Finite Group Databasebeta

← #44398865

Subgroups — #44398865

Order 29056.

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

normal subgroups only

This group has 214 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#28409743C211yesmaximal, characteristic
2#28409324C211yesmaximal, characteristic, complemented
3#28409319C211yesmaximal, characteristic
4#28408781C211yesmaximal, characteristic, complemented
5#28408764C211yesmaximal, characteristic, complemented
6#28408761C211yesmaximal, characteristic
7#494×C227C211yesmaximal, characteristic, complemented
8#20300279—21no—
9#20300268—21no—
10#20300264—22no—
11C4×Dic454C2211yescharacteristic
12C4×D908C2211yescharacteristic
13#20300250—21no—
14C4⋊C4×D227—21no—
15#20300221—21no—
16#20300219—21no—
17C4⋊(C227⋊C8)C2211yescharacteristic
18C4⋊(C227⋊C8)C2211yescharacteristic
19#20300211—21no—
20#20300208—21no—
21C2×Q16×C227—22no—
22C4⋊Q8×C227C2211yescharacteristic
23C4⋊Q8×C227C2211yescharacteristic
24C4×C8×C227C2211yescharacteristic
25C2×Dic454D411yescharacteristic
26C2×D908D411yescharacteristic
27C2×C4×D227—22no—
28C2×C4×D227—21no—
29C2×C227⋊C8—21no—
30C2×C227⋊C8—21no—
31C4⋊Dic227—22no—
32C4⋊Dic227D411yescharacteristic
33C4⋊Dic227D411yescharacteristic
34Dic227⋊C4—22no—
35Dic227⋊C4—21no—
36(C2×D227)⋊C4—22no—
37(C2×D227)⋊C4—21no—
38C4×Dic227—21no—
39C2×Q8×C227—21no—
40C2×Q8×C227D412yes—
41Q16×C227—44no—
42C2×C8×C227—21no—
43C4⋊C4×C227—21no—
44C4⋊C4×C227—21no—
45C4⋊C4×C227—21no—
46C4⋊C4×C227—21no—
47C42×C227C2311yescharacteristic
48C22×D227—21no—
49Dic454—21no—
50Dic454—21no—
51D908—21no—
52D908Q1612yescomplemented
53C2×Dic227—21no—
54C2×Dic227—21no—
55C2×Dic227—21no—
56C4×D227—41no—
57C4×D227—41no—
58C4×D227—82no—
59C227⋊C8—41no—
60C227⋊C8—41no—
61Q8×C227—42no—
62Q8×C227—42no—
63C2×C4×C227—22no—
64C2×C4×C227—41no—
65C2×C4×C227C2×D411yescharacteristic
66C2×C4×C227C2×D411yescharacteristic
67C2×C4×C227C2×D411yescharacteristic
68C8×C227—22no—
69C22×C227#12011yescharacteristic
70C2×D227—22no—
71C2×D227—41no—
72C2×D227—41no—
73C4×C227—21no—
74C4×C227—42no—
75C4×C227—81no—
76C4×C227C2×Q1611yescharacteristic
77C4×C227#13511yescharacteristic
78C4×C227#13711yescharacteristic
79C4×C227#13711yescharacteristic
80Dic227—41no—
81Dic227—41no—
82Dic227—41no—
83C2×C227#45111yescharacteristic
84C2×C227#45211yescharacteristic
85C2×C227#45311yescharacteristic
86D227—42no—
87C227#163211yescharacteristic, complemented, Sylow
88#1632—2271nomaximal, Sylow
89#554—2271no—
90#494D22711yescharacteristic, complemented
91#477—2271no—
92#476—2271no—
93#333—2271no—
94Q8⋊C8—2271no—
95D4⋊C8—2271no—
96C2×Q16—22no—
97C4⋊Q8C2×D22711yescharacteristic
98C4⋊Q8C2×D22711yescharacteristic
99#125—4541no—
100C22⋊Q8—4541no—
101C22⋊Q8—4542no—
102C4×Q8—2271no—
103C4×D4—4541no—
104C4×D4—2271no—
105C2×C4⋊C4—4541no—
106#107—4541no—
107#106—4541no—
108C4⋊C8—2271no—
109C4⋊C8—2271no—
110#103—4541no—
111#102—4541no—
112C4×C8C2×D22711yescharacteristic
113C2×Q8—21no—
114C2×Q8—2271no—
115C2×Q8C22⋊D22712yes—
116C2×D4—2271no—
117C22×C4—4542no—
118C22×C4—4541no—
119Q16—44no—
120C2×C8—21no—
121C2×C8—4541no—
122C2×C8—4541no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—4542no—
128C4⋊C4—4542no—
129C4⋊C4—4541no—
130C4⋊C4—2271no—
131C4⋊C4—2271no—
132C22⋊C4—4542no—
133C22⋊C4—4541no—
134C42—4541no—
135C42C22×D22711yescharacteristic
136C23—4541no—
137Q8—42no—
138Q8—42no—
139Q8—4541no—
140Q8—4541no—
141D4—4541no—
142D4—2272no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—18162no—
146C2×C4—9081no—
147C2×C4—9081no—
148C2×C4—4541no—
149C2×C4—4541no—
150C2×C4—4541no—
151C2×C4D4×D22711yescharacteristic
152C2×C4D4×D22711yescharacteristic
153C2×C4C2×C22⋊D22711yescharacteristic
154C8—22no—
155C8—9081no—
156C8—9081no—
157C22—9081no—
158C22—9081no—
159C22—4542no—
160C22#2030025611yescharacteristic
161C4—21no—
162C4—42no—
163C4—81no—
164C4—9081no—
165C4—9081no—
166C4—9081no—
167C4Q16×D22711yescharacteristic
168C4#2030031111yescharacteristic
169C4#2030032011yescharacteristic
170C4#2030032011yescharacteristic
171C2—9082no—
172C2#2840918611yescharacteristic
173C2#2840919311yescharacteristic
174C2#2840919911yescharacteristic
subgroupG/H
centerC22#20300256
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#494×C227C2
socleC22×C227#120
prime radicalC2×D908D4
hypercenter#494D227
hypocenterC227#1632
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #28409324C2semidirect
4. #28408781C2semidirect
5. #28408764C2semidirect
7. #494×C227C2semidirect
52. D908Q16semidirect
87. C227#1632semidirect
90. #494D227semidirect

Maximal subgroups

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

subgroupindex
#284097432
#284093242
#284093192
#284087812
#284087642
#284087612
#494×C2272
#1632227