Finite Group Databasebeta

← #13173078

Subgroups — #13173078

Order 1728.

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

normal subgroups only

This group has 686 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×#994C211yesmaximal, characteristic
2#11724469C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
4#10593662—312nomaximal
5#10592941—31nomaximal
6C32×#41⋊C3—21no—
7C32×#41⋊C3C2211yescharacteristic
8C3⋊(C3⋊#226)—21no—
9#88863—41nomaximal
10C32×H(5,𝔽2)D311yescharacteristic
11C3×#994D3112yes—
12#64265—312no—
13#64160—31no—
14#64160—31no—
15#64145—31no—
16C3⋊GU(2,𝔽3)—312no—
17#63664—31no—
18#63664—31no—
19#63660—31no—
20C33×Q8—41no—
21#6308—41no—
22C32×SL(2,𝔽3)D411yescharacteristic
23C3⋊(C3⋊(C3⋊C8))—41no—
24#5428—99no—
25#5242—94no—
26C32×#41—21no—
27C32×#41—31no—
28C32×#41C2×D311yescharacteristic
29C2×C32×D4—31no—
30C2×C32×D4—61no—
31C2×#634—31no—
32C3×#41⋊C3—212no—
33C3×#41⋊C3C2×D3112yes—
34C3⋊#226—612no—
35#3757—62no—
36#3756—61no—
37#3756—61no—
38#3756—1212no—
39#3754—61no—
40#3754—61no—
41C4×C3⋊Dic3—31no—
42#3748—31no—
43#3748—31no—
44C33×C4—41no—
45C33×C4—81no—
46C3⋊(C3⋊Dic3)—81no—
47C3×H(5,𝔽2)C3⋊D314yes—
48#994C3⋊D319yes—
49#984—99no—
50#951—94no—
51#951—94no—
52#936—94no—
53GU(2,𝔽3)—99no—
54#837—94no—
55#837—94no—
56#833—94no—
57C23×C32—32no—
58C32×Q8—412no—
59C32×Q8#6711yescharacteristic
60C32×Q8S411yescharacteristic
61C32×D4—31no—
62C32×D4—31no—
63C32×D4—61no—
64C32×D4—61no—
65C2×C32×C4—31no—
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#455—271noSylow
77C2×C33—41no—
78C3×#41—24no—
79C3×#41—34no—
80C3×#41C2×C3⋊D314yes—
81C2×C3×D4—34no—
82C2×C3×D4—64no—
83C2×Dic6—94no—
84#41⋊C3—29no—
85#41⋊C3C2×C3⋊D319yes—
86#226—189no—
87#217—188no—
88#216—184no—
89#216—184no—
90#216—369no—
91#214—184no—
92#214—184no—
93C4×Dic3—94no—
94#208—94no—
95#208—94no—
96C22×C32—31no—
97C22×C32—61no—
98C22×C32—122no—
99C32×C4—21no—
100C32×C4—31no—
101C32×C4—412no—
102C32×C4—812no—
103C32×C4C2×S411yescharacteristic
104C3⋊Dic3—121no—
105C3⋊Dic3—121no—
106C3⋊Dic3—2412no—
107H(5,𝔽2)C3⋊(C3⋊D3)11yescharacteristic
108#137—271no—
109#137—271no—
110#124—271no—
111#104—271no—
112#104—271no—
113#100—271no—
114C33—41noSylow
115C23×C3—38no—
116C3×Q8—49no—
117C3×Q8#63814yes—
118C3×Q8C3⋊S414yes—
119C3×D4—34no—
120C3×D4—34no—
121C3×D4—64no—
122C3×D4—64no—
123C2×C3×C4—34no—
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—412no—
135C2×C32—61no—
136C2×C32—121no—
137C2×C32GL(2,ℤ/4)11yescharacteristic
138#41—21no—
139#41—31no—
140#41C2×C3⋊(C3⋊D3)11yescharacteristic
141C2×Q8—271no—
142C2×D4—31no—
143C2×D4—61no—
144Q16—541no—
145Q16—541no—
146SD16—541no—
147SD16—541no—
148M16—271no—
149M16—271no—
150C22⋊C4—542no—
151C42—271no—
152C22×C3—34no—
153C22×C3—64no—
154C22×C3—128no—
155C3×C4—24no—
156C3×C4—34no—
157C3×C4—49no—
158C3×C4—89no—
159C3×C4C2×C3⋊S414yes—
160Dic3—364no—
161Dic3—364no—
162Dic3—729no—
163C32—412no—
164C32#542811yescharacteristic, complemented
165C23—32no—
166Q8—541no—
167Q8—541no—
168Q8#631211yescharacteristic
169Q8C3⋊(C3⋊S4)11yescharacteristic
170D4—31no—
171D4—31no—
172D4—61no—
173D4—61no—
174C2×C4—31no—
175C2×C4—61no—
176C2×C4—541no—
177C2×C4—541no—
178C8—541no—
179C8—541no—
180C2×C3—49no—
181C2×C3—64no—
182C2×C3—124no—
183C2×C3#6426914yes—
184C22—31no—
185C22—61no—
186C22—122no—
187C4—21no—
188C4—31no—
189C4—1081no—
190C4—1081no—
191C4C2×C3⋊(C3⋊S4)11yescharacteristic
192C3—49no—
193C3#1059366214yescomplemented
194C2—61no—
195C2—121no—
196C2#1172447311yescharacteristic
subgroupG/H
centerC2#11724473
commutator, cosocleC32×#41⋊C3C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC32×H(5,𝔽2)D3
socleC2×C32GL(2,ℤ/4)
prime radicalC32×#994C2
hypercenterQ8C3⋊(C3⋊S4)
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

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#5428semidirect
193. C3#10593662semidirect

Maximal subgroups

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

subgroupindex
C32×#9942
#117244692
C3⋊(C3⋊GU(2,𝔽3))2
#105936623
#105929413
#888634