Finite Group Databasebeta

← #13137397

Subgroups — #13137397

Order 1728.

415 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721060C211yesmaximal, characteristic
2#11721055C212yesmaximal, complemented
3C4×#6180C211yesmaximal, characteristic, complemented
4#63632⋊C3C212yesmaximal, complemented
5#11720316C211yesmaximal, characteristic, complemented
6#10590341—32nomaximal
7#3828⋊C3C2212yes—
8C42×H(3,𝔽3)C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11#3748⋊C3C2212yes—
12#88433C411yescharacteristic, complemented
13#88432C411yescharacteristic, complemented
14#88431C412yescomplemented
15#88430—22no—
16((C3⋊D3)⋊C3)⋊C8—22no—
17#63829—32no—
18#63817—32no—
19#63814—32no—
20#63632D312yescomplemented
21C3×#805—32no—
22C3×C4×C3⋊C8—32no—
23C3×C4×C3⋊C8—61no—
24#63577—32no—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27#638⋊C3—22no—
28C4⋊((C3⋊D3)⋊C3)C2×C412yes—
29C4×(C3⋊D3)⋊C3—22no—
30#634⋊C3C2×C412yes—
31#6180C2×C411yescharacteristic
32#6180C2×C411yescharacteristic
33#6180D411yescharacteristic
34#6180D411yescharacteristic
35(C3⋊(C3⋊C8))⋊C3—22no—
36#4824—92no—
37#4700—91no—
38#3828C2×D312yes—
39C3×#235—32no—
40C32×C42—21no—
41C32×C42C2×D312yes—
42#3748C2×D312yes—
43C3×#208—32no—
44C2×C3×C3⋊C8—32no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—61no—
47C2×C3×C3⋊C8—61no—
48#3719—32no—
49#3717—32no—
50#3716—32no—
51#3714—32no—
52#3712—62no—
53#3711—62no—
54#3710—62no—
55D3×C3⋊C8—62no—
56C22×H(3,𝔽3)#4111yescharacteristic
57C2×(C3⋊D3)⋊C3—22no—
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)C22×C411yescharacteristic
60C4×H(3,𝔽3)C2×D411yescharacteristic
61(C3⋊Dic3)⋊C3—22no—
62#950—92no—
63#948—92no—
64#907—92no—
65#904—91no—
66#901—92no—
67C3×#104—92no—
68C3×C4×C8—91no—
69#822—92no—
70#819—91no—
71#805—32no—
72#805—92no—
73C4×C3⋊C8—32no—
74C4×C3⋊C8—61no—
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77C2×C32×C4—21no—
78C2×C32×C4C22×D312yes—
79#638—22no—
80C4⋊(C3⋊D3)C4×D312yes—
81C4×C3⋊D3—22no—
82#634C4×D312yes—
83C3×#67—62no—
84C3×D12—32no—
85C3×C4×D3—62no—
86C3×Dic6—32no—
87C3⋊(C3⋊C8)—22no—
88C3×C3⋊C8—32no—
89C3×C3⋊C8—32no—
90C3×C3⋊C8—32no—
91C3×C3⋊C8—32no—
92C3×C3⋊C8—61no—
93C3×C3⋊C8—61no—
94C3×C3⋊C8—61no—
95C3×C3⋊C8—61no—
96C3×C3⋊C8—62no—
97#443—271noSylow
98C2×H(3,𝔽3)—21no—
99C2×H(3,𝔽3)C4×D411yescharacteristic
100(C3⋊D3)⋊C3—42no—
101C3×#41—92no—
102#235—32no—
103#235—92no—
104C3×M16—92no—
105C2×C3×C8—91no—
106C2×C3×C8—91no—
107C3×C42—32no—
108C3×C42—61no—
109C3×C42D3211yescharacteristic
110#216—92no—
111#215—92no—
112#214—92no—
113#213—92no—
114#208—32no—
115#208—92no—
116#208—182no—
117C2×C3⋊C8—32no—
118C2×C3⋊C8—32no—
119C2×C3⋊C8—61no—
120C2×C3⋊C8—61no—
121C2×C3⋊C8—182no—
122Dic12—91no—
123D24—91no—
124#204—92no—
125ΓL(1,𝔽25)—182no—
126ΓL(1,𝔽25)—182no—
127D3×C8—182no—
128D3×C8—182no—
129C22×C32—21no—
130C22×C32#23712yes—
131C2×C3⋊D3—22no—
132C2×C3×D3—62no—
133C32×C4—24no—
134C32×C4—24no—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4C2×C4×D312yes—
138C32×C4D3×D412yes—
139C3⋊Dic3—22no—
140C3×Dic3—62no—
141#135—271no—
142#131—272no—
143#108—271no—
144#104—92no—
145C4×C8—91no—
146H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
147C3×Q8—92no—
148C3×D4—92no—
149C3×D4—182no—
150C2×C3×C4—21no—
151C2×C3×C4—32no—
152C2×C3×C4—62no—
153C2×C3×C4—62no—
154C2×C3×C4—61no—
155C2×C3×C4—182no—
156C2×C3×C4C2×D3211yescharacteristic
157#67—62no—
158#67—182no—
159D12—32no—
160D12—92no—
161C4×D3—62no—
162C4×D3—182no—
163Dic6—32no—
164Dic6—92no—
165C3×C8—91no—
166C3×C8—91no—
167C3×C8—91no—
168C3×C8—91no—
169C3×C8—182no—
170C3⋊C8—32no—
171C3⋊C8—32no—
172C3⋊C8—32no—
173C3⋊C8—32no—
174C3⋊C8—61no—
175C3⋊C8—61no—
176C3⋊C8—61no—
177C3⋊C8—61no—
178C3⋊C8—62no—
179C3⋊C8—182no—
180C2×C32—22no—
181C2×C32—22no—
182C2×C32—21no—
183C2×C32#67⋊C412yes—
184C3⋊D3—42no—
185C3×D3—122no—
186#41—92no—
187Q16—271no—
188SD16—272no—
189D8—271no—
190M16—92no—
191M16—542no—
192C2×C8—91no—
193C2×C8—91no—
194C2×C8—542no—
195C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
196C22×C3—32no—
197C22×C3—61no—
198C22×C3—182no—
199C22×C3#380511yescharacteristic
200C2×D3—62no—
201C2×D3—182no—
202C3×C4—22no—
203C3×C4—32no—
204C3×C4—32no—
205C3×C4—64no—
206C3×C4—64no—
207C3×C4—61no—
208C3×C4—61no—
209C3×C4—182no—
210C3×C4C4×D3211yescharacteristic
211C3×C4#380211yescharacteristic
212Dic3—62no—
213Dic3—182no—
214C32—21no—
215C32#482412yescomplemented
216Q8—92no—
217D4—92no—
218D4—182no—
219C2×C4—21no—
220C2×C4—182no—
221C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
222C8—91no—
223C8—91no—
224C8—91no—
225C8—91no—
226C8—182no—
227C2×C3—21no—
228C2×C3—32no—
229C2×C3—62no—
230C2×C3—62no—
231C2×C3—61no—
232C2×C3—362no—
233C2×C3#6390211yescharacteristic
234D3—122no—
235D3—362no—
236C22—182no—
237C22#8867211yescharacteristic
238C4—22no—
239C4—182no—
240C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C4#8865511yescharacteristic
242C3—32no—
243C3—61no—
244C3#1059038911yescharacteristic
245C2—21no—
246C2—362no—
247C2#1172113811yescharacteristic
subgroupG/H
centerC4#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC2×C3#63902
prime radical#11721060C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#443
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721055C2semidirect
3. C4×#6180C2semidirect
4. #63632⋊C3C2semidirect
5. #11720316C2semidirect
12. #88433C4semidirect
13. #88432C4semidirect
14. #88431C4semidirect
20. #63632D3semidirect
146. H(3,𝔽3)#443semidirect
215. C32#4824semidirect

Maximal subgroups

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

subgroupindex
#117210602
#117210552
C4×#61802
#63632⋊C32
#117203162
#105903413