Finite Group Databasebeta

← #41012130

Subgroups — #41012130

Order 27008.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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#27213694C211yesmaximal, characteristic
2#27213168C212yesmaximal, complemented
3#27212894C212yesmaximal, complemented
4#27212890C211yesmaximal, characteristic, complemented
5#331×C211C211yesmaximal, characteristic, complemented
6C2×#16809961C412yes—
7C2×#16809945—22no—
8#20269977C2211yescharacteristic
9#20269964C412yes—
10#20269948—22no—
11D4×Dic211—22no—
12C42×D211C2211yescharacteristic
13C4⋊(C211⋊C8)C2212yes—
14Dic211⋊C8—22no—
15#127×C211C2211yescharacteristic
16C4⋊C8×C211C2212yes—
17#16809961—44no—
18C2×Dic422C2×C412yes—
19C2×C22⋊D211—22no—
20C22×Dic211—22no—
21C2×C4×D211C2×C411yescharacteristic
22C2×C4×D211D412yes—
23#16809945—44no—
24C2×C211⋊C8—22no—
25C4⋊Dic211—22no—
26C22⋊Dic211—24no—
27C4×Dic211C2×C411yescharacteristic
28C4×Dic211D412yes—
29C2×D4×C211—22no—
30C2×D4×C211C2×C412yes—
31C2×C8×C211—22no—
32C42×C211C2311yescharacteristic
33C22×D211C22⋊C411yescharacteristic
34Dic422—42no—
35C22⋊D211—44no—
36C2×Dic211—22no—
37C2×Dic211—44no—
38C2×Dic211C22⋊C412yes—
39C2×Dic211C22⋊C411yescharacteristic
40C4×D211—22no—
41C4×D211—22no—
42C4×D211—22no—
43C211⋊C8—42no—
44C23×C211—22no—
45D4×C211—42no—
46D4×C211—42no—
47D4×C211—42no—
48C2×C4×C211C22×C411yescharacteristic
49C2×C4×C211C2×D412yes—
50C8×C211—42no—
51C22×C211—42no—
52C22×C211—44no—
53C22×C211C2×C22⋊C411yescharacteristic
54C2×D211—21no—
55C2×D211—22no—
56C4×C211—21no—
57C4×C211AGL(1,ℤ/8)12yes—
58C4×C211#13712yes—
59Dic211—22no—
60Dic211—42no—
61Dic211#10012yes—
62C2×C211—82no—
63C2×C211C2×#10011yescharacteristic
64C2×C211#41712yes—
65D211—41no—
66C211#156011yescharacteristic, complemented, Sylow
67#1560—2111nomaximal, Sylow
68#532—2111no—
69#423—2112no—
70#333—2112no—
71#331—2111no—
72#331D21111yescharacteristic, complemented
73C2×#41—2112no—
74C2×M16—4222no—
75C4⋊Q8—2111no—
76#127C2×D21111yescharacteristic
77#124—2112no—
78#121—4222no—
79C4×D4—4222no—
80C2×C42—2111no—
81C4⋊C8—4222no—
82C4⋊C8—2112no—
83C4⋊C8C2×D21112yes—
84#41—8444no—
85C2×Q8—2112no—
86C2×D4—22no—
87C2×D4—4222no—
88C2×D4C4×D21112yes—
89C22×C4—4222no—
90C22×C4—2111no—
91C22×C4—2112no—
92M16—8444no—
93C2×C8—22no—
94C2×C8—4222no—
95C4⋊C4—4222no—
96C22⋊C4—4224no—
97C42—2111no—
98C42—2112no—
99C42C22×D21111yescharacteristic
100C23—22no—
101C23—2111no—
102Q8—8442no—
103D4—42no—
104D4—42no—
105D4—42no—
106D4—8444no—
107C2×C4—8444no—
108C2×C4—4222no—
109C2×C4—4222no—
110C2×C4—4222no—
111C2×C4—4222no—
112C2×C4—2112no—
113C2×C4—2111no—
114C2×C4C2×C4×D21111yescharacteristic
115C2×C4D4×D21112yes—
116C8—42no—
117C8—8442no—
118C22—42no—
119C22—44no—
120C22—4221no—
121C22—4222no—
122C22C22⋊C4×D21111yescharacteristic
123C4—21no—
124C4—8442no—
125C4—4222no—
126C4—2112no—
127C4#2027000112yes—
128C4#2027000612yes—
129C2—82no—
130C2—8441no—
131C2#100×D21111yescharacteristic
132C2#2721314412yes—
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#331×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213694C2
cosocleC42×C211C23
hypercenter#331D211
hypocenterC211#1560
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27213168C2semidirect
3. #27212894C2semidirect
4. #27212890C2semidirect
5. #331×C211C2semidirect
66. C211#1560semidirect
72. #331D211semidirect

Maximal subgroups

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

subgroupindex
#272136942
#272131682
#272128942
#272128902
#331×C2112
#1560211