Finite Group Databasebeta

← #27141018

Subgroups — #27141018

Order 12416.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261390C211yesmaximal, characteristic
2#20261131C211yesmaximal, characteristic, complemented
3#20261012C211yesmaximal, characteristic, complemented
4#20260480C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#461×C97C211yesmaximal, characteristic, complemented
8C2×#13095511—21no—
9#15690810C2211yescharacteristic
10#15690792—22no—
11#15690789—21no—
12#15690781—21no—
13C4×D388C2211yescharacteristic
14C4⋊C4×D97—21no—
15#15690743—21no—
16#15690741—22no—
17C4⋊#11715895C2211yescharacteristic
18#15690730—21no—
19C4×#11715895C2211yescharacteristic
20C2×SD16×C97—21no—
21C4⋊Q8×C97C2211yescharacteristic
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#103×C97—21no—
25#13095511—42no—
26#13095511—42no—
27C2×Dic194—21no—
28C2×D388D411yescharacteristic
29C2×C22⋊D97—22no—
30C2×C4×D97—22no—
31C2×C4×D97—21no—
32C2×#11715895—21no—
33C2×#11715895—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97D411yescharacteristic
36C4⋊Dic97D412yes—
37#13095490—22no—
38C22⋊Dic97—22no—
39#13095488—21no—
40C2×Q8×C97—21no—
41C2×D4×C97D411yescharacteristic
42C22×C4×C97—21no—
43SD16×C97—42no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C22⋊C4×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51C22×D97—21no—
52Q8×C97—41no—
53Q8×C97—41no—
54D4×C97—21no—
55D4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C22⋊D97—44no—
64C2×Dic97—21no—
65C2×Dic97—22no—
66D388—21no—
67D388SD1612yescomplemented
68C4×D97—44no—
69C4×D97—41no—
70C4×D97—41no—
71Dic194—41no—
72Dic194—41no—
73C8×C97—41no—
74#11715895—22no—
75#11715895—41no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97#12011yescharacteristic
80C2×D97—22no—
81C2×D97—41no—
82C2×D97—41no—
83C4×C97—21no—
84C4×C97—41no—
85C4×C97—81no—
86C4×C97C2×SD1611yescharacteristic
87C4×C97#13511yescharacteristic
88C4×C97AGL(1,ℤ/8)11yescharacteristic
89C4×C97AGL(1,ℤ/8)11yescharacteristic
90Dic97—41no—
91Dic97—42no—
92C2×C97—81no—
93C2×C97#44911yescharacteristic
94C2×C97#45011yescharacteristic
95C2×C97#45611yescharacteristic
96D97—42no—
97#1654—971nomaximal, Sylow
98C97#165411yescharacteristic, complemented, Sylow
99#547—971no—
100#498—971no—
101#474—971no—
102#461D9711yescharacteristic, complemented
103#332—971no—
104D4⋊C8—971no—
105D4⋊C8—971no—
106C2×#41—1941no—
107C2×SD16—21no—
108C4⋊Q8—971no—
109C4⋊Q8C2×D9711yescharacteristic
110#123—1942no—
111C22⋊Q8—1941no—
112#121—1941no—
113C4×D4—971no—
114C4×D4C2×D9711yescharacteristic
115C2×C4⋊C4—1941no—
116#107—1941no—
117#106—1942no—
118C4⋊C8—971no—
119C4⋊C8C2×D9711yescharacteristic
120#103—21no—
121#102—1941no—
122C4×C8—971no—
123#41—3882no—
124#41—3882no—
125C2×Q8—21no—
126C2×Q8—1941no—
127C2×D4—971no—
128C2×D4—1942no—
129C2×D4C22⋊D9711yescharacteristic
130C22×C4—21no—
131C22×C4—1942no—
132C22×C4—1941no—
133SD16—42no—
134C2×C8—21no—
135C2×C8—1941no—
136C2×C8—1941no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—972no—
140C4⋊C4—971no—
141C4⋊C4—1942no—
142C4⋊C4—1941no—
143C4⋊C4C22⋊D9711yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—1942no—
146C22⋊C4—1941no—
147C42C22×D9711yescharacteristic
148C23—21no—
149C23—1941no—
150Q8—41no—
151Q8—41no—
152Q8—3881no—
153Q8—3881no—
154D4—21no—
155D4—21no—
156D4—972no—
157D4—3884no—
158D4—1941no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—3884no—
164C2×C4—3881no—
165C2×C4—3881no—
166C2×C4—1941no—
167C2×C4—1942no—
168C2×C4D4×D9711yescharacteristic
169C2×C4D4×D9711yescharacteristic
170C2×C4C2×C22⋊D9711yescharacteristic
171C8—41no—
172C8—3881no—
173C8—1942no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—3881no—
178C22—3881no—
179C22—1942no—
180C22#1569077811yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—3881no—
185C4—3882no—
186C4SD16×D9711yescharacteristic
187C4#1569083011yescharacteristic
188C4#1569083611yescharacteristic
189C4#1569083711yescharacteristic
190C2—81no—
191C2—3882no—
192C2#2026087811yescharacteristic
193C2#2026088411yescharacteristic
194C2#2026091611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#461×C97C2
socleC22×C97#120
prime radical#20261390C2
hypercenter#461D97
hypocenterC97#1654
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #20261131C2semidirect
3. #20261012C2semidirect
4. #20260480C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #461×C97C2semidirect
67. D388SD16semidirect
98. C97#1654semidirect
102. #461D97semidirect

Maximal subgroups

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

subgroupindex
#202613902
#202611312
#202610122
#202604802
D4⋊#117158952
#202604622
#461×C972
#165497