Finite Group Databasebeta

← #13144730

Subgroups — #13144730

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×SD16×H(3,𝔽3)C211yesmaximal, characteristic
2Q8×#1084C211yesmaximal, characteristic, complemented
3C2×#88550C211yesmaximal, characteristic, complemented
4#11721846C211yesmaximal, characteristic
5#11720578C211yesmaximal, characteristic
6#11720549C211yesmaximal, characteristic, complemented
7#11720547C211yesmaximal, characteristic, complemented
8C3×#5166—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12SD16×H(3,𝔽3)—22no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#88552—21no—
15#88552—21no—
16#88550—22no—
17C4⋊#1084—21no—
18C4⋊#1084C2211yescharacteristic
19C4×#1084—21no—
20C4×#1084C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C2×C32×SD16D314yes—
23C3×Q8×Dic3—34no—
24C2×C3×#216—34no—
25C3×#936—34no—
26C3×#832—34no—
27C3×#816—34no—
28C3×Dic3⋊C8—34no—
29C23×H(3,𝔽3)—21no—
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)D412yes—
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×#1084—21no—
37C2×#1084—21no—
38C2×#1084D411yescharacteristic
39C2×#1084D411yescharacteristic
40#6229—21no—
41#6229—21no—
42C8×H(3,𝔽3)—21no—
43#6180—21no—
44C3×#464—91no—
45#5166—34no—
46C2×C32×Q8C2×D314yes—
47C2×C32×D4C2×D314yes—
48C2×C3×Dic6—34no—
49C32×SD16—28no—
50C2×C32×C8C2×D314yes—
51C3×#217—64no—
52C3×#217—64no—
53C3×#216—68no—
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)—41no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C4×H(3,𝔽3)—22no—
64C4×H(3,𝔽3)C2×D411yescharacteristic
65C4×H(3,𝔽3)#4111yescharacteristic
66#1084—21no—
67#1084—21no—
68#1084—41no—
69#1084—41no—
70C2×C3×Q16—91no—
71C2×C3×SD16—34no—
72C2×C3×SD16C3⋊D311yescharacteristic
73C3×#124—91no—
74C3×C4×Q8—91no—
75Q8×Dic3—34no—
76C2×#216—34no—
77#936—34no—
78C3×C4⋊C8—91no—
79C3×#103—91no—
80C3×#102—91no—
81#832—34no—
82#816—34no—
83Dic3⋊C8—34no—
84C23×C32—24no—
85C32×Q8—24no—
86C32×Q8#6718yes—
87C32×D4—24no—
88C32×D4—24no—
89C2×C32×C4—24no—
90C2×C32×C4C22×D314yes—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—64no—
94C2×C3×Dic3—64no—
95C3×Dic6—64no—
96C3×Dic6—64no—
97C32×C8—24no—
98C3×C3⋊C8—64no—
99#464—91noSylow
100C2×H(3,𝔽3)—81no—
101C2×H(3,𝔽3)#12111yescharacteristic
102C2×H(3,𝔽3)#13511yescharacteristic
103C2×H(3,𝔽3)#13711yescharacteristic
104C2×C3×Q8—34no—
105C2×C3×Q8—91no—
106C2×C3×Q8C2×C3⋊D311yescharacteristic
107C2×C3×D4—34no—
108C2×C3×D4C2×C3⋊D311yescharacteristic
109C2×Dic6—34no—
110C3×Q16—182no—
111C3×SD16—22no—
112C3×SD16—68no—
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#217—64no—
123#217—64no—
124#216—68no—
125C4⋊Dic3—34no—
126C4⋊Dic3—64no—
127C4×Dic3—34no—
128C4×Dic3—64no—
129C2×C3⋊C8—34no—
130C22×C32—44no—
131C22×C32—44no—
132C22×C32—44no—
133C22×C32D3×D414yes—
134C32×C4—28no—
135C32×C4#23714yes—
136C32×C4C2×#6714yes—
137C3×Dic3—64no—
138C3×Dic3—64no—
139C3×Dic3—124no—
140C3×Dic3—124no—
141C2×Q16—91no—
142C2×SD16#27211yescharacteristic
143#124—91no—
144C4×Q8—91no—
145C4⋊C8—91no—
146#103—91no—
147#102—91no—
148H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
149C23×C3—21no—
150C23×C3—64no—
151C3×Q8—21no—
152C3×Q8—38no—
153C3×Q8—64no—
154C3×Q8—181no—
155C3×Q8—181no—
156C3×Q8#63812yes—
157C3×D4—21no—
158C3×D4—21no—
159C3×D4—64no—
160C3×D4—64no—
161C2×C3×C4—21no—
162C2×C3×C4—34no—
163C2×C3×C4—64no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4—181no—
167C2×C3×C4—181no—
168C2×C3×C4C22×C3⋊D311yescharacteristic
169C2×Dic3—34no—
170C2×Dic3—34no—
171C2×Dic3—64no—
172C2×Dic3—64no—
173Dic6—64no—
174Dic6—64no—
175C3×C8—21no—
176C3×C8—64no—
177C3×C8—181no—
178C3⋊C8—64no—
179C2×C32—84no—
180C2×C32#91514yes—
181C2×C32#91614yes—
182C2×C32#93914yes—
183C2×Q8—91no—
184C2×Q8C2×#27211yescharacteristic
185C2×D4C2×#27211yescharacteristic
186Q16—182no—
187SD16—22no—
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—41no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3—124no—
201C22×C3—124no—
202C22×C3—124no—
203C22×C3D4×C3⋊D311yescharacteristic
204C3×C4—22no—
205C3×C4—34no—
206C3×C4—34no—
207C3×C4—68no—
208C3×C4—181no—
209C3×C4—181no—
210C3×C4—361no—
211C3×C4—361no—
212C3×C4#383011yescharacteristic
213C3×C4C2×#63811yescharacteristic
214Dic3—64no—
215Dic3—64no—
216Dic3—124no—
217Dic3—124no—
218C32#516614yescomplemented
219C23—21no—
220Q8—21no—
221Q8—181no—
222Q8—181no—
223Q8#623512yes—
224D4—21no—
225D4—21no—
226C2×C4—21no—
227C2×C4—91no—
228C2×C4—91no—
229C2×C4—181no—
230C2×C4—181no—
231C2×C4C22×#27211yescharacteristic
232C8—21no—
233C8—181no—
234C2×C3—34no—
235C2×C3—34no—
236C2×C3—34no—
237C2×C3—81no—
238C2×C3—244no—
239C2×C3#6412411yescharacteristic
240C2×C3#6412511yescharacteristic
241C2×C3#6414811yescharacteristic
242C22—41no—
243C22—41no—
244C22—41no—
245C22D4×#27211yescharacteristic
246C4—22no—
247C4—181no—
248C4—181no—
249C4—361no—
250C4—361no—
251C4#8874411yescharacteristic
252C4C2×#623511yescharacteristic
253C3—34no—
254C3#1059286511yescharacteristic
255C2—81no—
256C2#1172180111yescharacteristic
257C2#1172180211yescharacteristic
258C2#1172184911yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×SD16×H(3,𝔽3)C2
prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×C3×SD16C3⋊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×SD16

Complements

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

NH ≅ G/Ntype
2. Q8×#1084C2semidirect
3. C2×#88550C2semidirect
6. #11720549C2semidirect
7. #11720547C2semidirect
148. H(3,𝔽3)#464semidirect
218. C32#5166semidirect

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
Q8×#10842
C2×#885502
#117218462
#117205782
#117205492
#117205472
C3×#51663