Finite Group Databasebeta

← #41012034

Subgroups — #41012034

Order 27008.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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#27213702C211yesmaximal, characteristic
2#27213201C212yesmaximal
3#27212888C211yesmaximal, characteristic, complemented
4#27212887C212yesmaximal, complemented
5#330×C211C211yesmaximal, characteristic, complemented
6#20269976C412yes—
7#20269969C2211yescharacteristic
8#20269955—22no—
9#20269942—22no—
10#20269935C412yes—
11C42×D211C2211yescharacteristic
12Dic211⋊C8—22no—
13(C2×D211)⋊C8—22no—
14#20269888C2212yes—
15#125×C211C2211yescharacteristic
16#97×C211C2212yes—
17C2×Dic422—22no—
18C2×C4×D211C2×C411yescharacteristic
19C2×C4×D211D412yes—
20C2×C211⋊C8—22no—
21C4⋊Dic211C2×C412yes—
22Dic211⋊C4—22no—
23Dic211⋊C4—22no—
24(C2×D211)⋊C4—22no—
25C4×Dic211—22no—
26C4×Dic211C2×C411yescharacteristic
27C4×Dic211D412yes—
28C2×C8×C211—22no—
29C4⋊C4×C211—22no—
30C4⋊C4×C211C2×C412yes—
31C42×C211C2311yescharacteristic
32C22×D211C22⋊C411yescharacteristic
33Dic422—44no—
34C2×Dic211—22no—
35C2×Dic211C22⋊C412yes—
36C2×Dic211C22⋊C411yescharacteristic
37C4×D211—22no—
38C4×D211—42no—
39C211⋊C8—42no—
40C2×C4×C211—22no—
41C2×C4×C211C22×C411yescharacteristic
42C2×C4×C211C2×D412yes—
43C8×C211—42no—
44C22×C211C2×C22⋊C411yescharacteristic
45C2×D211—21no—
46C2×D211—22no—
47C4×C211—21no—
48C4×C211—22no—
49C4×C211—42no—
50Dic211—22no—
51Dic211—42no—
52Dic211#10112yes—
53C2×C211C2×#10111yescharacteristic
54C2×C211#42112yes—
55D211—41no—
56C211#153711yescharacteristic, complemented, Sylow
57#1537—2111nomaximal, Sylow
58#533—2111no—
59#432—2112no—
60#330—2111no—
61#330—2112no—
62#330D21111yescharacteristic, complemented
63C4⋊Q8—2112no—
64#125—2111no—
65#125C2×D21111yescharacteristic
66C22⋊Q8—4222no—
67C4×Q8—4222no—
68#117—2112no—
69C2×C42—2111no—
70C4⋊C8—4222no—
71C22⋊C8—4222no—
72#97—2112no—
73#97C2×D21112yes—
74C2×Q8—4222no—
75C22×C4—2112no—
76C22×C4—2111no—
77C2×C8—22no—
78C2×C8—4222no—
79C4⋊C4—22no—
80C4⋊C4—4222no—
81C4⋊C4—4222no—
82C4⋊C4—2112no—
83C4⋊C4C4×D21112yes—
84C22⋊C4—4222no—
85C42—4222no—
86C42—2111no—
87C42—2112no—
88C42C22×D21111yescharacteristic
89C23—2111no—
90Q8—8444no—
91C2×C4—22no—
92C2×C4—8442no—
93C2×C4—4222no—
94C2×C4—4222no—
95C2×C4—2112no—
96C2×C4—2111no—
97C2×C4C2×C4×D21111yescharacteristic
98C2×C4D4×D21112yes—
99C8—42no—
100C8—8442no—
101C22—4221no—
102C22—4222no—
103C22C22⋊C4×D21111yescharacteristic
104C4—21no—
105C4—22no—
106C4—42no—
107C4—8442no—
108C4—4222no—
109C4—2112no—
110C2—8441no—
111C2#101×D21111yescharacteristic
112C2#2721316112yes—
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#330×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×D211C22⋊C4
cosocleC42×C211C23
hypercenter#330D211
hypocenterC211#1537
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
3. #27212888C2semidirect
4. #27212887C2semidirect
5. #330×C211C2semidirect
56. C211#1537semidirect
62. #330D211semidirect

Maximal subgroups

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

subgroupindex
#272137022
#272132012
#272128882
#272128872
#330×C2112
#1537211