Finite Group Databasebeta

← #13144739

Subgroups — #13144739

Order 1728.

572 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 97 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×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721859C211yesmaximal, characteristic, complemented
3#11721842C211yesmaximal, characteristic
4#11721776C211yesmaximal, characteristic, complemented
5#11720584C211yesmaximal, characteristic, complemented
6#11720579C211yesmaximal, characteristic, complemented
7#11720552C211yesmaximal, characteristic, complemented
8C3×#5177—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11#88740C2211yescharacteristic
12SD16×H(3,𝔽3)—21no—
13SD16×H(3,𝔽3)—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#88550—21no—
16#88549—21no—
17#88546—21no—
18#88545—21no—
19#88538C2211yescharacteristic
20#88538C2211yescharacteristic
21#88538C2211yescharacteristic
22#88527—21no—
23C8×#272—21no—
24C2×C32×SD16D314yescomplemented
25C3×#942—34no—
26C3×#932—34no—
27C3×#901—34no—
28C3×#836—34no—
29C3×#833—34no—
30C3×#819—34no—
31C23×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)—21no—
34D4×H(3,𝔽3)—21no—
35D4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38#6235—21no—
39C4⋊#272D411yescharacteristic
40C4×#272—21no—
41#6229D411yescharacteristic
42C8×H(3,𝔽3)D411yescharacteristic
43C8×H(3,𝔽3)D411yescharacteristic
44#6180—21no—
45#6180—21no—
46#6180—21no—
47C3×#471—91no—
48#5177—34no—
49C2×C32×Q8C2×D314yes—
50C2×C32×D4C2×D314yes—
51C3×#235—34no—
52C32×SD16—24no—
53C32×SD16—24no—
54C2×C32×C8C2×D314yes—
55C3×#216—64no—
56C3×#215—64no—
57C3×#214—64no—
58C3×#213—64no—
59C3×#208—34no—
60C3×#208—34no—
61C3×#208—34no—
62C3×ΓL(1,𝔽25)—64no—
63C3×D3×C8—64no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)—81no—
66C22×H(3,𝔽3)#4111yescharacteristic
67C2×#272—21no—
68C4×H(3,𝔽3)—41no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)C2×D411yescharacteristic
71#1084—21no—
72C3×#137—91no—
73C3×AGL(1,ℤ/8)—91no—
74C2×C3×SD16—34no—
75C2×C3×SD16C3⋊D311yescharacteristic
76C3×#131—91no—
77#942—34no—
78#932—34no—
79#901—34no—
80C3×#108—91no—
81C3×#101—91no—
82C3×#100—91no—
83#836—34no—
84#833—34no—
85#819—34no—
86C23×C32—24no—
87C32×Q8—24no—
88C32×Q8—24no—
89C32×D4—24no—
90C32×D4—24no—
91C2×C32×C4—24no—
92C2×C32×C4C22×D314yes—
93C3×#67—64no—
94C3×D12—34no—
95C3×C4×D3—64no—
96C3×Dic6—34no—
97C32×C8#6714yes—
98C32×C8#6714yes—
99C3×C3⋊C8—64no—
100C3×C3⋊C8—64no—
101C3×C3⋊C8—64no—
102#471—91noSylow
103C2×H(3,𝔽3)—21no—
104C2×H(3,𝔽3)—81no—
105C2×H(3,𝔽3)#12111yescharacteristic
106#272—41no—
107C3×#41—91no—
108C2×C3×Q8—34no—
109C2×C3×Q8C2×C3⋊D311yescharacteristic
110C2×C3×D4—34no—
111C2×C3×D4C2×C3⋊D311yescharacteristic
112#235—34no—
113C3×Q16—181no—
114C3×SD16—21no—
115C3×SD16—21no—
116C3×SD16—64no—
117C3×SD16—64no—
118C3×SD16—181no—
119C3×SD16—181no—
120C3×D8—181no—
121C3×M16—91no—
122C3×M16—91no—
123C3×M16—91no—
124C3×M16—181no—
125C2×C3×C8—34no—
126C2×C3×C8—181no—
127C2×C3×C8C2×C3⋊D311yescharacteristic
128#216—64no—
129#215—64no—
130#214—64no—
131#213—64no—
132#208—34no—
133#208—34no—
134#208—34no—
135ΓL(1,𝔽25)—64no—
136D3×C8—64no—
137C22×C32—44no—
138C22×C32—84no—
139C22×C32#23714yes—
140C2×C3×D3—64no—
141C32×C4—44no—
142C32×C4D3×D414yes—
143C32×C4C2×#6714yes—
144C3×Dic3—64no—
145#137—91no—
146AGL(1,ℤ/8)—91no—
147C2×SD16#27211yescharacteristic, complemented
148#131—91no—
149#108—91no—
150#101—91no—
151#100—91no—
152H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
153C23×C3—21no—
154C23×C3—64no—
155C3×Q8—21no—
156C3×Q8—21no—
157C3×Q8—64no—
158C3×Q8—64no—
159C3×Q8—91no—
160C3×D4—21no—
161C3×D4—21no—
162C3×D4—64no—
163C3×D4—64no—
164C3×D4—91no—
165C3×D4—181no—
166C2×C3×C4—21no—
167C2×C3×C4—34no—
168C2×C3×C4—64no—
169C2×C3×C4—181no—
170C2×C3×C4C22×C3⋊D311yescharacteristic
171#67—64no—
172D12—34no—
173C4×D3—64no—
174Dic6—34no—
175C3×C8—34no—
176C3×C8—34no—
177C3×C8—181no—
178C3×C8—181no—
179C3×C8—181no—
180C3×C8#63811yescharacteristic
181C3×C8#63811yescharacteristic
182C3⋊C8—64no—
183C3⋊C8—64no—
184C3⋊C8—64no—
185C2×C32—24no—
186C2×C32—84no—
187C2×C32#93814yes—
188C3×D3—124no—
189#41—91no—
190C2×Q8C2×#27211yescharacteristic
191C2×D4C2×#27211yescharacteristic
192Q16—181no—
193SD16—21no—
194SD16—21no—
195SD16—181no—
196SD16—181no—
197D8—181no—
198M16—91no—
199M16—91no—
200M16—91no—
201M16—181no—
202C2×C8—181no—
203C2×C8C2×#27211yescharacteristic
204C22×C3—34no—
205C22×C3—41no—
206C22×C3—81no—
207C22×C3—124no—
208C22×C3—181no—
209C22×C3—244no—
210C22×C3#383011yescharacteristic
211C2×D3—64no—
212C3×C4—34no—
213C3×C4—34no—
214C3×C4—41no—
215C3×C4—124no—
216C3×C4—181no—
217C3×C4D4×C3⋊D311yescharacteristic
218C3×C4C2×#63811yescharacteristic
219Dic3—64no—
220C32#517714yescomplemented
221C23—21no—
222Q8—21no—
223Q8—21no—
224Q8—91no—
225D4—21no—
226D4—21no—
227D4—91no—
228D4—181no—
229C2×C4—21no—
230C2×C4—181no—
231C2×C4C22×#27211yescharacteristic
232C8—181no—
233C8—181no—
234C8—181no—
235C8#623511yescharacteristic
236C8#623511yescharacteristic
237C2×C3—21no—
238C2×C3—34no—
239C2×C3—64no—
240C2×C3—81no—
241C2×C3—244no—
242C2×C3—361no—
243C2×C3#6414711yescharacteristic
244D3—124no—
245C22—41no—
246C22—81no—
247C22—181no—
248C22#8874411yescharacteristic
249C4—41no—
250C4—181no—
251C4D4×#27211yescharacteristic
252C4C2×#623511yescharacteristic
253C3—34no—
254C3#1059287611yescharacteristic
255C2—21no—
256C2—81no—
257C2—361no—
258C2#1172184811yescharacteristic
subgroupG/H
center, socleC2×C3#64147
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×SD16×H(3,𝔽3)C2
prime radical#11721842C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterH(3,𝔽3)#471
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×SD16×H(3,𝔽3)C2semidirect
2. #11721859C2semidirect
4. #11721776C2semidirect
5. #11720584C2semidirect
6. #11720579C2semidirect
7. #11720552C2semidirect
24. C2×C32×SD16D3semidirect
147. C2×SD16#272semidirect
152. H(3,𝔽3)#471semidirect
220. C32#5177semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
#117218592
#117218422
#117217762
#117205842
#117205792
#117205522
C3×#51773