Finite Group Databasebeta

← #41013490

Subgroups — #41013490

Order 27008.

300 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 101 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×#20269936C211yesmaximal, characteristic
2C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
3#27212908C212yesmaximal, complemented
4#27212908C212yesmaximal, complemented
5#380×C211C211yesmaximal, characteristic
6C22×C211⋊C8C2212yes—
7#20269936—22no—
8#20269936C414yescomplemented
9C2×C4⋊Dic211C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C22⋊Dic211—21no—
12C2×C4×Dic211—21no—
13C22⋊(C211⋊C8)—22no—
14C23×C4×C211C2211yescharacteristic
15C2×C4⋊C4×C211C2212yes—
16C2×C22⋊C4×C211—21no—
17#95×C211—21no—
18C22×Dic211—21no—
19C22×Dic211—21no—
20C2×C211⋊C8—22no—
21C2×C211⋊C8—24no—
22C4⋊Dic211—21no—
23C4⋊Dic211—21no—
24C4⋊Dic211C2×C412yes—
25C4⋊Dic211C2×C412yes—
26C22⋊Dic211—24no—
27C4×Dic211—24no—
28C24×C211D411yescharacteristic
29C22×C4×C211—24no—
30C22×C4×C211—22no—
31C22×C4×C211C2×C412yes—
32C22×C4×C211D411yescharacteristic
33C22×C4×C211D412yes—
34C22×C4×C211C2311yescharacteristic
35C4⋊C4×C211—24no—
36C22⋊C4×C211—42no—
37C2×Dic211—22no—
38C2×Dic211—22no—
39C2×Dic211—42no—
40C2×Dic211—41no—
41C2×Dic211—41no—
42C211⋊C8—42no—
43C23×C211—22no—
44C23×C211—22no—
45C23×C211—21no—
46C23×C211—21no—
47C23×C211C22⋊C412yes—
48C23×C211C2×D411yescharacteristic
49C2×C4×C211—22no—
50C2×C4×C211—24no—
51C2×C4×C211—24no—
52C2×C4×C211—42no—
53C2×C4×C211—42no—
54C2×C4×C211—42no—
55C2×C4×C211C22⋊C412yes—
56C2×C4×C211C22×C411yescharacteristic
57C2×C4×C211C2×D411yescharacteristic
58C2×C4×C211#4114yes—
59C22×C211—22no—
60C22×C211—22no—
61C22×C211—22no—
62C22×C211—42no—
63C22×C211—42no—
64C22×C211C2×C22⋊C411yescharacteristic
65C22×C211#12312yes—
66C22×C211#13512yes—
67C22×C211AGL(1,ℤ/8)11yescharacteristic
68C22×C211#13711yescharacteristic
69C4×C211—22no—
70C4×C211—82no—
71C4×C211#11712yes—
72C4×C211#12312yes—
73Dic211—42no—
74Dic211—42no—
75C2×C211—42no—
76C2×C211#38111yescharacteristic
77C2×C211#41611yescharacteristic
78C2×C211#41711yescharacteristic
79C2×C211#48212yes—
80C2×C211#48512yes—
81C211#182511yescharacteristic, complemented, Sylow
82#1825—2111nomaximal, Sylow
83C2×#117—2111no—
84C2×C22⋊C8—2111no—
85#380D21111yescharacteristic
86#340—2112no—
87#340—2112no—
88C23×C4C2×D21111yescharacteristic
89C22×C8—2112no—
90#117—4222no—
91#117—2114no—
92C2×C4⋊C4—2111no—
93C2×C4⋊C4—2111no—
94C2×C4⋊C4C2×D21112yes—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—4221no—
97C2×C42—4221no—
98C22⋊C8—4222no—
99#95—21no—
100C24C22⋊D21111yescharacteristic
101C22×C4—24no—
102C22×C4—22no—
103C22×C4—4221no—
104C22×C4—4221no—
105C22×C4C4×D21112yes—
106C22×C4C22⋊D21111yescharacteristic
107C22×C4D84412yes—
108C22×C4C22×D21111yescharacteristic
109C2×C8—4222no—
110C2×C8—4224no—
111C4⋊C4—24no—
112C4⋊C4—4221no—
113C4⋊C4—4221no—
114C4⋊C4—2112no—
115C4⋊C4—2112no—
116C22⋊C4—42no—
117C22⋊C4—4224no—
118C42—4224no—
119C23—22no—
120C23—22no—
121C23—21no—
122C23—21no—
123C23(C2×D211)⋊C412yes—
124C23C2×C22⋊D21111yescharacteristic
125C2×C4—22no—
126C2×C4—24no—
127C2×C4—24no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—8442no—
132C2×C4—8441no—
133C2×C4—8441no—
134C2×C4—4222no—
135C2×C4—4222no—
136C2×C4(C2×D211)⋊C412yes—
137C2×C4C2×C4×D21111yescharacteristic
138C2×C4C2×D84411yescharacteristic
139C2×C4#1680996114yes—
140C8—8442no—
141C22—22no—
142C22—22no—
143C22—22no—
144C22—42no—
145C22—42no—
146C22C2×(C2×D211)⋊C411yescharacteristic
147C22#2026995812yes—
148C22#2026999712yes—
149C22#2027000311yescharacteristic
150C22#2027000811yescharacteristic
151C4—22no—
152C4—82no—
153C4—8442no—
154C4—8442no—
155C4#2026993312yes—
156C4#2026996112yes—
157C2—42no—
158C2#2721300711yescharacteristic
159C2#2721314011yescharacteristic
160C2#2721314511yescharacteristic
161C2#2721346712yes—
162C2#2721348412yes—
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#380×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenter#380D211
hypocenterC211#1825
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #27212908C2semidirect
4. #27212908C2semidirect
8. #20269936C4semidirect
81. C211#1825semidirect

Maximal subgroups

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

subgroupindex
C2×#202699362
C2×C22⋊(C211⋊C8)2
#272129082
#380×C2112
#1825211