Finite Group Databasebeta

← #44399033

Subgroups — #44399033

Order 29056.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#28409782C211yesmaximal, characteristic, complemented
2#28409422C211yesmaximal, characteristic
3#28409397C212yesmaximal, complemented
4#28408773C212yesmaximal, complemented
5#329×C227C211yesmaximal, characteristic, complemented
6C2×#16813871—21no—
7C2×#16813869—21no—
8#20300286—21no—
9#20300285C2211yescharacteristic
10#20300281—22no—
11#20300280C2212yes—
12#20300269—21no—
13#20300265—22no—
14#20300210—22no—
15#20300210—22no—
16#20300199C2211yescharacteristic
17#124×C227C2211yescharacteristic
18#97×C227C2212yes—
19C2×Dic454—21no—
20C2×D908—21no—
21C2×D908—22no—
22C22×Dic227—21no—
23#16813871—42no—
24#16813869—42no—
25C2×C227⋊C8—21no—
26C4⋊Dic227D412yes—
27C4⋊Dic227D412yes—
28Dic227⋊C4—22no—
29Dic227⋊C4—22no—
30Dic227⋊C4—22no—
31C22⋊Dic227—22no—
32C22⋊Dic227—21no—
33C4×Dic227—21no—
34C2×Q8×C227D411yescharacteristic
35C2×D4×C227D411yescharacteristic
36C2×C8×C227—22no—
37C22⋊C4×C227—22no—
38C42×C227C2311yescharacteristic
39C22×D227—41no—
40Dic454—81no—
41D908—41no—
42D908—41no—
43D908—44no—
44C2×Dic227—22no—
45C2×Dic227—22no—
46C2×Dic227—81no—
47C227⋊C8—22no—
48C23×C227—21no—
49Q8×C227—41no—
50D4×C227—41no—
51C2×C4×C227—21no—
52C2×C4×C227C2×D412yes—
53C2×C4×C227C2×D411yescharacteristic
54C8×C227—42no—
55C22×C227—41no—
56C22×C227—42no—
57C22×C227#12011yescharacteristic
58C2×D227—81no—
59C2×D227—82no—
60C4×C227—21no—
61C4×C227—22no—
62C4×C227—41no—
63Dic227—42no—
64Dic227—42no—
65C2×C227—81no—
66C2×C227#45311yescharacteristic
67C2×C227#45412yes—
68D227—161no—
69C227#165711yescharacteristic, complemented, Sylow
70#1657—2271nomaximal, Sylow
71#563—2271no—
72#496—2271no—
73#490—2272no—
74#330—2272no—
75#329D22711yescharacteristic, complemented
76C2×SD16—4541no—
77C2×D8—4541no—
78C4⋊Q8—4541no—
79#127—2271no—
80#126—4542no—
81#125—2272no—
82#124C2×D22711yescharacteristic
83#123—4541no—
84C22⋊Q8—4542no—
85#102—4542no—
86#102—4542no—
87#97—2271no—
88#97C2×D22712yes—
89C2×Q8—4541no—
90C2×Q8D90811yescharacteristic
91C2×D4—4542no—
92C2×D4—4541no—
93C2×D4D90811yescharacteristic
94C22×C4—4541no—
95SD16—9082no—
96D8—9082no—
97C2×C8—22no—
98C2×C8—4541no—
99C4⋊C4—4542no—
100C4⋊C4—4542no—
101C4⋊C4—4542no—
102C4⋊C4—2272no—
103C4⋊C4—2272no—
104C22⋊C4—22no—
105C22⋊C4—4542no—
106C22⋊C4—4541no—
107C42—4541no—
108C42C22×D22711yescharacteristic
109C23—21no—
110C23—9081no—
111Q8—41no—
112Q8—18161no—
113D4—41no—
114D4—9081no—
115D4—9081no—
116D4—9084no—
117C2×C4—21no—
118C2×C4—18161no—
119C2×C4—4542no—
120C2×C4—4542no—
121C2×C4D4×D22712yes—
122C2×C4C2×D90811yescharacteristic
123C8—42no—
124C8—4542no—
125C22—41no—
126C22—42no—
127C22—18161no—
128C22—18162no—
129C22#2030025811yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—9082no—
134C4—9082no—
135C2—81no—
136C2—36321no—
137C2#2840920211yescharacteristic
138C2#2840920612yes—
subgroupG/H
centerC22#20300258
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#329×C227C2
socleC22×C227#120
prime radical#28409422C2
hypercenter#329D227
hypocenterC227#1657
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #28409782C2semidirect
3. #28409397C2semidirect
4. #28408773C2semidirect
5. #329×C227C2semidirect
69. C227#1657semidirect
75. #329D227semidirect

Maximal subgroups

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

subgroupindex
#284097822
#284094222
#284093972
#284087732
#329×C2272
#1657227