Finite Group Databasebeta

← #37560081

Subgroups — #37560081

Order 23872.

290 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 107 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×#20238610C211yesmaximal, characteristic, complemented
2C22×Dic746C211yesmaximal, characteristic, complemented
3#26045780C211yesmaximal, characteristic, complemented
4#26045778C212yesmaximal, complemented
5#26045771C211yesmaximal, characteristic, complemented
6#26045770C212yesmaximal, complemented
7#26045766C211yesmaximal, characteristic, complemented
8#26045764C212yesmaximal, complemented
9C4×Dic746C211yesmaximal, characteristic, complemented
10#26045752C212yesmaximal, complemented
11C4×D4×C373C211yesmaximal, characteristic, complemented
12#20238610—24no—
13C2×Dic746—24no—
14C2×Dic746C2212yes—
15C2×Dic746C2211yescharacteristic
16C2×Dic746C2211yescharacteristic
17C2×D1492C2211yescharacteristic
18C2×C22⋊D373C2212yes—
19C2×C22⋊D373C2212yes—
20C22×Dic373C2212yes—
21C2×C4×D373C2212yes—
22C4⋊Dic373C2211yescharacteristic
23#20238589C2212yes—
24#20238589C2212yes—
25C22⋊Dic373C2212yes—
26#20238587C2212yes—
27#20238587C2212yes—
28#20238587C2212yes—
29C4×Dic373C2212yes—
30C2×D4×C373C2211yescharacteristic
31C22×C4×C373C2211yescharacteristic
32C22×C4×C373C2211yescharacteristic
33C4⋊C4×C373C2211yescharacteristic
34C22⋊C4×C373C2211yescharacteristic
35C22⋊C4×C373C2211yescharacteristic
36C42×C373C2211yescharacteristic
37C22×D373C2312yes—
38Dic746—22no—
39Dic746—24no—
40Dic746D414yescomplemented
41D1492—22no—
42C22⋊D373—24no—
43C22⋊D373—24no—
44C2×Dic373—24no—
45C2×Dic373C2312yes—
46C2×Dic373C2312yes—
47C2×Dic373C2312yes—
48C4×D373—24no—
49C23×C373C2311yescharacteristic
50C23×C373C2311yescharacteristic
51D4×C373—22no—
52C2×C4×C373—22no—
53C2×C4×C373—22no—
54C2×C4×C373C2311yescharacteristic
55C2×C4×C373C2311yescharacteristic
56C2×C4×C373C2311yescharacteristic
57C2×C4×C373C2311yescharacteristic
58C2×C4×C373C2311yescharacteristic
59C22×C373—21no—
60C22×C373—21no—
61C22×C373—21no—
62C22×C373—21no—
63C22×C373—21no—
64C22×C373#4112yescomplemented
65C22×C373C2411yescharacteristic
66C2×D373—22no—
67C2×D373—22no—
68C2×D373—22no—
69C4×C373—21no—
70C4×C373—21no—
71C4×C373—21no—
72C4×C373—21no—
73C4×C373C2×D412yes—
74Dic373—22no—
75Dic373—22no—
76Dic373C2×D414yes—
77C2×C373—22no—
78C2×C373—41no—
79C2×C373C22×D411yescharacteristic
80C2×C373C2×#4111yescharacteristic
81C2×C3732−1+411yescharacteristic
82D373—42no—
83C373#54811yescharacteristic, complemented, Sylow
84#548—3731nomaximal, Sylow
85C2×#41—3731no—
86C22×Q8—3731no—
87#124—3732no—
88#124—3731no—
89C22⋊Q8—3731no—
90C22⋊Q8—3732no—
91#121—3732no—
92#121—3731no—
93C4×Q8—3731no—
94C4×D4—3732no—
95C4×D4D37311yescharacteristic, complemented
96#41—7464no—
97C2×Q8—3731no—
98C2×Q8—3732no—
99C2×Q8—3731no—
100C2×Q8—7464no—
101C2×D4—3732no—
102C2×D4—3732no—
103C2×D4—3731no—
104C2×D4C2×D37311yescharacteristic
105C22×C4—3732no—
106C22×C4—3732no—
107C22×C4C2×D37311yescharacteristic
108C22×C4C2×D37311yescharacteristic
109C4⋊C4—3732no—
110C4⋊C4—3732no—
111C4⋊C4—3731no—
112C4⋊C4C2×D37311yescharacteristic
113C22⋊C4—3732no—
114C22⋊C4—3732no—
115C22⋊C4—3732no—
116C22⋊C4—3732no—
117C22⋊C4C2×D37311yescharacteristic
118C22⋊C4C2×D37311yescharacteristic
119C42—3732no—
120C42C2×D37311yescharacteristic
121C23—3732no—
122C23C22×D37311yescharacteristic
123C23C22×D37311yescharacteristic
124Q8—3734no—
125Q8—7464no—
126Q8—7462no—
127D4—22no—
128D4—7464no—
129D4—7464no—
130D4—7462no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—3732no—
134C2×C4—3732no—
135C2×C4—3732no—
136C2×C4—7464no—
137C2×C4—7464no—
138C2×C4C22×D37311yescharacteristic
139C2×C4C22×D37311yescharacteristic
140C2×C4C22×D37311yescharacteristic
141C2×C4C22×D37311yescharacteristic
142C2×C4C22×D37311yescharacteristic
143C22—21no—
144C22—21no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—7462no—
149C22—7462no—
150C22—7462no—
151C22#2023861012yescomplemented
152C22C23×D37311yescharacteristic
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—3734no—
158C4—7462no—
159C4—7462no—
160C4D4×D37312yes—
161C2—22no—
162C2—41no—
163C2—14922no—
164C2C2×D4×D37311yescharacteristic
165C2C2×#2023861011yescharacteristic
166C2#2604583911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
FittingC4×D4×C373C2
hypercenterC4×D4D373
hypocenterC373#548
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20238610C2semidirect
2. C22×Dic746C2semidirect
3. #26045780C2semidirect
4. #26045778C2semidirect
5. #26045771C2semidirect
6. #26045770C2semidirect
7. #26045766C2semidirect
8. #26045764C2semidirect
9. C4×Dic746C2semidirect
10. #26045752C2semidirect
11. C4×D4×C373C2semidirect
40. Dic746D4semidirect
64. C22×C373#41semidirect
83. C373#548semidirect
95. C4×D4D373semidirect
151. C22#20238610semidirect

Maximal subgroups

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

subgroupindex
C2×#202386102
C22×Dic7462
#260457802
#260457782
#260457712
#260457702
#260457662
#260457642
C4×Dic7462
#260457522
C4×D4×C3732
#548373