Finite Group Databasebeta

← #44401765

Subgroups — #44401765

Order 29056.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#28409762C211yesmaximal, characteristic, complemented
2#28409217C212yesmaximal
3#28409214C211yesmaximal, characteristic
4#28408830C212yesmaximal, complemented
5#356×C227C211yesmaximal, characteristic, complemented
6#20300336C2211yescharacteristic
7#20300321—22no—
8#20300319—21no—
9#20300287—22no—
10#20300286—21no—
11#20300286C2212yes—
12C4×Dic454—21no—
13#20300253—22no—
14#20300216—21no—
15#20300214—22no—
16#20300207C2211yescharacteristic
17C4⋊Q8×C227C2211yescharacteristic
18#101×C227C2212yes—
19#16813885—41no—
20#16813884—21no—
21#16813884—22no—
22C2×Dic454—21no—
23C2×Dic454—21no—
24C2×Dic454D412yes—
25Q8×D227—41no—
26Dic908—42no—
27#16813874—41no—
28#16813873—42no—
29#16813871—41no—
30#16813868—21no—
31C4⋊Dic227—21no—
32Dic227⋊C4—22no—
33Dic227⋊C4—22no—
34Dic227⋊C4—22no—
35C4×Dic227—21no—
36C4×Dic227—22no—
37C4×Dic227D412yes—
38C2×Q8×C227D411yescharacteristic
39C2×Q8×C227C2311yescharacteristic
40M16×C227—22no—
41C4⋊C4×C227—22no—
42C42×C227D411yescharacteristic
43Dic454—24no—
44Dic454—21no—
45Dic454—22no—
46Dic454—42no—
47Dic454—81no—
48D908—21no—
49D908—22no—
50C22⋊D227—41no—
51C2×Dic227—22no—
52C2×Dic227—22no—
53C4×D227—42no—
54C4×D227—41no—
55C227⋊C8—41no—
56Q8×C227—41no—
57Q8×C227—41no—
58C2×C4×C227—21no—
59C2×C4×C227—21no—
60C2×C4×C227C2×D412yes—
61C2×C4×C227C2×D411yescharacteristic
62C8×C227—42no—
63C22×C227#12011yescharacteristic
64C2×D227—41no—
65C4×C227—22no—
66C4×C227—22no—
67C4×C227—21no—
68C4×C227—41no—
69Dic227—24no—
70Dic227—41no—
71Dic227—42no—
72C2×C227—21no—
73C2×C227B(4,𝔽2)11yescharacteristic
74D227—81no—
75C227#220511yescharacteristic, complemented, Sylow
76#2205—2271nomaximal, Sylow
77Q82—2271no—
78#456—2271no—
79#456—2272no—
80#356—2272no—
81#356D22711yescharacteristic, complemented
822−1+4—2271no—
83#137—4542no—
84#137—4541no—
85C4⋊Q8—4542no—
86C4⋊Q8—4541no—
87C4⋊Q8—2272no—
88C4⋊Q8C2×D22711yescharacteristic
89C4×Q8—4542no—
90C4×Q8—4541no—
91#104—4542no—
92#104—4541no—
93#101—2271no—
94#101C2×D22712yes—
95#41—9081no—
96#41—4542no—
97#41—4541no—
98C2×Q8—9081no—
99C2×Q8—4541no—
100C2×Q8—4541no—
101C2×Q8—2272no—
102C2×Q8D90811yescharacteristic
103C2×Q8C22×D22711yescharacteristic
104Q16—9082no—
105Q16—9081no—
106SD16—9082no—
107SD16—9081no—
108M16—22no—
109M16—4541no—
110C4⋊C4—22no—
111C4⋊C4—4542no—
112C4⋊C4—4541no—
113C4⋊C4—4542no—
114C4⋊C4—4542no—
115C42—4542no—
116C42—4541no—
117C42—2272no—
118C42D90811yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—18161no—
122Q8—9082no—
123Q8—4542no—
124Q8—4541no—
125Q8—4544no—
126D4—9081no—
127D4—4542no—
128D4—4541no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—9082no—
132C2×C4—9081no—
133C2×C4—4542no—
134C2×C4—4542no—
135C2×C4D4×D22712yes—
136C2×C4C2×D90811yescharacteristic
137C8—42no—
138C8—9081no—
139C22—9081no—
140C22#2030025811yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—9081no—
146C4—9082no—
147C4—4544no—
148C2—21no—
149C2—18161no—
150C2#2840922011yescharacteristic
subgroupG/H
centerC2#28409220
commutator, cosocleC2×Q8×C227C23
FrattiniC2×Q8C22×D227
Fitting#356×C227C2
socleC2×C227B(4,𝔽2)
prime radical#20300336C22
hypercenter#356D227
hypocenterC227#2205
LCSG ▷ C2×Q8×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #28409762C2semidirect
4. #28408830C2semidirect
5. #356×C227C2semidirect
75. C227#2205semidirect
81. #356D227semidirect

Maximal subgroups

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

subgroupindex
#284097622
#284092172
#284092142
#284088302
#356×C2272
#2205227