Finite Group Databasebeta

← #41012442

Subgroups — #41012442

Order 27008.

202 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 53 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#27213777C211yesmaximal, characteristic
2#27213482C212yesmaximal
3#27213085C211yesmaximal, characteristic
4#27212884C212yesmaximal
5#324×C211C211yesmaximal, characteristic
6#20269976—22no—
7#20269968—22no—
8#20269956—21no—
9#20269956C2212yes—
10#20269936—21no—
11C2×C4⋊Dic211C2211yescharacteristic
12#20269911—22no—
13#20269902—22no—
14C211⋊#95—21no—
15C211⋊#95—21no—
16C211⋊#95C2211yescharacteristic
17C2×C4⋊C4×C211C2211yescharacteristic
18C22⋊C8×C211C2212yes—
19C2×Dic422—21no—
20C2×Dic422D412yes—
21C22×Dic211—21no—
22C22×Dic211—21no—
23C4⋊Dic211—21no—
24C4⋊Dic211—21no—
25C4⋊Dic211—21no—
26C4⋊Dic211—22no—
27C4⋊Dic211D412yes—
28Dic211⋊C4—22no—
29Dic211⋊C4—22no—
30C22⋊Dic211—22no—
31C4×Dic211—22no—
32C22×C4×C211D411yescharacteristic
33C22×C4×C211D411yescharacteristic
34C22×C4×C211C2311yescharacteristic
35C2×C8×C211—22no—
36C4⋊C4×C211—22no—
37Dic422—42no—
38Dic422—81no—
39C2×Dic211—22no—
40C2×Dic211—22no—
41C2×Dic211—41no—
42C2×Dic211—41no—
43C2×Dic211—42no—
44C2×Dic211—41no—
45C2×Dic211—41no—
46C23×C211C2×D411yescharacteristic
47C2×C4×C211—21no—
48C2×C4×C211—21no—
49C2×C4×C211—21no—
50C2×C4×C211—41no—
51C2×C4×C211—41no—
52C2×C4×C211Q1612yes—
53C2×C4×C211C2×D412yes—
54C8×C211—42no—
55C22×C211—21no—
56C22×C211—21no—
57C22×C211#12011yescharacteristic
58C22×C211C2×Q1611yescharacteristic
59C22×C211#13711yescharacteristic
60C4×C211—22no—
61C4×C211—22no—
62C4×C211—41no—
63Dic211—42no—
64Dic211—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#544—2111no—
75#484—2112no—
76#400—2111no—
77#328—2112no—
78#324D21111yescharacteristic
79C4⋊Q8—4222no—
80#125—4222no—
81C22⋊Q8—4221no—
82C22⋊Q8—2112no—
83#117—4221no—
84C2×C4⋊C4—2111no—
85C2×C4⋊C4C2×D21111yescharacteristic
86#107—4222no—
87#103—4222no—
88C22⋊C8C2×D21112yes—
89#95—4221no—
90#95—4221no—
91#95—2111no—
92C2×Q8—4221no—
93C2×Q8—2112no—
94C22×C4—4221no—
95C22×C4—4221no—
96C22×C4D84411yescharacteristic
97C22×C4D84411yescharacteristic
98C22×C4C22×D21111yescharacteristic
99C2×C8—22no—
100C4⋊C4—22no—
101C4⋊C4—4221no—
102C4⋊C4—4221no—
103C4⋊C4—4221no—
104C4⋊C4—4222no—
105C4⋊C4—4222no—
106C4⋊C4—4222no—
107C4⋊C4—2112no—
108C22⋊C4—4222no—
109C42—4222no—
110C23C2×D84411yescharacteristic
111Q8—8442no—
112Q8—16881no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—8441no—
119C2×C4—8441no—
120C2×C4—8442no—
121C2×C4—8441no—
122C2×C4—8441no—
123C2×C4—4222no—
124C2×C4—4222no—
125C2×C4Dic84412yes—
126C2×C4D4×D21112yes—
127C8—42no—
128C22—21no—
129C22—21no—
130C22#2026994711yescharacteristic
131C22C2×Dic84411yescharacteristic
132C22#2027001011yescharacteristic
133C4—22no—
134C4—22no—
135C4—41no—
136C4—8442no—
137C4—8442no—
138C4—16881no—
139C4—16881no—
140C2—21no—
141C2—21no—
142C2#2721330411yescharacteristic
143C2#2721332111yescharacteristic
144C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#324×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#324D211
hypocenterC211#1617
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

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
#272137772
#272134822
#272130852
#272128842
#324×C2112
#1617211