Finite Group Databasebeta

← #27150021

Subgroups — #27150021

Order 12416.

1068 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C24×Dic97C211yesmaximal, characteristic
2C2×#15690787C213yesmaximal, complemented
3#20260610C213yesmaximal, complemented
4#20260607C213yesmaximal, complemented
5#20260593C213yesmaximal, complemented
6#20260591C211yesmaximal, characteristic
7#379×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic, complemented
9C23×Dic97—212no—
10C23×Dic97C2213yes—
11C23×Dic97C2213yes—
12#15690787—212no—
13C2×C4⋊Dic97C2213yescomplemented
14C2×#13095490C2213yescomplemented
15C2×#13095490C2213yescomplemented
16C2×C22⋊Dic97C2213yes—
17C2×C22⋊Dic97C2213yes—
18#15690717C2216yescomplemented
19C25×C97C2211yescharacteristic
20C2×C22⋊C4×C97C2213yes—
21C2×C22⋊C4×C97C2213yes—
22C2×Dic194—26no—
23C22×Dic97—232no—
24C22×Dic97—26no—
25C22×Dic97—212no—
26C22×Dic97—212no—
27C22×Dic97D4112yes—
28C22×Dic97C2313yes—
29C22×Dic97C2311yescharacteristic
30C22×Dic97C2313yes—
31C22×Dic97C2311yescharacteristic
32C4⋊Dic97—26no—
33#13095490—26no—
34#13095490—26no—
35C22⋊Dic97—26no—
36C22⋊Dic97—26no—
37C24×C97—26no—
38C24×C97—26no—
39C24×C97Q814yescomplemented
40C24×C97C2313yes—
41C22×C4×C97C2311yescharacteristic
42C22×C4×C97C2313yes—
43C22⋊C4×C97—26no—
44C22⋊C4×C97—26no—
45C23×C97—224no—
46C23×C97—28no—
47C23×C97—224no—
48C23×C97—26no—
49C23×C97—26no—
50C23×C97C2×D416yes—
51C23×C97C2×Q816yes—
52C23×C97#4116yes—
53C23×C97C2411yescharacteristic
54C2×C4×C97—23no—
55C2×C4×C97—23no—
56C2×C4×C97—26no—
57C2×Dic97—224no—
58C2×Dic97—26no—
59C2×Dic97—224no—
60C2×Dic97—23no—
61C2×Dic97—23no—
62C2×Dic97—26no—
63C2×Dic97C2×D416yes—
64C2×Dic97C2×D416yes—
65Dic194—44no—
66C22×C97—232no—
67C22×C97—26no—
68C22×C97—212no—
69C22×C97—26no—
70C22×C97—212no—
71C22×C97C22⋊Q8112yescomplemented
72C22×C97C22×D413yes—
73C22×C97C22×Q811yescharacteristic
74C22×C97C2×#4113yes—
75C4×C97—41no—
76C4×C97—43no—
77Dic97—26no—
78Dic97—41no—
79Dic97—43no—
80Dic97#12014yes—
81C2×C97—212no—
82C2×C97C2×#12011yescharacteristic
83C2×C97C2×C22⋊Q813yes—
84C2×C97#52513yes—
85#2848—971nomaximal, Sylow
86C97#284811yescharacteristic, complemented, Sylow
87C24×C4—971no—
88C2×C22⋊Q8—973no—
89#385—973no—
90#385—973no—
91#380—973no—
92#379—971no—
93#379D9711yescharacteristic
94C25C2×D9711yescharacteristic
95C22×Q8—971no—
96C23×C4—973no—
97C23×C4—973no—
98C23×C4—19412no—
99C22⋊Q8—19412no—
100C2×C4⋊C4—973no—
101C2×C4⋊C4—973no—
102C2×C4⋊C4—973no—
103C2×C22⋊C4—973no—
104C2×C22⋊C4—973no—
105C2×C22⋊C4C2×D9713yes—
106C2×C22⋊C4C2×D9713yes—
107#95—976no—
108C24—26no—
109C24—26no—
110C24Dic19414yescomplemented
111C24C22×D9713yes—
112C2×Q8—1946no—
113C22×C4—971no—
114C22×C4—9712no—
115C22×C4—973no—
116C22×C4—973no—
117C22×C4—971no—
118C22×C4—19412no—
119C22×C4—19412no—
120C22×C4—19432no—
121C22×C4—1946no—
122C22×C4C22×D9713yes—
123C22×C4C22×D9711yescharacteristic
124C4⋊C4—1946no—
125C4⋊C4—1946no—
126C4⋊C4—1946no—
127C22⋊C4—26no—
128C22⋊C4—26no—
129C22⋊C4—1946no—
130C22⋊C4—1946no—
131C23—28no—
132C23—224no—
133C23—224no—
134C23—26no—
135C23—26no—
136C23D4×D9716yes—
137C23C2×Dic19416yes—
138C23#1309551016yes—
139C23C23×D9711yescharacteristic
140Q8—3884no—
141C2×C4—23no—
142C2×C4—23no—
143C2×C4—26no—
144C2×C4—976no—
145C2×C4—976no—
146C2×C4—19424no—
147C2×C4—1943no—
148C2×C4—1943no—
149C2×C4—1946no—
150C2×C4—19424no—
151C2×C4—1946no—
152C22—232no—
153C22—26no—
154C22—212no—
155C22—26no—
156C22—212no—
157C22#15690787112yescomplemented
158C22C2×D4×D9713yes—
159C22C22×Dic19411yescharacteristic
160C22C2×#1309551013yes—
161C4—41no—
162C4—43no—
163C4—974no—
164C4—3881no—
165C4—3883no—
166C4—1946no—
167C2—212no—
168C2#120×D9711yescharacteristic
169C2C2×#1569078713yes—
170C2#2026124113yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#379×C97C2
prime radicalC25×C97C22
hypercenter#379D97
hypocenterC97#2848
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#15690787C2semidirect
3. #20260610C2semidirect
4. #20260607C2semidirect
5. #20260593C2semidirect
8. C22×Dic194C22semidirect
13. C2×C4⋊Dic97C22semidirect
14. C2×#13095490C22semidirect
15. C2×#13095490C22semidirect
18. #15690717C22semidirect
39. C24×C97Q8semidirect
71. C22×C97C22⋊Q8semidirect
86. C97#2848semidirect
110. C24Dic194semidirect
157. C22#15690787semidirect

Maximal subgroups

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

subgroupindex
C24×Dic972
C2×#156907872
#202606102
#202606072
#202605932
#202605912
#379×C972
#284897