Finite Group Databasebeta

← #41012440

Subgroups — #41012440

Order 27008.

202 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213479C212yesmaximal
2#27213085C211yesmaximal, characteristic
3#27212883C212yesmaximal
4#27212872C211yesmaximal, characteristic
5#544×C211C211yesmaximal, characteristic
6C2×C4⋊Dic211C2211yescharacteristic
7#20269910—22no—
8#20269901—22no—
9C22⋊(C211⋊C8)C2212yes—
10C211⋊#95—21no—
11C211⋊#95—21no—
12C211⋊#95C2211yescharacteristic
13C4⋊Q8×C211—22no—
14#125×C211—22no—
15C22⋊Q8×C211—21no—
16C22⋊Q8×C211C2212yes—
17#117×C211—21no—
18C2×C4⋊C4×C211C2211yescharacteristic
19C22×Dic211—21no—
20C22×Dic211—21no—
21C2×C211⋊C8—22no—
22C4⋊Dic211—22no—
23C2×Q8×C211—21no—
24C2×Q8×C211D412yes—
25C22×C4×C211D411yescharacteristic
26C22×C4×C211D411yescharacteristic
27C22×C4×C211C2311yescharacteristic
28C4⋊C4×C211—21no—
29C4⋊C4×C211—21no—
30C4⋊C4×C211—21no—
31C4⋊C4×C211—22no—
32C4⋊C4×C211—22no—
33C4⋊C4×C211—22no—
34C4⋊C4×C211D412yes—
35C22⋊C4×C211—22no—
36C42×C211—22no—
37C2×Dic211—41no—
38C2×Dic211—41no—
39C2×Dic211—42no—
40C2×Dic211—41no—
41C2×Dic211—41no—
42C211⋊C8—42no—
43C23×C211C2×D411yescharacteristic
44Q8×C211—42no—
45Q8×C211—81no—
46C2×C4×C211—21no—
47C2×C4×C211—21no—
48C2×C4×C211—22no—
49C2×C4×C211—22no—
50C2×C4×C211—21no—
51C2×C4×C211—41no—
52C2×C4×C211—41no—
53C2×C4×C211Q1612yes—
54C2×C4×C211C2×D412yes—
55C22×C211—21no—
56C22×C211—21no—
57C22×C211#12011yescharacteristic
58C22×C211C2×Q1611yescharacteristic
59C22×C211#13711yescharacteristic
60C4×C211—22no—
61C4×C211—22no—
62C4×C211—42no—
63C4×C211—41no—
64C4×C211—42no—
65Dic211—81no—
66Dic211—81no—
67C2×C211—21no—
68C2×C211—21no—
69C2×C211#45111yescharacteristic
70C2×C211#45411yescharacteristic
71C2×C211#45811yescharacteristic
72C211#161711yescharacteristic, complemented, Sylow
73#1617—2111nomaximal, Sylow
74#544D21111yescharacteristic
75#484—2112no—
76#400—2111no—
77#328—2112no—
78#324—2111no—
79C4⋊Q8—22no—
80#125—22no—
81C22⋊Q8—21no—
82C22⋊Q8C2×D21112yes—
83#117—21no—
84C2×C4⋊C4—2111no—
85C2×C4⋊C4C2×D21111yescharacteristic
86#107—4222no—
87#103—4222no—
88C22⋊C8—2112no—
89#95—4221no—
90#95—4221no—
91#95—2111no—
92C2×Q8—21no—
93C2×Q8C22⋊D21112yes—
94C22×C4—4221no—
95C22×C4—4221no—
96C22×C4C22⋊D21111yescharacteristic
97C22×C4C22⋊D21111yescharacteristic
98C22×C4C22×D21111yescharacteristic
99C2×C8—4222no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—4222no—
107C4⋊C4C22⋊D21112yes—
108C22⋊C4—22no—
109C42—22no—
110C23C2×C22⋊D21111yescharacteristic
111Q8—42no—
112Q8—81no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—21no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—8441no—
121C2×C4—8441no—
122C2×C4—8442no—
123C2×C4—8441no—
124C2×C4—8441no—
125C2×C4#1680995212yes—
126C2×C4C2×C22⋊D21112yes—
127C8—8442no—
128C22—21no—
129C22—21no—
130C22#2026994611yescharacteristic
131C22C2×#1680995211yescharacteristic
132C22#2027000811yescharacteristic
133C4—22no—
134C4—22no—
135C4—42no—
136C4—41no—
137C4—42no—
138C4—16881no—
139C4—16881no—
140C2—21no—
141C2—21no—
142C2#2721330211yescharacteristic
143C2#2721331911yescharacteristic
144C2#2721333611yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#544×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#544D211
hypocenterC211#1617
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
72. C211#1617semidirect

Maximal subgroups

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

subgroupindex
#272134792
#272130852
#272128832
#272128722
#544×C2112
#1617211