Finite Group Databasebeta

← #13141064

Subgroups — #13141064

Order 1728.

412 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721869C211yesmaximal, characteristic
2#64160⋊C3C212yesmaximal
3C2×#88433C211yesmaximal, characteristic, complemented
4#11720321C212yesmaximal, complemented
5#11720316C211yesmaximal, characteristic, complemented
6#10591682—32nomaximal
7#41×H(3,𝔽3)C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9#88538—21no—
10#88538C2211yescharacteristic
11C2×#6180—21no—
12C2×#6180C2211yescharacteristic
13#3756⋊C3—22no—
14#3754⋊C3—22no—
15#3748⋊C3C2212yes—
16#88433—21no—
17#88433—21no—
18#64160D312yes—
19C3×#951—32no—
20C3×#948—32no—
21C3×#948—61no—
22C2×#3719—32no—
23#63577—32no—
24#63561—32no—
25#63560—32no—
26Q8×H(3,𝔽3)D411yescharacteristic
27D4×H(3,𝔽3)—21no—
28D4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×(C3⋊Dic3)⋊C3—22no—
32#634⋊C3—22no—
33#634⋊C3—22no—
34#6180—21no—
35#6180D411yescharacteristic
36#6180D411yescharacteristic
37(C3⋊(C3⋊C8))⋊C3—22no—
38#5204—92no—
39#5143—91no—
40C32×#41—21no—
41C32×#41C2×D312yes—
42C2×#634C2×D312yes—
43C2×C3×Dic6—32no—
44#3756—22no—
45#3754—22no—
46#3748C2×D312yes—
47C3×#216—62no—
48C3×#214—62no—
49C3×#208—32no—
50C3×#208—32no—
51C3×#208—62no—
52C3×#208—61no—
53C3×#208—62no—
54C2×C3×C3⋊C8—32no—
55C2×C3×C3⋊C8—62no—
56C2×C3×C3⋊C8—61no—
57C2×C3×C3⋊C8—62no—
58#3719—62no—
59#3719—62no—
60C22×H(3,𝔽3)—21no—
61C22×H(3,𝔽3)#4111yescharacteristic
62C4×H(3,𝔽3)—21no—
63C4×H(3,𝔽3)C2×D411yescharacteristic
64C4×H(3,𝔽3)C2×D411yescharacteristic
65(C3⋊Dic3)⋊C3—42no—
66C3×#137—92no—
67C3×#131—91no—
68#951—32no—
69#951—92no—
70#948—32no—
71#948—61no—
72C2×#216—92no—
73C2×Dic12—91no—
74#836—92no—
75#824—92no—
76#824—92no—
77#822—92no—
78#819—91no—
79C32×Q8—21no—
80C32×Q8#6712yes—
81C32×D4—22no—
82C32×D4—22no—
83C32×D4—21no—
84C32×D4#6712yes—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4C22×D312yes—
89C2×C3⋊Dic3—22no—
90#634—22no—
91#634—22no—
92C2×C3×Dic3—62no—
93C3×Dic6—62no—
94C3×Dic6—62no—
95C3⋊(C3⋊C8)—22no—
96C3×C3⋊C8—32no—
97C3×C3⋊C8—32no—
98C3×C3⋊C8—61no—
99C3×C3⋊C8—61no—
100C3×C3⋊C8—62no—
101C3×C3⋊C8—62no—
102C3×C3⋊C8—62no—
103#473—271noSylow
104C2×H(3,𝔽3)—21no—
105C2×H(3,𝔽3)—41no—
106C2×H(3,𝔽3)#12111yescharacteristic
107C3×#41—32no—
108C3×#41—61no—
109C3×#41D3211yescharacteristic
110C2×C3×Q8—92no—
111C2×Dic6—32no—
112C2×Dic6—92no—
113C3×Q16—182no—
114C3×SD16—182no—
115C3×M16—92no—
116C3×M16—91no—
117C3×M16—181no—
118C2×C3×C8—91no—
119C2×C3×C8—181no—
120#216—62no—
121#216—182no—
122#216—182no—
123#216—182no—
124#214—62no—
125#214—182no—
126#208—32no—
127#208—32no—
128#208—61no—
129#208—62no—
130#208—62no—
131#208—92no—
132C2×C3⋊C8—32no—
133C2×C3⋊C8—61no—
134C2×C3⋊C8—62no—
135C2×C3⋊C8—62no—
136Dic12—181no—
137Dic12—181no—
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—21no—
141C22×C32#23712yes—
142C32×C4—22no—
143C32×C4—22no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4D3×D412yes—
147C32×C4C2×#6712yes—
148C3⋊Dic3—42no—
149C3×Dic3—122no—
150#137—92no—
151C2×Q16—271no—
152#131—91no—
153#108—271no—
154#101—272no—
155H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
156C3×Q8—32no—
157C3×Q8—61no—
158C3×Q8—182no—
159C3×Q8—182no—
160C3×Q8#62511yescharacteristic
161C3×D4—21no—
162C3×D4—32no—
163C3×D4—62no—
164C3×D4—62no—
165C3×D4—61no—
166C3×D4#62511yescharacteristic
167C2×C3×C4—21no—
168C2×C3×C4—32no—
169C2×C3×C4—62no—
170C2×C3×C4—62no—
171C2×C3×C4—61no—
172C2×C3×C4—182no—
173C2×C3×C4C2×D3211yescharacteristic
174C2×Dic3—62no—
175C2×Dic3—182no—
176Dic6—62no—
177Dic6—62no—
178Dic6—182no—
179Dic6—182no—
180C3×C8—91no—
181C3×C8—91no—
182C3×C8—182no—
183C3×C8—181no—
184C3⋊C8—32no—
185C3⋊C8—32no—
186C3⋊C8—61no—
187C3⋊C8—61no—
188C3⋊C8—62no—
189C3⋊C8—62no—
190C3⋊C8—62no—
191C3⋊C8—182no—
192C2×C32—22no—
193C2×C32—21no—
194C2×C32—41no—
195C2×C32—42no—
196C2×C32—42no—
197C2×C32#93912yes—
198#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
199C2×Q8—92no—
200Q16—182no—
201Q16—541no—
202Q16—541no—
203SD16—182no—
204M16—91no—
205M16—92no—
206M16—181no—
207C2×C8—91no—
208C2×C8—181no—
209C22×C3—21no—
210C22×C3—32no—
211C22×C3—62no—
212C22×C3—62no—
213C22×C3—61no—
214C22×C3#380511yescharacteristic
215C3×C4—21no—
216C3×C4—32no—
217C3×C4—32no—
218C3×C4—62no—
219C3×C4—62no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—362no—
223C3×C4C2×#62511yescharacteristic
224C3×C4#381111yescharacteristic
225Dic3—122no—
226Dic3—362no—
227C32—21no—
228C32#520412yescomplemented
229Q8—182no—
230Q8—182no—
231Q8#619811yescharacteristic
232D4—21no—
233D4#619811yescharacteristic
234C2×C4—21no—
235C2×C4—182no—
236C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
237C8—91no—
238C8—91no—
239C8—182no—
240C8—181no—
241C2×C3—21no—
242C2×C3—32no—
243C2×C3—41no—
244C2×C3—62no—
245C2×C3—61no—
246C2×C3—121no—
247C2×C3—122no—
248C2×C3—122no—
249C2×C3#6398011yescharacteristic
250C22—21no—
251C22#8867211yescharacteristic
252C4—21no—
253C4—362no—
254C4C2×#619811yescharacteristic
255C4#8867711yescharacteristic
256C3—32no—
257C3—61no—
258C3#1059168311yescharacteristic
259C2—21no—
260C2—41no—
261C2#1172131811yescharacteristic
subgroupG/H
centerC2#11721318
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radical#41×H(3,𝔽3)C22
socleC2×C3#63980
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#473
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
3. C2×#88433C2semidirect
4. #11720321C2semidirect
5. #11720316C2semidirect
155. H(3,𝔽3)#473semidirect
228. C32#5204semidirect

Maximal subgroups

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

subgroupindex
#117218692
#64160⋊C32
C2×#884332
#117203212
#117203162
#105916823