Finite Group Databasebeta

← #41014501

Subgroups — #41014501

Order 27008.

350 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 71 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#27213944C212yesmaximal, complemented
2#27213114C211yesmaximal, characteristic, complemented
3#27213113C212yesmaximal, complemented
4C211⋊#338C211yesmaximal, characteristic, complemented
5#575×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2211yescharacteristic
7#20270008—22no—
8#20270007—22no—
9#20270003—22no—
10#20270002—22no—
11#20269997—24no—
12C2×#16809946C2212yes—
13C22×C211⋊C8C2211yescharacteristic
14C22⋊(C211⋊C8)—24no—
15(C2×D211)⋊C8—22no—
16C2×#41×C211C2212yes—
17#137×C211—22no—
18AGL(1,ℤ/8)×C211—22no—
19#135×C211—22no—
20C2×M16×C211C2211yescharacteristic
21#16809962—22no—
22#16809962—22no—
23C2×Dic422D411yescharacteristic
24C2×D844D411yescharacteristic
25C2×C22⋊D211—21no—
26C2×C4×D211—21no—
27#16809952—24no—
28#16809950—24no—
29#16809949—24no—
30#16809947—24no—
31#16809946—22no—
32#16809946—22no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—22no—
35C2×C211⋊C8D414yes—
36#41×C211—22no—
37#41×C211—22no—
38#41×C211—22no—
39#41×C211—22no—
40C2×Q8×C211D412yes—
41C2×D4×C211—22no—
42C2×D4×C211D412yes—
43C22×C4×C211—22no—
44C22×C4×C211C2311yescharacteristic
45Q16×C211—22no—
46SD16×C211—24no—
47D8×C211—22no—
48M16×C211—22no—
49C2×C8×C211D412yes—
50C22×D211—21no—
51Dic422—22no—
52D844—22no—
53C22⋊D211—42no—
54C2×Dic211—21no—
55C4×D211—41no—
56C4×D211—41no—
57C211⋊C8—24no—
58C211⋊C8—41no—
59C23×C211—22no—
60C23×C211#4111yescharacteristic
61Q8×C211—22no—
62Q8×C211—22no—
63D4×C211—22no—
64D4×C211—22no—
65D4×C211—42no—
66D4×C211—42no—
67C2×C4×C211—22no—
68C2×C4×C211—42no—
69C2×C4×C211—42no—
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D412yes—
72C2×C4×C211C2×D412yes—
73C2×C4×C211C2×D411yescharacteristic
74C8×C211—22no—
75C22×C211—41no—
76C22×C211—44no—
77C22×C211—42no—
78C22×C211#12111yescharacteristic
79C22×C211#12112yes—
80C2×D211—42no—
81C2×D211—41no—
82C4×C211—42no—
83C4×C211#12012yes—
84C4×C211#12011yescharacteristic
85C4×C211#12711yescharacteristic
86Dic211—41no—
87C2×C211—21no—
88C2×C211—22no—
89C2×C211—82no—
90C2×C211#39211yescharacteristic
91D211—81no—
92C211#201711yescharacteristic, complemented, Sylow
93#2017—2111nomaximal, Sylow
94#575—2112no—
95#575D21111yescharacteristic, complemented
96#408—2112no—
97#408—2111no—
98#338—2111no—
99C2×#41—2111no—
100C2×#41C2×D21112yes—
101#137—22no—
102#137—4222no—
103#137—4222no—
104AGL(1,ℤ/8)—22no—
105AGL(1,ℤ/8)—4222no—
106AGL(1,ℤ/8)—4222no—
107#135—22no—
108#135—4224no—
109C2×M16—2112no—
110C2×M16C2×D21111yescharacteristic
111C22×C8—2111no—
112C22⋊C8—4224no—
113C22⋊C8—4222no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118#41—4222no—
119#41—4222no—
120C2×Q8—2111no—
121C2×Q8C22⋊D21112yes—
122C2×D4—22no—
123C2×D4—4221no—
124C2×D4—2111no—
125C2×D4C22⋊D21112yes—
126C22×C4—22no—
127C22×C4—4221no—
128C22×C4C22×D21111yescharacteristic
129Q16—22no—
130Q16—4224no—
131SD16—24no—
132SD16—4224no—
133SD16—4224no—
134D8—22no—
135D8—4224no—
136M16—22no—
137M16—4222no—
138M16—4222no—
139C2×C8—4222no—
140C2×C8—4221no—
141C2×C8—2114no—
142C2×C8C22⋊D21112yes—
143C23—22no—
144C23—4221no—
145C23#1680996111yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—4222no—
149D4—22no—
150D4—22no—
151D4—42no—
152D4—42no—
153D4—8442no—
154D4—4222no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—8441no—
159C2×C4—8441no—
160C2×C4—4221no—
161C2×C4D4×D21111yescharacteristic
162C2×C4D4×D21112yes—
163C2×C4C2×C22⋊D21112yes—
164C2×C4C2×C22⋊D21111yescharacteristic
165C8—22no—
166C8—8441no—
167C8—4224no—
168C22—41no—
169C22—44no—
170C22—42no—
171C22—8442no—
172C22—8441no—
173C22#2026994811yescharacteristic
174C22#2026994912yes—
175C4—42no—
176C4—8441no—
177C4#2026994512yes—
178C4#2026994611yescharacteristic
179C4#2026997311yescharacteristic
180C2—22no—
181C2—21no—
182C2—82no—
183C2—16881no—
184C2#2721305411yescharacteristic
subgroupG/H
centerC4#20269973
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#575×C211C2
socleC2×C211#392
hypercenter#575D211
hypocenterC211#2017
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #27213944C2semidirect
2. #27213114C2semidirect
3. #27213113C2semidirect
4. C211⋊#338C2semidirect
5. #575×C211C2semidirect
92. C211#2017semidirect
95. #575D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
#272131142
#272131132
C211⋊#3382
#575×C2112
#2017211