Finite Group Databasebeta

← #41013792

Subgroups — #41013792

Order 27008.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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#27213624C211yesmaximal, characteristic
2#27213155C212yesmaximal
3C2×#20269896C211yesmaximal, characteristic
4#27212906C212yesmaximal
5C2×C4⋊Q8×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7Q8×Dic211—24no—
8#20269936—22no—
9#20269936C2212yes—
10#20269901—24no—
11#20269896—22no—
12C22×Q8×C211C2211yescharacteristic
13C22×Q8×C211C2211yescharacteristic
14C4⋊Q8×C211—22no—
15C4⋊Q8×C211C414yescomplemented
16C2×C4⋊C4×C211—22no—
17C2×C42×C211C2211yescharacteristic
18#16809946—22no—
19#16809946—22no—
20C2×C211⋊C8—22no—
21C4⋊Dic211—24no—
22C4⋊Dic211D414yes—
23C22⋊Dic211—22no—
24C4×Dic211—24no—
25C4×Dic211—22no—
26C2×Q8×C211—22no—
27C2×Q8×C211—21no—
28C2×Q8×C211—22no—
29C2×Q8×C211—22no—
30C2×Q8×C211—22no—
31C2×Q8×C211C2×C412yes—
32C2×Q8×C211C2×C412yes—
33C2×Q8×C211D412yes—
34C22×C4×C211—21no—
35C22×C4×C211—21no—
36C22×C4×C211D412yes—
37C22×C4×C211C2311yescharacteristic
38C4⋊C4×C211—28no—
39C42×C211—21no—
40C42×C211C2×C412yes—
41C2×Dic211—24no—
42C211⋊C8—42no—
43C23×C211C2×D411yescharacteristic
44Q8×C211—24no—
45Q8×C211—24no—
46Q8×C211—24no—
47Q8×C211—42no—
48C2×C4×C211—24no—
49C2×C4×C211—22no—
50C2×C4×C211—22no—
51C2×C4×C211—22no—
52C2×C4×C211—22no—
53C2×C4×C211—41no—
54C2×C4×C211—41no—
55C2×C4×C211C22⋊C414yes—
56C2×C4×C211C22×C411yescharacteristic
57C2×C4×C211C2×D412yes—
58C2×C4×C211C2×D411yescharacteristic
59C2×C4×C211#4112yes—
60C22×C211—22no—
61C22×C211C2×C22⋊C411yescharacteristic
62C22×C211#12011yescharacteristic
63C22×C211#12311yescharacteristic
64C4×C211—24no—
65C4×C211—24no—
66C4×C211—42no—
67C4×C211C4×D412yes—
68C4×C211#12112yes—
69Dic211—44no—
70C2×C211—21no—
71C2×C211—21no—
72C2×C211#38611yescharacteristic
73C2×C211#45612yes—
74C211#188611yescharacteristic, complemented, Sylow
75#1886—2111nomaximal, Sylow
76C2×C4⋊Q8D21111yescharacteristic
77#519—2111no—
78#419—2112no—
79C2×#101—2111no—
80#339—2112no—
81C22×Q8C2×D21111yescharacteristic
82C22×Q8C2×D21111yescharacteristic
83C2×M16—2112no—
84C4⋊Q8—22no—
85C4⋊Q8Dic21114yescomplemented
86C4×Q8—4224no—
87#117—4222no—
88#117—2112no—
89C2×C4⋊C4—22no—
90C2×C42C2×D21111yescharacteristic
91#103—4224no—
92#101—4222no—
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8C2×Dic21112yes—
99C2×Q8C2×Dic21112yes—
100C2×Q8C22⋊D21112yes—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4C22⋊D21112yes—
104C22×C4C22×D21111yescharacteristic
105M16—4222no—
106M16—4222no—
107C2×C8—4222no—
108C4⋊C4—28no—
109C4⋊C4—4224no—
110C4⋊C4—2114no—
111C22⋊C4—4222no—
112C42—21no—
113C42—4224no—
114C42—4222no—
115C42C2×Dic21112yes—
116C23C2×C22⋊D21111yescharacteristic
117Q8—24no—
118Q8—24no—
119Q8—24no—
120Q8—42no—
121C2×C4—24no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—4224no—
129C2×C4C22⋊Dic21114yes—
130C2×C4C22×Dic21111yescharacteristic
131C2×C4D4×D21112yes—
132C2×C4C2×C22⋊D21111yescharacteristic
133C2×C4#1680996112yes—
134C8—8442no—
135C22—22no—
136C22C2×C22⋊Dic21111yescharacteristic
137C22#2026994511yescharacteristic
138C22#2026995811yescharacteristic
139C4—24no—
140C4—24no—
141C4—42no—
142C4—8444no—
143C4D4×Dic21112yes—
144C4#2026994912yes—
145C2—21no—
146C2—21no—
147C2#2721302611yescharacteristic
148C2#2721332812yes—
subgroupG/H
centerC22#20269945
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×C4⋊Q8×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenterC2×C4⋊Q8D211
hypocenterC211#1886
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
15. C4⋊Q8×C211C4semidirect
74. C211#1886semidirect
85. C4⋊Q8Dic211semidirect

Maximal subgroups

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

subgroupindex
#272136242
#272131552
C2×#202698962
#272129062
C2×C4⋊Q8×C2112
#1886211