Finite Group Databasebeta

← #27143689

Subgroups — #27143689

Order 12416.

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

normal subgroups only

This group has 216 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
2#20261178C212yesmaximal, complemented
3#20260993C211yesmaximal, 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×Dic388C2211yescharacteristic, complemented
12#15690789—21no—
13Dic776—22no—
14#15690755—22no—
15#15690742C2211yescharacteristic, complemented
16#15690735—21no—
17C2×M16×C97C2211yescharacteristic
18M32×C97C2212yescomplemented
19#13095511—22no—
20#13095511—22no—
21#13095511—41no—
22#13095511—41no—
23C2×Dic194—21no—
24C2×Dic194—21no—
25C2×D388—21no—
26C2×C22⋊D97—41no—
27C2×C4×D97—41no—
28Dic388—21no—
29Dic388—22no—
30Dic388D412yes—
31#13095500—22no—
32#13095500—22no—
33D776—21no—
34D776D412yes—
35C4⋊Dic97—21no—
36#13095490—41no—
37C22⋊Dic97—41no—
38C22×C4×C97D411yescharacteristic
39M16×C97—22no—
40C2×C8×C97D411yescharacteristic
41C2×C8×C97C2311yescharacteristic
42C16×C97—22no—
43C23×C97D811yescharacteristic
44C22×D97—41no—
45C2×C4×C97—21no—
46C2×C4×C97—21no—
47C2×C4×C97D811yescharacteristic
48C2×C4×C97C2×D411yescharacteristic
49C22⋊D97—42no—
50C22⋊D97—42no—
51C2×Dic97—41no—
52C2×Dic97—41no—
53C2×Dic97—41no—
54D388—22no—
55D388—41no—
56C4×D97—42no—
57C4×D97—42no—
58Dic194—22no—
59Dic194—41no—
60Dic194—41no—
61Dic194—41no—
62C8×C97—21no—
63C8×C97C2×D412yes—
64C22×C97—21no—
65C22×C97—41no—
66C22×C97C2×D811yescharacteristic
67C2×D97—42no—
68C2×D97—82no—
69C4×C97—21no—
70C4×C97#12011yescharacteristic
71C4×C97AGL(1,ℤ/8)11yescharacteristic
72Dic97—42no—
73Dic97—81no—
74Dic97—81no—
75C2×C97—21no—
76C2×C97—41no—
77C2×C97#44711yescharacteristic
78D97—82no—
79#2191—971nomaximal, Sylow
80C97#219111yescharacteristic, complemented, Sylow
81#575—971no—
82#510—972no—
83#470—971no—
84#360—972no—
85#349D9711yescharacteristic, complemented
86C2×#41—1941no—
87#137—1942no—
88AGL(1,ℤ/8)—1942no—
89#135—972no—
90#135—1941no—
91C2×Q16—971no—
92C2×M16C2×D9711yescharacteristic
93C22⋊Q8—1941no—
94Q32—1942no—
95SD32—1942no—
96M32C2×D9712yescomplemented
97#106—971no—
98#103—1941no—
99#41—3881no—
100#41—3881no—
101#41—1942no—
102#41—1942no—
103C2×Q8—1941no—
104C2×Q8—1941no—
105C2×D4—3881no—
106C2×D4—1941no—
107C22×C4—3881no—
108C22×C4D38811yescharacteristic
109Q16—972no—
110Q16—1941no—
111Q16—1942no—
112SD16—1942no—
113SD16—1942no—
114D8—972no—
115D8—1941no—
116M16—22no—
117C2×C8D38811yescharacteristic
118C2×C8C22×D9711yescharacteristic
119C4⋊C4—3881no—
120C4⋊C4—1941no—
121C22⋊C4—3881no—
122C16—22no—
123C23—3881no—
124C23D77611yescharacteristic
125Q8—3881no—
126Q8—3881no—
127Q8—3881no—
128Q8—1942no—
129D4—3881no—
130D4—3882no—
131D4—3882no—
132D4—1942no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—3882no—
136C2×C4—3881no—
137C2×C4—3881no—
138C2×C4—3881no—
139C2×C4—3882no—
140C2×C4D77611yescharacteristic
141C2×C4C2×D38811yescharacteristic
142C8—21no—
143C8D4×D9712yes—
144C22—21no—
145C22—41no—
146C22—7762no—
147C22—3882no—
148C22C2×D77611yescharacteristic
149C4—21no—
150C4—7761no—
151C4—7761no—
152C4—3882no—
153C4#1569078011yescharacteristic
154C4#1569083811yescharacteristic
155C2—21no—
156C2—41no—
157C2—7762no—
158C2#2026087111yescharacteristic
subgroupG/H
centerC2#20260871
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#349×C97C2
socleC2×C97#447
prime radical#20261535C2
hypercenter#349D97
hypocenterC97#2191
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
2. #20261178C2semidirect
3. #20260993C2semidirect
4. #20260546C2semidirect
5. #349×C97C2semidirect
11. C2×Dic388C22semidirect
15. #15690742C22semidirect
18. M32×C97C22semidirect
80. C97#2191semidirect
85. #349D97semidirect
96. M32C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615352
#202611782
#202609932
#202605462
#349×C972
#219197