Finite Group Databasebeta

← #41014622

Subgroups — #41014622

Order 27008.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213727C211yesmaximal, characteristic
2C2×#20269905C212yesmaximal, complemented
3#27213145C212yesmaximal, complemented
4#27212915C211yesmaximal, characteristic, complemented
5#384×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2212yes—
7C2×#16809946C2212yes—
8#20269957—22no—
9#20269952—22no—
10#20269951—22no—
11#20269905—24no—
12#20269900—22no—
13#20269897—22no—
14C2×C4⋊C4×C211—22no—
15C2×C4⋊C4×C211C2211yescharacteristic
16C2×C4⋊C4×C211C2211yescharacteristic
17C2×C42×C211C2211yescharacteristic
18#16809962—22no—
19#16809962—22no—
20#16809962—22no—
21#16809962—22no—
22C2×Dic422D412yes—
23C2×D844—22no—
24C2×D844D412yes—
25C2×C22⋊D211—22no—
26C2×C4×D211—22no—
27#16809946—22no—
28#16809946—22no—
29C2×C211⋊C8—22no—
30C4⋊Dic211—22no—
31(C2×D211)⋊C4—24no—
32C22×C4×C211—21no—
33C22×C4×C211—21no—
34C22×C4×C211D412yes—
35C22×C4×C211C2311yescharacteristic
36C4⋊C4×C211—22no—
37C4⋊C4×C211—22no—
38C4⋊C4×C211—44no—
39C42×C211—21no—
40C42×C211D412yescomplemented
41C22×D211—22no—
42Dic422—22no—
43Dic422—22no—
44D844—22no—
45D844—22no—
46D844—82no—
47C22⋊D211—42no—
48C22⋊D211—42no—
49C2×Dic211—22no—
50C4×D211—42no—
51C4×D211—42no—
52C211⋊C8—42no—
53C23×C211C2×D411yescharacteristic
54C2×C4×C211—22no—
55C2×C4×C211—22no—
56C2×C4×C211—22no—
57C2×C4×C211—22no—
58C2×C4×C211—42no—
59C2×C4×C211—42no—
60C2×C4×C211—41no—
61C2×C4×C211C2×D412yes—
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211#4111yescharacteristic
64C2×C4×C211#4112yes—
65C22×C211—21no—
66C22×C211—21no—
67C22×C211#12011yescharacteristic
68C22×C211#12111yescharacteristic
69C22×C211#12411yescharacteristic
70C2×D211—42no—
71C2×D211—42no—
72C2×D211—42no—
73C4×C211—24no—
74C4×C211—42no—
75C4×C211—81no—
76C4×C211#12112yes—
77C4×C211#12312yes—
78Dic211—42no—
79C2×C211—21no—
80C2×C211—21no—
81C2×C211#39411yescharacteristic
82C2×C211#45311yescharacteristic
83C2×C211#45611yescharacteristic
84D211—82no—
85C211#203911yescharacteristic, complemented, Sylow
86#2039—2111nomaximal, Sylow
87#537—2111no—
88C2×#104—2112no—
89#417—2112no—
90#384D21111yescharacteristic, complemented
91#343—2111no—
92C2×#41—2112no—
93C2×M16—2112no—
94C22⋊Q8—4222no—
95#121—4222no—
96#121—4222no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D21111yescharacteristic
99C2×C4⋊C4C2×D21111yescharacteristic
100C2×C42C2×D21111yescharacteristic
101#104—4224no—
102#103—4222no—
103#102—4222no—
104#41—4222no—
105#41—4222no—
106#41—4222no—
107#41—4222no—
108C2×Q8—2112no—
109C2×D4—4222no—
110C2×D4—4222no—
111C2×D4—2112no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—4222no—
115C22×C4C22⋊D21112yes—
116C22×C4C22×D21111yescharacteristic
117M16—4222no—
118M16—4222no—
119C2×C8—4222no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—4222no—
124C22⋊C4—4224no—
125C42—21no—
126C42D84412yescomplemented
127C23—4222no—
128C23C2×C22⋊D21111yescharacteristic
129Q8—4222no—
130Q8—4222no—
131D4—8442no—
132D4—8442no—
133D4—16882no—
134D4—4222no—
135D4—4222no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—8442no—
144C2×C4—8442no—
145C2×C4—4222no—
146C2×C4D4×D21112yes—
147C2×C4C2×D84411yescharacteristic
148C2×C4#1680996211yescharacteristic
149C2×C4#1680996312yes—
150C8—8442no—
151C22—21no—
152C22—21no—
153C22—8442no—
154C22—8442no—
155C22—8442no—
156C22#2026994511yescharacteristic
157C22#2026995111yescharacteristic
158C22#2026996611yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—8442no—
163C4#2026995212yes—
164C4#2026996212yes—
165C2—21no—
166C2—21no—
167C2—16882no—
168C2#2721306311yescharacteristic
169C2#2721331311yescharacteristic
170C2#2721332811yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#384×C211C2
socleC22×C211#120
prime radical#27213727C2
hypercenter#384D211
hypocenterC211#2039
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#20269905C2semidirect
3. #27213145C2semidirect
4. #27212915C2semidirect
5. #384×C211C2semidirect
40. C42×C211D4semidirect
85. C211#2039semidirect
90. #384D211semidirect
126. C42D844semidirect

Maximal subgroups

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

subgroupindex
#272137272
C2×#202699052
#272131452
#272129152
#384×C2112
#2039211