Finite Group Databasebeta

← #41012599

Subgroups — #41012599

Order 27008.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#27213816C211yesmaximal, characteristic
2#27213492C211yesmaximal, characteristic, complemented
3#27213432C211yesmaximal, characteristic, complemented
4#27213352C211yesmaximal, characteristic, complemented
5#27212876C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic, complemented
7#331×C211C211yesmaximal, characteristic, complemented
8C2×#16809961—21no—
9C2×#16809951—21no—
10#20269977C2211yescharacteristic
11#20269973—21no—
12#20269949—22no—
13#20269948—21no—
14C4×Dic422C2211yescharacteristic
15C4×D844C2211yescharacteristic
16D4×Dic211—21no—
17#20269909—21no—
18#20269902—21no—
19#20269902—21no—
20#20269898—22no—
21C4×C211⋊C8C2211yescharacteristic
22#127×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4⋊C8×C211C2211yescharacteristic
25#16809961—42no—
26#16809961—42no—
27C2×Dic422—21no—
28C2×Dic422D411yescharacteristic
29C2×D844D411yescharacteristic
30C2×C22⋊D211—22no—
31C2×C22⋊D211—22no—
32C22×Dic211—22no—
33C2×C4×D211—21no—
34#16809951—42no—
35C2×C211⋊C8—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38C4⋊Dic211D411yescharacteristic
39C4⋊Dic211D411yescharacteristic
40Dic211⋊C4—21no—
41C22⋊Dic211—22no—
42(C2×D211)⋊C4—21no—
43C4×Dic211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211D412yes—
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51Dic422—21no—
52Dic422—41no—
53Dic422—41no—
54Dic422SD1612yescomplemented
55D844—21no—
56D844—21no—
57C22⋊D211—44no—
58C22⋊D211—82no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—44no—
63C2×Dic211—41no—
64C4×D211—41no—
65C4×D211—41no—
66C211⋊C8—41no—
67C23×C211—22no—
68D4×C211—42no—
69D4×C211—41no—
70D4×C211—41no—
71D4×C211—41no—
72D4×C211—41no—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C8×C211—41no—
77C8×C211—41no—
78C22×C211—42no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211#12011yescharacteristic
82C2×D211—41no—
83C2×D211—41no—
84C2×D211—41no—
85C4×C211—21no—
86C4×C211C2×SD1611yescharacteristic
87C4×C211#13511yescharacteristic
88C4×C211AGL(1,ℤ/8)11yescharacteristic
89C4×C211AGL(1,ℤ/8)11yescharacteristic
90Dic211—22no—
91Dic211—41no—
92Dic211—41no—
93Dic211—81no—
94C2×C211—82no—
95C2×C211#44911yescharacteristic
96C2×C211#45011yescharacteristic
97C2×C211#45511yescharacteristic
98D211—81no—
99C211#164211yescharacteristic, complemented, Sylow
100#1642—2111nomaximal, Sylow
101#550—2111no—
102#486—2111no—
103#475—2111no—
104#461—2111no—
105#331D21111yescharacteristic, complemented
106Q8⋊C8—2111no—
107D4⋊C8—2111no—
108C2×#41—4221no—
109C2×SD16—4221no—
110C4⋊Q8—2111no—
111#127—4221no—
112#127C2×D21111yescharacteristic
113#121—4222no—
114#121—4221no—
115C4×Q8—2111no—
116C4×D4—4221no—
117C4×D4—2111no—
118#106—4221no—
119C4⋊C8C2×D21111yescharacteristic
120C4⋊C8C2×D21111yescharacteristic
121#103—4221no—
122#103—4221no—
123#102—4222no—
124C4×C8—2111no—
125#41—8442no—
126#41—8442no—
127C2×Q8—4221no—
128C2×Q8—2111no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—4222no—
132C2×D4—4222no—
133C2×D4—2111no—
134C2×D4D84412yes—
135C22×C4—4222no—
136C22×C4—4221no—
137SD16—8442no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—4221no—
141C4⋊C4—4221no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4—2111no—
145C4⋊C4—2111no—
146C22⋊C4—4222no—
147C22⋊C4—4221no—
148C42—4221no—
149C42C22×D21111yescharacteristic
150C23—22no—
151C23—4221no—
152Q8—8441no—
153Q8—8441no—
154Q8—4221no—
155Q8—2112no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—8444no—
162D4—16882no—
163D4—4221no—
164D4—4221no—
165C2×C4—8444no—
166C2×C4—8441no—
167C2×C4—8441no—
168C2×C4—8441no—
169C2×C4—4221no—
170C2×C4—4221no—
171C2×C4—4221no—
172C2×C4D4×D21111yescharacteristic
173C2×C4D4×D21111yescharacteristic
174C2×C4C2×D84411yescharacteristic
175C8—41no—
176C8—41no—
177C8—8441no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—8441no—
182C22—8441no—
183C22—8441no—
184C22#2026994711yescharacteristic
185C4—21no—
186C4—8441no—
187C4—8441no—
188C4—16881no—
189C4—4222no—
190C4SD16×D21111yescharacteristic
191C4#2026999811yescharacteristic
192C4#2027000111yescharacteristic
193C4#2027000111yescharacteristic
194C2—82no—
195C2—16881no—
196C2#2721329211yescharacteristic
197C2#2721329811yescharacteristic
198C2#2721332611yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#331×C211C2
socleC22×C211#120
prime radical#27213816C2
hypercenter#331D211
hypocenterC211#1642
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27213492C2semidirect
3. #27213432C2semidirect
4. #27213352C2semidirect
5. #27212876C2semidirect
6. #27212873C2semidirect
7. #331×C211C2semidirect
54. Dic422SD16semidirect
99. C211#1642semidirect
105. #331D211semidirect

Maximal subgroups

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

subgroupindex
#272138162
#272134922
#272134322
#272133522
#272128762
#272128732
#331×C2112
#1642211