Finite Group Databasebeta

← #41013017

Subgroups — #41013017

Order 27008.

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

normal subgroups only

This group has 198 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#27213551C211yesmaximal, characteristic, complemented
2#27213542C211yesmaximal, characteristic, complemented
3#27213518C211yesmaximal, characteristic
4#27213498C211yesmaximal, characteristic, complemented
5#27213492C211yesmaximal, characteristic
6#27212867C211yesmaximal, characteristic, complemented
7#492×C211C211yesmaximal, characteristic
8C2×#16809950—22no—
9#20269977C2211yescharacteristic
10#20269977C2211yescharacteristic
11#20269911—22no—
12#20269908—22no—
13#20269900—22no—
14#20269898—22no—
15C4×C211⋊C8C2211yescharacteristic
16C4×C211⋊C8C2211yescharacteristic
17C2×SD16×C211—22no—
18C4⋊Q8×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×Q8×C211—21no—
31C2×D4×C211—21no—
32C2×D4×C211—21no—
33C2×D4×C211—21no—
34SD16×C211—44no—
35C2×C8×C211D412yes—
36C4⋊C4×C211—21no—
37C4⋊C4×C211—21no—
38C42×C211C2311yescharacteristic
39Dic422—42no—
40Dic422—42no—
41C2×Dic211—41no—
42C2×Dic211—41no—
43C211⋊C8—22no—
44C211⋊C8—22no—
45C23×C211—41no—
46Q8×C211—42no—
47D4×C211—42no—
48D4×C211—42no—
49D4×C211—42no—
50C2×C4×C211—41no—
51C2×C4×C211C2×D411yescharacteristic
52C2×C4×C211C2×D411yescharacteristic
53C2×C4×C211C2×D411yescharacteristic
54C8×C211SD1614yescomplemented
55C22×C211—81no—
56C22×C211—81no—
57C22×C211—81no—
58C22×C211#12711yescharacteristic
59C4×C211—81no—
60C4×C211C2×SD1612yes—
61C4×C211AGL(1,ℤ/8)12yes—
62C4×C211#13712yes—
63Dic211—81no—
64Dic211—81no—
65C2×C211—161no—
66C2×C211#49211yescharacteristic
67C2×C211#49611yescharacteristic
68C2×C211#49711yescharacteristic
69C211#171411yescharacteristic, complemented, Sylow
70#1714—2111nomaximal, Sylow
71#500—2111no—
72#498—2111no—
73#492—2111no—
74#492D21111yescharacteristic
75#487—2111no—
76#486—2111no—
77#322—2111no—
78C2×SD16—22no—
79C2×SD16—4222no—
80C4⋊Q8—2111no—
81C4⋊Q8—2111no—
82C4⋊Q8C2×D21111yescharacteristic
83#127C2×D21111yescharacteristic
84#107—4222no—
85#106—4222no—
86#103—4222no—
87#102—4222no—
88C4×C8—2111no—
89C4×C8—2111no—
90C4×C8C2×D21111yescharacteristic
91C2×Q8—21no—
92C2×Q8—4221no—
93C2×Q8—4221no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97SD16—44no—
98SD16—8444no—
99C2×C8—2112no—
100C2×C8—2112no—
101C2×C8C22⋊D21112yes—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—4221no—
105C4⋊C4—4221no—
106C4⋊C4—4221no—
107C4⋊C4—4221no—
108C42C22×D21111yescharacteristic
109C23—41no—
110Q8—42no—
111Q8—8442no—
112Q8—8442no—
113D4—42no—
114D4—42no—
115D4—42no—
116C2×C4—41no—
117C2×C4—8441no—
118C2×C4—8441no—
119C2×C4D4×D21111yescharacteristic
120C2×C4D4×D21111yescharacteristic
121C2×C4C2×C22⋊D21111yescharacteristic
122C8—4222no—
123C8—4222no—
124C8#1680995014yescomplemented
125C22—81no—
126C22—81no—
127C22—81no—
128C22#2026997311yescharacteristic
129C4—81no—
130C4—16881no—
131C4—16881no—
132C4C2×#1680995012yes—
133C4#2027000412yes—
134C4#2027000612yes—
135C2—161no—
136C2#2721351811yescharacteristic
137C2#2721353211yescharacteristic
138C2#2721353711yescharacteristic
subgroupG/H
centerC22#20269973
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#492×C211C2
socleC22×C211#127
prime radical#127×C211C22
hypercenter#492D211
hypocenterC211#1714
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #27213551C2semidirect
2. #27213542C2semidirect
4. #27213498C2semidirect
6. #27212867C2semidirect
54. C8×C211SD16semidirect
69. C211#1714semidirect
124. C8#16809950semidirect

Maximal subgroups

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

subgroupindex
#272135512
#272135422
#272135182
#272134982
#272134922
#272128672
#492×C2112
#1714211