Finite Group Databasebeta

← #10593194

Subgroups — #10593194

Order 576.

159 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 25 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
1C2×C3⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×C3×#226C211yesmaximal, characteristic, complemented
3C3⋊#859C211yesmaximal, characteristic, complemented
4#5425—31nomaximal
5#5187—31nomaximal
6C2×C3×SL(2,𝔽3)C2211yescharacteristic
7C3⋊(#41⋊C3)—21no—
8C3×#226—21no—
9#3724—41nomaximal
10C2×#239D311yescharacteristic
11C2×#41⋊C3—31no—
12C2×#226D311yescharacteristic, complemented
13C2×C3×Q16—31no—
14#946—31no—
15C2×#215—31no—
16#859—31no—
17#859—61no—
18#835—31no—
19#821—31no—
20(C2×D3)⋊C8—31no—
21C2×C3⋊Dic3—41no—
22C2×C3×Dic3—41no—
23C2×C3×Dic3—41no—
24C3×SL(2,𝔽3)D411yescharacteristic
25#452—91noSylow
26C2×C3×Q8—31no—
27C2×C3×Q8C2×D311yescharacteristic
28#239—21no—
29#239—61no—
30C2×D12—31no—
31C2×C4×D3—31no—
32#41⋊C3—61no—
33C2×SL(2,𝔽3)—21no—
34C2×SL(2,𝔽3)C2×D311yescharacteristic
35#226—21no—
36C3×Q16—61no—
37C3×Q16—61no—
38C2×C3×C8—31no—
39#215—62no—
40(C2×D3)⋊C4—61no—
41C4⋊Dic3—31no—
42Dic3⋊C4—61no—
43Dic3⋊C4—121no—
44C2×C3⋊C8—31no—
45C22×C32—41no—
46C3⋊Dic3—42no—
47C3×Dic3—81no—
48C3×Dic3—81no—
49C2×#41—31no—
50C2×Q16—31no—
51C2×SD16—91no—
52C22⋊Q8—91no—
53#103—91no—
54#102—91no—
55C22⋊C8—91no—
56C22×D3—31no—
57C3×Q8—31no—
58C3×Q8—61no—
59C3×Q8—61no—
60C3×Q8#6711yescharacteristic
61C2×C3×C4—31no—
62C2×C3×C4—61no—
63C2×C3×C4—121no—
64C2×Dic3—41no—
65C2×Dic3—61no—
66C2×Dic3—121no—
67C2×Dic3—241no—
68C2×Dic3S411yescharacteristic
69D12—32no—
70D12—61no—
71C4×D3—61no—
72C4×D3—61no—
73SL(2,𝔽3)—21no—
74SL(2,𝔽3)#6711yescharacteristic
75C3×C8—61no—
76C3⋊C8—61no—
77C2×C32—41no—
78C2×C32—41no—
79C2×C32—41no—
80#41—61no—
81#41—181no—
82C2×Q8—31no—
83C2×Q8D3211yescharacteristic
84C2×D4—91no—
85C22×C4—91no—
86Q16—61no—
87Q16—61no—
88SD16—182no—
89C2×C8—31no—
90C2×C8—91no—
91C4⋊C4—91no—
92C4⋊C4—181no—
93C22⋊C4—181no—
94C22×C3—41no—
95C22×C3—81no—
96C22×C3C2×S411yescharacteristic
97C2×D3—61no—
98C2×D3—61no—
99C2×D3—61no—
100C3×C4—31no—
101C3×C4—31no—
102C3×C4—121no—
103C3×C4—241no—
104Dic3—21no—
105Dic3—81no—
106Dic3—122no—
107Dic3—121no—
108Dic3—242no—
109C32—41noSylow
110C23—91no—
111Q8—31no—
112Q8—61no—
113Q8—61no—
114Q8#62511yescharacteristic
115D4—92no—
116D4—181no—
117C2×C4—31no—
118C2×C4—31no—
119C2×C4—61no—
120C2×C4—181no—
121C2×C4—181no—
122C2×C4—181no—
123C8—61no—
124C8—181no—
125C2×C3—41no—
126C2×C3—41no—
127C2×C3—41no—
128C2×C3—81no—
129C2×C3—81no—
130C2×C3—81no—
131C2×C3#98411yescharacteristic
132C2×C3#98511yescharacteristic
133C2×C3GL(2,ℤ/4)11yescharacteristic
134D3—121no—
135C22—181no—
136C22—181no—
137C22—181no—
138C22D3×S411yescharacteristic
139C4—31no—
140C4—31no—
141C4—61no—
142C4—121no—
143C4—361no—
144C3—41no—
145C3—81no—
146C3#542511yescharacteristic, complemented
147C2—361no—
148C2#6419611yescharacteristic
149C2#6419811yescharacteristic
150C2#6420911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22
FittingC2×C3×Q8C2×D3
socleC22×C3C2×S4
prime radicalC2×C3⋊(#41⋊C3)C2
hypocenterC3×SL(2,𝔽3)D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×#226C2semidirect
3. C3⋊#859C2semidirect
12. C2×#226D3semidirect
146. C3#5425semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(#41⋊C3)2
C2×C3×#2262
C3⋊#8592
#54253
#51873
#37244