Finite Group Databasebeta

← #41011959

Subgroups — #41011959

Order 27008.

264 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 85 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#27213012C211yesmaximal, characteristic, complemented
3#27212881C214yesmaximal, complemented
4#431×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
13C2×C4⋊Dic211—21no—
14C211⋊#95—21no—
15C211⋊#95C2212yes—
16C2×C42×C211C2211yescharacteristic
17C4⋊C8×C211—21no—
18C22⋊C8×C211C2212yes—
19C4×C8×C211—21no—
20#16809962—42no—
21#16809962—42no—
22C2×Dic422—21no—
23C2×D844—21no—
24C2×D844—21no—
25C2×D844C2×C412yes—
26C2×C22⋊D211—22no—
27C22×Dic211—22no—
28C2×C4×D211—22no—
29C4⋊Dic211—21no—
30C4⋊Dic211—42no—
31C4⋊Dic211C2×C412yes—
32(C2×D211)⋊C4—22no—
33(C2×D211)⋊C4—22no—
34C22×C4×C211D411yescharacteristic
35C22×C4×C211D411yescharacteristic
36C22×C4×C211C2311yescharacteristic
37C2×C8×C211—22no—
38C42×C211C2×C411yescharacteristic
39C42×C211C2×C411yescharacteristic
40C42×C211D411yescharacteristic
41C42×C211D411yescharacteristic
42C22×D211—22no—
43Dic422—42no—
44D844—42no—
45D844—42no—
46D844—42no—
47C22⋊D211—42no—
48C22⋊D211—42no—
49C2×Dic211—22no—
50C2×Dic211—42no—
51C2×Dic211—42no—
52C2×Dic211—42no—
53C4×D211—44no—
54C23×C211C2×D411yescharacteristic
55C2×C4×C211—41no—
56C2×C4×C211—41no—
57C2×C4×C211C22⋊C411yescharacteristic
58C2×C4×C211C22⋊C411yescharacteristic
59C2×C4×C211C22⋊C411yescharacteristic
60C2×C4×C211C22⋊C411yescharacteristic
61C2×C4×C211D812yes—
62C2×C4×C211SD1612yes—
63C2×C4×C211C22×C411yescharacteristic
64C2×C4×C211C2×D411yescharacteristic
65C8×C211—42no—
66C22×C211—21no—
67C22×C211—21no—
68C22×C211C2×C22⋊C411yescharacteristic
69C22×C211C2×D811yescharacteristic
70C22×C211C2×SD1611yescharacteristic
71C2×D211—42no—
72C2×D211—42no—
73C2×D211—42no—
74C4×C211—21no—
75C4×C211—21no—
76C4×C211—21no—
77C4×C211—21no—
78C4×C211#10212yes—
79C4×C211#10212yes—
80Dic211—42no—
81Dic211—82no—
82C2×C211—21no—
83C2×C211—21no—
84C2×C211#41011yescharacteristic
85C2×C211C2×#10211yescharacteristic
86C2×C211#42111yescharacteristic
87D211—82no—
88C211#151711yescharacteristic, complemented, Sylow
89#1517—2111nomaximal, Sylow
90#532—2111no—
91#431D21111yescharacteristic, complemented
92#382—2111no—
93#327—2114no—
94C2×#41—4221no—
95C4⋊Q8—2111no—
96#127—2111no—
97#124—4221no—
98#121—4221no—
99#121—2112no—
100C4×D4—4221no—
101C4×D4—2112no—
102C2×C4⋊C4—4221no—
103C2×C42C2×D21111yescharacteristic
104C4⋊C8—21no—
105C22⋊C8C2×D21112yes—
106C4×C8—21no—
107#95—4221no—
108#95—2112no—
109#41—8442no—
110#41—8442no—
111C2×Q8—4221no—
112C2×D4—4222no—
113C2×D4—4221no—
114C2×D4—4221no—
115C2×D4—2112no—
116C22×C4—4222no—
117C22×C4—4222no—
118C22×C4D84411yescharacteristic
119C22×C4D84411yescharacteristic
120C22×C4C22×D21111yescharacteristic
121C2×C8—22no—
122C4⋊C4—8442no—
123C4⋊C4—4221no—
124C4⋊C4—2112no—
125C22⋊C4—4222no—
126C22⋊C4—4222no—
127C42C4×D21111yescharacteristic
128C42C4×D21111yescharacteristic
129C42C22⋊D21111yescharacteristic
130C42C22⋊D21111yescharacteristic
131C23—4222no—
132C23C2×D84411yescharacteristic
133Q8—8442no—
134D4—8442no—
135D4—8442no—
136D4—8442no—
137D4—8442no—
138D4—8442no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—8444no—
142C2×C4—8442no—
143C2×C4—8442no—
144C2×C4—8442no—
145C2×C4—4222no—
146C2×C4(C2×D211)⋊C411yescharacteristic
147C2×C4(C2×D211)⋊C411yescharacteristic
148C2×C4(C2×D211)⋊C411yescharacteristic
149C2×C4(C2×D211)⋊C411yescharacteristic
150C2×C4D168812yes—
151C2×C4#1680995112yes—
152C2×C4C2×C4×D21111yescharacteristic
153C2×C4C2×C22⋊D21111yescharacteristic
154C8—42no—
155C22—21no—
156C22—21no—
157C22—8442no—
158C22—8442no—
159C22—8442no—
160C22C2×(C2×D211)⋊C411yescharacteristic
161C22C2×D168811yescharacteristic
162C22C2×#1680995111yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—8442no—
168C4—16882no—
169C4#2026989912yes—
170C4#2026989912yes—
171C2—21no—
172C2—21no—
173C2—16882no—
174C2#2721311911yescharacteristic
175C2C2×#2026989911yescharacteristic
176C2#2721316211yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#431×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213696C2
cosocleC22×C4×C211C23
hypercenter#431D211
hypocenterC211#1517
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #27213012C2semidirect
3. #27212881C2semidirect
4. #431×C211C2semidirect
6. #20269977C4semidirect
7. #20269974C4semidirect
12. C4×D844C4semidirect
88. C211#1517semidirect
91. #431D211semidirect

Maximal subgroups

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

subgroupindex
#272136962
#272130122
#272128812
#431×C2112
#1517211