Finite Group Databasebeta

← #27140818

Subgroups — #27140818

Order 12416.

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

normal subgroups only

This group has 240 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#20261450C211yesmaximal, characteristic, complemented
2#20261107C211yesmaximal, characteristic, complemented
3#20260927C211yesmaximal, characteristic, complemented
4#20260483C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#459×C97C211yesmaximal, characteristic, complemented
8C2×#13095499—22no—
9C2×#13095496—21no—
10#15690810C2211yescharacteristic
11#15690796—22no—
12#15690784—21no—
13#15690781—21no—
14C4×D388C2211yescharacteristic
15#15690769—21no—
16#15690768—21no—
17C4⋊#11715895C2211yescharacteristic
18#15690731—21no—
19C4×#11715895C2211yescharacteristic
20C2×D8×C97—21no—
21#127×C97C2211yescharacteristic
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194D412yes—
26C2×D388D411yescharacteristic
27C2×C22⋊D97—22no—
28C2×C4×D97—21no—
29#13095499—44no—
30#13095496—42no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97D411yescharacteristic
36C22⋊Dic97—22no—
37#13095488—22no—
38#13095488—21no—
39C4×Dic97—22no—
40C2×D4×C97—21no—
41C2×D4×C97—21no—
42C2×D4×C97—21no—
43C2×D4×C97D411yescharacteristic
44C22×C4×C97—21no—
45D8×C97—42no—
46C2×C8×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C22⋊C4×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51C23×C97—41no—
52C22×D97—21no—
53D4×C97—21no—
54D4×C97—21no—
55D4×C97—42no—
56D4×C97—41no—
57D4×C97—41no—
58D4×C97—41no—
59D4×C97—41no—
60C2×C4×C97—21no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C22⋊D97—82no—
67C2×Dic97—21no—
68C2×Dic97—22no—
69D388—21no—
70D388—21no—
71C4×D97—41no—
72C4×D97—41no—
73Dic194—42no—
74C8×C97—41no—
75#11715895—22no—
76#11715895—41no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97—81no—
81C22×C97—81no—
82C22×C97—81no—
83C22×C97#12011yescharacteristic
84C2×D97—41no—
85C2×D97—41no—
86C2×D97—41no—
87C4×C97—21no—
88C4×C97—41no—
89C4×C97#13511yescharacteristic
90C4×C97#13511yescharacteristic
91C4×C97AGL(1,ℤ/8)11yescharacteristic
92C4×C97AGL(1,ℤ/8)11yescharacteristic
93Dic97—41no—
94Dic97—42no—
95C2×C97—81no—
96C2×C97—161no—
97C2×C97#44911yescharacteristic
98C2×C97#44911yescharacteristic
99C2×C97#45311yescharacteristic
100D97—81no—
101#1624—971nomaximal, Sylow
102C97#162411yescharacteristic, complemented, Sylow
103#555—971no—
104#492—971no—
105#459—971no—
106#459D9711yescharacteristic, complemented
107#333—971no—
108D4⋊C8—971no—
109D4⋊C8—971no—
110C2×SD16—1942no—
111C2×D8—21no—
112C2×D8—1941no—
113C4⋊Q8—971no—
114#127C2×D9711yescharacteristic
115#124—1942no—
116#121—1941no—
117#121—1941no—
118C4×D4—971no—
119C4×D4C2×D9711yescharacteristic
120#117—1941no—
121#117—1941no—
122C4⋊C8—971no—
123C4⋊C8C2×D9711yescharacteristic
124#102—21no—
125#102—1941no—
126C4×C8—971no—
127C2×Q8—972no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—971no—
132C2×D4—1942no—
133C2×D4C22⋊D9711yescharacteristic
134C22×C4—21no—
135C22×C4—1941no—
136SD16—3884no—
137D8—42no—
138D8—3882no—
139C2×C8—21no—
140C2×C8—1941no—
141C2×C8—1941no—
142C4⋊C4—971no—
143C4⋊C4—1941no—
144C4⋊C4—1941no—
145C4⋊C4C22⋊D9711yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—1942no—
148C22⋊C4—1942no—
149C22⋊C4—1941no—
150C42—1942no—
151C42C22×D9711yescharacteristic
152C23—21no—
153C23—41no—
154C23—1941no—
155Q8—3882no—
156D4—21no—
157D4—21no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—7762no—
164D4—1941no—
165D4—1941no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3881no—
170C2×C4—3881no—
171C2×C4—1941no—
172C2×C4—1942no—
173C2×C4D4×D9711yescharacteristic
174C2×C4D4×D9711yescharacteristic
175C2×C4C2×C22⋊D9711yescharacteristic
176C8—41no—
177C8—3881no—
178C8—1942no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—3881no—
186C22—3881no—
187C22—3881no—
188C22#1569077811yescharacteristic
189C4—21no—
190C4—41no—
191C4—3881no—
192C4—3882no—
193C4#1569082911yescharacteristic
194C4#1569083011yescharacteristic
195C4#1569083411yescharacteristic
196C4#1569083611yescharacteristic
197C2—81no—
198C2—7761no—
199C2—161no—
200C2#2026087711yescharacteristic
201C2#2026087811yescharacteristic
202C2#2026090111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#459×C97C2
socleC22×C97#120
hypercenter#459D97
hypocenterC97#1624
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20261450C2semidirect
2. #20261107C2semidirect
3. #20260927C2semidirect
4. #20260483C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #459×C97C2semidirect
102. C97#1624semidirect
106. #459D97semidirect

Maximal subgroups

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

subgroupindex
#202614502
#202611072
#202609272
#202604832
D4⋊#117158952
#202604622
#459×C972
#162497