Finite Group Databasebeta

← #41013802

Subgroups — #41013802

Order 27008.

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

normal subgroups only

This group has 344 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, complemented
2#27213151C212yesmaximal
3C2×#20269896C211yesmaximal, characteristic, complemented
4#27212906C212yesmaximal, complemented
5C2×#124×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7Q8×Dic211—24no—
8#20269936—22no—
9#20269936C2212yes—
10#20269898—24no—
11#20269896—22no—
12C22×Q8×C211C2211yescharacteristic
13C22×D4×C211C2211yescharacteristic
14#124×C211—22no—
15#124×C211C414yescomplemented
16C2×C22⋊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—
26C24×C211—21no—
27C2×Q8×C211—22no—
28C2×Q8×C211—22no—
29C2×Q8×C211C2×C412yes—
30C2×Q8×C211D412yes—
31C2×D4×C211—22no—
32C2×D4×C211—21no—
33C2×D4×C211—22no—
34C2×D4×C211C2×C412yes—
35C22×C4×C211—21no—
36C22×C4×C211D412yes—
37C22×C4×C211C2311yescharacteristic
38C22⋊C4×C211—28no—
39C42×C211—21no—
40C42×C211C2×C412yes—
41C2×Dic211—24no—
42C211⋊C8—42no—
43C23×C211—22no—
44C23×C211—44no—
45C23×C211—41no—
46C23×C211—41no—
47C23×C211C2×D411yescharacteristic
48Q8×C211—24no—
49Q8×C211—24no—
50D4×C211—24no—
51D4×C211—42no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
54C2×C4×C211—22no—
55C2×C4×C211—42no—
56C2×C4×C211C22⋊C414yes—
57C2×C4×C211C22×C411yescharacteristic
58C2×C4×C211C2×D412yes—
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211#4112yes—
61C22×C211—22no—
62C22×C211—44no—
63C22×C211—42no—
64C22×C211—82no—
65C22×C211—82no—
66C22×C211C2×C22⋊C411yescharacteristic
67C22×C211#12011yescharacteristic
68C22×C211#12311yescharacteristic
69C4×C211—24no—
70C4×C211—42no—
71C4×C211C4×D412yes—
72C4×C211#12112yes—
73Dic211—44no—
74C2×C211—21no—
75C2×C211—21no—
76C2×C211—82no—
77C2×C211#38611yescharacteristic
78C2×C211#45412yes—
79C211#188811yescharacteristic, complemented, Sylow
80#1888—2111nomaximal, Sylow
81C2×#124D21111yescharacteristic
82#519—2111no—
83#418—2112no—
84C2×#101—2111no—
85#339—2112no—
86C22×Q8C2×D21111yescharacteristic
87C22×D4C2×D21111yescharacteristic
88C2×M16—2112no—
89#124—22no—
90#124Dic21114yescomplemented
91C4×Q8—4224no—
92#117—4222no—
93#117—2112no—
94C2×C22⋊C4—22no—
95C2×C42C2×D21111yescharacteristic
96#102—4224no—
97#101—4222no—
98C24—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8C2×Dic21112yes—
102C2×Q8C22⋊D21112yes—
103C2×D4—22no—
104C2×D4—21no—
105C2×D4—22no—
106C2×D4C2×Dic21112yes—
107C22×C4—21no—
108C22×C4C22⋊D21112yes—
109C22×C4C22×D21111yescharacteristic
110M16—4222no—
111M16—4222no—
112C2×C8—4222no—
113C4⋊C4—4224no—
114C4⋊C4—2114no—
115C22⋊C4—28no—
116C22⋊C4—4222no—
117C42—21no—
118C42—4224no—
119C42—4222no—
120C42C2×Dic21112yes—
121C23—22no—
122C23—44no—
123C23—41no—
124C23—41no—
125C23C2×C22⋊D21111yescharacteristic
126Q8—24no—
127Q8—24no—
128D4—24no—
129D4—42no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—42no—
134C2×C4—4224no—
135C2×C4C22⋊Dic21114yes—
136C2×C4C22×Dic21111yescharacteristic
137C2×C4D4×D21112yes—
138C2×C4C2×C22⋊D21111yescharacteristic
139C2×C4#1680996112yes—
140C8—8442no—
141C22—22no—
142C22—44no—
143C22—42no—
144C22—82no—
145C22—82no—
146C22C2×C22⋊Dic21111yescharacteristic
147C22#2026994511yescharacteristic
148C22#2026995811yescharacteristic
149C4—24no—
150C4—42no—
151C4—8444no—
152C4D4×Dic21112yes—
153C4#2026994912yes—
154C2—21no—
155C2—21no—
156C2—82no—
157C2#2721302611yescharacteristic
158C2#2721331812yes—
subgroupG/H
centerC22#20269945
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#124×C211C2
socleC22×C211#120
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#124D211
hypocenterC211#1888
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #27213624C2semidirect
3. C2×#20269896C2semidirect
4. #27212906C2semidirect
15. #124×C211C4semidirect
79. C211#1888semidirect
90. #124Dic211semidirect

Maximal subgroups

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

subgroupindex
#272136242
#272131512
C2×#202698962
#272129062
C2×#124×C2112
#1888211