Finite Group Databasebeta

← #27143684

Subgroups — #27143684

Order 12416.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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#20261535C211yesmaximal, characteristic, complemented
2#20261173C212yesmaximal, complemented
3#20260988C211yesmaximal, characteristic, complemented
4#20260546C212yesmaximal, complemented
5#349×C97C211yesmaximal, characteristic, complemented
6C2×#13095511—21no—
7#15690843—22no—
8#15690838—22no—
9#15690832—21no—
10#15690832C2212yes—
11C2×D776C2211yescharacteristic, complemented
12#15690784—21no—
13#15690755—22no—
14D1552—22no—
15#15690742C2211yescharacteristic, complemented
16#15690732—21no—
17C2×M16×C97C2211yescharacteristic
18M32×C97C2212yescomplemented
19#13095511—22no—
20#13095511—22no—
21#13095511—41no—
22#13095511—41no—
23C2×Dic194—21no—
24C2×D388—21no—
25C2×D388—21no—
26C2×C22⋊D97—41no—
27C2×C22⋊D97—41no—
28C2×C4×D97—41no—
29Dic388—21no—
30Dic388D412yescomplemented
31#13095500—22no—
32#13095500—22no—
33D776—21no—
34D776—22no—
35D776D412yescomplemented
36C4⋊Dic97—21no—
37#13095488—41no—
38C22×C4×C97D411yescharacteristic
39M16×C97—22no—
40C2×C8×C97D411yescharacteristic
41C2×C8×C97C2311yescharacteristic
42C16×C97—22no—
43C23×C97D811yescharacteristic
44C22×D97—41no—
45C22×D97—41no—
46C2×C4×C97—21no—
47C2×C4×C97—21no—
48C2×C4×C97D811yescharacteristic
49C2×C4×C97C2×D411yescharacteristic
50C22⋊D97—42no—
51C22⋊D97—42no—
52C22⋊D97—82no—
53C2×Dic97—41no—
54C2×Dic97—41no—
55D388—22no—
56D388—41no—
57D388—41no—
58D388—41no—
59C4×D97—42no—
60C4×D97—42no—
61Dic194—22no—
62Dic194—41no—
63C8×C97—21no—
64C8×C97C2×D412yes—
65C22×C97—21no—
66C22×C97—41no—
67C22×C97C2×D811yescharacteristic
68C2×D97—42no—
69C2×D97—82no—
70C2×D97—81no—
71C2×D97—81no—
72C2×D97—81no—
73C4×C97—21no—
74C4×C97#12011yescharacteristic
75C4×C97AGL(1,ℤ/8)11yescharacteristic
76Dic97—42no—
77Dic97—81no—
78C2×C97—21no—
79C2×C97—41no—
80C2×C97#44711yescharacteristic
81D97—82no—
82D97—161no—
83#2190—971nomaximal, Sylow
84C97#219011yescharacteristic, complemented, Sylow
85#575—971no—
86#509—972no—
87#469—971no—
88#360—972no—
89#349D9711yescharacteristic, complemented
90C2×#41—1941no—
91#137—1942no—
92AGL(1,ℤ/8)—1942no—
93#135—972no—
94#135—1941no—
95C2×D8—971no—
96C2×M16C2×D9711yescharacteristic
97#121—1941no—
98SD32—1942no—
99D16—1942no—
100M32C2×D9712yescomplemented
101#106—971no—
102#102—1941no—
103#41—3881no—
104#41—3881no—
105#41—1942no—
106#41—1942no—
107C2×Q8—1941no—
108C2×D4—3881no—
109C2×D4—3881no—
110C2×D4—1941no—
111C2×D4—1941no—
112C22×C4—3881no—
113C22×C4D38811yescharacteristic
114Q16—972no—
115Q16—1941no—
116SD16—1942no—
117SD16—1942no—
118D8—972no—
119D8—1941no—
120D8—1942no—
121M16—22no—
122C2×C8D38811yescharacteristic
123C2×C8C22×D9711yescharacteristic
124C4⋊C4—1941no—
125C22⋊C4—3881no—
126C16—22no—
127C23—3881no—
128C23—3881no—
129C23D77611yescharacteristic
130Q8—3881no—
131Q8—1942no—
132D4—7762no—
133D4—3881no—
134D4—3882no—
135D4—3882no—
136D4—3881no—
137D4—3881no—
138D4—1942no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—3882no—
142C2×C4—3881no—
143C2×C4—3881no—
144C2×C4—3882no—
145C2×C4D77611yescharacteristic
146C2×C4C2×D38811yescharacteristic
147C8—21no—
148C8D4×D9712yes—
149C22—21no—
150C22—41no—
151C22—7761no—
152C22—7761no—
153C22—7762no—
154C22—7761no—
155C22—3882no—
156C22C2×D77611yescharacteristic
157C4—21no—
158C4—7761no—
159C4—3882no—
160C4#1569078011yescharacteristic
161C4#1569083811yescharacteristic
162C2—21no—
163C2—41no—
164C2—7762no—
165C2—15521no—
166C2#2026087111yescharacteristic
subgroupG/H
centerC2#20260871
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#349×C97C2
socleC2×C97#447
hypercenter#349D97
hypocenterC97#2190
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
1. #20261535C2semidirect
2. #20261173C2semidirect
3. #20260988C2semidirect
4. #20260546C2semidirect
5. #349×C97C2semidirect
11. C2×D776C22semidirect
15. #15690742C22semidirect
18. M32×C97C22semidirect
30. Dic388D4semidirect
35. D776D4semidirect
84. C97#2190semidirect
89. #349D97semidirect
100. M32C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615352
#202611732
#202609882
#202605462
#349×C972
#219097