Finite Group Databasebeta

← #44401389

Subgroups — #44401389

Order 29056.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#28409603C211yesmaximal, characteristic
2#28409006C211yesmaximal, characteristic
3#28408823C214yesmaximal, complemented
4#413×C227C211yesmaximal, characteristic
5C22×Dic454C411yescharacteristic, complemented
6#20300286—21no—
7#20300275—21no—
8#20300269C412yescomplemented
9#20300269C2212yes—
10#20300267—21no—
11#20300267—21no—
12#20300247—21no—
13#20300247C411yescharacteristic, complemented
14#20300203—21no—
15#20300203C2212yes—
16C2×#41×C227C2211yescharacteristic
17C2×M16×C227—21no—
18#101×C227—21no—
19#100×C227C2212yes—
20C2×Dic454—21no—
21C2×Dic454—21no—
22C2×Dic454—41no—
23C2×Dic454—41no—
24C22×Dic227C2×C412yes—
25C4⋊Dic227—21no—
26C4⋊Dic227—41no—
27Dic227⋊C4—22no—
28Dic227⋊C4—22no—
29C22⋊Dic227—22no—
30C22⋊Dic227—22no—
31C22⋊Dic227—22no—
32C22⋊Dic227C2×C412yes—
33C4×Dic227—21no—
34C4×Dic227—41no—
35#41×C227—41no—
36#41×C227—41no—
37C2×Q8×C227D411yescharacteristic
38C2×D4×C227C2×C411yescharacteristic
39C2×D4×C227D411yescharacteristic
40C2×D4×C227C2311yescharacteristic
41C22×C4×C227C2×C411yescharacteristic
42C22×C4×C227D411yescharacteristic
43C22×C4×C227D411yescharacteristic
44M16×C227—22no—
45M16×C227—41no—
46C2×C8×C227—41no—
47Dic454—42no—
48Dic454—41no—
49Dic454—41no—
50C2×Dic227—22no—
51C2×Dic227—22no—
52C2×Dic227—42no—
53C2×Dic227—42no—
54C23×C227C22⋊C411yescharacteristic
55C23×C227C22×C411yescharacteristic
56C23×C227C2×D411yescharacteristic
57Q8×C227—41no—
58D4×C227—41no—
59D4×C227—41no—
60D4×C227—41no—
61C2×C4×C227—21no—
62C2×C4×C227—21no—
63C2×C4×C227—21no—
64C2×C4×C227—21no—
65C2×C4×C227—21no—
66C2×C4×C227—21no—
67C2×C4×C227C22⋊C411yescharacteristic
68C2×C4×C227C22⋊C411yescharacteristic
69C2×C4×C227C22⋊C411yescharacteristic
70C2×C4×C227C2×D411yescharacteristic
71C8×C227—42no—
72C22×C227—21no—
73C22×C227—21no—
74C22×C227—21no—
75C22×C227—41no—
76C22×C227—41no—
77C22×C227—41no—
78C22×C227C2×C22⋊C411yescharacteristic
79C4×C227—21no—
80C4×C227—21no—
81C4×C227—21no—
82C4×C227#9911yescharacteristic
83C4×C227#9911yescharacteristic
84Dic227—42no—
85Dic227—42no—
86Dic227—82no—
87C2×C227—21no—
88C2×C227—41no—
89C2×C227—41no—
90C2×C227—41no—
91C2×C227C2×#9911yescharacteristic
92C227#212311yescharacteristic, complemented, Sylow
93#2123—2271nomaximal, Sylow
94#536—2271no—
95#413D22711yescharacteristic
96#410—2271no—
97#352—2274no—
98C2×#41C2×D22711yescharacteristic
99C22×Q8—2271no—
100C2×M16—21no—
101C4⋊Q8—4541no—
102#124—4541no—
103#123—2272no—
104#123—2272no—
105C22⋊Q8—4541no—
106C22⋊Q8—4541no—
107#117—4541no—
108#117—2271no—
109#101—21no—
110#100C2×D22712yes—
111#99—4541no—
112#99—2272no—
113#41—41no—
114#41—41no—
115C2×Q8—9081no—
116C2×Q8—9081no—
117C2×Q8—4541no—
118C2×Q8—4541no—
119C2×Q8C22⋊D22711yescharacteristic
120C2×D4C4×D22711yescharacteristic
121C2×D4D90811yescharacteristic
122C2×D4C22×D22711yescharacteristic
123C22×C4—2272no—
124C22×C4C4×D22711yescharacteristic
125C22×C4C22⋊D22711yescharacteristic
126C22×C4D90811yescharacteristic
127M16—22no—
128M16—41no—
129C2×C8—41no—
130C4⋊C4—9081no—
131C4⋊C4—4542no—
132C4⋊C4—4541no—
133C4⋊C4—4542no—
134C22⋊C4—4542no—
135C22⋊C4—4542no—
136C22⋊C4—4542no—
137C22⋊C4—2272no—
138C42—9081no—
139C42—4541no—
140C23(C2×D227)⋊C411yescharacteristic
141C23C2×C4×D22711yescharacteristic
142C23C2×D90811yescharacteristic
143Q8—41no—
144Q8—9082no—
145Q8—9081no—
146Q8—9081no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—9082no—
157C2×C4—9082no—
158C2×C4—4542no—
159C2×C4—4542no—
160C2×C4(C2×D227)⋊C411yescharacteristic
161C2×C4(C2×D227)⋊C411yescharacteristic
162C2×C4(C2×D227)⋊C411yescharacteristic
163C2×C4C2×C22⋊D22711yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×(C2×D227)⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—18162no—
176C4—9082no—
177C4—9082no—
178C4#2030020211yescharacteristic
179C4#2030020211yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#2030020211yescharacteristic
subgroupG/H
centerC2C2×#20300202
commutatorC23×C227C22×C4
FrattiniC2×D4C22×D227
Fitting#413×C227C2
socleC2×C227C2×#99
prime radicalC2×#41×C227C22
cosocleC2×D4×C227C23
hypercenter#413D227
hypocenterC227#2123
LCSG ▷ C23×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #28408823C2semidirect
5. C22×Dic454C4semidirect
8. #20300269C4semidirect
13. #20300247C4semidirect
92. C227#2123semidirect

Maximal subgroups

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

subgroupindex
#284096032
#284090062
#284088232
#413×C2272
#2123227