Finite Group Databasebeta

← #41014749

Subgroups — #41014749

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#27213777C211yesmaximal, characteristic
2#27213188C212yesmaximal
3#27213011C211yesmaximal, characteristic
4#27212907C212yesmaximal
5#343×C211C211yesmaximal, characteristic
6#20269976—22no—
7#20269968—22no—
8#20269956—22no—
9#20269936C2212yes—
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4×Dic211C2211yescharacteristic
12#20269911—22no—
13#20269909—22no—
14C211⋊#95—22no—
15C2×M16×C211C2212yes—
16C2×C4⋊C4×C211C2211yescharacteristic
17C2×Dic422—22no—
18C22×Dic211—21no—
19C22×Dic211—21no—
20C4⋊Dic211—22no—
21C4⋊Dic211—22no—
22C4⋊Dic211D412yes—
23C4⋊Dic211D412yes—
24Dic211⋊C4—24no—
25C22⋊Dic211—22no—
26C4×Dic211—22no—
27C4×Dic211—21no—
28C4×Dic211Q812yescomplemented
29C22×C4×C211D412yes—
30C22×C4×C211C2311yescharacteristic
31M16×C211—22no—
32M16×C211Q814yescomplemented
33C2×C8×C211—22no—
34C4⋊C4×C211—22no—
35Dic422—82no—
36C2×Dic211—22no—
37C2×Dic211—22no—
38C2×Dic211—22no—
39C2×Dic211—41no—
40C2×Dic211—41no—
41C2×Dic211—42no—
42C2×Dic211—41no—
43C23×C211C2×D411yescharacteristic
44C2×C4×C211—22no—
45C2×C4×C211—42no—
46C2×C4×C211C2×D412yes—
47C2×C4×C211C2×Q811yescharacteristic
48C2×C4×C211C2×Q812yes—
49C2×C4×C211#4111yescharacteristic
50C8×C211—24no—
51C22×C211—21no—
52C22×C211—21no—
53C22×C211#12011yescharacteristic
54C22×C211C22⋊Q811yescharacteristic
55C22×C211C4⋊Q811yescharacteristic
56C4×C211—42no—
57C4×C211C22⋊Q812yes—
58C4×C211C4⋊Q812yes—
59Dic211—42no—
60Dic211—42no—
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#544—2111no—
71#428—2112no—
72#382—2111no—
73#343D21111yescharacteristic
74#339—2112no—
75C2×M16C2×D21112yes—
76C4⋊Q8—4222no—
77#125—4222no—
78C22⋊Q8—4222no—
79#117—2112no—
80C2×C4⋊C4—2111no—
81C2×C4⋊C4C2×D21111yescharacteristic
82C2×C42—2111no—
83#107—4222no—
84#106—4222no—
85#95—4222no—
86C2×Q8—4222no—
87C22×C4—4221no—
88C22×C4—4221no—
89C22×C4D84412yes—
90C22×C4C22×D21111yescharacteristic
91M16—22no—
92M16Dic42214yescomplemented
93C2×C8—22no—
94C4⋊C4—22no—
95C4⋊C4—4222no—
96C4⋊C4—4224no—
97C4⋊C4—4222no—
98C4⋊C4—2112no—
99C4⋊C4—2112no—
100C22⋊C4—4222no—
101C42—4222no—
102C42—4221no—
103C42—2112no—
104C23C2×D84411yescharacteristic
105Q8—16882no—
106C2×C4—22no—
107C2×C4—42no—
108C2×C4—8441no—
109C2×C4—8441no—
110C2×C4—8442no—
111C2×C4—8441no—
112C2×C4—4222no—
113C2×C4—4222no—
114C2×C4—4222no—
115C2×C4D4×D21112yes—
116C2×C4Q8×D21111yescharacteristic
117C2×C4C2×Dic42212yes—
118C2×C4#1680996111yescharacteristic
119C8—24no—
120C22—21no—
121C22—21no—
122C22#2026994711yescharacteristic
123C22#2026995711yescharacteristic
124C22#2026997711yescharacteristic
125C4—42no—
126C4—8442no—
127C4—8442no—
128C4—8442no—
129C4—16881no—
130C4#2026995412yes—
131C4#2026997612yes—
132C2—21no—
133C2—21no—
134C2#2721306711yescharacteristic
135C2#2721332111yescharacteristic
136C2#2721332611yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#343D211
hypocenterC211#2061
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
28. C4×Dic211Q8semidirect
32. M16×C211Q8semidirect
68. C211#2061semidirect
92. M16Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137772
#272131882
#272130112
#272129072
#343×C2112
#2061211