Finite Group Databasebeta

← #41013036

Subgroups — #41013036

Order 27008.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213518C211yesmaximal, characteristic, complemented
2#27213497C211yesmaximal, characteristic, complemented
3#27213492C211yesmaximal, characteristic, complemented
4#27213492C211yesmaximal, characteristic, complemented
5#27213492C211yesmaximal, characteristic, complemented
6#27212868C211yesmaximal, characteristic, complemented
7#493×C211C211yesmaximal, characteristic
8C2×#16809950—22no—
9#20269977C2211yescharacteristic
10#20269977C2211yescharacteristic
11#20269902—22no—
12#20269898—22no—
13#20269898—22no—
14#20269898—22no—
15C4×C211⋊C8C2211yescharacteristic
16C4×C211⋊C8C2211yescharacteristic
17C2×D8×C211—22no—
18#127×C211C2211yescharacteristic
19#127×C211C2211yescharacteristic
20C4×C8×C211C2211yescharacteristic
21C2×Dic422—21no—
22C2×Dic422—21no—
23#16809950—44no—
24C2×C211⋊C8D412yes—
25C2×C211⋊C8D412yes—
26C4⋊Dic211—21no—
27C4⋊Dic211—21no—
28C4⋊Dic211—21no—
29C4⋊Dic211—21no—
30C2×D4×C211—21no—
31C2×D4×C211—21no—
32C2×D4×C211—21no—
33C2×D4×C211—21no—
34C2×D4×C211—21no—
35C2×D4×C211—21no—
36D8×C211—44no—
37C2×C8×C211D412yes—
38C42×C211C2311yescharacteristic
39Dic422—42no—
40Dic422—42no—
41C2×Dic211—41no—
42C2×Dic211—41no—
43C211⋊C8—22no—
44C211⋊C8—22no—
45C23×C211—41no—
46C23×C211—41no—
47D4×C211—42no—
48D4×C211—42no—
49D4×C211—42no—
50D4×C211—42no—
51D4×C211—42no—
52D4×C211—42no—
53C2×C4×C211C2×D411yescharacteristic
54C2×C4×C211C2×D411yescharacteristic
55C2×C4×C211C2×D411yescharacteristic
56C8×C211—22no—
57C22×C211—81no—
58C22×C211—81no—
59C22×C211—81no—
60C22×C211—81no—
61C22×C211—81no—
62C22×C211—81no—
63C22×C211#12711yescharacteristic
64C4×C211#13512yes—
65C4×C211AGL(1,ℤ/8)12yes—
66C4×C211AGL(1,ℤ/8)12yes—
67Dic211—81no—
68Dic211—81no—
69C2×C211—161no—
70C2×C211—161no—
71C2×C211#49511yescharacteristic
72C2×C211#49611yescharacteristic
73C2×C211#49611yescharacteristic
74C211#171711yescharacteristic, complemented, Sylow
75#1717—2111nomaximal, Sylow
76#493D21111yescharacteristic
77#492—2111no—
78#487—2111no—
79#486—2111no—
80#486—2111no—
81#486—2111no—
82#322—2111no—
83C2×SD16—4222no—
84C2×D8—22no—
85C4⋊Q8—2111no—
86C4⋊Q8—2111no—
87#127C2×D21111yescharacteristic
88#127C2×D21111yescharacteristic
89#103—4222no—
90#102—4222no—
91#102—4222no—
92#102—4222no—
93C4×C8—2111no—
94C4×C8—2111no—
95C4×C8C2×D21111yescharacteristic
96C2×Q8—4221no—
97C2×Q8—4221no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104SD16—8444no—
105D8—44no—
106C2×C8—2112no—
107C2×C8—2112no—
108C2×C8C22⋊D21112yes—
109C4⋊C4—4221no—
110C4⋊C4—4221no—
111C4⋊C4—4221no—
112C4⋊C4—4221no—
113C42C22×D21111yescharacteristic
114C23—41no—
115C23—41no—
116Q8—8442no—
117Q8—8442no—
118D4—42no—
119D4—42no—
120D4—42no—
121D4—42no—
122D4—42no—
123D4—42no—
124C2×C4—8441no—
125C2×C4—8441no—
126C2×C4D4×D21111yescharacteristic
127C2×C4D4×D21111yescharacteristic
128C2×C4C2×C22⋊D21111yescharacteristic
129C8—22no—
130C8—4222no—
131C8—4222no—
132C22—81no—
133C22—81no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22—81no—
138C22#2026997311yescharacteristic
139C4—16881no—
140C4—16881no—
141C4#2026999612yes—
142C4#2027000112yes—
143C4#2027000312yes—
144C2—161no—
145C2—161no—
146C2#2721352911yescharacteristic
147C2#2721353311yescharacteristic
148C2#2721353411yescharacteristic
subgroupG/H
centerC22#20269973
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting, prime radical#493×C211C2
socleC22×C211#127
hypercenter#493D211
hypocenterC211#1717
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #27213518C2semidirect
2. #27213497C2semidirect
3. #27213492C2semidirect
4. #27213492C2semidirect
5. #27213492C2semidirect
6. #27212868C2semidirect
74. C211#1717semidirect

Maximal subgroups

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

subgroupindex
#272135182
#272134972
#272134922
#272128682
#493×C2112
#1717211