Finite Group Databasebeta

← #27140794

Subgroups — #27140794

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#20261470C211yesmaximal, characteristic, complemented
2#20261111C211yesmaximal, characteristic
3#20260932C211yesmaximal, characteristic, complemented
4#20260478C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260465C211yesmaximal, characteristic, complemented
7#460×C97C211yesmaximal, characteristic, complemented
8C2×#13095499—21no—
9C2×#13095496—22no—
10#15690807C2211yescharacteristic
11#15690799—21no—
12#15690798—21no—
13#15690795—22no—
14C4×Dic194C2211yescharacteristic
15C4×D388—21no—
16#15690772—21no—
17C4⋊#11715895C2211yescharacteristic
18#15690731—21no—
19C4×#11715895C2211yescharacteristic
20C2×SD16×C97—21no—
21#127×C97C2211yescharacteristic
22C4×Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194D411yescharacteristic
26C2×D388—21no—
27C2×D388—21no—
28C2×D388D412yes—
29C2×C4×D97—22no—
30#13095499—42no—
31#13095496—44no—
32C2×#11715895—21no—
33C2×#11715895—21no—
34C4⋊Dic97D411yescharacteristic
35#13095490—21no—
36#13095488—22no—
37#13095488—22no—
38#13095488—22no—
39C4×Dic97—21no—
40C2×Q8×C97D411yescharacteristic
41C2×D4×C97—21no—
42C2×D4×C97—21no—
43C2×D4×C97—21no—
44SD16×C97—42no—
45C2×C8×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C42×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—41no—
51C22×D97—22no—
52Q8×C97—21no—
53Q8×C97—21no—
54D4×C97—42no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—41no—
58D4×C97—41no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×Dic97—21no—
65C2×Dic97—21no—
66D388—42no—
67D388—41no—
68D388—41no—
69D388—41no—
70D388—41no—
71C4×D97—82no—
72Dic194—21no—
73Dic194—21no—
74C8×C97—41no—
75#11715895—22no—
76#11715895—41no—
77C22×C97—81no—
78C22×C97—81no—
79C22×C97—81no—
80C22×C97#12011yescharacteristic
81C2×D97—42no—
82C2×D97—42no—
83C2×D97—42no—
84C4×C97—21no—
85C4×C97—41no—
86C4×C97—41no—
87C4×C97#13511yescharacteristic
88C4×C97#13511yescharacteristic
89C4×C97AGL(1,ℤ/8)11yescharacteristic
90C4×C97AGL(1,ℤ/8)11yescharacteristic
91Dic97—41no—
92Dic97—41no—
93C2×C97—161no—
94C2×C97#44911yescharacteristic
95C2×C97#44911yescharacteristic
96C2×C97#45311yescharacteristic
97D97—82no—
98#1620—971nomaximal, Sylow
99C97#162011yescharacteristic, complemented, Sylow
100#559—971no—
101#493—971no—
102#460—971no—
103#460D9711yescharacteristic, complemented
104#331—971no—
105Q8⋊C8—971no—
106Q8⋊C8—971no—
107C2×SD16—21no—
108C2×SD16—1941no—
109C2×D8—1942no—
110#127—971no—
111#127C2×D9711yescharacteristic
112#124—1941no—
113#124—1941no—
114#123—1942no—
115C4×Q8—971no—
116C4×Q8C2×D9711yescharacteristic
117C4×D4—1941no—
118C4×D4—1941no—
119C4⋊C8—971no—
120C4⋊C8C2×D9711yescharacteristic
121#102—21no—
122#102—1941no—
123C4×C8—971no—
124C2×Q8—971no—
125C2×Q8C22⋊D9711yescharacteristic
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—972no—
130C2×D4—1941no—
131C2×D4—1941no—
132C22×C4—1942no—
133SD16—42no—
134SD16—3882no—
135D8—3884no—
136C2×C8—21no—
137C2×C8—1941no—
138C2×C8—1941no—
139C4⋊C4—21no—
140C4⋊C4—971no—
141C4⋊C4—1941no—
142C4⋊C4C22⋊D9711yescharacteristic
143C22⋊C4—1942no—
144C22⋊C4—1942no—
145C22⋊C4—1942no—
146C42—21no—
147C42—1941no—
148C42C22×D9711yescharacteristic
149C23—41no—
150C23—1942no—
151Q8—21no—
152Q8—21no—
153Q8—1941no—
154Q8—1941no—
155D4—42no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—3882no—
161D4—3881no—
162D4—3881no—
163D4—3881no—
164D4—3881no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—7762no—
168C2×C4—1941no—
169C2×C4—1941no—
170C2×C4D4×D9711yescharacteristic
171C2×C4D4×D9711yescharacteristic
172C2×C4C2×C22⋊D9711yescharacteristic
173C8—41no—
174C8—3881no—
175C8—1942no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—3882no—
180C22—3882no—
181C22—3882no—
182C22#1569077811yescharacteristic
183C4—21no—
184C4—41no—
185C4—41no—
186C4—3881no—
187C4—3881no—
188C4#1569083011yescharacteristic
189C4#1569083111yescharacteristic
190C4#1569083611yescharacteristic
191C4#1569083711yescharacteristic
192C2—7762no—
193C2—161no—
194C2#2026087811yescharacteristic
195C2#2026087911yescharacteristic
196C2#2026090111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#460×C97C2
socleC22×C97#120
prime radical#20261111C2
hypercenter#460D97
hypocenterC97#1620
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #20261470C2semidirect
3. #20260932C2semidirect
4. #20260478C2semidirect
5. Q8⋊#11715895C2semidirect
6. #20260465C2semidirect
7. #460×C97C2semidirect
99. C97#1620semidirect
103. #460D97semidirect

Maximal subgroups

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

subgroupindex
#202614702
#202611112
#202609322
#202604782
Q8⋊#117158952
#202604652
#460×C972
#162097