Finite Group Databasebeta

← #41014505

Subgroups — #41014505

Order 27008.

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

normal subgroups only

This group has 382 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
1C2×#20270003C212yesmaximal, complemented
2C2×#20269894C212yesmaximal, complemented
3#27213112C211yesmaximal, characteristic, complemented
4#27212911C211yesmaximal, characteristic, complemented
5C22×D8×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2211yescharacteristic
7#20270003—24no—
8#20270003—24no—
9C2×#16809950—22no—
10C2×#16809947—22no—
11C2×#16809946C2211yescharacteristic
12C2×#16809946C2212yes—
13#20269894—24no—
14(C2×D211)⋊C8—22no—
15C22×D4×C211C2212yes—
16C2×D8×C211—24no—
17C2×D8×C211—22no—
18C22×C8×C211C2211yescharacteristic
19#16809962—22no—
20#16809962—22no—
21C2×Dic422D411yescharacteristic
22C2×D844D411yescharacteristic
23C2×C22⋊D211—21no—
24C2×C4×D211—21no—
25#16809950—28no—
26#16809947—28no—
27#16809946—22no—
28#16809946—22no—
29#16809946D414yes—
30C2×C211⋊C8—22no—
31C2×C211⋊C8—21no—
32C24×C211—22no—
33C2×D4×C211—22no—
34C2×D4×C211—22no—
35C2×D4×C211—22no—
36C2×D4×C211—22no—
37C2×D4×C211—22no—
38C2×D4×C211D414yes—
39C22×C4×C211C2311yescharacteristic
40D8×C211—28no—
41C2×C8×C211—22no—
42C2×C8×C211D412yes—
43C22×D211—21no—
44Dic422—22no—
45D844—22no—
46C22⋊D211—42no—
47C2×Dic211—21no—
48C4×D211—41no—
49C4×D211—41no—
50C211⋊C8—24no—
51C211⋊C8—41no—
52C23×C211—24no—
53C23×C211—42no—
54C23×C211—42no—
55C23×C211—84no—
56C23×C211#4111yescharacteristic
57D4×C211—24no—
58D4×C211—24no—
59D4×C211—42no—
60D4×C211—42no—
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×D412yes—
63C2×C4×C211C2×D412yes—
64C2×C4×C211C2×D411yescharacteristic
65C8×C211—22no—
66C22×C211—22no—
67C22×C211—44no—
68C22×C211—84no—
69C22×C211—84no—
70C22×C211—84no—
71C22×C211#12111yescharacteristic
72C22×C211#12112yes—
73C2×D211—41no—
74C2×D211—42no—
75C4×C211#12012yes—
76C4×C211#12011yescharacteristic
77C4×C211#12711yescharacteristic
78Dic211—41no—
79C2×C211—22no—
80C2×C211—84no—
81C2×C211#39211yescharacteristic
82C2×C211#47212yes—
83D211—81no—
84C211#201811yescharacteristic, complemented, Sylow
85#2018—2111nomaximal, Sylow
86C2×AGL(1,ℤ/8)—2112no—
87C22×D8D21111yescharacteristic, complemented
88C2×#100—2112no—
89#408—2111no—
90#341—2111no—
91C2×#41—2111no—
92C22×D4C2×D21112yes—
93AGL(1,ℤ/8)—4224no—
94AGL(1,ℤ/8)—4224no—
95C2×SD16—4222no—
96C2×D8—24no—
97C2×D8—22no—
98C2×D8—4222no—
99C2×M16—2111no—
100C2×M16—2112no—
101C22×C8C2×D21111yescharacteristic
102#100—4224no—
103C22⋊C8—4222no—
104C24—22no—
105#41—4222no—
106#41—4222no—
107C2×Q8—2111no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—4221no—
114C2×D4—2111no—
115C2×D4C22⋊D21114yes—
116C22×C4—4221no—
117C22×C4C22×D21111yescharacteristic
118SD16—4228no—
119D8—28no—
120D8—4228no—
121M16—4222no—
122M16—4222no—
123M16—2114no—
124C2×C8—22no—
125C2×C8—4221no—
126C2×C8—4222no—
127C2×C8C22⋊D21112yes—
128C23—24no—
129C23—42no—
130C23—42no—
131C23—84no—
132C23—4221no—
133C23#1680996111yescharacteristic
134Q8—4222no—
135D4—24no—
136D4—24no—
137D4—42no—
138D4—42no—
139D4—8442no—
140D4—4222no—
141C2×C4—8441no—
142C2×C4—8441no—
143C2×C4—4221no—
144C2×C4D4×D21111yescharacteristic
145C2×C4D4×D21112yes—
146C2×C4C2×C22⋊D21112yes—
147C2×C4C2×C22⋊D21111yescharacteristic
148C8—22no—
149C8—8441no—
150C8—4224no—
151C22—22no—
152C22—44no—
153C22—84no—
154C22—84no—
155C22—84no—
156C22—8442no—
157C22—8441no—
158C22#2026994811yescharacteristic
159C22#2026994912yes—
160C4—8441no—
161C4#2026994512yes—
162C4#2026994611yescharacteristic
163C4#2026997311yescharacteristic
164C2—22no—
165C2—84no—
166C2—16881no—
167C2#2721305411yescharacteristic
168C2#2721341312yes—
subgroupG/H
centerC22#20269948
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC22×D8×C211C2
socleC22×C211#121
hypercenterC22×D8D211
hypocenterC211#2018
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#20270003C2semidirect
2. C2×#20269894C2semidirect
3. #27213112C2semidirect
4. #27212911C2semidirect
5. C22×D8×C211C2semidirect
84. C211#2018semidirect
87. C22×D8D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700032
C2×#202698942
#272131122
#272129112
C22×D8×C2112
#2018211