Finite Group Databasebeta

← #27140836

Subgroups — #27140836

Order 12416.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261400C211yesmaximal, characteristic, complemented
2#20260932C211yesmaximal, characteristic, complemented
3#20260927C211yesmaximal, characteristic
4#20260481C211yesmaximal, characteristic, complemented
5#20260467C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic, complemented
7#486×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97—21no—
9C2×#13095499—21no—
10C2×#13095496—21no—
11#15690809—21no—
12#15690788—22no—
13#15690786—21no—
14C4×Dic194C2211yescharacteristic
15C4×D388C2211yescharacteristic
16#15690772—21no—
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690731—21no—
20#15690731—21no—
21C4⋊Q8×C97C2211yescharacteristic
22#127×C97C2211yescharacteristic
23#102×C97—22no—
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—22no—
26C2×Dic194D411yescharacteristic
27Q8×D97—42no—
28Q8×D97—42no—
29C2×D388D411yescharacteristic
30C2×C4×D97—22no—
31C2×C4×D97—21no—
32#13095499—42no—
33#13095496—42no—
34C2×#11715895—21no—
35C2×#11715895—21no—
36C4⋊Dic97D411yescharacteristic
37C4⋊Dic97D411yescharacteristic
38#13095490—22no—
39#13095490—21no—
40#13095488—22no—
41#13095488—21no—
42C4×Dic97—21no—
43C2×Q8×C97—21no—
44C2×D4×C97—21no—
45C2×D4×C97—21no—
46C2×D4×C97—21no—
47C2×C8×C97—21no—
48C4⋊C4×C97—21no—
49C4⋊C4×C97D412yes—
50C42×C97C2311yescharacteristic
51C23×C97—41no—
52C22×D97—21no—
53Q8×C97—41no—
54Q8×C97—41no—
55D4×C97—42no—
56D4×C97—41no—
57D4×C97—41no—
58D4×C97—41no—
59D4×C97—41no—
60C2×C4×C97—22no—
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×Dic97—21no—
65C2×Dic97—21no—
66C2×Dic97—21no—
67D388—21no—
68D388SD1612yescomplemented
69C4×D97—44no—
70C4×D97—41no—
71C4×D97—41no—
72Dic194—21no—
73Dic194—44no—
74Dic194D812yescomplemented
75C8×C97—41no—
76#11715895—41no—
77#11715895—41no—
78C22×C97—81no—
79C22×C97—81no—
80C22×C97—81no—
81C22×C97#12011yescharacteristic
82C2×D97—22no—
83C2×D97—41no—
84C2×D97—41no—
85C4×C97—21no—
86C4×C97—42no—
87C4×C97C2×D811yescharacteristic
88C4×C97C2×SD1611yescharacteristic
89C4×C97AGL(1,ℤ/8)11yescharacteristic
90C4×C97#13711yescharacteristic
91Dic97—22no—
92Dic97—41no—
93Dic97—41no—
94C2×C97—161no—
95C2×C97#44711yescharacteristic
96C2×C97#44811yescharacteristic
97C2×C97#45411yescharacteristic
98D97—42no—
99#1627—971nomaximal, Sylow
100C97#162711yescharacteristic, complemented, Sylow
101D4×Q8—971no—
102#486D9711yescharacteristic, complemented
103#460—971no—
104#459—971no—
105#332—971no—
106Q8⋊C8—971no—
107D4⋊C8—971no—
108C22×Q8—1941no—
109C2×SD16—1941no—
110C2×D8—1941no—
111C4⋊Q8—1941no—
112C4⋊Q8C2×D9711yescharacteristic
113#127C2×D9711yescharacteristic
114C22⋊Q8—1941no—
115C22⋊Q8—1942no—
116C4×Q8—971no—
117C4×D4—971no—
118C4×D4—1941no—
119C4⋊C8—971no—
120C4⋊C8—971no—
121#102—22no—
122#102—1941no—
123#102—1941no—
124C4×C8C2×D9711yescharacteristic
125C2×Q8—21no—
126C2×Q8—971no—
127C2×Q8—3882no—
128C2×Q8—3882no—
129C2×Q8—1942no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—971no—
134C22×C4—1942no—
135C22×C4—1941no—
136SD16—3882no—
137D8—3882no—
138C2×C8—21no—
139C2×C8—1941no—
140C2×C8—1941no—
141C4⋊C4—21no—
142C4⋊C4—971no—
143C4⋊C4—971no—
144C4⋊C4—1942no—
145C4⋊C4—1941no—
146C4⋊C4C22⋊D9712yes—
147C22⋊C4—1942no—
148C22⋊C4—1941no—
149C42—1941no—
150C42C22×D9711yescharacteristic
151C23—41no—
152C23—1941no—
153Q8—41no—
154Q8—41no—
155Q8—972no—
156Q8—3884no—
157Q8—1941no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—972no—
164D4—1941no—
165C2×C4—22no—
166C2×C4—3884no—
167C2×C4—3881no—
168C2×C4—3881no—
169C2×C4—1941no—
170C2×C4—1941no—
171C2×C4—1941no—
172C2×C4D4×D9711yescharacteristic
173C2×C4D4×D9711yescharacteristic
174C2×C4C2×C22⋊D9711yescharacteristic
175C8—41no—
176C8—3881no—
177C8—3881no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—3881no—
182C22—3881no—
183C22—1942no—
184C22#1569077811yescharacteristic
185C4—21no—
186C4—42no—
187C4—3881no—
188C4—3881no—
189C4—1942no—
190C4D8×D9711yescharacteristic
191C4SD16×D9711yescharacteristic
192C4#1569083411yescharacteristic
193C4#1569083911yescharacteristic
194C2—161no—
195C2—3882no—
196C2#2026086711yescharacteristic
197C2#2026087211yescharacteristic
198C2#2026090711yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#486×C97C2
socleC22×C97#120
prime radical#20260927C2
hypercenter#486D97
hypocenterC97#1627
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #20261400C2semidirect
2. #20260932C2semidirect
4. #20260481C2semidirect
5. #20260467C2semidirect
6. #20260464C2semidirect
7. #486×C97C2semidirect
68. D388SD16semidirect
74. Dic194D8semidirect
100. C97#1627semidirect
102. #486D97semidirect

Maximal subgroups

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

subgroupindex
#202614002
#202609322
#202609272
#202604812
#202604672
#202604642
#486×C972
#162797