Finite Group Databasebeta

← #13173080

Subgroups — #13173080

Order 1728.

702 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C32×2−1+4⋊C3C211yesmaximal, characteristic
2#11724469C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
4#10593664—312nomaximal
5#10592945—31nomaximal
6C32×#41⋊C3C2211yescharacteristic
7C2×C32×SL(2,𝔽3)—21no—
8C3⋊(C3⋊#226)—21no—
9#88861—41nomaximal
10C32×2−1+4D311yescharacteristic
11C3×2−1+4⋊C3D3112yes—
12#64265—312no—
13#64160—31no—
14#64160—31no—
15#64153—31no—
16C3⋊GU(2,𝔽3)—312no—
17#63664—31no—
18#63664—31no—
19#63663—31no—
20C33×D4—41no—
21#6308—41no—
22C32×SL(2,𝔽3)D411yescharacteristic
23C3⋊(C3⋊(C3⋊C8))—41no—
24#5430—99no—
25#5246—94no—
26C32×#41—31no—
27C32×#41—61no—
28C32×#41C2×D311yescharacteristic
29C2×C32×Q8—21no—
30C2×C32×Q8—31no—
31C2×#634—31no—
32C3×#41⋊C3C2×D3112yes—
33C2×C3×SL(2,𝔽3)—212no—
34C3⋊#226—612no—
35#3756—61no—
36#3756—61no—
37#3754—61no—
38#3754—61no—
39#3754—1212no—
40#3750—62no—
41C4×C3⋊Dic3—31no—
42#3748—31no—
43#3748—31no—
44C22×C33—81no—
45C33×C4—41no—
46C3⋊(C3⋊Dic3)—81no—
47C3×2−1+4C3⋊D314yes—
482−1+4⋊C3C3⋊D319yes—
49#984—99no—
50#951—94no—
51#951—94no—
52#944—94no—
53GU(2,𝔽3)—99no—
54#837—94no—
55#837—94no—
56#836—94no—
57C32×Q8—31no—
58C32×Q8—61no—
59C32×Q8#6711yescharacteristic
60C32×D4—31no—
61C32×D4—412no—
62C32×D4—61no—
63C32×D4S411yescharacteristic
64C2×C32×C4—31no—
65C2×C32×C4—32no—
66C2×C32×C4—61no—
67C2×C3⋊Dic3—61no—
68C2×C3⋊Dic3—61no—
69#634—61no—
70#634—61no—
71#634—1212no—
72C3×SL(2,𝔽3)#67112yes—
73C3⋊(C3⋊C8)—61no—
74C3⋊(C3⋊C8)—61no—
75C3⋊(C3⋊C8)—1212no—
76#456—271noSylow
77C2×C33—41no—
78C2×C33—161no—
79C3×#41—34no—
80C3×#41—64no—
81C3×#41C2×C3⋊D314yes—
82C2×C3×Q8—24no—
83C2×C3×Q8—34no—
84C2×Dic6—94no—
85#41⋊C3C2×C3⋊D319yes—
86C2×SL(2,𝔽3)—29no—
87#226—189no—
88#216—184no—
89#216—184no—
90#214—184no—
91#214—184no—
92#214—369no—
93Dic3⋊C4—188no—
94C4×Dic3—94no—
95#208—94no—
96#208—94no—
97C22×C32—21no—
98C22×C32—31no—
99C22×C32—812no—
100C32×C4—31no—
101C32×C4—412no—
102C32×C4—61no—
103C32×C4C2×S411yescharacteristic
104C3⋊Dic3—62no—
105C3⋊Dic3—121no—
106C3⋊Dic3—2412no—
1072−1+4C3⋊(C3⋊D3)11yescharacteristic
108#137—271no—
109#137—271no—
110C4⋊Q8—271no—
111#104—271no—
112#104—271no—
113#101—271no—
114C33—41noSylow
115C3×Q8—34no—
116C3×Q8—64no—
117C3×Q8#63814yes—
118C3×D4—34no—
119C3×D4—49no—
120C3×D4—64no—
121C3×D4C3⋊S414yes—
122C2×C3×C4—34no—
123C2×C3×C4—38no—
124C2×C3×C4—64no—
125C2×Dic3—184no—
126C2×Dic3—184no—
127Dic6—184no—
128Dic6—184no—
129Dic6—369no—
130SL(2,𝔽3)#63819yes—
131C3⋊C8—184no—
132C3⋊C8—184no—
133C3⋊C8—369no—
134C2×C32—41no—
135C2×C32—412no—
136C2×C32—61no—
137C2×C32—1612no—
138C2×C32GL(2,ℤ/4)11yescharacteristic
139#41—31no—
140#41—61no—
141#41C2×C3⋊(C3⋊D3)11yescharacteristic
142C2×Q8—21no—
143C2×Q8—31no—
144C2×Q8—271no—
145Q16—541no—
146Q16—541no—
147SD16—541no—
148SD16—541no—
149M16—271no—
150M16—271no—
151C4⋊C4—542no—
152C42—271no—
153C22×C3—24no—
154C22×C3—34no—
155C22×C3—89no—
156C3×C4—34no—
157C3×C4—49no—
158C3×C4—64no—
159C3×C4C2×C3⋊S414yes—
160Dic3—188no—
161Dic3—364no—
162Dic3—729no—
163C32—412no—
164C32#543011yescharacteristic, complemented
165Q8—31no—
166Q8—61no—
167Q8—541no—
168Q8—541no—
169Q8#631211yescharacteristic
170D4—31no—
171D4—61no—
172D4C3⋊(C3⋊S4)11yescharacteristic
173C2×C4—31no—
174C2×C4—32no—
175C2×C4—61no—
176C2×C4—541no—
177C2×C4—541no—
178C8—541no—
179C8—541no—
180C2×C3—44no—
181C2×C3—49no—
182C2×C3—64no—
183C2×C3—169no—
184C2×C3#6426914yes—
185C22—21no—
186C22—31no—
187C4—31no—
188C4—61no—
189C4—542no—
190C4—1081no—
191C4C2×C3⋊(C3⋊S4)11yescharacteristic
192C3—49no—
193C3#1059366414yescomplemented
194C2—41no—
195C2—61no—
196C2#1172447311yescharacteristic
subgroupG/H
centerC2#11724473
commutator, cosocleC32×#41⋊C3C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC32×2−1+4D3
socleC2×C32GL(2,ℤ/4)
prime radicalC32×2−1+4⋊C3C2
hypercenterD4C3⋊(C3⋊S4)
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #11724469C2semidirect
3. C3⋊(C3⋊GU(2,𝔽3))C2semidirect
164. C32#5430semidirect
193. C3#10593664semidirect

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
#117244692
C3⋊(C3⋊GU(2,𝔽3))2
#105936643
#105929453
#888614