Finite Group Databasebeta

← #27143431

Subgroups — #27143431

Order 12416.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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#20260719C212yesmaximal
2#20260534C214yesmaximal, complemented
3#536×C97C211yesmaximal, characteristic
4C2×#13095495—22no—
5#15690729C2214yes—
6#15690727—22no—
7C2×#41×C97C2211yescharacteristic
8C22×Q8×C97C411yescharacteristic
9C4⋊Q8×C97C2212yes—
10#124×C97C412yes—
11#123×C97—22no—
12C22⋊Q8×C97—22no—
13#117×C97C411yescharacteristic
14#13095495—24no—
15#13095495—42no—
16C2×#11715895—42no—
17#41×C97—42no—
18C2×Q8×C97—21no—
19C2×Q8×C97—42no—
20C2×Q8×C97C2×C412yes—
21C2×Q8×C97C2311yescharacteristic
22C2×D4×C97C2×C411yescharacteristic
23C2×D4×C97D412yes—
24C22×C4×C97—21no—
25C22×C4×C97C2×C411yescharacteristic
26C22×C4×C97D412yes—
27C4⋊C4×C97—21no—
28C4⋊C4×C97—24no—
29C22⋊C4×C97—21no—
30C22⋊C4×C97—24no—
31C42×C97C2×C412yes—
32C23×C97C22⋊C412yes—
33C23×C97C22⋊C411yescharacteristic
34Q8×C97—41no—
35Q8×C97—42no—
36Q8×C97—42no—
37D4×C97—41no—
38D4×C97—42no—
39C2×C4×C97—22no—
40C2×C4×C97—22no—
41C2×C4×C97—22no—
42C2×C4×C97—22no—
43C2×C4×C97—22no—
44C2×C4×C97—42no—
45C2×C4×C97C22⋊C411yescharacteristic
46C2×C4×C97C22×C411yescharacteristic
47C2×C4×C97C2×D412yes—
48#11715895—44no—
49C22×C97—21no—
50C22×C97—22no—
51C22×C97—42no—
52C22×C97—41no—
53C22×C97C2×C22⋊C411yescharacteristic
54C4×C97—22no—
55C4×C97—21no—
56C4×C97—42no—
57C4×C97—42no—
58C4×C97#9912yes—
59C2×C97—21no—
60C2×C97—41no—
61C2×C97—42no—
62C2×C97C2×#9911yescharacteristic
63#2135—971nomaximal, Sylow
64C97#213511yescharacteristic, complemented, Sylow
65#536D9711yescharacteristic
66#413—972no—
67#356—974no—
68C2×#41C2×D9711yescharacteristic
69C22×Q8Dic9711yescharacteristic
70C2×M16—1942no—
71C4⋊Q8C2×D9712yes—
72#124Dic9712yes—
73#123—22no—
74C22⋊Q8—22no—
75#117Dic9711yescharacteristic
76#101—974no—
77#100—1942no—
78#41—42no—
79C2×Q8—21no—
80C2×Q8—42no—
81C2×Q8C2×Dic9712yes—
82C2×Q8C22×D9711yescharacteristic
83C2×D4C2×Dic9711yescharacteristic
84C2×D4C22⋊D9712yes—
85C22×C4—21no—
86C22×C4C2×Dic9711yescharacteristic
87C22×C4C22⋊D9712yes—
88M16—3882no—
89M16—1944no—
90C2×C8—3882no—
91C4⋊C4—21no—
92C4⋊C4—24no—
93C22⋊C4—21no—
94C22⋊C4—24no—
95C42C2×Dic9712yes—
96C23C22⋊Dic9712yes—
97C23C22⋊Dic9711yescharacteristic
98Q8—41no—
99Q8—42no—
100Q8—42no—
101D4—41no—
102D4—42no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—42no—
109C2×C4C22⋊Dic9711yescharacteristic
110C2×C4C22×Dic9711yescharacteristic
111C2×C4C2×C22⋊D9712yes—
112C8—3884no—
113C22—21no—
114C22—22no—
115C22—42no—
116C22—41no—
117C22C2×C22⋊Dic9711yescharacteristic
118C4—22no—
119C4—21no—
120C4—42no—
121C4—42no—
122C4#1569072512yes—
123C2—21no—
124C2—41no—
125C2—42no—
126C2C2×#1569072511yescharacteristic
subgroupG/H
centerC2C2×#15690725
commutatorC2×C4×C97C22×C4
FrattiniC2×Q8C22×D97
Fitting#536×C97C2
socleC2×C97C2×#99
prime radicalC2×#41×C97C22
cosocleC2×Q8×C97C23
hypercenter#536D97
hypocenterC97#2135
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #20260534C2semidirect
64. C97#2135semidirect

Maximal subgroups

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

subgroupindex
#202607192
#202605342
#536×C972
#213597