Finite Group Databasebeta

← #41013787

Subgroups — #41013787

Order 27008.

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

normal subgroups only

This group has 360 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#27213621C211yesmaximal, characteristic
2#27213155C212yesmaximal, complemented
3C2×#20269894C211yesmaximal, characteristic
4#27212906C212yesmaximal, complemented
5C2×#124×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7D4×Dic211—24no—
8#20269936C2212yes—
9C2×C22⋊Dic211—22no—
10#20269901—24no—
11#20269894—22no—
12C22×Q8×C211C2211yescharacteristic
13C22×D4×C211C2211yescharacteristic
14#124×C211—22no—
15#124×C211C414yescomplemented
16C2×C22⋊C4×C211—22no—
17C2×C42×C211C2211yescharacteristic
18C22×Dic211—24no—
19#16809946—22no—
20#16809946—22no—
21C2×C211⋊C8—22no—
22C4⋊Dic211D414yes—
23C22⋊Dic211—24no—
24C22⋊Dic211—22no—
25C4×Dic211—22no—
26C24×C211D412yes—
27C2×Q8×C211—21no—
28C2×Q8×C211—22no—
29C2×Q8×C211—22no—
30C2×Q8×C211C2×C412yescomplemented
31C2×D4×C211—22no—
32C2×D4×C211—22no—
33C2×D4×C211C2×C412yes—
34C2×D4×C211D412yes—
35C22×C4×C211—21no—
36C22×C4×C211—21no—
37C22×C4×C211C2311yescharacteristic
38C22⋊C4×C211—28no—
39C42×C211—21no—
40C42×C211C2×C412yescomplemented
41C2×Dic211—24no—
42C2×Dic211—84no—
43C211⋊C8—42no—
44C23×C211—22no—
45C23×C211—22no—
46C23×C211—44no—
47C23×C211C22⋊C414yes—
48C23×C211C2×D411yescharacteristic
49Q8×C211—24no—
50Q8×C211—42no—
51D4×C211—24no—
52D4×C211—24no—
53C2×C4×C211—22no—
54C2×C4×C211—22no—
55C2×C4×C211—42no—
56C2×C4×C211—41no—
57C2×C4×C211—41no—
58C2×C4×C211C22×C411yescharacteristic
59C2×C4×C211C2×D412yes—
60C2×C4×C211C2×D411yescharacteristic
61C2×C4×C211#4112yes—
62C22×C211—22no—
63C22×C211—24no—
64C22×C211—44no—
65C22×C211—44no—
66C22×C211—82no—
67C22×C211C2×C22⋊C411yescharacteristic
68C22×C211#12011yescharacteristic
69C22×C211#12311yescharacteristic
70C4×C211—42no—
71C4×C211—42no—
72C4×C211C4×D412yes—
73C4×C211#12112yes—
74Dic211—44no—
75C2×C211—21no—
76C2×C211—21no—
77C2×C211—44no—
78C2×C211#38611yescharacteristic
79C2×C211#45512yes—
80C211#188511yescharacteristic, complemented, Sylow
81#1885—2111nomaximal, Sylow
82C2×#124D21111yescharacteristic
83#518—2111no—
84#419—2112no—
85C2×#100—2111no—
86#339—2112no—
87C22×Q8C2×D21111yescharacteristic
88C22×D4C2×D21111yescharacteristic
89C2×M16—2112no—
90#124—22no—
91#124Dic21114yescomplemented
92C4×D4—4224no—
93#117—2112no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—4222no—
96C2×C42C2×D21111yescharacteristic
97#103—4224no—
98#100—4222no—
99C24C22⋊D21112yes—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8C2×Dic21112yescomplemented
104C2×D4—22no—
105C2×D4—22no—
106C2×D4C2×Dic21112yes—
107C2×D4C22⋊D21112yes—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—4224no—
111C22×C4C22×D21111yescharacteristic
112M16—4222no—
113M16—4222no—
114C2×C8—4222no—
115C4⋊C4—2114no—
116C22⋊C4—28no—
117C22⋊C4—4224no—
118C22⋊C4—4222no—
119C42—21no—
120C42—4222no—
121C42C2×Dic21112yescomplemented
122C23—22no—
123C23—22no—
124C23—44no—
125C23C22⋊Dic21114yes—
126C23C2×C22⋊D21111yescharacteristic
127Q8—24no—
128Q8—42no—
129D4—24no—
130D4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—42no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—16884no—
137C2×C4—4224no—
138C2×C4C22×Dic21111yescharacteristic
139C2×C4D4×D21112yes—
140C2×C4C2×C22⋊D21111yescharacteristic
141C2×C4#1680996112yes—
142C8—8442no—
143C22—22no—
144C22—24no—
145C22—44no—
146C22—44no—
147C22—82no—
148C22C2×C22⋊Dic21111yescharacteristic
149C22#2026994511yescharacteristic
150C22#2026995811yescharacteristic
151C4—42no—
152C4—42no—
153C4—8444no—
154C4D4×Dic21112yes—
155C4#2026994912yes—
156C2—21no—
157C2—21no—
158C2—44no—
159C2#2721302611yescharacteristic
160C2#2721332312yes—
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#1885
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
2. #27213155C2semidirect
4. #27212906C2semidirect
15. #124×C211C4semidirect
30. C2×Q8×C211C2×C4semidirect
40. C42×C211C2×C4semidirect
80. C211#1885semidirect
91. #124Dic211semidirect
103. C2×Q8C2×Dic211semidirect
121. C42C2×Dic211semidirect

Maximal subgroups

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

subgroupindex
#272136212
#272131552
C2×#202698942
#272129062
C2×#124×C2112
#1885211