Finite Group Databasebeta

← #41014858

Subgroups — #41014858

Order 27008.

204 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 59 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#27213194C211yesmaximal, characteristic, complemented
2#27213151C211yesmaximal, characteristic
3#27213151C211yesmaximal, characteristic
4#27212918C211yesmaximal, characteristic, complemented
5#27212917C211yesmaximal, characteristic, complemented
6#27212903C211yesmaximal, characteristic, complemented
7C2×#100×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269936C2211yescharacteristic
11#20269936C2211yescharacteristic
12#20269912—22no—
13#20269898—22no—
14#20269898—22no—
15C22×D4×C211C2211yescharacteristic
16C2×M16×C211C2211yescharacteristic
17C2×M16×C211C2211yescharacteristic
18#100×C211—22no—
19#16809946—21no—
20#16809946—21no—
21#16809946—21no—
22#16809946—21no—
23C2×C211⋊C8—21no—
24C2×C211⋊C8D412yes—
25C4⋊Dic211—21no—
26C4⋊Dic211—21no—
27C22⋊Dic211—21no—
28C22⋊Dic211—21no—
29C4×Dic211—21no—
30C4×Dic211—21no—
31C24×C211—21no—
32C2×D4×C211—21no—
33C2×D4×C211—21no—
34C2×D4×C211—21no—
35C2×D4×C211—21no—
36C2×D4×C211—21no—
37C2×D4×C211D412yes—
38C22×C4×C211C2311yescharacteristic
39M16×C211—21no—
40M16×C211—21no—
41M16×C211—21no—
42M16×C211—21no—
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C2×Dic211—41no—
46C2×Dic211—41no—
47C211⋊C8—22no—
48C211⋊C8—41no—
49C23×C211—22no—
50C23×C211—41no—
51C23×C211—41no—
52C23×C211—82no—
53C23×C211#4111yescharacteristic
54D4×C211—22no—
55D4×C211—22no—
56D4×C211—41no—
57D4×C211—41no—
58C2×C4×C211C2×D411yescharacteristic
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211#4111yescharacteristic
61C2×C4×C211#4111yescharacteristic
62C2×C4×C211#4111yescharacteristic
63C2×C4×C211#4111yescharacteristic
64C8×C211—41no—
65C8×C211—41no—
66C22×C211—41no—
67C22×C211—42no—
68C22×C211—82no—
69C22×C211—82no—
70C22×C211—82no—
71C22×C211#12311yescharacteristic
72C22×C211#12411yescharacteristic
73C22×C211#12611yescharacteristic
74C4×C211#12111yescharacteristic
75C4×C211#12311yescharacteristic
76C4×C211#12311yescharacteristic
77C4×C211#12611yescharacteristic
78Dic211—81no—
79Dic211—81no—
80C2×C211—21no—
81C2×C211—21no—
82C2×C211—21no—
83C2×C211—82no—
84C2×C211#39711yescharacteristic
85C211#208111yescharacteristic, complemented, Sylow
86#2081—2111nomaximal, Sylow
87#430—2111no—
88#418—2111no—
89#418—2111no—
90C2×#100D21111yescharacteristic
91#344—2111no—
92#344—2111no—
93H(3,ℤ/4)—2111no—
94C22×D4C2×D21111yescharacteristic
95C2×M16—2111no—
96C2×M16—2111no—
97C2×M16C2×D21111yescharacteristic
98C2×M16C2×D21111yescharacteristic
99#117—2111no—
100#117—2111no—
101#108—4222no—
102#102—4222no—
103#102—4222no—
104#100—22no—
105C24—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4C22⋊D21112yes—
112C22×C4C22×D21111yescharacteristic
113M16—21no—
114M16—21no—
115M16—21no—
116M16—21no—
117M16—4221no—
118M16—4221no—
119M16—4221no—
120M16—4221no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—4221no—
124C2×C8—2112no—
125C4⋊C4—4221no—
126C4⋊C4—4221no—
127C22⋊C4—4221no—
128C22⋊C4—4221no—
129C42—4221no—
130C42—4221no—
131C23—22no—
132C23—41no—
133C23—41no—
134C23—82no—
135C23#1680996211yescharacteristic
136D4—22no—
137D4—22no—
138D4—41no—
139D4—41no—
140C2×C4—8441no—
141C2×C4—8441no—
142C2×C4D4×D21111yescharacteristic
143C2×C4C2×C22⋊D21111yescharacteristic
144C2×C4#1680996111yescharacteristic
145C2×C4#1680996111yescharacteristic
146C2×C4#1680996111yescharacteristic
147C2×C4#1680996211yescharacteristic
148C8—41no—
149C8—41no—
150C8—8441no—
151C8—4222no—
152C22—41no—
153C22—42no—
154C22—82no—
155C22—82no—
156C22—82no—
157C22#2026996011yescharacteristic
158C22#2026996311yescharacteristic
159C22#2026997011yescharacteristic
160C4—16881no—
161C4—16881no—
162C4#2026994911yescharacteristic
163C4#2026995811yescharacteristic
164C4#2026995911yescharacteristic
165C4#2026997011yescharacteristic
166C2—21no—
167C2—21no—
168C2—21no—
169C2—82no—
170C2#2721307411yescharacteristic
subgroupG/H
centerC2#27213074
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#100×C211C2
socleC2×C211#397
prime radicalC22×D4×C211C22
hypercenterC2×#100D211
hypocenterC211#2081
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. #27213194C2semidirect
4. #27212918C2semidirect
5. #27212917C2semidirect
6. #27212903C2semidirect
85. C211#2081semidirect

Maximal subgroups

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

subgroupindex
#272131942
#272131512
#272129182
#272129172
#272129032
C2×#100×C2112
#2081211