Finite Group Databasebeta

← #41012601

Subgroups — #41012601

Order 27008.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213818C211yesmaximal, characteristic
2#27213493C211yesmaximal, characteristic, complemented
3#27213431C211yesmaximal, characteristic, complemented
4#27213352C211yesmaximal, characteristic, complemented
5#27212889C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic, complemented
7Q8⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×#16809963—21no—
9C2×#16809951—21no—
10#20269977C2211yescharacteristic
11#20269974—21no—
12#20269974C2211yescharacteristic
13#20269952—22no—
14#20269952—21no—
15C4×D844—21no—
16C4×D844C2211yescharacteristic
17#20269908—21no—
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269902—21no—
20#20269901—21no—
21#20269899—22no—
22C4×Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25#16809963—42no—
26#16809963—42no—
27C2×Dic422—21no—
28C2×D844—22no—
29C2×D844—22no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844D411yescharacteristic
33C2×D844D412yes—
34C2×C4×D211—22no—
35C2×C4×D211—21no—
36#16809951—42no—
37C2×C211⋊C8—21no—
38C4⋊Dic211—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211D411yescharacteristic
41(C2×D211)⋊C4—22no—
42(C2×D211)⋊C4—21no—
43C2×Q8×C211D411yescharacteristic
44C2×C8×C211—21no—
45C2×C8×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211D411yescharacteristic
48C42×C211—21no—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51C22×D211—22no—
52Dic422—41no—
53Dic422—41no—
54D844—21no—
55D844—21no—
56D844—42no—
57D844—44no—
58D844—41no—
59D844—41no—
60D844—41no—
61D844—41no—
62D844—82no—
63C2×Dic211—21no—
64C2×Dic211—41no—
65C4×D211—44no—
66C4×D211—41no—
67C4×D211—41no—
68C211⋊C8—41no—
69Q8×C211—21no—
70Q8×C211SD1612yescomplemented
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C8×C211—41no—
77C8×C211—41no—
78C22×C211#12011yescharacteristic
79C2×D211—41no—
80C2×D211—42no—
81C2×D211—42no—
82C2×D211—41no—
83C2×D211—42no—
84C2×D211—41no—
85C4×C211—22no—
86C4×C211—21no—
87C4×C211—41no—
88C4×C211C2×SD1611yescharacteristic
89C4×C211#13511yescharacteristic
90C4×C211AGL(1,ℤ/8)11yescharacteristic
91C4×C211AGL(1,ℤ/8)11yescharacteristic
92Dic211—41no—
93Dic211—81no—
94C2×C211#44911yescharacteristic
95C2×C211#45011yescharacteristic
96C2×C211#45511yescharacteristic
97D211—81no—
98D211—82no—
99C211#164211yescharacteristic, complemented, Sylow
100#1642—2111nomaximal, Sylow
101#550—2111no—
102#486—2111no—
103#475—2111no—
104#461—2111no—
105#331—2111no—
106Q8⋊C8D21111yescharacteristic, complemented
107D4⋊C8—2111no—
108C2×#41—4221no—
109C2×SD16—4221no—
110C4⋊Q8—2111no—
111#127—4221no—
112#127—2111no—
113#121—4222no—
114#121—4221no—
115C4×Q8C2×D21111yescharacteristic
116C4×D4—4221no—
117C4×D4—2111no—
118#106—4221no—
119C4⋊C8—2111no—
120C4⋊C8C2×D21111yescharacteristic
121#103—4221no—
122#103—4221no—
123#102—4222no—
124C4×C8C2×D21111yescharacteristic
125#41—8442no—
126#41—8442no—
127C2×Q8—4221no—
128C2×Q8D84411yescharacteristic
129C2×D4—4222no—
130C2×D4—4222no—
131C2×D4—4221no—
132C2×D4—4221no—
133C2×D4—2112no—
134C2×D4—2111no—
135C22×C4—4222no—
136C22×C4—4221no—
137SD16—8442no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—4221no—
141C4⋊C4—21no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4—2111no—
145C4⋊C4D84411yescharacteristic
146C22⋊C4—4222no—
147C22⋊C4—4221no—
148C42—21no—
149C42C22×D21111yescharacteristic
150C23—4222no—
151C23—4221no—
152Q8—21no—
153Q8—8441no—
154Q8—8441no—
155Q8#1680995112yescomplemented
156D4—8442no—
157D4—8444no—
158D4—8441no—
159D4—8441no—
160D4—8441no—
161D4—8441no—
162D4—16882no—
163D4—4221no—
164D4—4221no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—8444no—
168C2×C4—8441no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—4221no—
172C2×C4D4×D21111yescharacteristic
173C2×C4D4×D21111yescharacteristic
174C2×C4C2×D84411yescharacteristic
175C8—41no—
176C8—41no—
177C8—8441no—
178C22—8442no—
179C22—8441no—
180C22—8442no—
181C22—8441no—
182C22—8442no—
183C22—8441no—
184C22#2026994711yescharacteristic
185C4—21no—
186C4—22no—
187C4—41no—
188C4—8441no—
189C4—16881no—
190C4C2×#1680995111yescharacteristic
191C4#2027000011yescharacteristic
192C4#2027000411yescharacteristic
193C4#2027000511yescharacteristic
194C2—16882no—
195C2—16881no—
196C2#2721329411yescharacteristic
197C2#2721329911yescharacteristic
198C2#2721332511yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingQ8⋊C8×C211C2
socleC22×C211#120
prime radical#27213818C2
hypercenterQ8⋊C8D211
hypocenterC211#1642
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #27213493C2semidirect
3. #27213431C2semidirect
4. #27213352C2semidirect
5. #27212889C2semidirect
6. #27212875C2semidirect
7. Q8⋊C8×C211C2semidirect
70. Q8×C211SD16semidirect
99. C211#1642semidirect
106. Q8⋊C8D211semidirect
155. Q8#16809951semidirect

Maximal subgroups

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

subgroupindex
#272138182
#272134932
#272134312
#272133522
#272128892
#272128752
Q8⋊C8×C2112
#1642211