Finite Group Databasebeta

← #13138805

Subgroups — #13138805

Order 1728.

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

normal subgroups only

This group has 499 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
1C2×#3744⋊C3C212yesmaximal, complemented
2#11721113C212yesmaximal, complemented
3C2×#88433C211yesmaximal, characteristic, complemented
4C2×#88432C211yesmaximal, characteristic
5C8×#1084C211yesmaximal, characteristic, complemented
6#10590862—32nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
8C2×#634⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12#3744⋊C3—24no—
13#88450—24no—
14#88433—22no—
15#88432—22no—
16C2×#3744D312yescomplemented
17C2×C3×#204—32no—
18#63864—32no—
19#63860—32no—
20C2×#3719—32no—
21C2×#3714—32no—
22C3×C8×Dic3—32no—
23C3×C8×Dic3—61no—
24C22×(C3⋊D3)⋊C3—22no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084D412yes—
27C2×(C3⋊Dic3)⋊C3—22no—
28C4⋊((C3⋊D3)⋊C3)—22no—
29C4⋊((C3⋊D3)⋊C3)—22no—
30#634⋊C3—22no—
31#634⋊C3—22no—
32C8×H(3,𝔽3)D412yes—
33#6180D412yes—
34#5168—92no—
35#4696—91no—
36C2×C4⋊(C3⋊D3)C2×D312yes—
37C2×#634C2×D312yes—
38C2×C3×D12—32no—
39C2×C3×Dic6—32no—
40C2×C32×C8—21no—
41C2×C32×C8C2×D312yes—
42#3744—24no—
43C3×C4×Dic3—32no—
44C3×C4×Dic3—61no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—61no—
47C3×#204—64no—
48#3722—64no—
49(C2×D3)⋊Dic3—64no—
50#3719—64no—
51#3714—64no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C2×(C3⋊D3)⋊C3—42no—
54C2×(C3⋊D3)⋊C3—44no—
55C4×H(3,𝔽3)C2×D411yescharacteristic
56C4×H(3,𝔽3)C2×D411yescharacteristic
57#1084—22no—
58(C3⋊Dic3)⋊C3—42no—
59C2×C3×SD16—92no—
60#947—92no—
61C2×#216—92no—
62#936—92no—
63C2×#213—92no—
64C2×Dic12—91no—
65C2×D24—91no—
66C2×#204—32no—
67C2×#204—92no—
68#875—92no—
69C3×C4×C8—91no—
70C8×Dic3—32no—
71C8×Dic3—61no—
72C22×C3⋊D3—22no—
73C22×C3×D3—62no—
74C2×C32×C4—21no—
75C2×C32×C4C22×D312yes—
76C2×C3⋊Dic3—22no—
77C4⋊(C3⋊D3)—22no—
78C4⋊(C3⋊D3)—22no—
79#634—22no—
80#634—22no—
81C2×C3×Dic3—34no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C3×D12—62no—
85C3×D12—62no—
86C3×Dic6—62no—
87C3×Dic6—62no—
88C32×C8—22no—
89C32×C8#6714yes—
90C3×C3⋊C8—34no—
91C3×C3⋊C8—62no—
92#495—271noSylow
93C2×H(3,𝔽3)#12711yescharacteristic
94C2×H(3,𝔽3)#13512yes—
95(C3⋊D3)⋊C3—82no—
96C2×C3×Q8—92no—
97C2×C3×D4—92no—
98C2×D12—32no—
99C2×D12—92no—
100C2×Dic6—32no—
101C2×Dic6—92no—
102C3×SD16—184no—
103C2×C3×C8—32no—
104C2×C3×C8—61no—
105C2×C3×C8—91no—
106C2×C3×C8D3211yescharacteristic
107C3×C42—91no—
108#217—184no—
109#216—184no—
110#213—184no—
111(C2×D3)⋊C4—184no—
112(C2×D3)⋊C4—184no—
113C4×Dic3—32no—
114C4×Dic3—61no—
115C2×C3⋊C8—32no—
116C2×C3⋊C8—61no—
117Dic12—182no—
118D24—182no—
119#204—64no—
120#204—184no—
121C22×C32—21no—
122C22×C32D3×D412yes—
123C2×C3⋊D3—44no—
124C2×C3⋊D3—42no—
125C2×C3×D3—124no—
126C2×C3×D3—122no—
127C32×C4—21no—
128C32×C4—21no—
129C32×C4D3×D412yes—
130C32×C4C2×#6712yes—
131C3⋊Dic3—42no—
132C3×Dic3—64no—
133C3×Dic3—64no—
134C3×Dic3—122no—
135C2×Q16—271no—
136C2×SD16—92no—
137C2×D8—271no—
138#124—272no—
139C4×C8—91no—
140H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
141C23×C3—182no—
142C22×D3—62no—
143C22×D3—182no—
144C3×Q8—182no—
145C3×Q8—182no—
146C3×D4—182no—
147C3×D4—182no—
148C2×C3×C4—32no—
149C2×C3×C4—61no—
150C2×C3×C4—92no—
151C2×C3×C4—182no—
152C2×C3×C4C2×D3211yescharacteristic
153C2×Dic3—34no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—182no—
157D12—62no—
158D12—62no—
159D12—182no—
160D12—182no—
161Dic6—62no—
162Dic6—62no—
163Dic6—182no—
164Dic6—182no—
165C3×C8—34no—
166C3×C8—62no—
167C3×C8—92no—
168C3×C8#62512yes—
169C3⋊C8—34no—
170C3⋊C8—62no—
171C2×C32—22no—
172C2×C32—21no—
173C2×C32#91614yes—
174C2×C32#94012yes—
175C3⋊D3—82no—
176C3×D3—242no—
177C2×Q8—92no—
178C2×D4—92no—
179Q16—542no—
180SD16—184no—
181D8—542no—
182C2×C8—91no—
183C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
184C22⋊C4—544no—
185C42—91no—
186C22×C3—32no—
187C22×C3—61no—
188C22×C3—362no—
189C22×C3—364no—
190C22×C3#380211yescharacteristic
191C2×D3—124no—
192C2×D3—122no—
193C2×D3—362no—
194C2×D3—364no—
195C3×C4—32no—
196C3×C4—32no—
197C3×C4—61no—
198C3×C4—61no—
199C3×C4—182no—
200C3×C4—362no—
201C3×C4#380211yescharacteristic
202C3×C4C2×#62511yescharacteristic
203Dic3—64no—
204Dic3—64no—
205Dic3—122no—
206Dic3—362no—
207C32—21no—
208C32#516812yescomplemented
209C23—182no—
210Q8—182no—
211Q8—182no—
212D4—182no—
213D4—182no—
214C2×C4—92no—
215C2×C4—182no—
216C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
217C8—92no—
218C8#619812yes—
219C2×C3—34no—
220C2×C3—32no—
221C2×C3—62no—
222C2×C3—61no—
223C2×C3—722no—
224C2×C3#6380912yes—
225C2×C3#6391411yescharacteristic
226D3—242no—
227D3—722no—
228C22—364no—
229C22—362no—
230C22#8865511yescharacteristic
231C4—182no—
232C4—362no—
233C4#8865511yescharacteristic
234C4C2×#619811yescharacteristic
235C3—32no—
236C3—61no—
237C3#1059086711yescharacteristic
238C2—722no—
239C2#1172084312yes—
240C2#1172114411yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#88432C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#495
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#3744⋊C3C2semidirect
2. #11721113C2semidirect
3. C2×#88433C2semidirect
5. C8×#1084C2semidirect
16. C2×#3744D3semidirect
140. H(3,𝔽3)#495semidirect
208. C32#5168semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
#117211132
C2×#884332
C2×#884322
C8×#10842
#105908623