Finite Group Databasebeta

← #27142283

Subgroups — #27142283

Order 12416.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261299C211yesmaximal, characteristic
2#20260719C211yesmaximal, characteristic, complemented
3C2×#15690726C212yesmaximal, complemented
4#20260507C212yesmaximal, complemented
5#568×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7C2×#13095495C2212yes—
8#15690806—21no—
9#15690799—21no—
10#15690784—22no—
11#15690778C414yes—
12#15690769C2211yescharacteristic
13#15690729—21no—
14#15690727—21no—
15#15690726—24no—
16C2×#41×C97C2211yescharacteristic
17#131×C97—24no—
18C2×M16×C97—22no—
19C2×M16×C97C2212yes—
20C23×D97C2×C412yes—
21C2×D388—21no—
22C2×D388—21no—
23C2×D388—22no—
24C2×D388—22no—
25C2×C22⋊D97—24no—
26#13095495—22no—
27#13095495—22no—
28C2×#11715895—22no—
29C4⋊Dic97—21no—
30C22⋊Dic97C2×C412yes—
31#13095488—24no—
32C4×Dic97—21no—
33#41×C97—22no—
34#41×C97—22no—
35C2×Q8×C97D411yescharacteristic
36C2×D4×C97C2×C412yes—
37C2×D4×C97D411yescharacteristic
38C22×C4×C97D412yes—
39C22×C4×C97C2311yescharacteristic
40M16×C97—24no—
41M16×C97—22no—
42M16×C97D414yescomplemented
43C2×C8×C97—24no—
44C2×C8×C97—22no—
45C23×C97C22⋊C412yes—
46C23×C97C22×C411yescharacteristic
47C22×D97—22no—
48C22×D97—24no—
49C22×D97—82no—
50Q8×C97—22no—
51D4×C97—24no—
52D4×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97—22no—
55C2×C4×C97C22⋊C412yes—
56C2×C4×C97C2×D412yes—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97#4112yes—
60C22⋊D97—84no—
61C2×Dic97—22no—
62D388—41no—
63D388—42no—
64D388—41no—
65C8×C97—24no—
66#11715895—42no—
67C22×C97—22no—
68C22×C97—41no—
69C22×C97—42no—
70C22×C97C2×C22⋊C411yescharacteristic
71C22×C97C4×D412yes—
72C2×D97—44no—
73C2×D97—42no—
74C2×D97—84no—
75C4×C97—22no—
76C4×C97#12011yescharacteristic
77C4×C97#12112yes—
78C4×C97#12311yescharacteristic
79Dic97—42no—
80C2×C97—21no—
81C2×C97—22no—
82C2×C97—42no—
83C2×C97#38611yescharacteristic
84D97—82no—
85#1914—971nomaximal, Sylow
86C97#191411yescharacteristic, complemented, Sylow
87#568D9711yescharacteristic, complemented
88#535—971no—
89#413—971no—
90C2×#100—972no—
91#344—972no—
92C2×#41C2×D9711yescharacteristic
93C22×D4—971no—
94#131—24no—
95C2×M16—22no—
96C2×M16—972no—
97C2×M16C2×D9712yes—
98#127—1941no—
99#124—1941no—
100#121—1942no—
101#120—974no—
102#117—971no—
103#101—1941no—
104#100—1941no—
105#100—1944no—
106C24—972no—
107#41—22no—
108#41—22no—
109C2×Q8D38811yescharacteristic
110C2×D4—1944no—
111C2×D4—1942no—
112C2×D4—1941no—
113C2×D4—1941no—
114C2×D4—1942no—
115C2×D4C4×D9712yes—
116C2×D4D38811yescharacteristic
117C22×C4C22⋊D9712yes—
118C22×C4C22×D9711yescharacteristic
119M16—24no—
120M16—22no—
121M16—1942no—
122M16—1942no—
123M16C22⋊D9714yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—1942no—
127C4⋊C4—1941no—
128C22⋊C4—972no—
129C22⋊C4—1944no—
130C42—1941no—
131C23—7762no—
132C23—1944no—
133C23—1942no—
134C23#1309548812yes—
135C23C2×C4×D9711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—7764no—
140D4—3881no—
141D4—3882no—
142D4—3881no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—1942no—
146C2×C4#1309548812yes—
147C2×C4C2×C22⋊D9712yes—
148C2×C4C2×C22⋊D9711yescharacteristic
149C2×C4C2×D38811yescharacteristic
150C2×C4#1309551112yes—
151C8—24no—
152C8—3882no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—7764no—
157C22—3884no—
158C22—3882no—
159C22C2×#1309548811yescharacteristic
160C22C4×C22⋊D9712yes—
161C4—22no—
162C4—3882no—
163C4#1569077911yescharacteristic
164C4#1569078412yes—
165C4#1569079311yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—7762no—
170C2#2026061411yescharacteristic
subgroupG/H
centerC2#20260614
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#568×C97C2
socleC2×C97#386
prime radical#20261299C2
cosocleC22×C4×C97C23
hypercenter#568D97
hypocenterC97#1914
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20260719C2semidirect
3. C2×#15690726C2semidirect
4. #20260507C2semidirect
5. #568×C97C2semidirect
42. M16×C97D4semidirect
86. C97#1914semidirect
87. #568D97semidirect
123. M16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612992
#202607192
C2×#156907262
#202605072
#568×C972
#191497