Finite Group Databasebeta

← #13138878

Subgroups — #13138878

Order 1728.

415 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 79 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#64110⋊C3C212yesmaximal, complemented
2#11721060C211yesmaximal, characteristic
3C8×#1084C211yesmaximal, characteristic, complemented
4#11720316C211yesmaximal, characteristic, complemented
5#11720305C212yesmaximal, complemented
6#10590923—32nomaximal
7#3828⋊C3C2212yes—
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C4×#1084C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11#3748⋊C3C2212yes—
12#3743⋊C3—22no—
13C8×(C3⋊D3)⋊C3—22no—
14#88433C411yescharacteristic, complemented
15#88432C411yescharacteristic, complemented
16#88431C412yescomplemented
17#64110D312yescomplemented
18C3×#901—32no—
19#63829—32no—
20C3×C8×Dic3—32no—
21C3×C8×Dic3—61no—
22#63577—32no—
23#63570—32no—
24#63569—32no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—21no—
27#638⋊C3—22no—
28C4⋊((C3⋊D3)⋊C3)C2×C412yes—
29C4×(C3⋊D3)⋊C3—22no—
30#634⋊C3C2×C412yes—
31C8×H(3,𝔽3)D411yescharacteristic
32C8×H(3,𝔽3)D411yescharacteristic
33#6180C2×C411yescharacteristic
34#6180C2×C411yescharacteristic
35(C3⋊(C3⋊C8))⋊C3—22no—
36#5144—92no—
37#4700—91no—
38#3828C2×D312yes—
39C3×#235—32no—
40C2×C32×C8—21no—
41C2×C32×C8C2×D312yes—
42#3748C2×D312yes—
43#3743—22no—
44C8×C3⋊D3—22no—
45C3×C4×Dic3—32no—
46C3×C4×Dic3—61no—
47C3×#208—32no—
48C2×C3×C3⋊C8—32no—
49C2×C3×C3⋊C8—61no—
50C3×ΓL(1,𝔽25)—62no—
51C3×D3×C8—62no—
52#3719—32no—
53#3717—32no—
54#3716—32no—
55#3714—32no—
56C22×H(3,𝔽3)#4111yescharacteristic
57C2×(C3⋊D3)⋊C3—22no—
58C4×H(3,𝔽3)C22×C411yescharacteristic
59C4×H(3,𝔽3)C2×D411yescharacteristic
60#1084—22no—
61(C3⋊Dic3)⋊C3—22no—
62C3×#131—92no—
63#950—92no—
64#904—91no—
65#901—32no—
66#901—92no—
67C3×C4×C8—91no—
68#837—92no—
69#825—92no—
70#822—92no—
71#819—91no—
72C8×Dic3—32no—
73C8×Dic3—61no—
74#805—92no—
75C2×C32×C4—21no—
76C2×C32×C4C22×D312yes—
77#638—22no—
78C4⋊(C3⋊D3)C4×D312yes—
79C4×C3⋊D3—22no—
80#634C4×D312yes—
81C3×#67—62no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C3×D12—32no—
85C3×C4×D3—62no—
86C3×Dic6—32no—
87C32×C8—21no—
88C32×C8—21no—
89C32×C8#6712yes—
90C32×C8#6712yes—
91C3⋊(C3⋊C8)—22no—
92C3×C3⋊C8—32no—
93C3×C3⋊C8—32no—
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—
105C3×M16—182no—
106C2×C3×C8—32no—
107C2×C3×C8—61no—
108C2×C3×C8—91no—
109C2×C3×C8—182no—
110C2×C3×C8D3211yescharacteristic
111C3×C42—91no—
112#216—92no—
113#215—92no—
114#214—92no—
115#213—92no—
116C4×Dic3—32no—
117C4×Dic3—61no—
118#208—32no—
119#208—92no—
120C2×C3⋊C8—32no—
121C2×C3⋊C8—61no—
122Dic12—91no—
123D24—91no—
124#204—92no—
125ΓL(1,𝔽25)—62no—
126ΓL(1,𝔽25)—182no—
127D3×C8—62no—
128D3×C8—182no—
129C22×C32—21no—
130C22×C32#23512yes—
131C2×C3⋊D3—22no—
132C2×C3×D3—62no—
133C32×C4—21no—
134C32×C4—21no—
135C32×C4C2×C4×D312yes—
136C32×C4C2×#6712yes—
137C3⋊Dic3—22no—
138C3×Dic3—64no—
139C3×Dic3—64no—
140C3×Dic3—62no—
141#135—271no—
142#131—92no—
143#108—271no—
144#104—272no—
145C4×C8—91no—
146H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
147C3×Q8—92no—
148C3×D4—92no—
149C3×D4—182no—
150C2×C3×C4—32no—
151C2×C3×C4—61no—
152C2×C3×C4—182no—
153C2×C3×C4—181no—
154C2×C3×C4C2×D3211yescharacteristic
155#67—62no—
156#67—182no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159D12—32no—
160D12—92no—
161C4×D3—62no—
162C4×D3—182no—
163Dic6—32no—
164Dic6—92no—
165C3×C8—32no—
166C3×C8—32no—
167C3×C8—61no—
168C3×C8—61no—
169C3×C8—91no—
170C3×C8—91no—
171C3×C8—182no—
172C3×C8#62511yescharacteristic
173C3×C8#62511yescharacteristic
174C3⋊C8—32no—
175C3⋊C8—32no—
176C3⋊C8—61no—
177C3⋊C8—61no—
178C3⋊C8—62no—
179C3⋊C8—182no—
180C2×C32—22no—
181C2×C32—22no—
182C2×C32—21no—
183C2×C32C4×#6712yes—
184C3⋊D3—42no—
185C3×D3—122no—
186#41—92no—
187Q16—271no—
188SD16—272no—
189D8—271no—
190M16—92no—
191M16—182no—
192C2×C8—91no—
193C2×C8—182no—
194C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
195C42—91no—
196C22×C3—32no—
197C22×C3—61no—
198C22×C3—182no—
199C22×C3#379811yescharacteristic
200C2×D3—62no—
201C2×D3—182no—
202C3×C4—32no—
203C3×C4—32no—
204C3×C4—61no—
205C3×C4—61no—
206C3×C4—182no—
207C3×C4—182no—
208C3×C4C4×D3211yescharacteristic
209C3×C4C2×#62511yescharacteristic
210Dic3—64no—
211Dic3—64no—
212Dic3—62no—
213Dic3—182no—
214C32—21no—
215C32#514412yescomplemented
216Q8—92no—
217D4—92no—
218D4—182no—
219C2×C4—181no—
220C2×C4—182no—
221C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
222C8—91no—
223C8—91no—
224C8—182no—
225C8#619811yescharacteristic
226C8#619811yescharacteristic
227C2×C3—21no—
228C2×C3—32no—
229C2×C3—62no—
230C2×C3—62no—
231C2×C3—61no—
232C2×C3—362no—
233C2×C3C4×#62511yescharacteristic
234D3—122no—
235D3—362no—
236C22—182no—
237C22#8865011yescharacteristic
238C4—182no—
239C4—182no—
240C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C4C2×#619811yescharacteristic
242C3—32no—
243C3—61no—
244C3#1059092111yescharacteristic
245C2—21no—
246C2—362no—
247C2C4×#619811yescharacteristic
subgroupG/H
centerC8#6198
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C4×#625
prime radical#11721060C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#443
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C8 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #64110⋊C3C2semidirect
3. C8×#1084C2semidirect
4. #11720316C2semidirect
5. #11720305C2semidirect
14. #88433C4semidirect
15. #88432C4semidirect
16. #88431C4semidirect
17. #64110D3semidirect
146. H(3,𝔽3)#443semidirect
215. C32#5144semidirect

Maximal subgroups

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

subgroupindex
#64110⋊C32
#117210602
C8×#10842
#117203162
#117203052
#105909233