Finite Group Databasebeta

← #41012598

Subgroups — #41012598

Order 27008.

250 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 61 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#27213492C211yesmaximal, characteristic, complemented
2#27213431C211yesmaximal, characteristic, complemented
3#27213351C211yesmaximal, characteristic, complemented
4#27212890C211yesmaximal, characteristic, complemented
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
7#550×C211C211yesmaximal, characteristic
8C2×#16809950—21no—
9#20269977C2211yescharacteristic
10#20269908—21no—
11C4⋊(C211⋊C8)C2211yescharacteristic
12C4⋊(C211⋊C8)C2211yescharacteristic
13#20269901—21no—
14#20269900—21no—
15#20269898—22no—
16C4×C211⋊C8C2211yescharacteristic
17C2×#41×C211—21no—
18#127×C211—21no—
19#127×C211C2211yescharacteristic
20#121×C211—22no—
21#121×C211—21no—
22C4×Q8×C211C2211yescharacteristic
23C4×D4×C211—21no—
24C4×D4×C211C2211yescharacteristic
25C2×Dic422—21no—
26#16809950—42no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32#41×C211—42no—
33#41×C211—42no—
34C2×Q8×C211D411yescharacteristic
35C2×D4×C211—22no—
36C2×D4×C211—22no—
37C2×D4×C211—21no—
38C2×D4×C211—21no—
39C2×D4×C211D411yescharacteristic
40C2×D4×C211D412yes—
41C22×C4×C211—22no—
42C22×C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211D411yescharacteristic
45C4⋊C4×C211D411yescharacteristic
46C22⋊C4×C211—22no—
47C22⋊C4×C211—21no—
48C42×C211—21no—
49C42×C211C2311yescharacteristic
50Dic422—41no—
51Dic422—41no—
52C2×Dic211—41no—
53C211⋊C8—41no—
54C211⋊C8—41no—
55C211⋊C8—41no—
56C23×C211—21no—
57C23×C211—22no—
58Q8×C211—21no—
59Q8×C211SD1612yescomplemented
60D4×C211—21no—
61D4×C211—21no—
62D4×C211—42no—
63D4×C211—44no—
64D4×C211—41no—
65D4×C211—41no—
66D4×C211—41no—
67D4×C211—41no—
68D4×C211—82no—
69C2×C4×C211—21no—
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211—44no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C22×C211—41no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211—41no—
82C22×C211—42no—
83C22×C211—41no—
84C22×C211#12011yescharacteristic
85C4×C211—22no—
86C4×C211—21no—
87C4×C211—41no—
88C4×C211—41no—
89C4×C211C2×SD1611yescharacteristic
90C4×C211#13511yescharacteristic
91C4×C211AGL(1,ℤ/8)11yescharacteristic
92C4×C211AGL(1,ℤ/8)11yescharacteristic
93Dic211—81no—
94C2×C211—81no—
95C2×C211—82no—
96C2×C211#44911yescharacteristic
97C2×C211#45011yescharacteristic
98C2×C211#45511yescharacteristic
99C211#164211yescharacteristic, complemented, Sylow
100#1642—2111nomaximal, Sylow
101#550D21111yescharacteristic
102#486—2111no—
103#475—2111no—
104#461—2111no—
105#331—2111no—
106Q8⋊C8—2111no—
107D4⋊C8—2111no—
108C2×#41—21no—
109C2×SD16—4221no—
110C4⋊Q8—2111no—
111#127—21no—
112#127C2×D21111yescharacteristic
113#121—22no—
114#121—21no—
115C4×Q8C2×D21111yescharacteristic
116C4×D4—21no—
117C4×D4C2×D21111yescharacteristic
118#106—4221no—
119C4⋊C8—2111no—
120C4⋊C8—2111no—
121#103—4221no—
122#103—4221no—
123#102—4222no—
124C4×C8—2111no—
125#41—42no—
126#41—42no—
127C2×Q8—4221no—
128C2×Q8C22⋊D21111yescharacteristic
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4C22⋊D21111yescharacteristic
134C2×D4C22⋊D21112yes—
135C22×C4—22no—
136C22×C4—21no—
137SD16—8442no—
138C2×C8—4221no—
139C2×C8—4221no—
140C2×C8—4221no—
141C4⋊C4—21no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4C22⋊D21111yescharacteristic
145C4⋊C4C22⋊D21111yescharacteristic
146C22⋊C4—22no—
147C22⋊C4—21no—
148C42—21no—
149C42C22×D21111yescharacteristic
150C23—22no—
151C23—21no—
152Q8—21no—
153Q8—8441no—
154Q8—8441no—
155Q8#1680995012yescomplemented
156D4—21no—
157D4—21no—
158D4—42no—
159D4—44no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4—82no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—44no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—8441no—
172C2×C4C2×C22⋊D21111yescharacteristic
173C2×C4C2×C22⋊D21111yescharacteristic
174C2×C4C2×C22⋊D21111yescharacteristic
175C8—8441no—
176C8—8441no—
177C8—8441no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22#2026994611yescharacteristic
185C4—21no—
186C4—22no—
187C4—41no—
188C4—41no—
189C4—16881no—
190C4C2×#1680995011yescharacteristic
191C4#2026999711yescharacteristic
192C4#2027000211yescharacteristic
193C4#2027000311yescharacteristic
194C2—82no—
195C2—81no—
196C2#2721329111yescharacteristic
197C2#2721329711yescharacteristic
198C2#2721332411yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting, prime radical#550×C211C2
socleC22×C211#120
hypercenter#550D211
hypocenterC211#1642
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #27213492C2semidirect
2. #27213431C2semidirect
3. #27213351C2semidirect
4. #27212890C2semidirect
5. Q8⋊(C211⋊C8)C2semidirect
6. D4⋊(C211⋊C8)C2semidirect
59. Q8×C211SD16semidirect
99. C211#1642semidirect
155. Q8#16809950semidirect

Maximal subgroups

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

subgroupindex
#272134922
#272134312
#272133512
#272128902
Q8⋊(C211⋊C8)2
D4⋊(C211⋊C8)2
#550×C2112
#1642211