Finite Group Databasebeta

← #41014570

Subgroups — #41014570

Order 27008.

244 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 65 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#27213703C211yesmaximal, characteristic, complemented
2#27213163C211yesmaximal, characteristic
3#27213120C211yesmaximal, characteristic
4#27212918C211yesmaximal, characteristic, complemented
5#27212906C211yesmaximal, characteristic, complemented
6#27212903C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9#20269977—21no—
10#20269969—21no—
11#20269956—22no—
12C4×Dic422—22no—
13#20269936C2211yescharacteristic
14#20269936C2211yescharacteristic
15#20269936C2211yescharacteristic
16#20269904—22no—
17#20269892—22no—
18C2×#41×C211C2211yescharacteristic
19C2×M16×C211C2211yescharacteristic
20C2×C42×C211C2211yescharacteristic
21#104×C211—21no—
22#104×C211—21no—
23C2×Dic422—22no—
24#16809946—21no—
25#16809946—21no—
26C2×C211⋊C8—21no—
27C4⋊Dic211—22no—
28C4⋊Dic211—21no—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31Dic211⋊C4—24no—
32C22⋊Dic211—21no—
33C22⋊Dic211—21no—
34C22⋊Dic211D412yes—
35C4×Dic211—21no—
36C4×Dic211—21no—
37C4×Dic211D412yes—
38#41×C211—21no—
39#41×C211—21no—
40#41×C211—21no—
41#41×C211—21no—
42C2×Q8×C211Q811yescharacteristic, complemented
43C2×D4×C211—21no—
44C2×D4×C211Q811yescharacteristic, complemented
45C22×C4×C211—21no—
46C22×C4×C211—21no—
47C22×C4×C211C2311yescharacteristic
48M16×C211—21no—
49M16×C211—21no—
50C2×C8×C211—21no—
51C42×C211—21no—
52C42×C211D411yescharacteristic
53C42×C211D411yescharacteristic
54Dic422—44no—
55C2×Dic211—22no—
56C2×Dic211—22no—
57C2×Dic211—41no—
58C211⋊C8—41no—
59C23×C211—21no—
60C23×C211C2×D411yescharacteristic
61Q8×C211—21no—
62Q8×C211—21no—
63D4×C211—21no—
64D4×C211—21no—
65D4×C211—41no—
66D4×C211—41no—
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211—42no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×Q811yescharacteristic
76C2×C4×C211#4111yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C8×C211—41no—
80C22×C211—41no—
81C22×C211—41no—
82C22×C211—81no—
83C22×C211#12011yescharacteristic
84C22×C211C22⋊Q811yescharacteristic
85C22×C211#12411yescharacteristic
86C4×C211—22no—
87C4×C211—41no—
88C4×C211—41no—
89C4×C211C22⋊Q811yescharacteristic
90C4×C211C22⋊Q811yescharacteristic
91C4×C211#12411yescharacteristic
92C4×C211#12411yescharacteristic
93Dic211—42no—
94Dic211—42no—
95Dic211—81no—
96C2×C211—21no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211—81no—
100C2×C211#39311yescharacteristic
101C211#203111yescharacteristic, complemented, Sylow
102#2031—2111nomaximal, Sylow
103#533—2111no—
104#421—2111no—
105C2×#104D21111yescharacteristic
106#410—2111no—
107#344—2111no—
108#339—2111no—
109H(3,ℤ/4)—2111no—
110C2×#41C2×D21111yescharacteristic
111C2×M16—2111no—
112C2×M16C2×D21111yescharacteristic
113C4⋊Q8—4221no—
114#125—4221no—
115C22⋊Q8—4222no—
116C4×Q8—4222no—
117#117—2111no—
118#117—2111no—
119#117—2111no—
120C2×C42C2×D21111yescharacteristic
121#104—21no—
122#104—21no—
123#104—4222no—
124#99—4222no—
125#41—21no—
126#41—21no—
127#41—21no—
128#41—21no—
129C2×Q8—4222no—
130C2×Q8Dic42211yescharacteristic, complemented
131C2×D4—21no—
132C2×D4Dic42211yescharacteristic, complemented
133C22×C4—21no—
134C22×C4—21no—
135C22×C4C22×D21111yescharacteristic
136M16—21no—
137M16—21no—
138M16—4221no—
139M16—4221no—
140C2×C8—21no—
141C2×C8—4221no—
142C4⋊C4—4224no—
143C4⋊C4—4222no—
144C4⋊C4—4221no—
145C4⋊C4—4221no—
146C4⋊C4—4221no—
147C22⋊C4—4221no—
148C22⋊C4—4221no—
149C22⋊C4—2112no—
150C42—21no—
151C42—4221no—
152C42—4221no—
153C42—2112no—
154C42C22⋊D21111yescharacteristic
155C42C22⋊D21111yescharacteristic
156C23—21no—
157C23D4×D21111yescharacteristic
158Q8—21no—
159Q8—21no—
160Q8—8444no—
161D4—21no—
162D4—21no—
163D4—41no—
164D4—41no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—8441no—
172C2×C4—4222no—
173C2×C4—4222no—
174C2×C4D4×D21111yescharacteristic
175C2×C4C2×C22⋊D21111yescharacteristic
176C2×C4C2×Dic42211yescharacteristic
177C2×C4#1680996111yescharacteristic
178C2×C4#1680996111yescharacteristic
179C2×C4#1680996211yescharacteristic
180C8—41no—
181C8—8441no—
182C22—41no—
183C22—41no—
184C22—81no—
185C22#2026994511yescharacteristic
186C22#2026995411yescharacteristic
187C22#2026996311yescharacteristic
188C4—22no—
189C4—41no—
190C4—41no—
191C4—8442no—
192C4—8442no—
193C4—16881no—
194C4#2026995411yescharacteristic
195C4#2026995611yescharacteristic
196C4#2026996311yescharacteristic
197C4#2026996411yescharacteristic
198C2—21no—
199C2—21no—
200C2—21no—
201C2—81no—
202C2#2721305811yescharacteristic
subgroupG/H
centerC4#20269963
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC2×C211#393
prime radicalC2×#41×C211C22
hypercenterC2×#104D211
hypocenterC211#2031
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. #27213703C2semidirect
4. #27212918C2semidirect
5. #27212906C2semidirect
6. #27212903C2semidirect
42. C2×Q8×C211Q8semidirect
44. C2×D4×C211Q8semidirect
101. C211#2031semidirect
130. C2×Q8Dic422semidirect
132. C2×D4Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137032
#272131632
#272131202
#272129182
#272129062
#272129032
C2×#104×C2112
#2031211