Finite Group Databasebeta

← #41012895

Subgroups — #41012895

Order 27008.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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#27213542C211yesmaximal, characteristic, complemented
2#27213436C211yesmaximal, characteristic
3#27213224C211yesmaximal, characteristic
4#27212898C211yesmaximal, characteristic, complemented
5#27212894C211yesmaximal, characteristic, complemented
6#27212892C211yesmaximal, characteristic, complemented
7#461×C211C211yesmaximal, characteristic
8C2×#16809946—21no—
9#20269977C2211yescharacteristic
10#20269908—21no—
11#20269908—21no—
12#20269908—21no—
13C4⋊(C211⋊C8)C2211yescharacteristic
14C4⋊(C211⋊C8)C2211yescharacteristic
15#20269898—21no—
16C22⋊(C211⋊C8)—21no—
17C4×C211⋊C8C2211yescharacteristic
18C2×SD16×C211—21no—
19C4⋊Q8×C211C2211yescharacteristic
20C4×D4×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22#103×C211—21no—
23C2×Dic422—21no—
24#16809946—42no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8D411yescharacteristic
28C2×C211⋊C8D411yescharacteristic
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C2×Q8×C211—21no—
32C2×D4×C211D411yescharacteristic
33C22×C4×C211—21no—
34SD16×C211—42no—
35C2×C8×C211—21no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211D411yescharacteristic
39C22⋊C4×C211—21no—
40C42×C211C2311yescharacteristic
41Dic422—81no—
42C2×Dic211—41no—
43C211⋊C8—21no—
44C211⋊C8—41no—
45C211⋊C8—41no—
46C211⋊C8SD1612yescomplemented
47C23×C211—21no—
48Q8×C211—41no—
49Q8×C211—41no—
50D4×C211—21no—
51D4×C211—21no—
52C2×C4×C211—21no—
53C2×C4×C211—41no—
54C2×C4×C211—41no—
55C2×C4×C211—41no—
56C2×C4×C211C2×D411yescharacteristic
57C2×C4×C211C2×D411yescharacteristic
58C2×C4×C211#4111yescharacteristic
59C8×C211—41no—
60C22×C211—41no—
61C22×C211—41no—
62C22×C211—41no—
63C22×C211#12111yescharacteristic
64C4×C211—21no—
65C4×C211—41no—
66C4×C211—81no—
67C4×C211C2×SD1611yescharacteristic
68C4×C211AGL(1,ℤ/8)11yescharacteristic
69C4×C211#13711yescharacteristic
70C4×C211#13711yescharacteristic
71Dic211—81no—
72C2×C211—81no—
73C2×C211#46111yescharacteristic
74C2×C211#46811yescharacteristic
75C2×C211#47111yescharacteristic
76C211#169111yescharacteristic, complemented, Sylow
77#1691—2111nomaximal, Sylow
78#498—2111no—
79#476—2111no—
80#461D21111yescharacteristic
81#436—2111no—
82#335—2111no—
83#333—2111no—
84#332—2111no—
85C2×SD16—21no—
86C2×M16—4221no—
87C4⋊Q8—2111no—
88C4⋊Q8C2×D21111yescharacteristic
89C4×D4C2×D21111yescharacteristic
90#106—4221no—
91#106—4221no—
92#106—4221no—
93C4⋊C8—2111no—
94C4⋊C8—2111no—
95C4⋊C8C2×D21111yescharacteristic
96#103—21no—
97#102—4221no—
98C22⋊C8—4221no—
99C4×C8—2111no—
100C2×Q8—21no—
101C2×Q8—4221no—
102C2×D4C22⋊D21111yescharacteristic
103C22×C4—21no—
104SD16—42no—
105M16—8442no—
106C2×C8—21no—
107C2×C8—4221no—
108C2×C8—4221no—
109C2×C8—2111no—
110C2×C8—2111no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—4221no—
114C4⋊C4—4221no—
115C4⋊C4C22⋊D21111yescharacteristic
116C22⋊C4—21no—
117C42C22×D21111yescharacteristic
118C23—21no—
119Q8—41no—
120Q8—41no—
121Q8—16881no—
122D4—21no—
123D4—21no—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—8441no—
129C2×C4D4×D21111yescharacteristic
130C2×C4C2×C22⋊D21111yescharacteristic
131C2×C4#1680996111yescharacteristic
132C8—41no—
133C8—8441no—
134C8—8441no—
135C8—4221no—
136C8—2112no—
137C22—41no—
138C22—41no—
139C22—41no—
140C22#2026994911yescharacteristic
141C4—21no—
142C4—41no—
143C4—81no—
144C4—16881no—
145C4SD16×D21111yescharacteristic
146C4#2027000311yescharacteristic
147C4#2027000611yescharacteristic
148C4#2027000711yescharacteristic
149C2—81no—
150C2#2721334911yescharacteristic
151C2#2721339011yescharacteristic
152C2#2721341111yescharacteristic
subgroupG/H
centerC22#20269949
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#461×C211C2
socleC22×C211#121
prime radicalC2×D4×C211D4
hypercenter#461D211
hypocenterC211#1691
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #27213542C2semidirect
4. #27212898C2semidirect
5. #27212894C2semidirect
6. #27212892C2semidirect
46. C211⋊C8SD16semidirect
76. C211#1691semidirect

Maximal subgroups

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

subgroupindex
#272135422
#272134362
#272132242
#272128982
#272128942
#272128922
#461×C2112
#1691211