Finite Group Databasebeta

← #27157924

Subgroups — #27157924

Order 12416.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261740C211yesmaximal, characteristic
2#20261671C211yesmaximal, characteristic
3#20261605C214yesmaximal
4#20261290C211yesmaximal, characteristic
5C2×#13095521C2212yes—
6C2×C4×#87604C2212yes—
7#15690877—21no—
8#15690877—21no—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4—21no—
11#15690810C411yescharacteristic, complemented
12#15690801C412yescomplemented
13#15690790—21no—
14#15690786—21no—
15Q8×Dic97—21no—
16#15690775—21no—
17#15690768C2212yes—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21#13095521—24no—
22C2×#11715897—22no—
23C4×#87604—22no—
24C4×#87604Q814yescomplemented
25C2×Dic194—21no—
26C2×C4×D97D411yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97C2311yescharacteristic
29C4⋊Dic97—21no—
30C4⋊Dic97—22no—
31C4⋊Dic97C2×C412yes—
32#13095490—22no—
33#13095488—22no—
34C4×Dic97—22no—
35C4×Dic97C2×C411yescharacteristic
36C4×Dic97D411yescharacteristic
37C4×Dic97D411yescharacteristic
38C2×Q8×C97—21no—
39C4⋊C4×C97—21no—
40C4⋊C4×C97C2×C412yes—
41C42×C97C2×C411yescharacteristic
42C22×D97C2×D411yescharacteristic
43C2×#87604—24no—
44C2×#87604—42no—
45C2×#87604—42no—
46Q8×C97—42no—
47C2×C4×C97—22no—
48C2×C4×C97C22⋊C411yescharacteristic
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22×C411yescharacteristic
51C2×Dic97—22no—
52C2×Dic97C22⋊C411yescharacteristic
53C2×Dic97C22⋊C411yescharacteristic
54C2×Dic97C2×D411yescharacteristic
55C4×D97—41no—
56C4×D97—41no—
57C4×D97C2×Q812yes—
58C4×D97#4112yes—
59Dic194—42no—
60#11715897—42no—
61C22×C97C2×C22⋊C411yescharacteristic
62C2×D97—21no—
63C2×D97—21no—
64C2×D97C22⋊Q812yes—
65#87604—44no—
66C4×C97—21no—
67C4×C97—21no—
68C4×C97—42no—
69C4×C97C4×Q812yes—
70Dic97—21no—
71Dic97—21no—
72Dic97—42no—
73Dic97#12411yescharacteristic
74Dic97C4⋊Q811yescharacteristic
75C2×C97#39111yescharacteristic
76C2×C97#42111yescharacteristic
77C2×C97#42111yescharacteristic
78D97—22no—
79#1995—971nomaximal, Sylow
80C97#199511yescharacteristic, complemented, Sylow
81#533—971no—
82#423—971no—
83#376—971no—
84#339—974no—
85C2×M16—972no—
86C4⋊Q8—971no—
87C4⋊Q8#8760411yescharacteristic, complemented
88#125—972no—
89C22⋊Q8—1941no—
90C22⋊Q8—1941no—
91C4×Q8—1941no—
92C4×Q8—1941no—
93#117—972no—
94C2×C42—971no—
95C2×C42—972no—
96C4⋊C8—1941no—
97C4⋊C8—1941no—
98#95—1941no—
99#95—1941no—
100C2×Q8—21no—
101C2×Q8—1941no—
102C22×C4—971no—
103C22×C4—971no—
104C22×C4—971no—
105C22×C4—1942no—
106M16—1944no—
107C2×C8—1942no—
108C4⋊C4—21no—
109C4⋊C4—972no—
110C4⋊C4—1942no—
111C4⋊C4—1942no—
112C4⋊C4—1941no—
113C4⋊C4C2×#8760412yes—
114C22⋊C4—1942no—
115C42—971no—
116C42—971no—
117C42—971no—
118C42—974no—
119C42—1942no—
120C42—1942no—
121C42C2×#8760411yescharacteristic
122C23—971no—
123Q8—42no—
124Q8—3882no—
125C2×C4—22no—
126C2×C4—971no—
127C2×C4—972no—
128C2×C4—972no—
129C2×C4—971no—
130C2×C4—971no—
131C2×C4—3881no—
132C2×C4—3881no—
133C2×C4—3882no—
134C2×C4—3882no—
135C2×C4—1944no—
136C2×C4—1942no—
137C2×C4C22⋊#8760411yescharacteristic
138C2×C4C22⋊#8760411yescharacteristic
139C2×C4C22×#8760411yescharacteristic
140C8—3882no—
141C22—972no—
142C22—1941no—
143C22—1941no—
144C22C2×C22⋊#8760411yescharacteristic
145C4—21no—
146C4—21no—
147C4—42no—
148C4—971no—
149C4—971no—
150C4—3884no—
151C4—3882no—
152C4—1941no—
153C4—1941no—
154C4Q8×#8760412yes—
155C2—1942no—
156C2(C2×#87604)⋊Q811yescharacteristic
157C2#2026173311yescharacteristic
158C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1995
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #15690810C4semidirect
12. #15690801C4semidirect
19. C4⋊Q8×C97C4semidirect
24. C4×#87604Q8semidirect
80. C97#1995semidirect
87. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617402
#202616712
#202616052
#202612902
#199597