Finite Group Databasebeta

← #41014000

Subgroups — #41014000

Order 27008.

186 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 75 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#27213182C211yesmaximal, characteristic
2#27213173C211yesmaximal, characteristic
3#27213104C211yesmaximal, characteristic
4#27212910C211yesmaximal, characteristic
5#27212906C211yesmaximal, characteristic
6#27212901C211yesmaximal, characteristic
7#339×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11#20269936C2211yescharacteristic
12#20269910C411yescharacteristic, complemented
13#20269910C411yescharacteristic, complemented
14#20269908C411yescharacteristic, complemented
15#20269908C411yescharacteristic, complemented
16C4⋊(C211⋊C8)—21no—
17C4⋊(C211⋊C8)—21no—
18#20269887—21no—
19C8×Dic211—21no—
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22C2×C42×C211C2211yescharacteristic
23#16809946—21no—
24#16809946—21no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8C2×C411yescharacteristic
31C2×C211⋊C8C2×C411yescharacteristic
32C2×C211⋊C8D411yescharacteristic
33C2×C211⋊C8D411yescharacteristic
34C4⋊Dic211C2×C411yescharacteristic
35C4⋊Dic211C2×C411yescharacteristic
36C22⋊Dic211—21no—
37C4×Dic211—21no—
38C22×C4×C211—21no—
39C22×C4×C211C2311yescharacteristic
40M16×C211—21no—
41M16×C211Q812yes—
42C2×C8×C211—21no—
43C4⋊C4×C211C2×C411yescharacteristic
44C4⋊C4×C211C2×C411yescharacteristic
45C22⋊C4×C211—21no—
46C42×C211—21no—
47C42×C211—21no—
48C42×C211—21no—
49C2×Dic211—21no—
50C2×Dic211—21no—
51C211⋊C8—21no—
52C211⋊C8—21no—
53C211⋊C8—21no—
54C211⋊C8—21no—
55C211⋊C8—41no—
56C23×C211#4111yescharacteristic
57C2×C4×C211—22no—
58C2×C4×C211—21no—
59C2×C4×C211—21no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211C22×C411yescharacteristic
63C2×C4×C211C2×D411yescharacteristic
64C2×C4×C211C2×Q811yescharacteristic
65C2×C4×C211#4111yescharacteristic
66C2×C4×C211#4111yescharacteristic
67C2×C4×C211#4111yescharacteristic
68C8×C211—22no—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211C4×D411yescharacteristic
72C22×C211#12511yescharacteristic
73C22×C211#12611yescharacteristic
74C4×C211—42no—
75C4×C211—41no—
76C4×C211—41no—
77C4×C211#11711yescharacteristic
78C4×C211C4×Q811yescharacteristic
79C4×C211C22⋊Q811yescharacteristic
80C4×C211#12311yescharacteristic
81Dic211—41no—
82Dic211—41no—
83C2×C211—21no—
84C2×C211—21no—
85C2×C211#38711yescharacteristic
86C2×C211#44311yescharacteristic
87C2×C211#44411yescharacteristic
88C211#192211yescharacteristic, complemented, Sylow
89#1922—2111nomaximal, Sylow
90#427—2111no—
91#424—2111no—
92#405—2111no—
93#341—2111no—
94#339—2111no—
95#339D21111yescharacteristic
96#336—2111no—
97C2×M16—2111no—
98C2×M16C2×D21111yescharacteristic
99C22×C8—2111no—
100C22×C8—2111no—
101#117—2111no—
102#117C2×D21111yescharacteristic
103C2×C42C2×D21111yescharacteristic
104#107—2111no—
105#107—2111no—
106#106—2111no—
107#106—2111no—
108C4⋊C8—4221no—
109C4⋊C8—4221no—
110#97—4221no—
111C4×C8—4221no—
112C22×C4—21no—
113C22×C4C22×D21111yescharacteristic
114M16—21no—
115M16—4221no—
116M16—4221no—
117M16Dic42212yes—
118C2×C8—21no—
119C2×C8—4221no—
120C2×C8—4221no—
121C2×C8—4221no—
122C2×C8—4221no—
123C2×C8—4221no—
124C2×C8—2111no—
125C2×C8—2111no—
126C2×C8—2111no—
127C2×C8—2111no—
128C4⋊C4—2111no—
129C4⋊C4—2111no—
130C4⋊C4C4×D21111yescharacteristic
131C4⋊C4C4×D21111yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—4221no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42—4221no—
138C23#1680996111yescharacteristic
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—4221no—
145C2×C4—4221no—
146C2×C4C2×C4×D21111yescharacteristic
147C2×C4D4×D21111yescharacteristic
148C2×C4C2×Dic42211yescharacteristic
149C2×C4#1680996111yescharacteristic
150C2×C4#1680996211yescharacteristic
151C2×C4#1680996311yescharacteristic
152C8—22no—
153C8—8441no—
154C8—4221no—
155C8—4221no—
156C8—4221no—
157C8—4221no—
158C22—21no—
159C22—21no—
160C22(C22⋊D211)⋊C411yescharacteristic
161C22#2026996811yescharacteristic
162C22#2026997011yescharacteristic
163C4—42no—
164C4—41no—
165C4—41no—
166C4—8441no—
167C4—8441no—
168C4#2026993511yescharacteristic
169C4C4×Dic42211yescharacteristic
170C4#2026995411yescharacteristic
171C4#2026996211yescharacteristic
172C2—21no—
173C2—21no—
174C2#2721303311yescharacteristic
175C2#2721326011yescharacteristic
176C2#2721326511yescharacteristic
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211C4×D4
prime radicalC23×C211#41
cosocleC22×C4×C211C23
hypercenter#339D211
hypocenterC211#1922
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
12. #20269910C4semidirect
13. #20269910C4semidirect
14. #20269908C4semidirect
15. #20269908C4semidirect
88. C211#1922semidirect

Maximal subgroups

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

subgroupindex
#272131822
#272131732
#272131042
#272129102
#272129062
#272129012
#339×C2112
#1922211