Finite Group Databasebeta

← #41011946

Subgroups — #41011946

Order 27008.

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

normal subgroups only

This group has 252 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#27213696C211yesmaximal, characteristic
2#27213014C211yesmaximal, characteristic, complemented
3#27212881C214yesmaximal, complemented
4#432×C211C211yesmaximal, characteristic, complemented
5C2×#16809962—21no—
6#20269977C411yescharacteristic, complemented
7#20269974C411yescharacteristic, complemented
8#20269965—21no—
9#20269951—21no—
10#20269951C2212yes—
11C4×D844—21no—
12C4×D844C412yescomplemented
13C211⋊#95—21no—
14C211⋊#95—21no—
15C211⋊#95C2212yes—
16C2×C42×C211C2211yescharacteristic
17C4⋊C8×C211—21no—
18C22⋊C8×C211C2212yes—
19#97×C211—21no—
20#16809962—44no—
21C2×Dic422—21no—
22C2×D844—21no—
23C2×D844—21no—
24C2×D844C2×C412yes—
25C2×C22⋊D211—22no—
26C22×Dic211—22no—
27C2×C4×D211—22no—
28C4⋊Dic211—21no—
29C4⋊Dic211C2×C412yes—
30(C2×D211)⋊C4—22no—
31(C2×D211)⋊C4—22no—
32C22×C4×C211D411yescharacteristic
33C22×C4×C211D411yescharacteristic
34C22×C4×C211C2311yescharacteristic
35C2×C8×C211—22no—
36C42×C211C2×C411yescharacteristic
37C42×C211C2×C411yescharacteristic
38C42×C211D411yescharacteristic
39C42×C211D411yescharacteristic
40C22×D211—22no—
41Dic422—42no—
42D844—42no—
43D844—42no—
44D844—42no—
45C22⋊D211—44no—
46C2×Dic211—22no—
47C2×Dic211—42no—
48C2×Dic211—42no—
49C2×Dic211—42no—
50C4×D211—44no—
51C23×C211C2×D411yescharacteristic
52C2×C4×C211—22no—
53C2×C4×C211—41no—
54C2×C4×C211—41no—
55C2×C4×C211C22⋊C411yescharacteristic
56C2×C4×C211C22⋊C411yescharacteristic
57C2×C4×C211C22⋊C411yescharacteristic
58C2×C4×C211C22⋊C411yescharacteristic
59C2×C4×C211C22×C411yescharacteristic
60C2×C4×C211C2×D411yescharacteristic
61C8×C211—42no—
62C22×C211—21no—
63C22×C211—21no—
64C22×C211C2×C22⋊C411yescharacteristic
65C22×C211AGL(1,ℤ/8)12yes—
66C2×D211—42no—
67C2×D211—42no—
68C2×D211—42no—
69C4×C211—21no—
70C4×C211—21no—
71C4×C211—21no—
72C4×C211—21no—
73C4×C211—21no—
74C4×C211—21no—
75Dic211—42no—
76Dic211—82no—
77C2×C211—22no—
78C2×C211#41011yescharacteristic
79C2×C211#41811yescharacteristic
80C2×C211#42111yescharacteristic
81D211—82no—
82C211#151411yescharacteristic, complemented, Sylow
83#1514—2111nomaximal, Sylow
84#532—2111no—
85#432D21111yescharacteristic, complemented
86#383—2111no—
87#327—2114no—
88C2×#41—4221no—
89C4⋊Q8—2111no—
90#127—2111no—
91#124—4221no—
92#121—4221no—
93#121—2112no—
94C4×D4—4221no—
95C4×D4—2112no—
96C2×C42C2×D21111yescharacteristic
97C4⋊C8—21no—
98C22⋊C8C2×D21112yes—
99#97—21no—
100#95—4221no—
101#95—4221no—
102#95—2112no—
103#41—8444no—
104C2×Q8—4221no—
105C2×D4—4222no—
106C2×D4—4221no—
107C2×D4—4221no—
108C2×D4—2112no—
109C22×C4—4222no—
110C22×C4—4222no—
111C22×C4D84411yescharacteristic
112C22×C4D84411yescharacteristic
113C22×C4C22×D21111yescharacteristic
114C2×C8—22no—
115C4⋊C4—4221no—
116C4⋊C4—2112no—
117C22⋊C4—4222no—
118C22⋊C4—4222no—
119C42C4×D21111yescharacteristic
120C42C4×D21111yescharacteristic
121C42C22⋊D21111yescharacteristic
122C42C22⋊D21111yescharacteristic
123C23—4222no—
124C23C2×D84411yescharacteristic
125Q8—8442no—
126D4—8442no—
127D4—8442no—
128D4—8442no—
129D4—8444no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—8444no—
134C2×C4—8442no—
135C2×C4—8442no—
136C2×C4—8442no—
137C2×C4—4222no—
138C2×C4(C2×D211)⋊C411yescharacteristic
139C2×C4(C2×D211)⋊C411yescharacteristic
140C2×C4(C2×D211)⋊C411yescharacteristic
141C2×C4(C2×D211)⋊C411yescharacteristic
142C2×C4C2×C4×D21111yescharacteristic
143C2×C4C2×C22⋊D21111yescharacteristic
144C8—42no—
145C22—21no—
146C22—21no—
147C22—8442no—
148C22—8442no—
149C22—8442no—
150C22C2×(C2×D211)⋊C411yescharacteristic
151C22#2027000512yes—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—8442no—
159C4—16882no—
160C2—22no—
161C2—16882no—
162C2#2721311911yescharacteristic
163C2#2721315211yescharacteristic
164C2#2721316211yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#432×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213696C2
cosocleC22×C4×C211C23
hypercenter#432D211
hypocenterC211#1514
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #27213014C2semidirect
3. #27212881C2semidirect
4. #432×C211C2semidirect
6. #20269977C4semidirect
7. #20269974C4semidirect
12. C4×D844C4semidirect
82. C211#1514semidirect
85. #432D211semidirect

Maximal subgroups

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

subgroupindex
#272136962
#272130142
#272128812
#432×C2112
#1514211