Finite Group Databasebeta

← #41013789

Subgroups — #41013789

Order 27008.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#20269975C211yesmaximal, characteristic
2#27213624C211yesmaximal, characteristic
3#27213156C212yesmaximal
4#27212907C212yesmaximal
5C2×#101×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7#20269975—22no—
8#20269975C414yescomplemented
9Q8×Dic211—24no—
10#20269936—22no—
11#20269936C2212yes—
12C2×Dic211⋊C4—22no—
13C2×C4×Dic211C2211yescharacteristic
14#20269902—24no—
15C22×Q8×C211C2211yescharacteristic
16C2×M16×C211C2212yes—
17#101×C211—22no—
18C2×Dic422—21no—
19C2×Dic422—22no—
20C2×Dic422—22no—
21C2×Dic422C2×C412yes—
22C22×Dic211—21no—
23C22×Dic211—21no—
24C4⋊Dic211—24no—
25C4⋊Dic211D414yes—
26Dic211⋊C4—28no—
27C22⋊Dic211—22no—
28C4×Dic211—24no—
29C4×Dic211—21no—
30C4×Dic211—22no—
31C4×Dic211C2×C412yes—
32C2×Q8×C211—22no—
33C2×Q8×C211—22no—
34C2×Q8×C211C2×C412yes—
35C2×Q8×C211D412yes—
36C22×C4×C211D412yes—
37C22×C4×C211C2311yescharacteristic
38M16×C211—22no—
39M16×C211—22no—
40C2×C8×C211—22no—
41Dic422—24no—
42Dic422—42no—
43C2×Dic211—24no—
44C2×Dic211—22no—
45C2×Dic211—22no—
46C2×Dic211—24no—
47C2×Dic211—41no—
48C2×Dic211—41no—
49C23×C211C2×D411yescharacteristic
50Q8×C211—24no—
51Q8×C211—24no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
54C2×C4×C211C22⋊C414yes—
55C2×C4×C211C22×C411yescharacteristic
56C2×C4×C211C2×D412yes—
57C2×C4×C211C2×D411yescharacteristic
58C2×C4×C211#4112yes—
59C8×C211—42no—
60C22×C211—22no—
61C22×C211C2×C22⋊C411yescharacteristic
62C22×C211#12011yescharacteristic
63C22×C211#12311yescharacteristic
64C4×C211—24no—
65C4×C211C4×D412yes—
66C4×C211#12112yes—
67Dic211—24no—
68Dic211—42no—
69Dic211—44no—
70C2×C211—21no—
71C2×C211—21no—
72C2×C211#38611yescharacteristic
73C2×C211#45612yes—
74C211#188611yescharacteristic, complemented, Sylow
75#1886—2111nomaximal, Sylow
76C2×C4⋊Q8—2111no—
77#519—2111no—
78#419—2112no—
79C2×#101D21111yescharacteristic
80#339—2112no—
81C22×Q8—2111no—
82C22×Q8C2×D21111yescharacteristic
83C2×M16C2×D21112yes—
84C4⋊Q8—4222no—
85C4⋊Q8—2114no—
86C4×Q8—4224no—
87#117—4222no—
88#117—2112no—
89C2×C4⋊C4—4222no—
90C2×C42—2111no—
91#103—4224no—
92#101—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—4222no—
96C2×Q8—4222no—
97C2×Q8—4221no—
98C2×Q8—2112no—
99C2×Q8C4×D21112yes—
100C2×Q8C22⋊D21112yes—
101C22×C4—4221no—
102C22×C4—4221no—
103C22×C4D84412yes—
104C22×C4C22×D21111yescharacteristic
105M16—22no—
106M16—22no—
107C2×C8—22no—
108C4⋊C4—4224no—
109C4⋊C4—4228no—
110C4⋊C4—2114no—
111C22⋊C4—4222no—
112C42—4224no—
113C42—4222no—
114C42—4221no—
115C42—2112no—
116C23C2×D84411yescharacteristic
117Q8—24no—
118Q8—24no—
119Q8—8442no—
120Q8—4224no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—8441no—
124C2×C4—8441no—
125C2×C4—4222no—
126C2×C4—4224no—
127C2×C4—4224no—
128C2×C4—4222no—
129C2×C4(C2×D211)⋊C414yes—
130C2×C4C2×C4×D21111yescharacteristic
131C2×C4D4×D21112yes—
132C2×C4C2×C22⋊D21111yescharacteristic
133C2×C4#1680996112yes—
134C8—42no—
135C22—22no—
136C22C2×(C2×D211)⋊C411yescharacteristic
137C22#2026994711yescharacteristic
138C22#2026996111yescharacteristic
139C4—24no—
140C4—8444no—
141C4—8442no—
142C4—4224no—
143C4(C22⋊D211)⋊C412yes—
144C4#2026994912yes—
145C2—21no—
146C2—21no—
147C2#2721302311yescharacteristic
148C2#2721333112yes—
subgroupG/H
centerC22#20269947
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenterC2×#101D211
hypocenterC211#1886
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
8. #20269975C4semidirect
74. C211#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#202699752
#272136242
#272131562
#272129072
C2×#101×C2112
#1886211