Finite Group Databasebeta

← #44399003

Subgroups — #44399003

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#28409752C211yesmaximal, characteristic
2#28409428C211yesmaximal, characteristic, complemented
3#28409239C211yesmaximal, characteristic, complemented
4#28408779C211yesmaximal, characteristic, complemented
5#28408762C211yesmaximal, characteristic, complemented
6D4⋊(C227⋊C8)C211yesmaximal, characteristic
7#474×C227C211yesmaximal, characteristic
8C2×#16813874—21no—
9#20300288C2211yescharacteristic
10#20300281—22no—
11#20300275—21no—
12#20300267—21no—
13C4×Dic454C2211yescharacteristic
14#20300253—21no—
15#20300247—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—
25C2×Dic454—21no—
26C2×Dic454D411yescharacteristic
27#16813874—42no—
28C2×C227⋊C8—21no—
29C2×C227⋊C8—21no—
30C4⋊Dic227—21no—
31C4⋊Dic227D412yes—
32C4⋊Dic227D411yescharacteristic
33Dic227⋊C4—22no—
34Dic227⋊C4—21no—
35C22⋊Dic227—22no—
36C22⋊Dic227—22no—
37C4×Dic227—22no—
38C4×Dic227—21no—
39C2×Q8×C227—21no—
40C2×D4×C227D411yescharacteristic
41C22×C4×C227—21no—
42C2×C8×C227—21no—
43C4⋊C4×C227—21no—
44C4⋊C4×C227—21no—
45C4⋊C4×C227D411yescharacteristic
46C22⋊C4×C227—21no—
47C42×C227C2311yescharacteristic
48Dic454—21no—
49Dic454—21no—
50Dic454—41no—
51Dic454—41no—
52C2×Dic227—21no—
53C2×Dic227—21no—
54C2×Dic227—22no—
55C227⋊C8—22no—
56C227⋊C8—41no—
57C23×C227—21no—
58Q8×C227—41no—
59Q8×C227—41no—
60D4×C227—21no—
61D4×C227—21no—
62C2×C4×C227—21no—
63C2×C4×C227—41no—
64C2×C4×C227—41no—
65C2×C4×C227—41no—
66C2×C4×C227C2×D411yescharacteristic
67C2×C4×C227C2×D411yescharacteristic
68C2×C4×C227C2×D411yescharacteristic
69C8×C227—41no—
70C22×C227—41no—
71C22×C227—41no—
72C22×C227—41no—
73C22×C227#12011yescharacteristic
74C4×C227—21no—
75C4×C227—41no—
76C4×C227—81no—
77C4×C227#13511yescharacteristic
78C4×C227#13511yescharacteristic
79C4×C227AGL(1,ℤ/8)11yescharacteristic
80C4×C227#13711yescharacteristic
81Dic227—41no—
82Dic227—42no—
83Dic227—41no—
84C2×C227—81no—
85C2×C227#44911yescharacteristic
86C2×C227#45211yescharacteristic
87C2×C227#45611yescharacteristic
88C227#165211yescharacteristic, complemented, Sylow
89#1652—2271nomaximal, Sylow
90#556—2271no—
91#498—2271no—
92#474D22711yescharacteristic
93#462—2271no—
94#332—2271no—
95Q8⋊C8—2271no—
96D4⋊C8—2271no—
97C2×Q16—4541no—
98C4⋊Q8—2271no—
99C4⋊Q8C2×D22711yescharacteristic
100#126—4542no—
101#124—4541no—
102C22⋊Q8—4541no—
103C4×Q8—4541no—
104C4×Q8—2271no—
105C4×D4C2×D22711yescharacteristic
106#117—4541no—
107#107—21no—
108#106—4542no—
109C4⋊C8—2271no—
110C4⋊C8C2×D22711yescharacteristic
111#103—4541no—
112#102—21no—
113C4×C8—2271no—
114C2×Q8—21no—
115C2×Q8—4541no—
116C2×Q8—2271no—
117C2×D4C22⋊D22711yescharacteristic
118C22×C4—21no—
119Q16—9082no—
120C2×C8—21no—
121C2×C8—4541no—
122C2×C8—4541no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—4542no—
126C4⋊C4—4541no—
127C4⋊C4—4541no—
128C4⋊C4—2271no—
129C4⋊C4—2272no—
130C4⋊C4C22⋊D22711yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—4542no—
133C22⋊C4—4542no—
134C42—4542no—
135C42—4541no—
136C42C22×D22711yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—9081no—
141Q8—9081no—
142Q8—4541no—
143Q8—4541no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—4541no—
151C2×C4—4541no—
152C2×C4—4542no—
153C2×C4D4×D22711yescharacteristic
154C2×C4D4×D22711yescharacteristic
155C2×C4C2×C22⋊D22711yescharacteristic
156C8—41no—
157C8—9081no—
158C8—4542no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2030025611yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—9081no—
167C4—9082no—
168C4—9081no—
169C4#2030030711yescharacteristic
170C4#2030030811yescharacteristic
171C4#2030031211yescharacteristic
172C4#2030031911yescharacteristic
173C2—81no—
174C2#2840917411yescharacteristic
175C2#2840919211yescharacteristic
176C2#2840921311yescharacteristic
subgroupG/H
centerC22#20300256
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#474×C227C2
socleC22×C227#120
prime radicalC2×D4×C227D4
hypercenter#474D227
hypocenterC227#1652
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. #28409239C2semidirect
4. #28408779C2semidirect
5. #28408762C2semidirect
88. C227#1652semidirect

Maximal subgroups

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

subgroupindex
#284097522
#284094282
#284092392
#284087792
#284087622
D4⋊(C227⋊C8)2
#474×C2272
#1652227