Finite Group Databasebeta

← #41014701

Subgroups — #41014701

Order 27008.

264 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 67 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#27213174C211yesmaximal, characteristic, complemented
2#27213142C212yesmaximal, complemented
3#27213120C212yesmaximal, complemented
4#27212903C211yesmaximal, characteristic, complemented
5#575×C211C211yesmaximal, characteristic
6C22×C211⋊C8C2211yescharacteristic
7#20269936C2212yes—
8#20269936C2211yescharacteristic
9Dic211⋊C8—22no—
10#20269901—22no—
11#20269898—22no—
12#20269892—24no—
13C2×#41×C211C2212yes—
14#137×C211—22no—
15AGL(1,ℤ/8)×C211—22no—
16#135×C211—22no—
17C2×M16×C211C2211yescharacteristic
18C2×C211⋊C8—22no—
19C2×C211⋊C8—21no—
20C4⋊Dic211—22no—
21C4⋊Dic211—21no—
22C22⋊Dic211—22no—
23C22⋊Dic211D412yes—
24C4×Dic211—21no—
25C4×Dic211—22no—
26#41×C211—22no—
27#41×C211—22no—
28#41×C211—22no—
29#41×C211—22no—
30C2×Q8×C211D412yes—
31C2×D4×C211—22no—
32C2×D4×C211D412yes—
33C22×C4×C211—22no—
34C22×C4×C211C2311yescharacteristic
35Q16×C211—22no—
36SD16×C211—24no—
37D8×C211—22no—
38M16×C211—22no—
39C2×C8×C211D412yes—
40C2×Dic211—22no—
41C2×Dic211—42no—
42C211⋊C8—41no—
43C23×C211—22no—
44C23×C211C2×D411yescharacteristic
45Q8×C211—22no—
46Q8×C211—22no—
47D4×C211—22no—
48D4×C211—22no—
49D4×C211—42no—
50D4×C211—42no—
51C2×C4×C211—22no—
52C2×C4×C211—42no—
53C2×C4×C211—42no—
54C2×C4×C211C2×D412yes—
55C2×C4×C211C2×D411yescharacteristic
56C2×C4×C211#4111yescharacteristic
57C2×C4×C211#4112yes—
58C8×C211—22no—
59C22×C211—41no—
60C22×C211—42no—
61C22×C211—82no—
62C22×C211#12111yescharacteristic
63C22×C211#12112yes—
64C4×C211—42no—
65C4×C211#12011yescharacteristic
66C4×C211#12312yes—
67C4×C211#12411yescharacteristic
68Dic211—42no—
69Dic211—82no—
70C2×C211—21no—
71C2×C211—22no—
72C2×C211—82no—
73C2×C211#39411yescharacteristic
74C211#205311yescharacteristic, complemented, Sylow
75#2053—2111nomaximal, Sylow
76#575D21111yescharacteristic
77#424—2111no—
78#416—2112no—
79#410—2112no—
80H(3,ℤ/4)—2111no—
81C2×#41C2×D21112yes—
82#137—22no—
83AGL(1,ℤ/8)—22no—
84#135—22no—
85C2×M16C2×D21111yescharacteristic
86C22×C8—2111no—
87#117—2112no—
88#117—2111no—
89C4⋊C8—4222no—
90#103—4222no—
91#102—4222no—
92#99—4224no—
93#41—22no—
94#41—22no—
95#41—22no—
96#41—22no—
97C2×Q8C22⋊D21112yes—
98C2×D4—22no—
99C2×D4C22⋊D21112yes—
100C22×C4—22no—
101C22×C4C22×D21111yescharacteristic
102Q16—22no—
103SD16—24no—
104D8—22no—
105M16—22no—
106C2×C8—4222no—
107C2×C8—4221no—
108C2×C8C22⋊D21112yes—
109C4⋊C4—4222no—
110C4⋊C4—4221no—
111C22⋊C4—4222no—
112C22⋊C4—2112no—
113C42—4222no—
114C42—4221no—
115C23—22no—
116C23D4×D21111yescharacteristic
117Q8—22no—
118Q8—22no—
119D4—22no—
120D4—22no—
121D4—42no—
122D4—42no—
123C2×C4—22no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—8442no—
127C2×C4—4222no—
128C2×C4C2×C22⋊D21112yes—
129C2×C4C2×C22⋊D21111yescharacteristic
130C2×C4#1680996111yescharacteristic
131C2×C4#1680996112yes—
132C8—22no—
133C8—8441no—
134C22—41no—
135C22—42no—
136C22—82no—
137C22#2026994811yescharacteristic
138C22#2026994912yes—
139C4—42no—
140C4—8442no—
141C4—16882no—
142C4#2026994611yescharacteristic
143C4#2026995812yes—
144C4#2026996411yescharacteristic
145C2—21no—
146C2—22no—
147C2—82no—
148C2#2721306011yescharacteristic
subgroupG/H
centerC4#20269964
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#575×C211C2
socleC2×C211#394
hypercenter#575D211
hypocenterC211#2053
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #27213174C2semidirect
2. #27213142C2semidirect
3. #27213120C2semidirect
4. #27212903C2semidirect
74. C211#2053semidirect

Maximal subgroups

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

subgroupindex
#272131742
#272131422
#272131202
#272129032
#575×C2112
#2053211