Finite Group Databasebeta

← #41011950

Subgroups — #41011950

Order 27008.

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

normal subgroups only

This group has 208 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#27213703C211yesmaximal, characteristic
2#27213014C211yesmaximal, characteristic
3#27212884C214yesmaximal
4#432×C211C211yesmaximal, characteristic
5#20269977C411yescharacteristic, complemented
6#20269977C411yescharacteristic, complemented
7#20269969—21no—
8#20269956—21no—
9#20269956C2212yes—
10C4×Dic422—21no—
11C4×Dic422C412yescomplemented
12#20269936—21no—
13C211⋊#95—21no—
14C211⋊#95—21no—
15C211⋊#95C2212yes—
16C2×C42×C211C2211yescharacteristic
17C4⋊C8×C211—21no—
18C22⋊C8×C211C2212yes—
19#97×C211—21no—
20C2×Dic422—21no—
21C2×Dic422C2×C412yes—
22C22×Dic211—22no—
23C4⋊Dic211—21no—
24C4⋊Dic211—21no—
25C4⋊Dic211—21no—
26C4⋊Dic211C2×C412yes—
27Dic211⋊C4—22no—
28Dic211⋊C4—22no—
29C22⋊Dic211—22no—
30C4×Dic211—22no—
31C22×C4×C211D411yescharacteristic
32C22×C4×C211D411yescharacteristic
33C22×C4×C211C2311yescharacteristic
34C2×C8×C211—22no—
35C42×C211C2×C411yescharacteristic
36C42×C211C2×C411yescharacteristic
37C42×C211D411yescharacteristic
38C42×C211D411yescharacteristic
39Dic422—42no—
40Dic422—42no—
41C2×Dic211—22no—
42C2×Dic211—22no—
43C2×Dic211—42no—
44C2×Dic211—42no—
45C2×Dic211—42no—
46C23×C211C2×D411yescharacteristic
47C2×C4×C211—22no—
48C2×C4×C211—41no—
49C2×C4×C211—41no—
50C2×C4×C211C22⋊C411yescharacteristic
51C2×C4×C211C22⋊C411yescharacteristic
52C2×C4×C211C22⋊C411yescharacteristic
53C2×C4×C211C22⋊C411yescharacteristic
54C2×C4×C211C22×C411yescharacteristic
55C2×C4×C211C2×D411yescharacteristic
56C8×C211—42no—
57C22×C211—21no—
58C22×C211—21no—
59C22×C211C2×C22⋊C411yescharacteristic
60C22×C211#13712yes—
61C4×C211—21no—
62C4×C211—21no—
63C4×C211—21no—
64C4×C211—21no—
65C4×C211—21no—
66C4×C211—21no—
67Dic211—42no—
68Dic211—42no—
69Dic211—82no—
70C2×C211—22no—
71C2×C211#41011yescharacteristic
72C2×C211#41911yescharacteristic
73C2×C211#42111yescharacteristic
74C211#151511yescharacteristic, complemented, Sylow
75#1515—2111nomaximal, Sylow
76#533—2111no—
77#432D21111yescharacteristic
78#383—2111no—
79#328—2114no—
80C4⋊Q8—2111no—
81C4⋊Q8—2111no—
82#125—4221no—
83C22⋊Q8—4221no—
84C22⋊Q8—2112no—
85C4×Q8—4221no—
86C4×Q8—2112no—
87#117—4221no—
88C2×C42C2×D21111yescharacteristic
89C4⋊C8—21no—
90C22⋊C8C2×D21112yes—
91#97—21no—
92#95—4221no—
93#95—4221no—
94#95—2112no—
95C2×Q8—4221no—
96C2×Q8—2112no—
97C22×C4—4222no—
98C22×C4D84411yescharacteristic
99C22×C4D84411yescharacteristic
100C22×C4C22×D21111yescharacteristic
101C2×C8—22no—
102C4⋊C4—4221no—
103C4⋊C4—4221no—
104C4⋊C4—4221no—
105C4⋊C4—4222no—
106C4⋊C4—4222no—
107C4⋊C4—2112no—
108C22⋊C4—4222no—
109C42—4222no—
110C42C4×D21111yescharacteristic
111C42C4×D21111yescharacteristic
112C42C22⋊D21111yescharacteristic
113C42C22⋊D21111yescharacteristic
114C23C2×D84411yescharacteristic
115Q8—8442no—
116Q8—8442no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—8442no—
121C2×C4—8442no—
122C2×C4—8442no—
123C2×C4—4222no—
124C2×C4—4222no—
125C2×C4(C2×D211)⋊C411yescharacteristic
126C2×C4(C2×D211)⋊C411yescharacteristic
127C2×C4(C2×D211)⋊C411yescharacteristic
128C2×C4(C2×D211)⋊C411yescharacteristic
129C2×C4C2×C4×D21111yescharacteristic
130C2×C4C2×C22⋊D21111yescharacteristic
131C8—42no—
132C22—21no—
133C22—21no—
134C22C2×(C2×D211)⋊C411yescharacteristic
135C22#2027001012yes—
136C4—21no—
137C4—21no—
138C4—21no—
139C4—21no—
140C4—21no—
141C4—21no—
142C4—8442no—
143C4—8442no—
144C4—16882no—
145C2—22no—
146C2#2721311911yescharacteristic
147C2#2721315611yescharacteristic
148C2#2721316211yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#432×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenter#432D211
hypocenterC211#1515
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
5. #20269977C4semidirect
6. #20269977C4semidirect
11. C4×Dic422C4semidirect
74. C211#1515semidirect

Maximal subgroups

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

subgroupindex
#272137032
#272130142
#272128842
#432×C2112
#1515211