Finite Group Databasebeta

← #44399005

Subgroups — #44399005

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#28409756C211yesmaximal, characteristic
2#28409428C211yesmaximal, characteristic, complemented
3#28409313C211yesmaximal, characteristic
4#28409243C211yesmaximal, characteristic, complemented
5#28408762C211yesmaximal, characteristic, complemented
6#28408759C211yesmaximal, characteristic
7#332×C227C211yesmaximal, characteristic, complemented
8C2×Dic908—21no—
9#20300288C2211yescharacteristic
10#20300283—22no—
11#20300277—21no—
12#20300268—21no—
13Q8×Dic227—21no—
14C4×Dic454C2211yescharacteristic
15C4×D908C2211yescharacteristic
16#20300246—21no—
17#20300222—21no—
18#20300219—22no—
19#20300213—21no—
20#20300210—21no—
21C4×C227⋊C8C2211yescharacteristic
22C4⋊Q8×C227C2211yescharacteristic
23C4⋊C8×C227C2211yescharacteristic
24C4⋊C8×C227C2211yescharacteristic
25C2×Dic454—21no—
26C2×Dic454D411yescharacteristic
27C2×D908D411yescharacteristic
28C2×C4×D227—21no—
29Dic908—42no—
30C2×C227⋊C8—21no—
31C4⋊Dic227—22no—
32C4⋊Dic227—21no—
33C4⋊Dic227—21no—
34C4⋊Dic227D411yescharacteristic
35C4⋊Dic227D411yescharacteristic
36Dic227⋊C4—21no—
37(C2×D227)⋊C4—22no—
38(C2×D227)⋊C4—22no—
39(C2×D227)⋊C4—21no—
40C4×Dic227—22no—
41C4×Dic227—21no—
42C2×Q8×C227—21no—
43C2×C8×C227—21no—
44C2×C8×C227—21no—
45C4⋊C4×C227—21no—
46C4⋊C4×C227D412yes—
47C42×C227C2311yescharacteristic
48C22×D227—21no—
49Dic454—21no—
50Dic454—21no—
51Dic454—41no—
52Dic454—41no—
53D908—21no—
54D908—21no—
55C2×Dic227—21no—
56C2×Dic227—21no—
57C2×Dic227—21no—
58C2×Dic227—41no—
59C4×D227—41no—
60C4×D227—41no—
61C227⋊C8—22no—
62Q8×C227—41no—
63Q8×C227—41no—
64C2×C4×C227—22no—
65C2×C4×C227C2×D411yescharacteristic
66C2×C4×C227C2×D411yescharacteristic
67C2×C4×C227C2×D411yescharacteristic
68C8×C227—41no—
69C8×C227—41no—
70C22×C227#12011yescharacteristic
71C2×D227—41no—
72C2×D227—41no—
73C2×D227—41no—
74C4×C227—21no—
75C4×C227—42no—
76C4×C227#13511yescharacteristic
77C4×C227#13511yescharacteristic
78C4×C227AGL(1,ℤ/8)11yescharacteristic
79C4×C227#13711yescharacteristic
80Dic227—41no—
81Dic227—41no—
82Dic227—41no—
83Dic227—81no—
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#462—2271no—
94#332D22711yescharacteristic, complemented
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×D4—2271no—
106#117—4541no—
107#107—4541no—
108#106—4542no—
109C4⋊C8C2×D22711yescharacteristic
110C4⋊C8C2×D22711yescharacteristic
111#103—4541no—
112#102—4541no—
113C4×C8—2271no—
114C2×Q8—21no—
115C2×Q8—4541no—
116C2×Q8—2271no—
117C2×D4—2271no—
118C22×C4—4541no—
119Q16—9082no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—4541no—
123C4⋊C4—21no—
124C4⋊C4—4542no—
125C4⋊C4—4541no—
126C4⋊C4—4541no—
127C4⋊C4—4541no—
128C4⋊C4—2271no—
129C4⋊C4—2271no—
130C4⋊C4D90812yes—
131C22⋊C4—4542no—
132C22⋊C4—4542no—
133C22⋊C4—4541no—
134C42—4542no—
135C42—4541no—
136C42C22×D22711yescharacteristic
137C23—4541no—
138Q8—41no—
139Q8—41no—
140Q8—9081no—
141Q8—9081no—
142Q8—4541no—
143Q8—4541no—
144D4—4541no—
145D4—4541no—
146C2×C4—22no—
147C2×C4—9081no—
148C2×C4—9081no—
149C2×C4—9081no—
150C2×C4—4541no—
151C2×C4—4541no—
152C2×C4—4541no—
153C2×C4D4×D22711yescharacteristic
154C2×C4D4×D22711yescharacteristic
155C2×C4C2×D90811yescharacteristic
156C8—41no—
157C8—41no—
158C8—4542no—
159C22—9081no—
160C22—9081no—
161C22—9081no—
162C22#2030025811yescharacteristic
163C4—21no—
164C4—42no—
165C4—18161no—
166C4—9081no—
167C4—9081no—
168C4—9081no—
169C4#2030030711yescharacteristic
170C4#2030031111yescharacteristic
171C4#2030031511yescharacteristic
172C4#2030031711yescharacteristic
173C2—18161no—
174C2#2840918011yescharacteristic
175C2#2840919511yescharacteristic
176C2#2840921711yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#332×C227C2
socleC22×C227#120
prime radicalC2×D908D4
hypercenter#332D227
hypocenterC227#1652
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #28409428C2semidirect
4. #28409243C2semidirect
5. #28408762C2semidirect
7. #332×C227C2semidirect
88. C227#1652semidirect
94. #332D227semidirect

Maximal subgroups

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

subgroupindex
#284097562
#284094282
#284093132
#284092432
#284087622
#284087592
#332×C2272
#1652227