Finite Group Databasebeta

← #44399927

Subgroups — #44399927

Order 29056.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#20300219C212yesmaximal, complemented
2C2×C22⋊(C227⋊C8)C211yesmaximal, characteristic
3#28408892C211yesmaximal, characteristic
4#28408794C212yesmaximal, complemented
5C22×C4⋊C4×C227C211yesmaximal, characteristic
6C22×C227⋊C8C2212yes—
7C2×C4⋊Dic227C2211yescharacteristic
8C2×C4⋊Dic227C2211yescharacteristic
9C2×C22⋊Dic227—21no—
10#20300219—24no—
11C22⋊(C227⋊C8)C414yescomplemented
12C227⋊#95—21no—
13C23×C4×C227—21no—
14C23×C4×C227C2211yescharacteristic
15C2×C4⋊C4×C227—21no—
16C2×C4⋊C4×C227—24no—
17C2×C4⋊C4×C227C2212yes—
18C22×Dic227—21no—
19C22×Dic227—21no—
20C2×C227⋊C8—22no—
21C2×C227⋊C8—22no—
22C2×C227⋊C8C2×C414yescomplemented
23C4⋊Dic227—22no—
24C4⋊Dic227—22no—
25C22⋊Dic227—42no—
26C24×C227D411yescharacteristic
27C22×C4×C227—22no—
28C22×C4×C227—22no—
29C22×C4×C227—24no—
30C22×C4×C227—22no—
31C22×C4×C227—42no—
32C22×C4×C227—42no—
33C22×C4×C227C2×C412yes—
34C22×C4×C227D411yescharacteristic
35C22×C4×C227Q812yes—
36C22×C4×C227C2311yescharacteristic
37C4⋊C4×C227—24no—
38C4⋊C4×C227—22no—
39C4⋊C4×C227D414yes—
40C2×Dic227—41no—
41C2×Dic227—41no—
42C2×Dic227—41no—
43C2×Dic227—41no—
44C2×Dic227—41no—
45C2×Dic227—41no—
46C227⋊C8—24no—
47C23×C227—21no—
48C23×C227—21no—
49C23×C227—21no—
50C23×C227—21no—
51C23×C227C4⋊C412yes—
52C23×C227SD1612yes—
53C23×C227SD1612yes—
54C23×C227C2×D411yescharacteristic
55C2×C4×C227—21no—
56C2×C4×C227—21no—
57C2×C4×C227—24no—
58C2×C4×C227—24no—
59C2×C4×C227—24no—
60C2×C4×C227—48no—
61C2×C4×C227—42no—
62C2×C4×C227—42no—
63C2×C4×C227C4⋊C412yes—
64C2×C4×C227C22×C411yescharacteristic
65C2×C4×C227C2×D412yes—
66C2×C4×C227C2×Q811yescharacteristic
67C2×C4×C227#4112yes—
68C22×C227—22no—
69C22×C227—22no—
70C22×C227—22no—
71C22×C227—22no—
72C22×C227—22no—
73C22×C227—22no—
74C22×C227#10614yescomplemented
75C22×C227C2×C4⋊C411yescharacteristic
76C22×C227#12011yescharacteristic
77C22×C227#12311yescharacteristic
78C22×C227C2×SD1611yescharacteristic
79C22×C227C2×SD1611yescharacteristic
80C22×C227AGL(1,ℤ/8)11yescharacteristic
81C22×C227#13711yescharacteristic
82C4×C227—22no—
83C4×C227—44no—
84C4×C227C4×D412yes—
85C4×C227C22⋊Q812yes—
86Dic227—81no—
87Dic227—81no—
88C2×C227—24no—
89C2×C227#38511yescharacteristic
90C2×C227C2×#10611yescharacteristic
91C2×C227#42811yescharacteristic
92C2×C227#44811yescharacteristic
93C2×C227#45011yescharacteristic
94C2×C227#48111yescharacteristic
95C2×C227#48311yescharacteristic
96C227#185311yescharacteristic, complemented, Sylow
97#1853—2271nomaximal, Sylow
98C22×C4⋊C4D22711yescharacteristic
99C2×#106—2272no—
100C2×C22⋊C8—2271no—
101#380—2271no—
102#340—2272no—
103C23×C4—21no—
104C23×C4C2×D22711yescharacteristic
105C22×C8—2272no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—2271no—
109C2×C4⋊C4—2271no—
110C2×C4⋊C4C2×D22712yes—
111C2×C22⋊C4—4541no—
112#106—4544no—
113C22⋊C8—2274no—
114#95—4541no—
115C24C22⋊D22711yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—4541no—
123C22×C4—4541no—
124C22×C4C4×D22712yes—
125C22×C4C22⋊D22711yescharacteristic
126C22×C4Dic45412yes—
127C22×C4C22×D22711yescharacteristic
128C2×C8—4542no—
129C2×C8—4542no—
130C2×C8—2274no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—4542no—
134C4⋊C4—4542no—
135C4⋊C4C22⋊D22714yes—
136C22⋊C4—9082no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23Dic227⋊C412yes—
142C23#1681387112yes—
143C23#1681387212yes—
144C23C2×C22⋊D22711yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—48no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—9081no—
154C2×C4—9081no—
155C2×C4—9081no—
156C2×C4—9081no—
157C2×C4—9081no—
158C2×C4—9081no—
159C2×C4Dic227⋊C412yes—
160C2×C4C2×C4×D22711yescharacteristic
161C2×C4C2×C22⋊D22712yes—
162C2×C4C2×Dic45411yescharacteristic
163C2×C4#1681388412yes—
164C8—4544no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#2030021914yescomplemented
172C22C2×Dic227⋊C411yescharacteristic
173C22#2030025711yescharacteristic
174C22#2030027111yescharacteristic
175C22C2×#1681387111yescharacteristic
176C22C2×#1681387211yescharacteristic
177C22#2030031411yescharacteristic
178C22#2030031911yescharacteristic
179C4—22no—
180C4—44no—
181C4—18161no—
182C4—18161no—
183C4C4×C22⋊D22712yes—
184C4#2030026712yes—
185C2—24no—
186C2#2840890511yescharacteristic
187C2C2×#2030021911yescharacteristic
188C2#2840907211yescharacteristic
189C2#2840917111yescharacteristic
190C2#2840918311yescharacteristic
191C2#2840935011yescharacteristic
192C2#2840936211yescharacteristic
subgroupG/H
centerC23C2×C22⋊D227
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
FittingC22×C4⋊C4×C227C2
socleC23×C227C2×D4
prime radicalC24×C227D4
cosocleC22×C4×C227C23
hypercenterC22×C4⋊C4D227
hypocenterC227#1853
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#20300219C2semidirect
4. #28408794C2semidirect
11. C22⋊(C227⋊C8)C4semidirect
22. C2×C227⋊C8C2×C4semidirect
74. C22×C227#106semidirect
96. C227#1853semidirect
171. C22#20300219semidirect

Maximal subgroups

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

subgroupindex
C2×#203002192
C2×C22⋊(C227⋊C8)2
#284088922
#284087942
C22×C4⋊C4×C2272
#1853227