Finite Group Databasebeta

← #13136142

Subgroups — #13136142

Order 1728.

154 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 29 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#11720651C211yesmaximal, characteristic
2#131⋊C27C211yesmaximal, characteristic, complemented
3#11720109C211yesmaximal, characteristic, complemented
4#10588824—31nomaximal
5#41⋊C27C2211yescharacteristic
6#88391C411yescharacteristic, complemented
7#88390C411yescharacteristic, complemented
8C8×D27—41nomaximal
9C9×#131D311yescharacteristic
10#63513—31no—
11#63464—31no—
12#63396—31no—
13#63384—31no—
14#63381—31no—
15C8×Dic9—31no—
16C4×D27—41no—
17Q8⋊C27C2×C411yescharacteristic
18C8×C27—41no—
19C27⋊C8—41no—
20#5144—91no—
21C9×#41C2×D311yescharacteristic
22#3697—31no—
23C9×M16—31no—
24C2×C8×C9—31no—
25#3675—31no—
26#3674—31no—
27#3673—31no—
28#3672—31no—
29C4×Dic9—31no—
30#3667—31no—
31C2×C9⋊C8—31no—
32#3663—61no—
33C8×D9—61no—
34C2×D27—41no—
35C4×C27—41no—
36Dic27—41no—
37C3×#131D911yescharacteristic
38#950—91no—
39#901—91no—
40#837—91no—
41#825—91no—
42#822—91no—
43C8×Dic3—91no—
44Q8×C9C4×D311yescharacteristic
45D4×C9—31no—
46C2×C4×C9—31no—
47#611—61no—
48C2×Dic9—61no—
49D36—31no—
50C4×D9—61no—
51Dic18—31no—
52C8×C9—31no—
53C8×C9S411yescharacteristic
54C9⋊C8—31no—
55C9⋊C8—31no—
56C9⋊C8—61no—
57#443—271noSylow
58C2×C27—41no—
59D27—42no—
60C3×#41C2×D911yescharacteristic
61#235—91no—
62C3×M16—31no—
63C2×C3×C8—31no—
64#216—91no—
65#215—91no—
66#214—91no—
67#213—91no—
68C4×Dic3—91no—
69#208—91no—
70C2×C3⋊C8—91no—
71ΓL(1,𝔽25)—181no—
72D3×C8—181no—
73C22×C9—31no—
74C2×D9—61no—
75C4×C9—31no—
76C4×C9C2×S411yescharacteristic
77Dic9—62no—
78Dic9—61no—
79#135—271no—
80#131—271no—
81#131D2711yescharacteristic, complemented
82#108—271no—
83#104—271no—
84#104—271no—
85C4×C8—271no—
86C27—41noSylow
87C3×Q8C4×D911yescharacteristic
88C3×D4—31no—
89C2×C3×C4—31no—
90#67—181no—
91C2×Dic3—181no—
92D12—91no—
93C4×D3—181no—
94Dic6—91no—
95C3×C8—31no—
96C3×C8(C22⋊C9)⋊C211yescharacteristic
97C3⋊C8—91no—
98C3⋊C8—91no—
99C3⋊C8—181no—
100C2×C9—61no—
101C2×C9C4×S411yescharacteristic
102D9—121no—
103#41—271no—
104#41C2×D2711yescharacteristic
105Q16—271no—
106SD16—271no—
107SD16—271no—
108D8—271no—
109M16—31no—
110M16—271no—
111M16—541no—
112C2×C8—31no—
113C2×C8—271no—
114C2×C8—541no—
115C42—271no—
116C22×C3—31no—
117C2×D3—181no—
118C3×C4—31no—
119C3×C4C2×(C22⋊C9)⋊C211yescharacteristic
120Dic3—182no—
121Dic3—181no—
122C9#540411yescharacteristic
123Q8—271no—
124Q8C4×D2711yescharacteristic
125D4—31no—
126D4—271no—
127D4—541no—
128C2×C4—31no—
129C2×C4—541no—
130C2×C4—541no—
131C8—31no—
132C8—271no—
133C8—271no—
134C8—541no—
135C8(C22⋊C27)⋊C211yescharacteristic
136C2×C3—61no—
137C2×C3C4×(C22⋊C9)⋊C211yescharacteristic
138D3—361no—
139C22—31no—
140C22—541no—
141C4—31no—
142C4—542no—
143C4—541no—
144C4C2×(C22⋊C27)⋊C211yescharacteristic
145C3#1058950611yescharacteristic
146C2—61no—
147C2—1081no—
148C2C4×(C22⋊C27)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC8(C22⋊C27)⋊C2
commutator, hypocenterQ8⋊C27C2×C4
FrattiniC4×C9C2×S4
FittingC9×#131D3
socleC2×C3C4×(C22⋊C9)⋊C2
prime radical#11720651C2
cosocle#41⋊C27C22
LCSG ▷ Q8⋊C27
UCSC1 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #131⋊C27C2semidirect
3. #11720109C2semidirect
6. #88391C4semidirect
7. #88390C4semidirect
81. #131D27semidirect

Maximal subgroups

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

subgroupindex
#117206512
#131⋊C272
#117201092
#105888243
C8×D274