Finite Group Databasebeta

← #27143710

Subgroups — #27143710

Order 12416.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#20261536C211yesmaximal, characteristic
2#20261178C211yesmaximal, characteristic, complemented
3C2×#15690754C211yesmaximal, characteristic, complemented
4#20261011C211yesmaximal, characteristic, complemented
5#20260541C211yesmaximal, characteristic, complemented
6#20260520C211yesmaximal, characteristic, complemented
7#361×C97C211yesmaximal, characteristic, complemented
8#15690855—21no—
9#15690843—21no—
10#15690836—21no—
11#15690834—22no—
12#15690832C2211yescharacteristic
13#15690829—21no—
14C2×Dic388C2211yescharacteristic, complemented
15D8×D97—21no—
16#15690816C2211yescharacteristic
17Dic776—21no—
18#15690755—21no—
19#15690754—22no—
20C2×#13095486C2211yescharacteristic, complemented
21#15690728—21no—
22C2×D8×C97C2211yescharacteristic
23M32×C97C2211yescharacteristic, complemented
24#108×C97C2211yescharacteristic, complemented
25#13095511—21no—
26#13095511D411yescharacteristic
27#13095510—22no—
28#13095510—41no—
29C2×Dic194—21no—
30C2×C22⋊D97—42no—
31D4×D97—22no—
32D4×D97—41no—
33Dic388—21no—
34Dic388—21no—
35Dic388—41no—
36Dic388D411yescharacteristic
37#13095500—21no—
38#13095500—41no—
39#13095499—22no—
40D776D411yescharacteristic
41#13095496—22no—
42#13095495D411yescharacteristic
43#13095494—21no—
44C8×D97—21no—
45#13095486—21no—
46C2×D4×C97—21no—
47D8×C97—21no—
48D8×C97D412yes—
49M16×C97—21no—
50C2×C8×C97C2311yescharacteristic
51C16×C97—21no—
52C23×C97—41no—
53C22×D97—42no—
54D4×C97—22no—
55D4×C97—41no—
56C2×C4×C97C2×D411yescharacteristic
57C22⋊D97—21no—
58C22⋊D97—42no—
59C22⋊D97—42no—
60C22⋊D97—41no—
61C2×Dic97—42no—
62C2×Dic97—41no—
63D388—21no—
64D388D811yescharacteristic
65C4×D97—21no—
66C4×D97—41no—
67Dic194—21no—
68Dic194—41no—
69Dic194—41no—
70Dic194D811yescharacteristic
71C8×C97—41no—
72C8×C97C2×D411yescharacteristic
73C8×C97C2×D411yescharacteristic
74#11715895—21no—
75C22×C97—42no—
76C22×C97—161no—
77C22×C97AGL(1,ℤ/8)11yescharacteristic
78C2×D97—21no—
79C2×D97—41no—
80C2×D97—162no—
81C4×C97#12011yescharacteristic
82C4×C97C2×D811yescharacteristic
83Dic97—21no—
84Dic97—41no—
85Dic97—81no—
86C2×C97—21no—
87C2×C97—82no—
88C2×C97#44711yescharacteristic
89D97—41no—
90D97—81no—
91#2194—971nomaximal, Sylow
92C97#219411yescharacteristic, complemented, Sylow
93#576—971no—
94#510—971no—
95C2×SD32—971no—
96#473—971no—
97#361D9711yescharacteristic, complemented
98#359—971no—
99#350—971no—
100H(5,𝔽2)—1941no—
101#137—1941no—
102AGL(1,ℤ/8)—1941no—
103AGL(1,ℤ/8)—1942no—
104#135—971no—
105#135—1941no—
106C2×Q16—971no—
107C2×D8—1941no—
108C2×D8C2×D9711yescharacteristic
109#131—971no—
110Q32—1941no—
111SD32—1941no—
112SD32—1942no—
113M32C2×D9711yescharacteristic, complemented
114C2×C16—971no—
115#108C2×D9711yescharacteristic, complemented
116#101—1941no—
117#41—971no—
118#41—3881no—
119#41—1942no—
120#41—1941no—
121C2×Q8—1941no—
122C2×D4—21no—
123C2×D4—3881no—
124C2×D4—3882no—
125C2×D4—1942no—
126Q16—971no—
127Q16—3881no—
128Q16—1941no—
129Q16—1941no—
130SD16—3881no—
131SD16—1942no—
132SD16—1941no—
133D8—21no—
134D8—971no—
135D8—1942no—
136D8D38812yes—
137M16—21no—
138M16—971no—
139M16—1941no—
140C2×C8—1941no—
141C2×C8C22×D9711yescharacteristic
142C16—21no—
143C16—1941no—
144C23—41no—
145C23—3882no—
146Q8—971no—
147Q8—3881no—
148Q8—3881no—
149Q8—1941no—
150D4—22no—
151D4—41no—
152D4—971no—
153D4—3882no—
154D4—3881no—
155D4—3882no—
156D4—1941no—
157D4—1941no—
158C2×C4—3881no—
159C2×C4—3882no—
160C2×C4—3881no—
161C2×C4—1941no—
162C2×C4D4×D9711yescharacteristic
163C8—41no—
164C8—1941no—
165C8D4×D9711yescharacteristic
166C8C2×D38811yescharacteristic
167C22—42no—
168C22—161no—
169C22—15522no—
170C22—3881no—
171C22—1941no—
172C22#1569083711yescharacteristic
173C4—7761no—
174C4—3881no—
175C4—1941no—
176C4#1569078011yescharacteristic
177C4D8×D9711yescharacteristic
178C2—21no—
179C2—82no—
180C2—7761no—
181C2—3881no—
182C2#2026087011yescharacteristic
subgroupG/H
centerC2#20260870
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#361×C97C2
socleC2×C97#447
prime radical#20261536C2
hypercenter#361D97
hypocenterC97#2194
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #20261178C2semidirect
3. C2×#15690754C2semidirect
4. #20261011C2semidirect
5. #20260541C2semidirect
6. #20260520C2semidirect
7. #361×C97C2semidirect
14. C2×Dic388C22semidirect
20. C2×#13095486C22semidirect
23. M32×C97C22semidirect
24. #108×C97C22semidirect
92. C97#2194semidirect
97. #361D97semidirect
113. M32C2×D97semidirect
115. #108C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615362
#202611782
C2×#156907542
#202610112
#202605412
#202605202
#361×C972
#219497