Finite Group Databasebeta

← #13144759

Subgroups — #13144759

Order 1728.

616 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 111 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×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721864C211yesmaximal, characteristic
3Q8×#1084C211yesmaximal, characteristic, complemented
4C2×#88549C211yesmaximal, characteristic
5#11720583C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic
7#11720547C211yesmaximal, characteristic, complemented
8C3×#5185—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12Q16×H(3,𝔽3)—22no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#88549—22no—
15#88542—21no—
16#88542—21no—
17C4⋊#1084—21no—
18C4⋊#1084C2211yescharacteristic
19C4×#1084—21no—
20C4×#1084C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C2×C32×Q16D314yescomplemented
23C3×#947—34no—
24C3×Q8×Dic3—34no—
25C2×C3×#215—34no—
26C3×#835—34no—
27C3×#821—34no—
28C3×Dic3⋊C8—34no—
29C22×#272—21no—
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)D412yes—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—21no—
38C2×#1084D411yescharacteristic
39C2×#1084D411yescharacteristic
40C4⋊#272—21no—
41C4⋊#272—21no—
42C8×H(3,𝔽3)—21no—
43#6180—21no—
44C3×#464—91no—
45#5185—34no—
46C2×C32×Q8C2×D314yes—
47C2×C32×Q8C2×D314yes—
48C2×C3×D12—34no—
49C32×Q16—28no—
50C2×C32×C8C2×D314yes—
51C3×#215—68no—
52C3×(C2×D3)⋊C4—64no—
53C3×(C2×D3)⋊C4—64no—
54C3×C4⋊Dic3—34no—
55C3×C4⋊Dic3—64no—
56C3×C4×Dic3—34no—
57C3×C4×Dic3—64no—
58C2×C3×C3⋊C8—34no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C2×#272—41no—
61C2×#272—41no—
62C2×#272—41no—
63C4×H(3,𝔽3)—22no—
64C4×H(3,𝔽3)—41no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)#4111yescharacteristic
67#1084—21no—
68#1084—21no—
69#1084—41no—
70C2×C3×Q16—34no—
71C2×C3×Q16C3⋊D311yescharacteristic
72C2×C3×SD16—91no—
73C3×#124—91no—
74C3×C4×Q8—91no—
75#947—34no—
76Q8×Dic3—34no—
77C2×#215—34no—
78C3×C4⋊C8—91no—
79C3×#103—91no—
80C3×#102—91no—
81#835—34no—
82#821—34no—
83Dic3⋊C8—34no—
84C22×C3×D3—64no—
85C32×Q8—24no—
86C32×Q8—24no—
87C32×Q8—24no—
88C32×Q8#6718yes—
89C2×C32×C4—24no—
90C2×C32×C4—24no—
91C2×C32×C4C22×D314yes—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—34no—
94C2×C3×Dic3—64no—
95C3×D12—64no—
96C3×D12—64no—
97C32×C8—24no—
98C3×C3⋊C8—64no—
99#464—91noSylow
100C2×H(3,𝔽3)#12111yescharacteristic
101C2×H(3,𝔽3)#13511yescharacteristic
102C2×H(3,𝔽3)#13711yescharacteristic
103#272—81no—
104C2×C3×Q8—34no—
105C2×C3×Q8—34no—
106C2×C3×Q8C2×C3⋊D311yescharacteristic
107C2×C3×Q8C2×C3⋊D311yescharacteristic
108C2×C3×D4—91no—
109C2×D12—34no—
110C3×Q16—22no—
111C3×Q16—68no—
112C3×SD16—182no—
113C2×C3×C8—34no—
114C2×C3×C8—91no—
115C2×C3×C8C2×C3⋊D311yescharacteristic
116C3×C4⋊C4—91no—
117C3×C4⋊C4—181no—
118C3×C22⋊C4—181no—
119C3×C22⋊C4—181no—
120C3×C42—91no—
121C3×C42—181no—
122#215—68no—
123(C2×D3)⋊C4—64no—
124(C2×D3)⋊C4—64no—
125C4⋊Dic3—34no—
126C4⋊Dic3—64no—
127C4×Dic3—34no—
128C4×Dic3—64no—
129C2×C3⋊C8—34no—
130C22×C32D3×D414yes—
131C2×C3×D3—124no—
132C2×C3×D3—124no—
133C2×C3×D3—124no—
134C32×C4—28no—
135C32×C4—44no—
136C32×C4#23714yes—
137C32×C4C2×#6714yes—
138C3×Dic3—64no—
139C3×Dic3—64no—
140C3×Dic3—124no—
141C2×Q16#27211yescharacteristic, complemented
142C2×SD16—91no—
143#124—91no—
144C4×Q8—91no—
145C4⋊C8—91no—
146#103—91no—
147#102—91no—
148H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
149C23×C3—181no—
150C22×D3—64no—
151C3×Q8—21no—
152C3×Q8—21no—
153C3×Q8—21no—
154C3×Q8—38no—
155C3×Q8—64no—
156C3×Q8—64no—
157C3×Q8—64no—
158C3×Q8#63812yes—
159C3×D4—181no—
160C3×D4—181no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—34no—
164C2×C3×C4—64no—
165C2×C3×C4—64no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—181no—
169C2×C3×C4C22×C3⋊D311yescharacteristic
170C2×Dic3—34no—
171C2×Dic3—34no—
172C2×Dic3—64no—
173D12—64no—
174D12—64no—
175C3×C8—21no—
176C3×C8—64no—
177C3×C8—181no—
178C3⋊C8—64no—
179C2×C32#91814yes—
180C2×C32#91914yes—
181C2×C32#93914yes—
182C3×D3—244no—
183C2×Q8C2×#27211yescharacteristic
184C2×Q8C2×#27211yescharacteristic
185C2×D4—91no—
186Q16—22no—
187SD16—182no—
188C2×C8—91no—
189C2×C8C2×#27211yescharacteristic
190C4⋊C4—91no—
191C4⋊C4—181no—
192C22⋊C4—181no—
193C22⋊C4—181no—
194C42—91no—
195C42—181no—
196C22×C3—34no—
197C22×C3—361no—
198C22×C3—361no—
199C22×C3—361no—
200C22×C3D4×C3⋊D311yescharacteristic
201C2×D3—124no—
202C2×D3—124no—
203C2×D3—124no—
204C3×C4—22no—
205C3×C4—34no—
206C3×C4—34no—
207C3×C4—41no—
208C3×C4—68no—
209C3×C4—124no—
210C3×C4—181no—
211C3×C4—181no—
212C3×C4—361no—
213C3×C4#383011yescharacteristic
214C3×C4C2×#63811yescharacteristic
215Dic3—64no—
216Dic3—64no—
217Dic3—124no—
218C32#518514yescomplemented
219C23—181no—
220Q8—21no—
221Q8—21no—
222Q8—21no—
223Q8#623512yes—
224D4—181no—
225D4—181no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—181no—
231C2×C4C22×#27211yescharacteristic
232C8—21no—
233C8—181no—
234C2×C3—34no—
235C2×C3—34no—
236C2×C3—34no—
237C2×C3—721no—
238C2×C3#6412711yescharacteristic
239C2×C3#6412811yescharacteristic
240C2×C3#6414811yescharacteristic
241D3—244no—
242C22—361no—
243C22—361no—
244C22—361no—
245C22D4×#27211yescharacteristic
246C4—22no—
247C4—41no—
248C4—181no—
249C4—181no—
250C4—361no—
251C4#8874411yescharacteristic
252C4C2×#623511yescharacteristic
253C3—34no—
254C3#1059288411yescharacteristic
255C2—721no—
256C2#1172180711yescharacteristic
257C2#1172180811yescharacteristic
258C2#1172184911yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×Q16×H(3,𝔽3)C2
prime radicalC2×C4⋊#272C22
hypercenterC2×C3×Q16C3⋊D3
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C2×Q16×H(3,𝔽3)C2semidirect
3. Q8×#1084C2semidirect
5. #11720583C2semidirect
7. #11720547C2semidirect
22. C2×C32×Q16D3semidirect
141. C2×Q16#272semidirect
148. H(3,𝔽3)#464semidirect
218. C32#5185semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×H(3,𝔽3)2
#117218642
Q8×#10842
C2×#885492
#117205832
#117205542
#117205472
C3×#51853