Finite Group Databasebeta

← #41013801

Subgroups — #41013801

Order 27008.

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

normal subgroups only

This group has 344 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×#20269966C211yesmaximal, characteristic
2#27213150C212yesmaximal
3C2×#20269896C211yesmaximal, characteristic, complemented
4#27212905C212yesmaximal, complemented
5#519×C211C211yesmaximal, characteristic, complemented
6C22×D844C2211yescharacteristic
7C2×#16809946C2212yes—
8#20269966—22no—
9#20269966C414yescomplemented
10C2×(C2×D211)⋊C4—22no—
11C2×C4×Dic211C2211yescharacteristic
12#20269897—24no—
13#20269896—22no—
14C22×Q8×C211C2211yescharacteristic
15C4×Q8×C211—24no—
16#117×C211—22no—
17#117×C211C2212yes—
18C23×D211—21no—
19C2×D844—22no—
20C2×D844—21no—
21C2×D844—22no—
22C2×D844C2×C412yes—
23C22×Dic211—21no—
24#16809946—22no—
25#16809946—22no—
26C2×C211⋊C8—22no—
27(C2×D211)⋊C4—28no—
28C4×Dic211—21no—
29C4×Dic211C2×C412yes—
30C2×Q8×C211—22no—
31C2×Q8×C211—22no—
32C2×Q8×C211C2×C412yes—
33C2×Q8×C211D412yes—
34C22×C4×C211D412yes—
35C22×C4×C211C2311yescharacteristic
36C4⋊C4×C211—24no—
37C4⋊C4×C211D414yes—
38C22⋊C4×C211—22no—
39C42×C211—24no—
40C42×C211—22no—
41C22×D211—22no—
42C22×D211—44no—
43C22×D211—41no—
44C22×D211—41no—
45D844—24no—
46D844—42no—
47C2×Dic211—22no—
48C2×Dic211—42no—
49C211⋊C8—42no—
50C23×C211C2×D411yescharacteristic
51Q8×C211—24no—
52Q8×C211—24no—
53C2×C4×C211—22no—
54C2×C4×C211—22no—
55C2×C4×C211—24no—
56C2×C4×C211C22⋊C414yes—
57C2×C4×C211C22×C411yescharacteristic
58C2×C4×C211C2×D412yes—
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211#4112yes—
61C22×C211—22no—
62C22×C211C2×C22⋊C411yescharacteristic
63C22×C211#12011yescharacteristic
64C22×C211#12311yescharacteristic
65C2×D211—44no—
66C2×D211—42no—
67C2×D211—82no—
68C2×D211—82no—
69C4×C211—24no—
70C4×C211—44no—
71C4×C211C4×D412yes—
72C4×C211#12112yes—
73Dic211—42no—
74C2×C211—21no—
75C2×C211—21no—
76C2×C211#38611yescharacteristic
77C2×C211#45412yes—
78D211—82no—
79C211#188811yescharacteristic, complemented, Sylow
80#1888—2111nomaximal, Sylow
81C2×#124—2111no—
82#519D21111yescharacteristic, complemented
83#418—2112no—
84C2×#101—2111no—
85#339—2112no—
86C22×Q8C2×D21111yescharacteristic
87C22×D4—2111no—
88C2×M16—2112no—
89#124—4222no—
90#124—2114no—
91C4×Q8—24no—
92#117—22no—
93#117C2×D21112yes—
94C2×C22⋊C4—4222no—
95C2×C42—2111no—
96#102—4224no—
97#101—4222no—
98C24—4221no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8C4×D21112yes—
102C2×Q8D84412yes—
103C2×D4—4222no—
104C2×D4—4221no—
105C2×D4—4222no—
106C2×D4—2112no—
107C22×C4—4221no—
108C22×C4C22⋊D21112yes—
109C22×C4C22×D21111yescharacteristic
110M16—4222no—
111M16—4222no—
112C2×C8—4222no—
113C4⋊C4—24no—
114C4⋊C4C22⋊D21114yes—
115C22⋊C4—22no—
116C22⋊C4—4228no—
117C42—24no—
118C42—22no—
119C42—4221no—
120C42—2112no—
121C23—8444no—
122C23—8441no—
123C23—8441no—
124C23—4222no—
125C23C2×C22⋊D21111yescharacteristic
126Q8—24no—
127Q8—24no—
128D4—8442no—
129D4—4224no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—24no—
133C2×C4—8442no—
134C2×C4—4222no—
135C2×C4(C2×D211)⋊C414yes—
136C2×C4C2×C4×D21111yescharacteristic
137C2×C4C2×C22⋊D21112yes—
138C2×C4C2×D84411yescharacteristic
139C2×C4#1680996212yes—
140C8—8442no—
141C22—22no—
142C22—8444no—
143C22—8442no—
144C22—16882no—
145C22—16882no—
146C22C2×(C2×D211)⋊C411yescharacteristic
147C22#2026994611yescharacteristic
148C22#2026996011yescharacteristic
149C4—24no—
150C4—44no—
151C4—8442no—
152C4C4×C22⋊D21112yes—
153C4#2026995112yes—
154C2—21no—
155C2—21no—
156C2—16882no—
157C2#2721302511yescharacteristic
158C2#2721331912yes—
subgroupG/H
centerC22#20269946
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#519×C211C2
socleC22×C211#120
prime radicalC22×D844C22
cosocleC22×C4×C211C23
hypercenter#519D211
hypocenterC211#1888
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #519

Complements

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

NH ≅ G/Ntype
3. C2×#20269896C2semidirect
4. #27212905C2semidirect
5. #519×C211C2semidirect
9. #20269966C4semidirect
79. C211#1888semidirect
82. #519D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699662
#272131502
C2×#202698962
#272129052
#519×C2112
#1888211