Finite Group Databasebeta

← #41012369

Subgroups — #41012369

Order 27008.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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
2#27213287C212yesmaximal, complemented
3#27213066C211yesmaximal, characteristic, complemented
4#27212881C212yesmaximal, complemented
5#324×C211C211yesmaximal, characteristic, complemented
6C2×#16809962—21no—
7C22×Dic422C2211yescharacteristic
8C2×#16809951—22no—
9#20269957—22no—
10#20269952—22no—
11#20269951—21no—
12#20269951C2212yes—
13C2×Dic211⋊C4—21no—
14#20269902—22no—
15C211⋊#95—21no—
16C211⋊#95C2211yescharacteristic
17C2×C4⋊C4×C211C2211yescharacteristic
18C22⋊C8×C211C2212yes—
19#16809962—42no—
20#16809962—42no—
21C2×Dic422—21no—
22C2×Dic422—22no—
23C2×Dic422—42no—
24C2×D844—21no—
25C2×D844—21no—
26C2×D844D412yes—
27C2×C22⋊D211—22no—
28C22×Dic211—21no—
29C22×Dic211—21no—
30C2×C4×D211—22no—
31#16809951—44no—
32C4⋊Dic211—21no—
33C4⋊Dic211D412yes—
34Dic211⋊C4—42no—
35(C2×D211)⋊C4—22no—
36(C2×D211)⋊C4—22no—
37C22×C4×C211D411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211C2311yescharacteristic
40C2×C8×C211—22no—
41C4⋊C4×C211—22no—
42C22×D211—22no—
43Dic422—24no—
44Dic422—42no—
45Dic422—42no—
46D844—42no—
47D844—42no—
48D844—81no—
49C22⋊D211—42no—
50C22⋊D211—42no—
51C2×Dic211—22no—
52C2×Dic211—22no—
53C2×Dic211—22no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C2×Dic211—41no—
57C2×Dic211—41no—
58C4×D211—44no—
59C23×C211C2×D411yescharacteristic
60C2×C4×C211—21no—
61C2×C4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211SD1612yescomplemented
66C2×C4×C211C2×D412yes—
67C8×C211—42no—
68C22×C211—21no—
69C22×C211—21no—
70C22×C211#12011yescharacteristic
71C22×C211C2×SD1611yescharacteristic
72C22×C211AGL(1,ℤ/8)11yescharacteristic
73C2×D211—42no—
74C2×D211—42no—
75C2×D211—42no—
76C4×C211—22no—
77C4×C211—22no—
78C4×C211—41no—
79Dic211—42no—
80Dic211—42no—
81Dic211—42no—
82C2×C211—21no—
83C2×C211—21no—
84C2×C211#45011yescharacteristic
85C2×C211#45611yescharacteristic
86C2×C211B(4,𝔽2)11yescharacteristic
87D211—82no—
88C211#160411yescharacteristic, complemented, Sylow
89#1604—2111nomaximal, Sylow
90#537—2111no—
91#448—2112no—
92#395—2111no—
93#327—2112no—
94#324D21111yescharacteristic, complemented
95C2×#41—4221no—
96C22×Q8—2111no—
97C2×SD16—4222no—
98C22⋊Q8—4222no—
99#121—4222no—
100#121—4221no—
101#121—2112no—
102C2×C4⋊C4—4221no—
103C2×C4⋊C4C2×D21111yescharacteristic
104#103—4222no—
105C22⋊C8C2×D21112yes—
106#95—4221no—
107#95—2111no—
108#41—8442no—
109#41—8442no—
110C2×Q8—8442no—
111C2×Q8—4221no—
112C2×Q8—4222no—
113C2×D4—4221no—
114C2×D4—4221no—
115C2×D4—4222no—
116C2×D4—2112no—
117C22×C4—4222no—
118C22×C4—4221no—
119C22×C4—4221no—
120C22×C4D84411yescharacteristic
121C22×C4D84411yescharacteristic
122C22×C4C22×D21111yescharacteristic
123SD16—8444no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—8442no—
127C4⋊C4—4221no—
128C4⋊C4—2112no—
129C22⋊C4—4222no—
130C22⋊C4—4222no—
131C23—4222no—
132C23C2×D84411yescharacteristic
133Q8—8442no—
134Q8—8442no—
135Q8—4224no—
136D4—8442no—
137D4—8442no—
138D4—8442no—
139D4—8442no—
140D4—16881no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—8444no—
147C2×C4—8441no—
148C2×C4—8441no—
149C2×C4—8441no—
150C2×C4—8441no—
151C2×C4—4222no—
152C2×C4—4222no—
153C2×C4—4222no—
154C2×C4#1680995112yescomplemented
155C2×C4D4×D21112yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—8442no—
160C22—8442no—
161C22—8442no—
162C22#2026994711yescharacteristic
163C22C2×#1680995111yescharacteristic
164C22#2027000511yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—8442no—
169C4—8442no—
170C4—8442no—
171C2—21no—
172C2—21no—
173C2—16882no—
174C2#2721329911yescharacteristic
175C2#2721333111yescharacteristic
176C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#324×C211C2
socleC22×C211#120
prime radical#27213727C2
hypercenter#324D211
hypocenterC211#1604
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27213287C2semidirect
3. #27213066C2semidirect
4. #27212881C2semidirect
5. #324×C211C2semidirect
65. C2×C4×C211SD16semidirect
88. C211#1604semidirect
94. #324D211semidirect
154. C2×C4#16809951semidirect

Maximal subgroups

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

subgroupindex
#272137272
#272132872
#272130662
#272128812
#324×C2112
#1604211