Finite Group Databasebeta

← #41014744

Subgroups — #41014744

Order 27008.

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

normal subgroups only

This group has 246 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#27213015C211yesmaximal, characteristic
4#27212907C212yesmaximal
5#343×C211C211yesmaximal, characteristic
6#20269976—22no—
7#20269968—22no—
8#20269956—22no—
9#20269936C2212yes—
10C2×C4⋊Dic211C2211yescharacteristic
11C2×Dic211⋊C4—22no—
12C2×C4×Dic211C2211yescharacteristic
13#20269911—22no—
14#20269909—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—
25Dic211⋊C4—44no—
26C22⋊Dic211—22no—
27C4×Dic211—22no—
28C4×Dic211—21no—
29C4×Dic211Q812yescomplemented
30C22×C4×C211D412yes—
31C22×C4×C211C2311yescharacteristic
32M16×C211—22no—
33M16×C211Q814yescomplemented
34C2×C8×C211—22no—
35C4⋊C4×C211—22no—
36Dic422—82no—
37C2×Dic211—22no—
38C2×Dic211—22no—
39C2×Dic211—22no—
40C2×Dic211—22no—
41C2×Dic211—22no—
42C2×Dic211—42no—
43C2×Dic211—41no—
44C23×C211C2×D411yescharacteristic
45C2×C4×C211—22no—
46C2×C4×C211—42no—
47C2×C4×C211C2×D412yes—
48C2×C4×C211C2×Q811yescharacteristic
49C2×C4×C211C2×Q812yes—
50C2×C4×C211#4111yescharacteristic
51C8×C211—24no—
52C22×C211—21no—
53C22×C211—21no—
54C22×C211#12011yescharacteristic
55C22×C211C22⋊Q811yescharacteristic
56C22×C211C4⋊Q811yescharacteristic
57C4×C211—42no—
58C4×C211C22⋊Q812yes—
59C4×C211C4⋊Q812yes—
60Dic211—24no—
61Dic211—42no—
62Dic211—42no—
63Dic211—81no—
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#384—2111no—
74#343D21111yescharacteristic
75#339—2112no—
76C2×M16C2×D21112yes—
77C4⋊Q8—4222no—
78#125—4222no—
79C22⋊Q8—4222no—
80#117—2112no—
81C2×C4⋊C4—4222no—
82C2×C4⋊C4—2111no—
83C2×C4⋊C4C2×D21111yescharacteristic
84C2×C42—2111no—
85#107—4222no—
86#106—4222no—
87C2×Q8—4222no—
88C22×C4—4221no—
89C22×C4—4221no—
90C22×C4D84412yes—
91C22×C4C22×D21111yescharacteristic
92M16—22no—
93M16Dic42214yescomplemented
94C2×C8—22no—
95C4⋊C4—22no—
96C4⋊C4—8444no—
97C4⋊C4—4222no—
98C4⋊C4—4224no—
99C4⋊C4—4222no—
100C4⋊C4—2112no—
101C4⋊C4—2112no—
102C22⋊C4—4222no—
103C42—4222no—
104C42—4221no—
105C42—2112no—
106C23C2×D84411yescharacteristic
107Q8—16882no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—8442no—
111C2×C4—8441no—
112C2×C4—4222no—
113C2×C4—4222no—
114C2×C4—4222no—
115C2×C4—4222no—
116C2×C4—4222no—
117C2×C4D4×D21112yes—
118C2×C4Q8×D21111yescharacteristic
119C2×C4C2×Dic42212yes—
120C2×C4#1680996111yescharacteristic
121C8—24no—
122C22—21no—
123C22—21no—
124C22#2026994711yescharacteristic
125C22#2026995711yescharacteristic
126C22#2026997711yescharacteristic
127C4—42no—
128C4—8442no—
129C4—8442no—
130C4—16881no—
131C4—4224no—
132C4#2026995412yes—
133C4#2026997612yes—
134C2—21no—
135C2—21no—
136C2#2721306711yescharacteristic
137C2#2721331611yescharacteristic
138C2#2721333111yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#343D211
hypocenterC211#2060
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
29. C4×Dic211Q8semidirect
33. M16×C211Q8semidirect
69. C211#2060semidirect
93. M16Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137772
#272131882
#272130152
#272129072
#343×C2112
#2060211