Finite Group Databasebeta

← #41013786

Subgroups — #41013786

Order 27008.

360 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20269964C211yesmaximal, characteristic
2#27213154C212yesmaximal, complemented
3C2×#20269894C211yesmaximal, characteristic
4#27212905C212yesmaximal, complemented
5#518×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7C2×#16809946C2212yes—
8#20269964—22no—
9#20269964C414yescomplemented
10C2×C22⋊Dic211—22no—
11C2×C4×Dic211C2211yescharacteristic
12#20269900—24no—
13#20269894—22no—
14C22×D4×C211C2211yescharacteristic
15C4×D4×C211—24no—
16#117×C211C2212yes—
17C2×C22⋊C4×C211—22no—
18C2×Dic422—21no—
19C2×Dic422—22no—
20C2×Dic422—22no—
21C2×Dic422C2×C412yescomplemented
22C22×Dic211—21no—
23C22×Dic211—21no—
24#16809946—22no—
25#16809946—22no—
26C2×C211⋊C8—22no—
27C22⋊Dic211—28no—
28C4×Dic211—21no—
29C4×Dic211C2×C412yescomplemented
30C24×C211D412yes—
31C2×D4×C211—22no—
32C2×D4×C211—22no—
33C2×D4×C211C2×C412yes—
34C2×D4×C211D412yes—
35C22×C4×C211—24no—
36C22×C4×C211C2311yescharacteristic
37C4⋊C4×C211D414yes—
38C22⋊C4×C211—24no—
39C22⋊C4×C211—22no—
40C42×C211—22no—
41Dic422—24no—
42Dic422—42no—
43C2×Dic211—22no—
44C2×Dic211—22no—
45C2×Dic211—42no—
46C2×Dic211—41no—
47C2×Dic211—41no—
48C211⋊C8—42no—
49C23×C211—22no—
50C23×C211—22no—
51C23×C211—44no—
52C23×C211C22⋊C414yes—
53C23×C211C2×D411yescharacteristic
54D4×C211—24no—
55D4×C211—24no—
56C2×C4×C211—24no—
57C2×C4×C211—84no—
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—44no—
71C4×C211C4×D412yes—
72C4×C211#12112yes—
73Dic211—42no—
74Dic211—42no—
75C2×C211—21no—
76C2×C211—21no—
77C2×C211—44no—
78C2×C211#38611yescharacteristic
79C2×C211#45512yes—
80C211#188511yescharacteristic, complemented, Sylow
81#1885—2111nomaximal, Sylow
82C2×#124—2111no—
83#518D21111yescharacteristic
84#419—2112no—
85C2×#100—2111no—
86#339—2112no—
87C22×Q8—2111no—
88C22×D4C2×D21111yescharacteristic
89C2×M16—2112no—
90#124—4222no—
91#124—2114no—
92C4×D4—24no—
93#117C2×D21112yes—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—4222no—
96C2×C42—2111no—
97#103—4224no—
98#100—4222no—
99C24C22⋊D21112yes—
100C2×Q8—4222no—
101C2×Q8—4222no—
102C2×Q8—4221no—
103C2×Q8—2112no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4C4×D21112yes—
107C2×D4D84412yes—
108C22×C4—24no—
109C22×C4—4221no—
110C22×C4—4221no—
111C22×C4C22×D21111yescharacteristic
112M16—4222no—
113M16—4222no—
114C2×C8—4222no—
115C4⋊C4C22⋊D21114yes—
116C22⋊C4—24no—
117C22⋊C4—22no—
118C22⋊C4—4228no—
119C42—22no—
120C42—4221no—
121C42—2112no—
122C23—22no—
123C23—22no—
124C23—44no—
125C23(C2×D211)⋊C414yes—
126C23C2×C22⋊D21111yescharacteristic
127Q8—8442no—
128Q8—4224no—
129D4—24no—
130D4—24no—
131C2×C4—24no—
132C2×C4—84no—
133C2×C4—8442no—
134C2×C4—8441no—
135C2×C4—8441no—
136C2×C4—4222no—
137C2×C4—4222no—
138C2×C4C2×C4×D21111yescharacteristic
139C2×C4C2×C22⋊D21112yes—
140C2×C4C2×D84411yescharacteristic
141C2×C4#1680996212yes—
142C8—8442no—
143C22—22no—
144C22—24no—
145C22—44no—
146C22—44no—
147C22—82no—
148C22C2×(C2×D211)⋊C411yescharacteristic
149C22#2026994611yescharacteristic
150C22#2026996011yescharacteristic
151C4—44no—
152C4—8442no—
153C4—8442no—
154C4C4×C22⋊D21112yes—
155C4#2026995112yes—
156C2—21no—
157C2—21no—
158C2—44no—
159C2#2721302511yescharacteristic
160C2#2721332412yes—
subgroupG/H
centerC22#20269946
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#518×C211C2
socleC22×C211#120
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenter#518D211
hypocenterC211#1885
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #27213154C2semidirect
4. #27212905C2semidirect
9. #20269964C4semidirect
21. C2×Dic422C2×C4semidirect
29. C4×Dic211C2×C4semidirect
80. C211#1885semidirect

Maximal subgroups

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

subgroupindex
C2×#202699642
#272131542
C2×#202698942
#272129052
#518×C2112
#1885211