Finite Group Databasebeta

← #44399002

Subgroups — #44399002

Order 29056.

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

normal subgroups only

This group has 200 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#28409755C211yesmaximal, characteristic
2#28409428C211yesmaximal, characteristic, complemented
3#28409309C211yesmaximal, characteristic
4#28408777C211yesmaximal, characteristic, complemented
5Q8⋊(C227⋊C8)C211yesmaximal, characteristic, complemented
6#28408759C211yesmaximal, characteristic
7#462×C227C211yesmaximal, characteristic, complemented
8#20300288C2211yescharacteristic
9#20300283—22no—
10#20300276—21no—
11#20300264—21no—
12C4×Dic454—21no—
13C4×D908C2211yescharacteristic
14#20300246—21no—
15#20300221—21no—
16#20300219—22no—
17C4⋊(C227⋊C8)C2211yescharacteristic
18#20300208—21no—
19C4×C227⋊C8C2211yescharacteristic
20C2×Q16×C227—21no—
21C4⋊Q8×C227C2211yescharacteristic
22C4×Q8×C227C2211yescharacteristic
23C4⋊C8×C227C2211yescharacteristic
24#103×C227—21no—
25C2×Dic454—21no—
26C2×D908D411yescharacteristic
27C2×C4×D227—21no—
28C2×C227⋊C8—21no—
29C2×C227⋊C8—21no—
30C4⋊Dic227—21no—
31C4⋊Dic227D412yes—
32C4⋊Dic227D411yescharacteristic
33Dic227⋊C4—22no—
34(C2×D227)⋊C4—22no—
35(C2×D227)⋊C4—22no—
36(C2×D227)⋊C4—21no—
37C4×Dic227—22no—
38C2×Q8×C227—21no—
39C2×Q8×C227D411yescharacteristic
40Q16×C227—42no—
41C2×C8×C227—21no—
42C4⋊C4×C227—21no—
43C4⋊C4×C227—21no—
44C4⋊C4×C227—21no—
45C4⋊C4×C227D411yescharacteristic
46C42×C227—21no—
47C42×C227C2311yescharacteristic
48C22×D227—21no—
49Dic454—41no—
50Dic454—41no—
51D908—21no—
52D908—21no—
53C2×Dic227—21no—
54C2×Dic227—22no—
55C4×D227—41no—
56C4×D227—41no—
57C227⋊C8—22no—
58C227⋊C8—41no—
59Q8×C227—21no—
60Q8×C227—21no—
61Q8×C227—41no—
62Q8×C227—41no—
63C2×C4×C227—21no—
64C2×C4×C227—21no—
65C2×C4×C227—41no—
66C2×C4×C227C2×D411yescharacteristic
67C2×C4×C227C2×D411yescharacteristic
68C2×C4×C227C2×D411yescharacteristic
69C8×C227—41no—
70C22×C227#12011yescharacteristic
71C2×D227—41no—
72C2×D227—41no—
73C2×D227—41no—
74C4×C227—21no—
75C4×C227—41no—
76C4×C227—41no—
77C4×C227—81no—
78C4×C227#13511yescharacteristic
79C4×C227#13511yescharacteristic
80C4×C227AGL(1,ℤ/8)11yescharacteristic
81C4×C227#13711yescharacteristic
82Dic227—41no—
83Dic227—42no—
84C2×C227#44911yescharacteristic
85C2×C227#45211yescharacteristic
86C2×C227#45611yescharacteristic
87D227—81no—
88C227#165211yescharacteristic, complemented, Sylow
89#1652—2271nomaximal, Sylow
90#556—2271no—
91#498—2271no—
92#474—2271no—
93#462D22711yescharacteristic, complemented
94#332—2271no—
95Q8⋊C8—2271no—
96D4⋊C8—2271no—
97C2×Q16—21no—
98C4⋊Q8—2271no—
99C4⋊Q8C2×D22711yescharacteristic
100#126—4542no—
101#124—4541no—
102C22⋊Q8—4541no—
103C4×Q8—4541no—
104C4×Q8C2×D22711yescharacteristic
105C4×D4—2271no—
106#117—4541no—
107#107—4541no—
108#106—4542no—
109C4⋊C8—2271no—
110C4⋊C8C2×D22711yescharacteristic
111#103—21no—
112#102—4541no—
113C4×C8—2271no—
114C2×Q8—21no—
115C2×Q8—4541no—
116C2×Q8C22⋊D22711yescharacteristic
117C2×D4—2271no—
118C22×C4—4541no—
119Q16—42no—
120C2×C8—21no—
121C2×C8—4541no—
122C2×C8—4541no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—4542no—
127C4⋊C4—4541no—
128C4⋊C4—2272no—
129C4⋊C4—2271no—
130C4⋊C4C22⋊D22711yescharacteristic
131C22⋊C4—4542no—
132C22⋊C4—4542no—
133C22⋊C4—4541no—
134C42—21no—
135C42—4542no—
136C42C22×D22711yescharacteristic
137C23—4541no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—9081no—
143Q8—9081no—
144D4—4541no—
145D4—4541no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—9081no—
150C2×C4—9081no—
151C2×C4—4541no—
152C2×C4—4542no—
153C2×C4D4×D22711yescharacteristic
154C2×C4D4×D22711yescharacteristic
155C2×C4C2×C22⋊D22711yescharacteristic
156C8—41no—
157C8—9081no—
158C8—4542no—
159C22—9081no—
160C22—9081no—
161C22—9081no—
162C22#2030025611yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—9081no—
168C4—9082no—
169C4#2030030811yescharacteristic
170C4#2030031111yescharacteristic
171C4#2030031411yescharacteristic
172C4#2030032011yescharacteristic
173C2—18161no—
174C2#2840917511yescharacteristic
175C2#2840919311yescharacteristic
176C2#2840921311yescharacteristic
subgroupG/H
centerC22#20300256
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#462×C227C2
socleC22×C227#120
prime radicalC2×D908D4
hypercenter#462D227
hypocenterC227#1652
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #28409428C2semidirect
4. #28408777C2semidirect
5. Q8⋊(C227⋊C8)C2semidirect
7. #462×C227C2semidirect
88. C227#1652semidirect
93. #462D227semidirect

Maximal subgroups

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

subgroupindex
#284097552
#284094282
#284093092
#284087772
Q8⋊(C227⋊C8)2
#284087592
#462×C2272
#1652227