Finite Group Databasebeta

← #41014743

Subgroups — #41014743

Order 27008.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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#27213777C211yesmaximal, characteristic
2#27213186C212yesmaximal
3#27212915C211yesmaximal, characteristic
4#27212906C212yesmaximal
5#384×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7#20269976—22no—
8#20269968—22no—
9#20269956—22no—
10#20269936C2212yes—
11#20269910—22no—
12#20269908—22no—
13C2×C4⋊C4×C211—22no—
14C2×C4⋊C4×C211C2211yescharacteristic
15C2×C4⋊C4×C211C2211yescharacteristic
16C2×C42×C211C2211yescharacteristic
17C2×Dic422—22no—
18#16809946—22no—
19#16809946Q814yescomplemented
20C2×C211⋊C8—22no—
21C4⋊Dic211—22no—
22C4⋊Dic211D412yes—
23C4⋊Dic211D412yes—
24Dic211⋊C4—24no—
25C22⋊Dic211—22no—
26C4×Dic211—22no—
27C22×C4×C211—21no—
28C22×C4×C211—21no—
29C22×C4×C211D412yes—
30C22×C4×C211C2311yescharacteristic
31C4⋊C4×C211—22no—
32C4⋊C4×C211—22no—
33C4⋊C4×C211—44no—
34C42×C211—21no—
35C42×C211Q812yescomplemented
36Dic422—82no—
37C2×Dic211—22no—
38C2×Dic211—22no—
39C211⋊C8—24no—
40C23×C211C2×D411yescharacteristic
41C2×C4×C211—22no—
42C2×C4×C211—22no—
43C2×C4×C211—22no—
44C2×C4×C211—22no—
45C2×C4×C211—42no—
46C2×C4×C211—42no—
47C2×C4×C211—41no—
48C2×C4×C211C2×D412yes—
49C2×C4×C211C2×Q812yes—
50C2×C4×C211C2×Q811yescharacteristic
51C2×C4×C211#4111yescharacteristic
52C22×C211—21no—
53C22×C211—21no—
54C22×C211#12011yescharacteristic
55C22×C211C22⋊Q811yescharacteristic
56C22×C211C4⋊Q811yescharacteristic
57C4×C211—24no—
58C4×C211—42no—
59C4×C211—81no—
60C4×C211C22⋊Q812yes—
61C4×C211C4⋊Q812yes—
62Dic211—42no—
63Dic211—42no—
64C2×C211—21no—
65C2×C211—21no—
66C2×C211#39511yescharacteristic
67C2×C211#45311yescharacteristic
68C2×C211#45611yescharacteristic
69C211#206011yescharacteristic, complemented, Sylow
70#2060—2111nomaximal, Sylow
71#544—2111no—
72#428—2112no—
73#384D21111yescharacteristic
74#343—2111no—
75#339—2112no—
76C2×M16—2112no—
77C4⋊Q8—4222no—
78#125—4222no—
79C22⋊Q8—4222no—
80#117—2112no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4C2×D21111yescharacteristic
83C2×C4⋊C4C2×D21111yescharacteristic
84C2×C42C2×D21111yescharacteristic
85#107—4222no—
86#106—4222no—
87C2×Q8—4222no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4C22⋊D21112yes—
91C22×C4C22×D21111yescharacteristic
92M16—4222no—
93M16—2114no—
94C2×C8—4222no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—44no—
98C4⋊C4—4222no—
99C4⋊C4—4224no—
100C4⋊C4—2112no—
101C4⋊C4—2112no—
102C22⋊C4—4222no—
103C42—21no—
104C42—4222no—
105C42Dic42212yescomplemented
106C23C2×C22⋊D21111yescharacteristic
107Q8—16882no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—4222no—
116C2×C4—4222no—
117C2×C4D4×D21112yes—
118C2×C4Q8×D21112yes—
119C2×C4C2×Dic42211yescharacteristic
120C2×C4#1680996211yescharacteristic
121C8—4224no—
122C22—21no—
123C22—21no—
124C22#2026994511yescharacteristic
125C22#2026995611yescharacteristic
126C22#2026997511yescharacteristic
127C4—24no—
128C4—42no—
129C4—81no—
130C4—8442no—
131C4—8442no—
132C4#2026995512yes—
133C4#2026997612yes—
134C2—21no—
135C2—21no—
136C2#2721306611yescharacteristic
137C2#2721331311yescharacteristic
138C2#2721332811yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#384×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#384D211
hypocenterC211#2060
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
19. #16809946Q8semidirect
35. C42×C211Q8semidirect
69. C211#2060semidirect
105. C42Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137772
#272131862
#272129152
#272129062
#384×C2112
#2060211