Finite Group Databasebeta

← #41014624

Subgroups — #41014624

Order 27008.

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

normal subgroups only

This group has 290 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#27213011C211yesmaximal, characteristic, complemented
4#27212915C211yesmaximal, characteristic, complemented
5#537×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7C2×C4⋊Dic211C2211yescharacteristic
8C2×C4×Dic211C2211yescharacteristic
9#20269904—24no—
10#20269901—22no—
11#20269898—22no—
12C211⋊#95—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—
24C4×Dic211—21no—
25C4×Dic211D412yescomplemented
26#41×C211—22no—
27#41×C211—22no—
28#41×C211—22no—
29#41×C211—22no—
30C2×Q8×C211D412yes—
31C2×D4×C211—22no—
32C2×D4×C211—22no—
33C2×D4×C211D412yes—
34C22×C4×C211—22no—
35C22×C4×C211D412yes—
36C22×C4×C211C2311yescharacteristic
37C4⋊C4×C211—22no—
38C4⋊C4×C211—22no—
39C22⋊C4×C211—24no—
40C2×Dic211—22no—
41C2×Dic211—41no—
42C2×Dic211—41no—
43C2×Dic211—42no—
44C2×Dic211—41no—
45C211⋊C8—42no—
46C23×C211—22no—
47C23×C211C2×D411yescharacteristic
48Q8×C211—22no—
49Q8×C211—22no—
50D4×C211—22no—
51D4×C211—22no—
52D4×C211—42no—
53D4×C211—42no—
54D4×C211—82no—
55C2×C4×C211—22no—
56C2×C4×C211—22no—
57C2×C4×C211—42no—
58C2×C4×C211—42no—
59C2×C4×C211—42no—
60C2×C4×C211C2×D411yescharacteristic
61C2×C4×C211C2×D412yes—
62C2×C4×C211#4112yes—
63C2×C4×C211#4111yescharacteristic
64C22×C211—21no—
65C22×C211—21no—
66C22×C211—42no—
67C22×C211—42no—
68C22×C211—42no—
69C22×C211#12011yescharacteristic
70C22×C211#12111yescharacteristic
71C22×C211#12411yescharacteristic
72C4×C211—42no—
73C4×C211—42no—
74C4×C211#12112yes—
75C4×C211#12312yes—
76Dic211—42no—
77Dic211—81no—
78C2×C211—21no—
79C2×C211—21no—
80C2×C211—82no—
81C2×C211#39411yescharacteristic
82C2×C211#45411yescharacteristic
83C2×C211#45511yescharacteristic
84C211#204011yescharacteristic, complemented, Sylow
85#2040—2111nomaximal, Sylow
86#537D21111yescharacteristic
87C2×#104—2112no—
88#417—2112no—
89#382—2111no—
90#343—2111no—
91C2×#41C2×D21112yes—
92C2×M16—2112no—
93C22⋊Q8—22no—
94#121—22no—
95#121—22no—
96C2×C4⋊C4—2111no—
97C2×C4⋊C4C2×D21111yescharacteristic
98C2×C42—2111no—
99#104—4224no—
100#103—4222no—
101#102—4222no—
102#95—4222no—
103#41—22no—
104#41—22no—
105#41—22no—
106#41—22no—
107C2×Q8C22⋊D21112yes—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4C22⋊D21112yes—
111C22×C4—22no—
112C22×C4—4221no—
113C22×C4—4221no—
114C22×C4C22⋊D21112yes—
115C22×C4C22×D21111yescharacteristic
116M16—4222no—
117M16—4222no—
118C2×C8—4222no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—4222no—
122C22⋊C4—24no—
123C42—4221no—
124C42—2112no—
125C23—22no—
126C23C2×C22⋊D21111yescharacteristic
127Q8—22no—
128Q8—22no—
129D4—22no—
130D4—22no—
131D4—42no—
132D4—42no—
133D4—82no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—8441no—
140C2×C4—8441no—
141C2×C4—8442no—
142C2×C4—8441no—
143C2×C4—4222no—
144C2×C4D4×D21111yescharacteristic
145C2×C4C2×C22⋊D21112yes—
146C2×C4#1680996112yes—
147C2×C4#1680996111yescharacteristic
148C8—8442no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#2026994611yescharacteristic
155C22#2026994811yescharacteristic
156C22#2026996411yescharacteristic
157C4—42no—
158C4—42no—
159C4—8442no—
160C4—16881no—
161C4#2026994912yes—
162C4#2026995812yes—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2721306011yescharacteristic
167C2#2721331911yescharacteristic
168C2#2721332411yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#537×C211C2
socleC22×C211#120
hypercenter#537D211
hypocenterC211#2040
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. #27213011C2semidirect
4. #27212915C2semidirect
25. C4×Dic211D4semidirect
84. C211#2040semidirect

Maximal subgroups

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

subgroupindex
C2×#202699042
#272131472
#272130112
#272129152
#537×C2112
#2040211