Finite Group Databasebeta

← #64309

Subgroups — #64309

Order 288.

312 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 108 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#3830C211yesmaximal, characteristic, complemented
2C3×#237C212yesmaximal, complemented
3C2×#627C212yesmaximal, complemented
4#3805C212yesmaximal, complemented
5#3804C214yesmaximal, complemented
6#3798C211yesmaximal, characteristic, complemented
7#3797C211yesmaximal, characteristic, complemented
8D3×Dic6C212yesmaximal, complemented
9#1010—32nomaximal
10C32×D4C2211yescharacteristic
11#638C2212yes—
12C2×C3⋊Dic3C2212yes—
13C4×C3⋊D3C2211yescharacteristic
14#634C2211yescharacteristic
15C3×#67C2214yes—
16C2×C3×Dic3C2214yes—
17C3×C4×D3C2212yes—
18C3×Dic6C2212yes—
19#627C2214yes—
20#627C2212yes—
21#627C2211yescharacteristic
22#626C2212yes—
23#625C2211yescharacteristic
24#624C2212yes—
25D3×Dic3C2214yes—
26C3×#41—32no—
27D3×Q8—32no—
28D3×Q8—32no—
29#237—32no—
30#237—34no—
31#237—34no—
32#237—61no—
33#237D312yescomplemented
34#235—34no—
35#235—32no—
36C2×Dic6—34no—
37C2×Dic6—32no—
38C22×C32C2312yes—
39C2×C3⋊D3C2311yescharacteristic
40C2×C3×D3C2312yes—
41C32×C4C2311yescharacteristic
42C3⋊Dic3C2312yes—
43C3⋊Dic3C2311yescharacteristic
44C3×Dic3C2314yes—
45C3×Dic3C2312yes—
462−1+4—91noSylow
47C3×Q8—32no—
48C3×D4—21no—
49C3×D4—34no—
50C3×D4C2×D312yes—
51C2×C3×C4—34no—
52C2×C3×C4—32no—
53#67—34no—
54#67—32no—
55#67—32no—
56#67—62no—
57#67C2×D314yes—
58C2×Dic3—34no—
59C2×Dic3—34no—
60C2×Dic3—62no—
61C2×Dic3C2×D314yes—
62D12—32no—
63C4×D3—34no—
64C4×D3—32no—
65C4×D3—34no—
66C4×D3—61no—
67C4×D3C2×D312yes—
68Dic6—32no—
69Dic6—32no—
70Dic6—34no—
71Dic6—34no—
72Dic6—34no—
73Dic6—61no—
74Dic6C2×D312yes—
75C2×C32—22no—
76C2×C32C2411yescharacteristic
77C3⋊D3—21no—
78C3×D3—22no—
79#41—32no—
80#41—91no—
81#41—94no—
82#41—92no—
83#41—91no—
84C2×Q8—92no—
85C2×Q8—92no—
86C2×Q8—91no—
87C22×C3—22no—
88C22×C3—32no—
89C22×C3C22×D314yes—
90C2×D3—32no—
91C2×D3—61no—
92C2×D3C22×D312yes—
93C3×C4—21no—
94C3×C4—32no—
95C3×C4—34no—
96C3×C4C22×D312yes—
97Dic3—34no—
98Dic3—32no—
99Dic3—62no—
100Dic3—61no—
101Dic3C22×D312yes—
102Dic3C22×D314yes—
103C322−1+411yescharacteristic, complemented, Sylow
104Q8—32no—
105Q8—91no—
106Q8—94no—
107Q8—92no—
108Q8—91no—
109D4—34no—
110D4—92no—
111D4—91no—
112D4—92no—
113D4D3211yescharacteristic
114C2×C4—34no—
115C2×C4—32no—
116C2×C4—92no—
117C2×C4—94no—
118C2×C4—92no—
119C2×C4—91no—
120C2×C3—24no—
121C2×C3—21no—
122C2×C3—42no—
123C2×C3—62no—
124C2×C3C23×D312yes—
125D3—22no—
126D3—62no—
127D3—121no—
128C22—32no—
129C22—91no—
130C22C2×D3212yes—
131C4—32no—
132C4—34no—
133C4—92no—
134C4—91no—
135C4C2×D3211yescharacteristic
136C3—21no—
137C3#101012yescomplemented
138C2—22no—
139C2—62no—
140C2—181no—
141C2C22×D3211yescharacteristic
subgroupG/H
center, FrattiniC2C22×D32
commutator, socle, cosocleC2×C32C24
FittingC32×D4C22
hypercenterD4D32
hypocenterC322−1+4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #3830C2semidirect
2. C3×#237C2semidirect
3. C2×#627C2semidirect
4. #3805C2semidirect
5. #3804C2semidirect
6. #3798C2semidirect
7. #3797C2semidirect
8. D3×Dic6C2semidirect
33. #237D3semidirect
103. C322−1+4semidirect
137. C3#1010semidirect

Maximal subgroups

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

subgroupindex
#38302
C3×#2372
C2×#6272
#38052
#38042
#37982
#37972
D3×Dic62
#10103