Finite Group Databasebeta

← #41014704

Subgroups — #41014704

Order 27008.

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

normal subgroups only

This group has 264 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#27213946C211yesmaximal, characteristic
2#27213143C212yesmaximal, complemented
3#27213119C212yesmaximal, complemented
4#27212903C211yesmaximal, characteristic, complemented
5#424×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2212yes—
7#20270010—22no—
8#20270005—22no—
9#20269999—22no—
10#20269936C2212yes—
11#20269902—22no—
12#20269899—22no—
13#20269891—24no—
14C2×M16×C211C2211yescharacteristic
15C22×C8×C211C2211yescharacteristic
16#117×C211C2211yescharacteristic
17C4⋊C8×C211—22no—
18#16809962—22no—
19#16809962—22no—
20#16809962—22no—
21#16809962—22no—
22C2×Dic422D412yes—
23C2×D844D412yes—
24C2×C22⋊D211—22no—
25C2×C4×D211—22no—
26Dic844—22no—
27#16809951—24no—
28D1688—22no—
29C4⋊Dic211—22no—
30C22⋊Dic211—22no—
31C4×Dic211—22no—
32C22×C4×C211C2311yescharacteristic
33M16×C211—22no—
34C2×C8×C211—21no—
35C2×C8×C211—22no—
36C2×C8×C211D412yes—
37C4⋊C4×C211—21no—
38C22⋊C4×C211D412yes—
39C42×C211—21no—
40C22×D211—22no—
41Dic422—22no—
42Dic422—22no—
43D844—22no—
44D844—22no—
45C22⋊D211—42no—
46C22⋊D211—42no—
47C2×Dic211—22no—
48C2×Dic211—42no—
49C4×D211—42no—
50C4×D211—42no—
51C23×C211C2×D411yescharacteristic
52C2×C4×C211—22no—
53C2×C4×C211C2×D412yes—
54C2×C4×C211C2×D411yescharacteristic
55C2×C4×C211#4111yescharacteristic
56C2×C4×C211#4112yes—
57C8×C211—22no—
58C8×C211—41no—
59C22×C211—41no—
60C22×C211#12111yescharacteristic
61C22×C211#12112yes—
62C2×D211—42no—
63C2×D211—82no—
64C4×C211—42no—
65C4×C211#12011yescharacteristic
66C4×C211#12312yes—
67C4×C211#12411yescharacteristic
68Dic211—42no—
69Dic211—82no—
70C2×C211—22no—
71C2×C211—21no—
72C2×C211#39411yescharacteristic
73D211—82no—
74C211#205311yescharacteristic, complemented, Sylow
75#2053—2111nomaximal, Sylow
76#575—2111no—
77#424D21111yescharacteristic, complemented
78#416—2112no—
79#410—2112no—
80H(3,ℤ/4)—2111no—
81C2×#41—2112no—
82#137—4222no—
83AGL(1,ℤ/8)—4222no—
84#135—4222no—
85C2×M16C2×D21111yescharacteristic
86C22×C8C2×D21111yescharacteristic
87#117—2112no—
88#117C2×D21111yescharacteristic
89C4⋊C8—22no—
90#103—4222no—
91#102—4222no—
92#99—4224no—
93#41—4222no—
94#41—4222no—
95#41—4222no—
96#41—4222no—
97C2×Q8—2112no—
98C2×D4—4222no—
99C2×D4—2112no—
100C22×C4—4222no—
101C22×C4C22×D21111yescharacteristic
102Q16—4222no—
103SD16—4224no—
104D8—4222no—
105M16—22no—
106C2×C8—22no—
107C2×C8—21no—
108C2×C8C22⋊D21112yes—
109C4⋊C4—21no—
110C4⋊C4—4222no—
111C22⋊C4—4222no—
112C22⋊C4D84412yes—
113C42—21no—
114C42—4222no—
115C23—4222no—
116C23C2×D84411yescharacteristic
117Q8—4222no—
118Q8—4222no—
119D4—8442no—
120D4—8442no—
121D4—4222no—
122D4—4222no—
123C2×C4—22no—
124C2×C4—8442no—
125C2×C4—8442no—
126C2×C4—8442no—
127C2×C4—4222no—
128C2×C4D4×D21112yes—
129C2×C4C2×C22⋊D21111yescharacteristic
130C2×C4#1680996211yescharacteristic
131C2×C4#1680996312yes—
132C8—22no—
133C8—41no—
134C22—41no—
135C22—8442no—
136C22—16882no—
137C22#2026995111yescharacteristic
138C22#2026995212yes—
139C4—42no—
140C4—8442no—
141C4—16882no—
142C4#2026994511yescharacteristic
143C4#2026996212yes—
144C4#2026996611yescharacteristic
145C2—21no—
146C2—22no—
147C2—16882no—
148C2#2721306311yescharacteristic
subgroupG/H
centerC4#20269966
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#424×C211C2
socleC2×C211#394
prime radical#27213946C2
hypercenter#424D211
hypocenterC211#2053
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #27213143C2semidirect
3. #27213119C2semidirect
4. #27212903C2semidirect
5. #424×C211C2semidirect
74. C211#2053semidirect
77. #424D211semidirect

Maximal subgroups

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

subgroupindex
#272139462
#272131432
#272131192
#272129032
#424×C2112
#2053211