Finite Group Databasebeta

← #27144235

Subgroups — #27144235

Order 12416.

1224 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 249 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, complemented
2C2×#15690779C211yesmaximal, characteristic
3C2×#15690770C213yesmaximal, complemented
4#20260612C213yesmaximal, complemented
5#20260607C213yesmaximal, complemented
6C22⋊C4×Dic97C213yesmaximal, complemented
7#379×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97C2213yes—
9C23×Dic97—212no—
10C23×Dic97C2213yes—
11C23×Dic97C2213yes—
12C22×C4×D97C2211yescharacteristic, complemented
13#15690779C418yescomplemented
14#15690770—212no—
15C2×#13095490C2213yescomplemented
16C2×C22⋊Dic97C2213yes—
17C2×#13095488C2213yescomplemented
18C2×C4×Dic97C2213yescomplemented
19#15690717C2216yescomplemented
20C25×C97C2211yescharacteristic
21C2×C22⋊C4×C97C2213yes—
22C2×C22⋊C4×C97C2213yes—
23C23×D97C2311yescharacteristic
24C2×C22⋊D97—26no—
25C2×C22⋊D97—26no—
26C2×C22⋊D97C2×C4112yescomplemented
27C22×Dic97—232no—
28C22×Dic97—26no—
29C22×Dic97—212no—
30C22×Dic97—212no—
31C22×Dic97D4112yescomplemented
32C22×Dic97C2313yes—
33C22×Dic97C2313yes—
34C22×Dic97C2311yescharacteristic
35C2×C4×D97—26no—
36#13095490—26no—
37C22⋊Dic97C2×C4112yescomplemented
38#13095488—26no—
39C4×Dic97—26no—
40C24×C97—26no—
41C24×C97—26no—
42C24×C97C2×C414yescomplemented
43C24×C97C2313yes—
44C22×C4×C97C2313yes—
45C22×C4×C97C2311yescharacteristic
46C22⋊C4×C97—26no—
47C22⋊C4×C97—26no—
48C23×C97—224no—
49C23×C97—28no—
50C23×C97—224no—
51C23×C97—26no—
52C23×C97—26no—
53C23×C97C22×C416yes—
54C23×C97C2×D416yes—
55C23×C97#4116yes—
56C23×C97C2411yescharacteristic
57C22×D97—23no—
58C22×D97—23no—
59C22×D97C22×C412yescomplemented
60C2×C4×C97—23no—
61C2×C4×C97—23no—
62C2×C4×C97—26no—
63C22⋊D97—224no—
64C2×Dic97—224no—
65C2×Dic97—23no—
66C2×Dic97—26no—
67C2×Dic97—23no—
68C2×Dic97—224no—
69C2×Dic97C22×C416yescomplemented
70C2×Dic97C2×D416yes—
71C2×Dic97C2×D416yes—
72C4×D97—44no—
73C22×C97—232no—
74C22×C97—212no—
75C22×C97—26no—
76C22×C97—26no—
77C22×C97—212no—
78C22×C97C4×D4112yescomplemented
79C22×C97C23×C411yescharacteristic
80C22×C97C22×D413yes—
81C22×C97C2×#4113yes—
82C2×D97—26no—
83C2×D97—41no—
84C2×D97—43no—
85C4×C97—41no—
86C4×C97—43no—
87Dic97—26no—
88Dic97—26no—
89Dic97#12014yescomplemented
90C2×C97—212no—
91C2×C97C2×C4×D413yes—
92C2×C97C2×#12011yescharacteristic
93C2×C97#52513yes—
94D97—42no—
95C4×#120—971nomaximal, Sylow
96C97C4×#12011yescharacteristic, complemented, Sylow
97C24×C4—971no—
98C2×#120—971no—
99C2×C4×D4—973no—
100#386—973no—
101#385—973no—
102#379D9711yescharacteristic, complemented
103C4×C22⋊C4—973no—
104C25C2×D9711yescharacteristic
105C22×D4—973no—
106C23×C4—973no—
107C23×C4—973no—
108C23×C4—971no—
109C23×C4—19412no—
110#120—978no—
111C4×D4—19412no—
112C2×C4⋊C4—973no—
113C2×C22⋊C4—973no—
114C2×C22⋊C4—973no—
115C2×C22⋊C4C2×D9713yes—
116C2×C22⋊C4C2×D9713yes—
117C2×C42—973no—
118#95—976no—
119C24—26no—
120C24—26no—
121C24—971no—
122C24C4×D9714yescomplemented
123C24C22×D9713yes—
124C2×D4—9712no—
125C2×D4—1946no—
126C2×D4—1946no—
127C22×C4—971no—
128C22×C4—9712no—
129C22×C4—973no—
130C22×C4—973no—
131C22×C4—1946no—
132C22×C4—19412no—
133C22×C4—19412no—
134C22×C4—19432no—
135C22×C4—1946no—
136C22×C4C22×D9713yes—
137C22×C4C22×D9711yescharacteristic
138C4⋊C4—1946no—
139C22⋊C4—26no—
140C22⋊C4—26no—
141C22⋊C4—9712no—
142C22⋊C4—1946no—
143C42—1946no—
144C23—28no—
145C23—224no—
146C23—224no—
147C23—26no—
148C23—26no—
149C23—972no—
150C23—1943no—
151C23—1943no—
152C23C2×C4×D9716yes—
153C23D4×D9716yes—
154C23#1309551016yes—
155C23C23×D9711yescharacteristic
156D4—19424no—
157C2×C4—23no—
158C2×C4—23no—
159C2×C4—26no—
160C2×C4—976no—
161C2×C4—976no—
162C2×C4—976no—
163C2×C4—3884no—
164C2×C4—19424no—
165C2×C4—19424no—
166C2×C4—1943no—
167C2×C4—1946no—
168C2×C4—1943no—
169C22—232no—
170C22—212no—
171C22—26no—
172C22—26no—
173C22—212no—
174C22—3881no—
175C22—3883no—
176C22—1946no—
177C22#15690770112yescomplemented
178C22C22×C4×D9711yescharacteristic
179C22C2×D4×D9713yes—
180C22C2×#1309551013yes—
181C4—41no—
182C4—43no—
183C4—974no—
184C4—1946no—
185C4—1946no—
186C2—212no—
187C2—3882no—
188C2C2×#1569077013yes—
189C2#120×D9711yescharacteristic
190C2#2026124113yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#379×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690779C2
hypercenter#379D97
hypocenterC97C4×#120
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C24×Dic97C2semidirect
3. C2×#15690770C2semidirect
4. #20260612C2semidirect
5. #20260607C2semidirect
6. C22⋊C4×Dic97C2semidirect
7. #379×C97C2semidirect
12. C22×C4×D97C22semidirect
13. #15690779C4semidirect
15. C2×#13095490C22semidirect
17. C2×#13095488C22semidirect
18. C2×C4×Dic97C22semidirect
19. #15690717C22semidirect
26. C2×C22⋊D97C2×C4semidirect
31. C22×Dic97D4semidirect
37. C22⋊Dic97C2×C4semidirect
42. C24×C97C2×C4semidirect
59. C22×D97C22×C4semidirect
69. C2×Dic97C22×C4semidirect
78. C22×C97C4×D4semidirect
89. Dic97#120semidirect
96. C97C4×#120semidirect
102. #379D97semidirect
122. C24C4×D97semidirect
177. C22#15690770semidirect

Maximal subgroups

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

subgroupindex
C24×Dic972
C2×#156907792
C2×#156907702
#202606122
#202606072
C22⋊C4×Dic972
#379×C972
C4×#12097