Finite Group Databasebeta

← #41014745

Subgroups — #41014745

Order 27008.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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#27213189C212yesmaximal
2#27213011C211yesmaximal, characteristic
3#27212915C211yesmaximal, characteristic
4#27212905C212yesmaximal
5#544×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7C2×C4⋊Dic211C2211yescharacteristic
8C2×C4×Dic211C2211yescharacteristic
9#20269910—22no—
10#20269908—22no—
11C211⋊#95—22no—
12C4⋊Q8×C211—22no—
13#125×C211—22no—
14C22⋊Q8×C211—22no—
15#117×C211C2212yes—
16C2×C4⋊C4×C211C2211yescharacteristic
17C22×Dic211—21no—
18C22×Dic211—21no—
19#16809946—22no—
20#16809946Q814yescomplemented
21C2×C211⋊C8—22no—
22C4⋊Dic211—22no—
23C4×Dic211—21no—
24C4×Dic211Q812yescomplemented
25C2×Q8×C211—22no—
26C22×C4×C211D412yes—
27C22×C4×C211C2311yescharacteristic
28C4⋊C4×C211—22no—
29C4⋊C4×C211—24no—
30C4⋊C4×C211—22no—
31C4⋊C4×C211D412yes—
32C4⋊C4×C211D412yes—
33C22⋊C4×C211—22no—
34C42×C211—22no—
35C2×Dic211—22no—
36C2×Dic211—41no—
37C2×Dic211—41no—
38C2×Dic211—42no—
39C2×Dic211—41no—
40C211⋊C8—24no—
41C23×C211C2×D411yescharacteristic
42Q8×C211—82no—
43C2×C4×C211—22no—
44C2×C4×C211—22no—
45C2×C4×C211—22no—
46C2×C4×C211—42no—
47C2×C4×C211C2×D412yes—
48C2×C4×C211C2×Q812yes—
49C2×C4×C211C2×Q811yescharacteristic
50C2×C4×C211#4111yescharacteristic
51C22×C211—21no—
52C22×C211—21no—
53C22×C211#12011yescharacteristic
54C22×C211C22⋊Q811yescharacteristic
55C22×C211C4⋊Q811yescharacteristic
56C4×C211—42no—
57C4×C211—42no—
58C4×C211—42no—
59C4×C211C22⋊Q812yes—
60C4×C211C4⋊Q812yes—
61Dic211—42no—
62Dic211—81no—
63C2×C211—21no—
64C2×C211—21no—
65C2×C211#39511yescharacteristic
66C2×C211#45411yescharacteristic
67C2×C211#45511yescharacteristic
68C211#206111yescharacteristic, complemented, Sylow
69#2061—2111nomaximal, Sylow
70#544D21111yescharacteristic
71#428—2112no—
72#382—2111no—
73#343—2111no—
74#339—2112no—
75C2×M16—2112no—
76C4⋊Q8—22no—
77#125—22no—
78C22⋊Q8—22no—
79#117C2×D21112yes—
80C2×C4⋊C4—2111no—
81C2×C4⋊C4C2×D21111yescharacteristic
82C2×C42—2111no—
83#107—4222no—
84#106—4222no—
85#95—4222no—
86C2×Q8—22no—
87C22×C4—4221no—
88C22×C4—4221no—
89C22×C4C22⋊D21112yes—
90C22×C4C22×D21111yescharacteristic
91M16—4222no—
92M16—2114no—
93C2×C8—4222no—
94C4⋊C4—22no—
95C4⋊C4—24no—
96C4⋊C4—22no—
97C4⋊C4—4222no—
98C4⋊C4C22⋊D21112yes—
99C4⋊C4C22⋊D21112yes—
100C22⋊C4—22no—
101C42—22no—
102C42—4221no—
103C42—2112no—
104C23C2×C22⋊D21111yescharacteristic
105Q8—82no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—8441no—
111C2×C4—8441no—
112C2×C4—8442no—
113C2×C4—8441no—
114C2×C4—4222no—
115C2×C4C2×C22⋊D21112yes—
116C2×C4Q8×D21112yes—
117C2×C4Q8×D21111yescharacteristic
118C2×C4#1680996311yescharacteristic
119C8—4224no—
120C22—21no—
121C22—21no—
122C22#2026994611yescharacteristic
123C22#2026995311yescharacteristic
124C22#2026997511yescharacteristic
125C4—42no—
126C4—42no—
127C4—42no—
128C4—8442no—
129C4—16881no—
130C4#2026995312yes—
131C4#2026997512yes—
132C2—21no—
133C2—21no—
134C2#2721306511yescharacteristic
135C2#2721331911yescharacteristic
136C2#2721332411yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#544×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#544D211
hypocenterC211#2061
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
20. #16809946Q8semidirect
24. C4×Dic211Q8semidirect
68. C211#2061semidirect

Maximal subgroups

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

subgroupindex
#272131892
#272130112
#272129152
#272129052
#544×C2112
#2061211