234C211yesmaximal, characteristic, complemented
4#28408779C211yesmaximal, characteristic, complemented
5D4⋊(C227⋊C8)C211yesmaximal, characteristic, complemented
6#28408759C211yesmaximal, characteristic, complemented
7#474×C227C211yesmaximal, characteristic, complemented
8C2×#16813883—21no—
9C2×#16813871—21no—
10#20300288C2211yescharacteristic
11#20300272—22no—
12#20300268—21no—
13#20300262—21no—
14C4×D908C2211yescharacteristic
15C2×C4⋊Dic227—21no—
16#20300219—22no—
17C4⋊(C227⋊C8)C2211yescharacteristic
18#20300212—21no—
19C4×C227⋊C8C2211yescharacteristic
20C4⋊Q8×C227C2211yescharacteristic
21C4×D4×C227C2211yescharacteristic
22#107×C227—21no—
23C4⋊C8×C227C2211yescharacteristic
24#102×C227—21no—
25#16813883—42no—
26#16813883—42no—
27C2×Dic454—21no—
28C2×D908D411yescharacteristic
29C2×C22⋊D227—22no—
30C22×Dic227—22no—
31C2×C4×D227—21no—
32#16813871—42no—
33C2×C227⋊C8—21no—
34C2×C227⋊C8—21no—
35C4⋊Dic227—22no—
36C4⋊Dic227—21no—
37C4⋊Dic227D412yes—
38C4⋊Dic227D411yescharacteristic
39(C2×D227)⋊C4—22no—
40(C2×D227)⋊C4—21no—
41C2×Q8×C227—21no—
42C2×D4×C227D411yescharacteristic
43C22×C4×C227—21no—
44C2×C8×C227—21no—
45C4⋊C4×C227—21no—
46C4⋊C4×C227—21no—
47C4⋊C4×C227D411yescharacteristic
48C22⋊C4×C227—21no—
49C42×C227C2311yescharacteristic
50C22×D227—21no—
51Dic454—41no—
52Dic454—41no—
53D908—21no—
54D908—21no—
55C22⋊D227—44no—
56C2×Dic227—21no—
57C2×Dic227—22no—
58C2×Dic227—44no—
59C4×D227—41no—
60C4×D227—41no—
61C227⋊C8—22no—
62C227⋊C8—41no—
63C23×C227—21no—
64Q8×C227—41no—
65Q8×C227—41no—
66D4×C227—21no—
67D4×C227SD1612yescomplemented
68C2×C4×C227—21no—
69C2×C4×C227—41no—
70C2×C4×C227—41no—
71C2×C4×C227—41no—
72C2×C4×C227C2×D411yescharacteristic
73C2×C4×C227C2×D411yescharacteristic
74C2×C4×C227C2×D411yescharacteristic
75C8×C227—41no—
76C22×C227—22no—
77C22×C227—41no—
78C22×C227—41no—
79C22×C227#12011yescharacteristic
80C2×D227—41no—
81C2×D227—41no—
82C2×D227—41no—
83C4×C227—21no—
84C4×C227—41no—
85C4×C227—81no—
86C4×C227C2×SD1611yescharacteristic
87C4×C227#13511yescharacteristic
88C4×C227AGL(1,ℤ/8)11yescharacteristic
89C4×C227AGL(1,ℤ/8)11yescharacteristic
90Dic227—41no—
91Dic227—42no—
92C2×C227—42no—
93C2×C227#44911yescharacteristic
94C2×C227#45011yescharacteristic
95C2×C227#45611yescharacteristic
96D227—81no—
97C227#165411yescharacteristic, complemented, Sylow
98#1654—2271nomaximal, Sylow
99#547—2271no—
100#498—2271no—
101#474D22711yescharacteristic, complemented
102#461—2271no—
103#332—2271no—
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×D4C2×D22711yescharacteristic
115C2×C4⋊C4—4541no—
116#107—21no—
117#106—4542no—
118C4⋊C8—2271no—
119C4⋊C8C2×D22711yescharacteristic
120#103—4541no—
121#102—21no—
122C4×C8—2271no—
123#41—9082no—
124#41—9082no—
125C2×Q8—21no—
126C2×Q8—4541no—
127C2×D4—4542no—
128C2×D4—2271no—
129C2×D4C22⋊D22711yescharacteristic
130C22×C4—21no—
131C22×C4—4542no—
132C22×C4—4541no—
133SD16—9082no—
134C2×C8—21no—
135C2×C8—4541no—
136C2×C8—4541no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—4542no—
140C4⋊C4—4541no—
141C4⋊C4—2271no—
142C4⋊C4—2272no—
143C4⋊C4C22⋊D22711yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—4542no—
146C22⋊C4—4541no—
147C42C22×D22711yescharacteristic
148C23—21no—
149C23—4541no—
150Q8—41no—
151Q8—41no—
152Q8—9081no—
153Q8—9081no—
154D4—21no—
155D4—9084no—
156D4—4541no—
157D4—4541no—
158D4#1681387112yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—9084no—
164C2×C4—9081no—
165C2×C4—9081no—
166C2×C4—4541no—
167C2×C4—4542no—
168C2×C4D4×D22711yescharacteristic
169C2×C4D4×D22711yescharacteristic
170C2×C4C2×C22⋊D22711yescharacteristic
171C8—41no—
172C8—9081no—
173C8—4542no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—9081no—
178C22—9081no—
179C22—9081no—
180C22#2030025611yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—9081no—
185C4—9082no—
186C4C2×#1681387111yescharacteristic
187C4#2030030711yescharacteristic
188C4#2030031211yescharacteristic
189C4#2030031311yescharacteristic
190C2—42no—
191C2—18161no—
192C2#2840917411yescharacteristic
193C2#2840918211yescharacteristic
194C2#2840921311yescharacteristic
subgroupG/H
centerC22#20300256
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#474×C227C2
socleC22×C227#120
prime radical#28409686C2
hypercenter#474D227
hypocenterC227#1654
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #28409428C2semidirect
3. #28409234C2semidirect
4. #28408779C2semidirect
5. D4⋊(C227⋊C8)C2semidirect
6. #28408759C2semidirect
7. #474×C227C2semidirect
67. D4×C227SD16semidirect
97. C227#1654semidirect
101. #474D227semidirect
158. D4#16813871semidirect

Maximal subgroups

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

subgroupindex
#284096862
#284094282
#284092342
#284087792
D4⋊(C227⋊C8)2
#284087592
#474×C2272
#1654227