Finite Group Databasebeta

← #13141239

Subgroups — #13141239

Order 1728.

286 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 35 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#11722586C211yesmaximal, characteristic, complemented
2C32×#393C311yesmaximal, characteristic
3#393⋊C9C313yesmaximal, complemented
4C22×#1094—41nomaximal
5C22×C32×Q8C2×C311yescharacteristic
6C2×C32×C22⋊C4—31no—
7C22⋊(Q8⋊C9)C2×C313yescomplemented
8C32×#95—31no—
9C2×#1094—42no—
10C2×#1094—41no—
11#6204—82no—
12C3×#393—31no—
13C3×#393C3211yescharacteristic
14C24×C32A411yescharacteristic
15C2×C32×Q8—62no—
16C22×C32×C4—31no—
17C22×C32×C4—31no—
18C22×C22⋊C9—43no—
19C32×C22⋊C4—62no—
20C22×3−1+2—161no—
21#1094—41no—
22C22×C3×Q8—31no—
23C22×C3×Q8C2×C3211yescharacteristic
24C2×C3×C22⋊C4—33no—
25C2×C3×C22⋊C4—31no—
26C3×#95—31no—
27C3×#95—33no—
28C23×C32—61no—
29C23×C32—61no—
30C23×C32SL(2,𝔽3)12yes—
31C23×C32C2×A411yescharacteristic
32C32×Q8—22no—
33C32×Q8—62no—
34C2×C32×C4—32no—
35C2×C32×C4—62no—
36C2×C32×C4—61no—
37C2×C32×C4—62no—
38C2×C22⋊C9—46no—
39C2×C22⋊C9—43no—
40Q8⋊C9—86no—
41#3933−1+211yescharacteristic, complemented, Sylow
42C2×3−1+2—162no—
43C2×3−1+2—161no—
44C24×C3—31no—
45C24×C3C3×A411yescharacteristic
46C2×C3×Q8—62no—
47C2×C3×Q8—66no—
48C22×C3×C4—33no—
49C22×C3×C4—31no—
50C22×C3×C4—33no—
51C22×C3×C4—31no—
52C3×C22⋊C4—62no—
53C3×C22⋊C4—66no—
54C22×C32—31no—
55C22×C32—31no—
56C22×C32—41no—
57C22×C32—62no—
58C22×C32—121no—
59C22×C32C2×SL(2,𝔽3)11yescharacteristic
60C32×C4—62no—
61C32×C4—121no—
62C22×C9—163no—
63C22⋊C9—43no—
64C22×Q8C2×3−1+211yescharacteristic, complemented
65C2×C22⋊C4—31no—
66#95—31no—
673−1+2—161noSylow
68C23×C3—32no—
69C23×C3—31no—
70C23×C3—61no—
71C23×C3—61no—
72C23×C3—63no—
73C23×C3—63no—
74C23×C3C3×SL(2,𝔽3)12yes—
75C23×C3C2×C3×A411yescharacteristic
76C3×Q8—22no—
77C3×Q8—62no—
78C3×Q8—62no—
79C3×Q8—66no—
80C2×C3×C4—36no—
81C2×C3×C4—32no—
82C2×C3×C4—66no—
83C2×C3×C4—62no—
84C2×C3×C4—63no—
85C2×C3×C4—61no—
86C2×C3×C4—62no—
87C2×C3×C4—66no—
88C2×C32—31no—
89C2×C32—31no—
90C2×C32—42no—
91C2×C32#120⋊C311yescharacteristic
92C2×C9—166no—
93C2×C9—163no—
94C24#109411yescharacteristic
95C2×Q8—62no—
96C22×C4—31no—
97C22×C4—31no—
98C22⋊C4—62no—
99C22×C3—31no—
100C22×C3—33no—
101C22×C3—33no—
102C22×C3—31no—
103C22×C3—31no—
104C22×C3—41no—
105C22×C3—62no—
106C22×C3—66no—
107C22×C3—121no—
108C22×C3—121no—
109C22×C3—123no—
110C22×C3C2×C3×SL(2,𝔽3)11yescharacteristic
111C3×C4—66no—
112C3×C4—62no—
113C3×C4—123no—
114C3×C4—121no—
115C32#393⋊C311yescharacteristic
116C9—163no—
117C23—61no—
118C23—61no—
119C23#620412yescomplemented
120C23C2×#109411yescharacteristic
121Q8—22no—
122Q8—62no—
123C2×C4—32no—
124C2×C4—62no—
125C2×C4—61no—
126C2×C4—62no—
127C2×C3—33no—
128C2×C3—33no—
129C2×C3—31no—
130C2×C3—31no—
131C2×C3—31no—
132C2×C3—42no—
133C2×C3—122no—
134C2×C3C3×#120⋊C311yescharacteristic
135C22—31no—
136C22—31no—
137C22—41no—
138C22—62no—
139C22—121no—
140C22C2×#620411yescharacteristic
141C4—62no—
142C4—121no—
143C3—31no—
144C3C3×#393⋊C311yescharacteristic
145C2—31no—
146C2—31no—
147C2—42no—
148C2#1172134711yescharacteristic
subgroupG/H
centerC2×C3C3×#120⋊C3
commutator, cosocleC22×C3×Q8C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#393C3
prime radicalC24×C32A4
hypercenterC2×C32#120⋊C3
hypocenterC22×Q8C2×3−1+2
LCSG ▷ C22×C3×Q8 ▷ C22×Q8
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. #11722586C2semidirect
3. #393⋊C9C3semidirect
7. C22⋊(Q8⋊C9)C2×C3semidirect
41. #3933−1+2semidirect
64. C22×Q8C2×3−1+2semidirect
119. C23#6204semidirect

Maximal subgroups

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

subgroupindex
#117225862
C32×#3933
#393⋊C93
C22×#10944