Finite Group Databasebeta

← #41012480

Subgroups — #41012480

Order 27008.

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

normal subgroups only

This group has 240 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#27213861C211yesmaximal, characteristic, complemented
2#27213518C211yesmaximal, characteristic, complemented
3#27213338C211yesmaximal, characteristic, complemented
4#27212894C211yesmaximal, characteristic, complemented
5D4⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic, complemented
7#459×C211C211yesmaximal, characteristic, complemented
8C2×#16809950—22no—
9C2×#16809947—21no—
10#20269977C2211yescharacteristic
11#20269963—22no—
12#20269951—21no—
13#20269948—21no—
14C4×D844C2211yescharacteristic
15#20269936—21no—
16#20269935—21no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18#20269898—21no—
19C4×C211⋊C8C2211yescharacteristic
20C2×D8×C211—21no—
21#127×C211C2211yescharacteristic
22C4×D4×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24#102×C211—21no—
25C2×Dic422D412yes—
26C2×D844D411yescharacteristic
27C2×C22⋊D211—22no—
28C2×C4×D211—21no—
29#16809950—44no—
30#16809947—42no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211D411yescharacteristic
36C22⋊Dic211—22no—
37(C2×D211)⋊C4—22no—
38(C2×D211)⋊C4—21no—
39C4×Dic211—22no—
40C2×D4×C211—21no—
41C2×D4×C211—21no—
42C2×D4×C211—21no—
43C2×D4×C211D411yescharacteristic
44C22×C4×C211—21no—
45D8×C211—42no—
46C2×C8×C211—21no—
47C4⋊C4×C211D411yescharacteristic
48C22⋊C4×C211—21no—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51Dic422—42no—
52D844—21no—
53D844—21no—
54C22⋊D211—82no—
55C2×Dic211—21no—
56C2×Dic211—22no—
57C4×D211—41no—
58C4×D211—41no—
59C211⋊C8—22no—
60C211⋊C8—41no—
61C23×C211—21no—
62C23×C211—41no—
63D4×C211—21no—
64D4×C211—21no—
65D4×C211—42no—
66D4×C211—41no—
67D4×C211—41no—
68D4×C211—41no—
69D4×C211—41no—
70C2×C4×C211—21no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C8×C211—41no—
77C22×C211—41no—
78C22×C211—41no—
79C22×C211—41no—
80C22×C211—81no—
81C22×C211—81no—
82C22×C211—81no—
83C22×C211#12011yescharacteristic
84C2×D211—41no—
85C2×D211—41no—
86C2×D211—41no—
87C4×C211—21no—
88C4×C211—41no—
89C4×C211#13511yescharacteristic
90C4×C211#13511yescharacteristic
91C4×C211AGL(1,ℤ/8)11yescharacteristic
92C4×C211AGL(1,ℤ/8)11yescharacteristic
93Dic211—41no—
94Dic211—42no—
95C2×C211—81no—
96C2×C211—161no—
97C2×C211#44911yescharacteristic
98C2×C211#44911yescharacteristic
99C2×C211#45311yescharacteristic
100D211—81no—
101C211#162411yescharacteristic, complemented, Sylow
102#1624—2111nomaximal, Sylow
103#555—2111no—
104#492—2111no—
105#459—2111no—
106#459D21111yescharacteristic, complemented
107#333—2111no—
108D4⋊C8—2111no—
109D4⋊C8—2111no—
110C2×SD16—4222no—
111C2×D8—21no—
112C2×D8—4221no—
113C4⋊Q8—2111no—
114#127C2×D21111yescharacteristic
115#124—4222no—
116#121—4221no—
117#121—4221no—
118C4×D4—2111no—
119C4×D4C2×D21111yescharacteristic
120#117—4221no—
121#117—4221no—
122C4⋊C8—2111no—
123C4⋊C8C2×D21111yescharacteristic
124#102—21no—
125#102—4221no—
126C4×C8—2111no—
127C2×Q8—2112no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—4222no—
132C2×D4—2111no—
133C2×D4C22⋊D21111yescharacteristic
134C22×C4—21no—
135C22×C4—4221no—
136SD16—8444no—
137D8—42no—
138D8—8442no—
139C2×C8—21no—
140C2×C8—4221no—
141C2×C8—4221no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4—2111no—
145C4⋊C4C22⋊D21111yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—4222no—
148C22⋊C4—4222no—
149C22⋊C4—4221no—
150C42—4222no—
151C42C22×D21111yescharacteristic
152C23—21no—
153C23—41no—
154C23—4221no—
155Q8—8442no—
156D4—21no—
157D4—21no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—16882no—
164D4—4221no—
165D4—4221no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—4221no—
172C2×C4—4222no—
173C2×C4D4×D21111yescharacteristic
174C2×C4D4×D21111yescharacteristic
175C2×C4C2×C22⋊D21111yescharacteristic
176C8—41no—
177C8—8441no—
178C8—4222no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—8441no—
186C22—8441no—
187C22—8441no—
188C22#2026994511yescharacteristic
189C4—21no—
190C4—41no—
191C4—8442no—
192C4—8441no—
193C4#2026999611yescharacteristic
194C4#2026999711yescharacteristic
195C4#2027000111yescharacteristic
196C4#2027000311yescharacteristic
197C2—81no—
198C2—161no—
199C2—16881no—
200C2#2721328811yescharacteristic
201C2#2721328911yescharacteristic
202C2#2721331211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#459×C211C2
socleC22×C211#120
hypercenter#459D211
hypocenterC211#1624
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #27213861C2semidirect
2. #27213518C2semidirect
3. #27213338C2semidirect
4. #27212894C2semidirect
5. D4⋊(C211⋊C8)C2semidirect
6. #27212873C2semidirect
7. #459×C211C2semidirect
101. C211#1624semidirect
106. #459D211semidirect

Maximal subgroups

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

subgroupindex
#272138612
#272135182
#272133382
#272128942
D4⋊(C211⋊C8)2
#272128732
#459×C2112
#1624211