Finite Group Databasebeta

← #41014619

Subgroups — #41014619

Order 27008.

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

normal subgroups only

This group has 306 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
1C2×#20269904C212yesmaximal, complemented
2#27213147C212yesmaximal, complemented
3#27213015C211yesmaximal, characteristic, complemented
4#27212915C211yesmaximal, characteristic, complemented
5#537×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7C2×C4⋊Dic211C2211yescharacteristic
8C2×Dic211⋊C4—22no—
9C2×C4×Dic211C2211yescharacteristic
10#20269904—24no—
11#20269901—22no—
12#20269898—22no—
13C2×#41×C211C2212yes—
14C22⋊Q8×C211—22no—
15#121×C211—22no—
16#121×C211—22no—
17C2×C4⋊C4×C211C2211yescharacteristic
18C22×Dic211—21no—
19C22×Dic211—21no—
20#16809946—22no—
21#16809946—22no—
22C2×C211⋊C8—22no—
23C4⋊Dic211—22no—
24Dic211⋊C4—44no—
25C4×Dic211—21no—
26C4×Dic211D412yescomplemented
27#41×C211—22no—
28#41×C211—22no—
29#41×C211—22no—
30#41×C211—22no—
31C2×Q8×C211D412yes—
32C2×D4×C211—22no—
33C2×D4×C211—22no—
34C2×D4×C211D412yes—
35C22×C4×C211—22no—
36C22×C4×C211D412yes—
37C22×C4×C211C2311yescharacteristic
38C4⋊C4×C211—22no—
39C4⋊C4×C211—22no—
40C22⋊C4×C211—24no—
41C2×Dic211—22no—
42C2×Dic211—22no—
43C2×Dic211—22no—
44C2×Dic211—42no—
45C2×Dic211—41no—
46C211⋊C8—42no—
47C23×C211—22no—
48C23×C211C2×D411yescharacteristic
49Q8×C211—22no—
50Q8×C211—22no—
51D4×C211—22no—
52D4×C211—22no—
53D4×C211—42no—
54D4×C211—42no—
55D4×C211—82no—
56C2×C4×C211—22no—
57C2×C4×C211—22no—
58C2×C4×C211—42no—
59C2×C4×C211—42no—
60C2×C4×C211—42no—
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×D412yes—
63C2×C4×C211#4112yes—
64C2×C4×C211#4111yescharacteristic
65C22×C211—21no—
66C22×C211—21no—
67C22×C211—42no—
68C22×C211—42no—
69C22×C211—42no—
70C22×C211#12011yescharacteristic
71C22×C211#12111yescharacteristic
72C22×C211#12411yescharacteristic
73C4×C211—42no—
74C4×C211—42no—
75C4×C211#12112yes—
76C4×C211#12312yes—
77Dic211—24no—
78Dic211—81no—
79C2×C211—21no—
80C2×C211—21no—
81C2×C211—82no—
82C2×C211#39411yescharacteristic
83C2×C211#45311yescharacteristic
84C2×C211#45611yescharacteristic
85C211#203911yescharacteristic, complemented, Sylow
86#2039—2111nomaximal, Sylow
87#537D21111yescharacteristic
88C2×#104—2112no—
89#417—2112no—
90#384—2111no—
91#343—2111no—
92C2×#41C2×D21112yes—
93C2×M16—2112no—
94C22⋊Q8—22no—
95#121—22no—
96#121—22no—
97C2×C4⋊C4—4222no—
98C2×C4⋊C4—2111no—
99C2×C4⋊C4C2×D21111yescharacteristic
100C2×C42—2111no—
101#104—4224no—
102#103—4222no—
103#102—4222no—
104#41—22no—
105#41—22no—
106#41—22no—
107#41—22no—
108C2×Q8C22⋊D21112yes—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4C22⋊D21112yes—
112C22×C4—22no—
113C22×C4—4221no—
114C22×C4—4221no—
115C22×C4C22⋊D21112yes—
116C22×C4C22×D21111yescharacteristic
117M16—4222no—
118M16—4222no—
119C2×C8—4222no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—8444no—
123C4⋊C4—4222no—
124C22⋊C4—24no—
125C42—4221no—
126C42—2112no—
127C23—22no—
128C23C2×C22⋊D21111yescharacteristic
129Q8—22no—
130Q8—22no—
131D4—22no—
132D4—22no—
133D4—42no—
134D4—42no—
135D4—82no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—8442no—
142C2×C4—8441no—
143C2×C4—4222no—
144C2×C4—4222no—
145C2×C4—4222no—
146C2×C4D4×D21111yescharacteristic
147C2×C4C2×C22⋊D21112yes—
148C2×C4#1680996112yes—
149C2×C4#1680996111yescharacteristic
150C8—8442no—
151C22—21no—
152C22—21no—
153C22—42no—
154C22—42no—
155C22—42no—
156C22#2026994611yescharacteristic
157C22#2026994811yescharacteristic
158C22#2026996411yescharacteristic
159C4—42no—
160C4—42no—
161C4—16881no—
162C4—4224no—
163C4#2026994912yes—
164C4#2026995812yes—
165C2—21no—
166C2—21no—
167C2—82no—
168C2#2721306011yescharacteristic
169C2#2721331411yescharacteristic
170C2#2721332911yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#537×C211C2
socleC22×C211#120
hypercenter#537D211
hypocenterC211#2039
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#20269904C2semidirect
2. #27213147C2semidirect
3. #27213015C2semidirect
4. #27212915C2semidirect
26. C4×Dic211D4semidirect
85. C211#2039semidirect

Maximal subgroups

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

subgroupindex
C2×#202699042
#272131472
#272130152
#272129152
#537×C2112
#2039211