Finite Group Databasebeta

← #41014838

Subgroups — #41014838

Order 27008.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C2×#20269911C211yesmaximal, characteristic, complemented
2#27213147C212yesmaximal
3#27212914C212yesmaximal, complemented
4#27212911C211yesmaximal, characteristic, complemented
5#408×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7C2×C4⋊Dic211C2212yes—
8#20269911—22no—
9#20269901—22no—
10#20269898—22no—
11C2×#41×C211C2211yescharacteristic
12C2×M16×C211C2211yescharacteristic
13C22×C8×C211C2211yescharacteristic
14C22⋊C8×C211—22no—
15C22×Dic211—22no—
16#16809946—22no—
17#16809946—22no—
18C2×C211⋊C8—22no—
19C4⋊Dic211—22no—
20C4⋊Dic211—22no—
21#41×C211—22no—
22#41×C211—22no—
23C2×Q8×C211D411yescharacteristic
24C2×D4×C211—21no—
25C2×D4×C211D411yescharacteristic
26C22×C4×C211—21no—
27C22×C4×C211C2311yescharacteristic
28M16×C211—22no—
29C2×C8×C211—22no—
30C2×C8×C211—21no—
31C2×C8×C211D412yes—
32C2×Dic211—42no—
33C2×Dic211—82no—
34C211⋊C8—42no—
35C23×C211—21no—
36C23×C211#4111yescharacteristic
37Q8×C211—22no—
38D4×C211—22no—
39D4×C211—42no—
40C2×C4×C211—21no—
41C2×C4×C211—41no—
42C2×C4×C211—41no—
43C2×C4×C211C2×D411yescharacteristic
44C2×C4×C211C2×D411yescharacteristic
45C2×C4×C211#4112yes—
46C2×C4×C211#4112yes—
47C8×C211—22no—
48C8×C211—41no—
49C22×C211—21no—
50C22×C211—21no—
51C22×C211—41no—
52C22×C211—41no—
53C22×C211—41no—
54C22×C211#12111yescharacteristic
55C22×C211#12612yes—
56C4×C211—41no—
57C4×C211#12311yescharacteristic
58C4×C211#12312yes—
59C4×C211#12411yescharacteristic
60Dic211—82no—
61C2×C211—22no—
62C2×C211—81no—
63C2×C211#39711yescharacteristic
64C2×C211#47211yescharacteristic
65C2×C211#47311yescharacteristic
66C211#207811yescharacteristic, complemented, Sylow
67#2078—2111nomaximal, Sylow
68C2×#107—2111no—
69#417—2112no—
70#408D21111yescharacteristic
71#343—2112no—
72#341—2111no—
73C2×#41C2×D21111yescharacteristic
74C2×M16—2112no—
75C2×M16C2×D21111yescharacteristic
76C22×C8C2×D21111yescharacteristic
77C2×C4⋊C4—2112no—
78#107—4222no—
79#103—4222no—
80#102—4222no—
81C22⋊C8—22no—
82#41—22no—
83#41—22no—
84C2×Q8D84411yescharacteristic
85C2×D4—21no—
86C2×D4D84411yescharacteristic
87C22×C4—21no—
88C22×C4—4222no—
89C22×C4C22×D21111yescharacteristic
90M16—22no—
91M16—4222no—
92M16—4222no—
93C2×C8—22no—
94C2×C8—21no—
95C2×C8—4222no—
96C2×C8C22⋊D21112yes—
97C4⋊C4—4222no—
98C4⋊C4—4222no—
99C23—21no—
100C23#1680996211yescharacteristic
101Q8—22no—
102D4—22no—
103D4—42no—
104C2×C4—21no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—8442no—
108C2×C4—16882no—
109C2×C4C2×C22⋊D21111yescharacteristic
110C2×C4C2×D84411yescharacteristic
111C2×C4#1680996112yes—
112C2×C4#1680996112yes—
113C8—22no—
114C8—41no—
115C8—8442no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—41no—
121C22#2026995111yescharacteristic
122C22#2026997012yes—
123C4—41no—
124C4—16882no—
125C4#2026995811yescharacteristic
126C4#2026996112yes—
127C4#2026996411yescharacteristic
128C2—22no—
129C2—81no—
130C2#2721307111yescharacteristic
131C2#2721341611yescharacteristic
132C2#2721342111yescharacteristic
subgroupG/H
centerC22#20269951
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#408×C211C2
socleC22×C211#121
prime radicalC2×#41×C211C22
hypercenter#408D211
hypocenterC211#2078
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#20269911C2semidirect
3. #27212914C2semidirect
4. #27212911C2semidirect
66. C211#2078semidirect

Maximal subgroups

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

subgroupindex
C2×#202699112
#272131472
#272129142
#272129112
#408×C2112
#2078211