Finite Group Databasebeta

← #44401763

Subgroups — #44401763

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#28409215C212yesmaximal
3#28408831C211yesmaximal, characteristic, complemented
4#28408830C212yesmaximal, complemented
5#456×C227C211yesmaximal, characteristic
6#20300318—22no—
7#20300287—22no—
8#20300286—21no—
9#20300286C2212yes—
10C4×Dic454—21no—
11#20300253—22no—
12#20300215—22no—
13#20300207C2212yes—
142−1+4×C227C2211yescharacteristic
15#137×C227—21no—
16C4⋊Q8×C227C2211yescharacteristic
17#104×C227—21no—
18#101×C227C2211yescharacteristic
19C2×Dic454—21no—
20C2×Dic454—21no—
21C2×Dic454D412yes—
22#16813874—42no—
23#16813872—42no—
24#16813868—22no—
25C4⋊Dic227—21no—
26Dic227⋊C4—22no—
27Dic227⋊C4—22no—
28Dic227⋊C4—22no—
29C4×Dic227—21no—
30C4×Dic227—22no—
31C4×Dic227D412yes—
32#41×C227—22no—
33#41×C227—21no—
34#41×C227—41no—
35C2×Q8×C227—41no—
36C2×Q8×C227D411yescharacteristic
37C2×Q8×C227C2311yescharacteristic
38Q16×C227—41no—
39SD16×C227—41no—
40M16×C227—21no—
41C4⋊C4×C227—22no—
42C42×C227D411yescharacteristic
43Dic454—24no—
44Dic454—42no—
45Dic454—81no—
46C2×Dic227—22no—
47C2×Dic227—22no—
48C227⋊C8—42no—
49Q8×C227—21no—
50Q8×C227—22no—
51Q8×C227—41no—
52Q8×C227—41no—
53D4×C227—21no—
54D4×C227—22no—
55D4×C227—41no—
56C2×C4×C227—21no—
57C2×C4×C227—21no—
58C2×C4×C227—41no—
59C2×C4×C227—42no—
60C2×C4×C227C2×D412yes—
61C2×C4×C227C2×D411yescharacteristic
62C8×C227—41no—
63C22×C227—41no—
64C22×C227#12011yescharacteristic
65C4×C227—22no—
66C4×C227—21no—
67C4×C227—22no—
68C4×C227—41no—
69C4×C227—41no—
70Dic227—24no—
71Dic227—42no—
72C2×C227—21no—
73C2×C227—81no—
74C2×C227B(4,𝔽2)11yescharacteristic
75C227#220511yescharacteristic, complemented, Sylow
76#2205—2271nomaximal, Sylow
77Q82—2271no—
78#456—2272no—
79#456D22711yescharacteristic
80#356—2271no—
81#356—2272no—
822−1+4C2×D22711yescharacteristic
83#137—21no—
84#137—4542no—
85C4⋊Q8—4542no—
86C4⋊Q8—4541no—
87C4⋊Q8—2272no—
88C4⋊Q8C2×D22711yescharacteristic
89C4×Q8—4542no—
90C4×Q8—4541no—
91#104—21no—
92#104—4542no—
93#101—2272no—
94#101C2×D22711yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—4541no—
100C2×Q8—4541no—
101C2×Q8—2272no—
102C2×Q8C22⋊D22711yescharacteristic
103C2×Q8C22×D22711yescharacteristic
104Q16—41no—
105Q16—9082no—
106SD16—41no—
107SD16—9082no—
108M16—21no—
109M16—4542no—
110C4⋊C4—22no—
111C4⋊C4—4542no—
112C4⋊C4—4541no—
113C4⋊C4—4542no—
114C4⋊C4—4542no—
115C42—4542no—
116C42—4541no—
117C42—2272no—
118C42C22⋊D22711yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—18161no—
124Q8—9082no—
125Q8—4544no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—4542no—
134C2×C4—4542no—
135C2×C4D4×D22712yes—
136C2×C4C2×C22⋊D22711yescharacteristic
137C8—41no—
138C8—9082no—
139C22—41no—
140C22#2030025611yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—9082no—
147C4—4544no—
148C2—21no—
149C2—81no—
150C2#2840921811yescharacteristic
subgroupG/H
centerC2#28409218
commutator, cosocleC2×Q8×C227C23
FrattiniC2×Q8C22×D227
Fitting#456×C227C2
socleC2×C227B(4,𝔽2)
prime radical2−1+4×C227C22
hypercenter#456D227
hypocenterC227#2205
LCSG ▷ C2×Q8×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #28409762C2semidirect
3. #28408831C2semidirect
4. #28408830C2semidirect
75. C227#2205semidirect

Maximal subgroups

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

subgroupindex
#284097622
#284092152
#284088312
#284088302
#456×C2272
#2205227