Finite Group Databasebeta

← #27142904

Subgroups — #27142904

Order 12416.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#20261361C211yesmaximal, characteristic
2#20260596C211yesmaximal, characteristic
3#20260596C212yesmaximal
4#20260501C212yesmaximal
5#342×C97C211yesmaximal, characteristic
6C23×Dic97C2211yescharacteristic
7#15690789—22no—
8#15690787—24no—
9C2×C22⋊Dic97C2211yescharacteristic
10C2×C22⋊Dic97C2212yes—
11#15690717—24no—
12#15690717—22no—
13C2×C22⋊C4×C97C2212yes—
14C2×C22⋊C4×C97C2211yescharacteristic
15C2×Dic194—22no—
16C22×Dic97—22no—
17C22×Dic97—21no—
18C22×Dic97—21no—
19C22×Dic97—22no—
20C22×Dic97—21no—
21C22×Dic97D414yes—
22C22×Dic97Q812yescomplemented
23C4⋊Dic97—22no—
24#13095490—24no—
25C22⋊Dic97—22no—
26C22⋊Dic97—22no—
27C22⋊Dic97—22no—
28C24×C97C2311yescharacteristic
29C22×C4×C97—22no—
30C22×C4×C97D412yes—
31C22⋊C4×C97—22no—
32C22⋊C4×C97—22no—
33C22⋊C4×C97—22no—
34C23×C97—42no—
35C23×C97C2×D412yes—
36C23×C97C2×D411yescharacteristic
37C23×C97C2×Q811yescharacteristic
38C23×C97#4111yescharacteristic
39C23×C97#4112yes—
40C2×C4×C97—22no—
41C2×C4×C97—42no—
42C2×C4×C97—44no—
43C2×C4×C97—42no—
44C2×Dic97—24no—
45C2×Dic97—24no—
46C2×Dic97—22no—
47C2×Dic97—41no—
48C2×Dic97—42no—
49C2×Dic97—41no—
50C2×Dic97—44no—
51C2×Dic97—42no—
52C2×Dic97—41no—
53Dic194—82no—
54C22×C97—24no—
55C22×C97—22no—
56C22×C97—41no—
57C22×C97—41no—
58C22×C97—42no—
59C22×C97#12011yescharacteristic
60C22×C97C22⋊Q812yes—
61C22×C97C22⋊Q811yescharacteristic
62C22×C97#12412yes—
63C22×C97#12411yescharacteristic
64C4×C97—42no—
65C4×C97—82no—
66Dic97—42no—
67Dic97—44no—
68Dic97—81no—
69C2×C97—21no—
70C2×C97—22no—
71C2×C97—22no—
72C2×C97—21no—
73C2×C97#39311yescharacteristic
74C2×C97#45812yes—
75#2030—971nomaximal, Sylow
76C97#203011yescharacteristic, complemented, Sylow
77#543—971no—
78#381—972no—
79#381—971no—
80#342—972no—
81#342D9711yescharacteristic
82C23×C4—971no—
83C22⋊Q8—1944no—
84C22⋊Q8—1942no—
85C2×C22⋊C4—972no—
86C2×C22⋊C4—971no—
87C2×C22⋊C4C2×D9712yes—
88C2×C22⋊C4C2×D9711yescharacteristic
89#95—1944no—
90#95—1942no—
91C24C22×D9711yescharacteristic
92C2×Q8—1942no—
93C22×C4—22no—
94C22×C4—974no—
95C22×C4—972no—
96C22×C4—1941no—
97C22×C4—1942no—
98C22×C4—1941no—
99C22×C4—1941no—
100C22×C4—1942no—
101C22×C4D38812yes—
102C4⋊C4—1944no—
103C4⋊C4—1942no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—1942no—
108C22⋊C4—1942no—
109C22⋊C4—1942no—
110C23—42no—
111C23D4×D9712yes—
112C23C2×D38811yescharacteristic
113C23Q8×D9711yescharacteristic
114C23#1309551011yescharacteristic
115C23#1309551112yes—
116Q8—7762no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—3882no—
122C2×C4—3881no—
123C2×C4—3881no—
124C2×C4—3882no—
125C2×C4—3881no—
126C2×C4—3884no—
127C2×C4—1944no—
128C2×C4—1944no—
129C2×C4—1942no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#1569078011yescharacteristic
136C22#1569078812yes—
137C22#1569079011yescharacteristic
138C22#1569079612yes—
139C22#1569079811yescharacteristic
140C4—42no—
141C4—82no—
142C4—7761no—
143C4—3882no—
144C4—3884no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#2026064811yescharacteristic
150C2#2026092612yes—
subgroupG/H
centerC22#15690780
commutator, prime radical, cosocleC24×C97C23
FrattiniC24C22×D97
Fitting#342×C97C2
socleC22×C97#120
hypercenter#342D97
hypocenterC97#2030
LCSG ▷ C24×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
22. C22×Dic97Q8semidirect
76. C97#2030semidirect

Maximal subgroups

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

subgroupindex
#202613612
#202605962
#202605012
#342×C972
#203097