Finite Group Databasebeta

← #13144003

Subgroups — #13144003

Order 1728.

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

normal subgroups only

This group has 528 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
1#11721864C211yesmaximal, characteristic
2#11721758C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
4#11720583C211yesmaximal, characteristic, complemented
5#11720554C211yesmaximal, characteristic
6C8×#1084C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic
8C3×#4800—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13#88542—21no—
14#88542—21no—
15C4⋊#1084—21no—
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×#947—34no—
21C3×#890—34no—
22C32×#103D314yescomplemented
23C3×#835—34no—
24C3×#821—34no—
25C3×C8×Dic3—34no—
26C3×#809—34no—
27C22×#272—21no—
28Q8×H(3,𝔽3)—21no—
29Q8×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—21no—
34C2×#1084D411yescharacteristic
35C2×#1084D411yescharacteristic
36C4⋊#272—21no—
37C4⋊#272—21no—
38C8×H(3,𝔽3)—21no—
39#6180—21no—
40C3×#488—91no—
41#4800—34no—
42C2×C32×Q8C2×D314yes—
43C2×C3×D12—34no—
44C2×C32×C8C2×D314yes—
45C32×C4⋊C4C2×D314yes—
46C3×(C2×D3)⋊C4—64no—
47C3×(C2×D3)⋊C4—64no—
48C3×C4⋊Dic3—34no—
49C3×C4⋊Dic3—64no—
50C3×Dic3⋊C4—64no—
51C3×C4×Dic3—34no—
52C2×C3×C3⋊C8—34no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×#272—41no—
55C2×#272—41no—
56C2×#272—41no—
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)#4111yescharacteristic
60C4×H(3,𝔽3)#4111yescharacteristic
61#1084—21no—
62#1084—21no—
63#1084—41no—
64C3×#125—91no—
65C3×#124—91no—
66#947—34no—
67#890—34no—
68C3×#103—34no—
69C3×#103—91no—
70C3×#103C3⋊D311yescharacteristic
71C3×#102—91no—
72C3×#102—91no—
73C3×C4×C8—91no—
74#835—34no—
75#821—34no—
76C8×Dic3—34no—
77#809—34no—
78C22×C3×D3—64no—
79C32×Q8—24no—
80C32×Q8—24no—
81C2×C32×C4—24no—
82C2×C32×C4—24no—
83C2×C32×C4C22×D314yes—
84C2×C3×Dic3—34no—
85C2×C3×Dic3—34no—
86C2×C3×Dic3—64no—
87C3×D12—64no—
88C3×D12—64no—
89C32×C8—24no—
90C3×C3⋊C8—64no—
91#488—91noSylow
92C2×H(3,𝔽3)#12411yescharacteristic
93C2×H(3,𝔽3)#13511yescharacteristic
94C2×H(3,𝔽3)#13511yescharacteristic
95#272—81no—
96C2×C3×Q8—34no—
97C2×C3×Q8C2×C3⋊D311yescharacteristic
98C2×C3×D4—91no—
99C2×D12—34no—
100C2×C3×C8—34no—
101C2×C3×C8—91no—
102C2×C3×C8C2×C3⋊D311yescharacteristic
103C3×C4⋊C4—34no—
104C3×C4⋊C4—91no—
105C3×C4⋊C4—181no—
106C3×C4⋊C4—181no—
107C3×C4⋊C4C2×C3⋊D311yescharacteristic
108C3×C22⋊C4—181no—
109C3×C22⋊C4—181no—
110C3×C42—91no—
111(C2×D3)⋊C4—64no—
112(C2×D3)⋊C4—64no—
113C4⋊Dic3—34no—
114C4⋊Dic3—64no—
115Dic3⋊C4—64no—
116C4×Dic3—34no—
117C2×C3⋊C8—34no—
118C22×C32D3×D414yes—
119C2×C3×D3—124no—
120C2×C3×D3—124no—
121C2×C3×D3—124no—
122C32×C4—44no—
123C32×C4—44no—
124C32×C4#23514yes—
125C32×C4#23714yes—
126C3×Dic3—64no—
127C3×Dic3—64no—
128C3×Dic3—124no—
129#125—91no—
130#124—91no—
131#103—91no—
132#103#27211yescharacteristic, complemented
133#102—91no—
134#102—91no—
135C4×C8—91no—
136H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
137C23×C3—181no—
138C22×D3—64no—
139C3×Q8—21no—
140C3×Q8—21no—
141C3×Q8—64no—
142C3×Q8—64no—
143C3×D4—181no—
144C3×D4—181no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—34no—
148C2×C3×C4—64no—
149C2×C3×C4—64no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4—181no—
153C2×C3×C4C22×C3⋊D311yescharacteristic
154C2×Dic3—34no—
155C2×Dic3—34no—
156C2×Dic3—64no—
157D12—64no—
158D12—64no—
159C3×C8—21no—
160C3×C8—64no—
161C3×C8—181no—
162C3⋊C8—64no—
163C2×C32#88514yes—
164C2×C32#91614yes—
165C2×C32#91914yes—
166C3×D3—244no—
167C2×Q8C2×#27211yescharacteristic
168C2×D4—91no—
169C2×C8—91no—
170C2×C8C2×#27211yescharacteristic
171C4⋊C4—91no—
172C4⋊C4—181no—
173C4⋊C4—181no—
174C4⋊C4C2×#27211yescharacteristic
175C22⋊C4—181no—
176C22⋊C4—181no—
177C42—91no—
178C22×C3—34no—
179C22×C3—361no—
180C22×C3—361no—
181C22×C3—361no—
182C22×C3D4×C3⋊D311yescharacteristic
183C2×D3—124no—
184C2×D3—124no—
185C2×D3—124no—
186C3×C4—34no—
187C3×C4—34no—
188C3×C4—41no—
189C3×C4—41no—
190C3×C4—124no—
191C3×C4—124no—
192C3×C4—181no—
193C3×C4—181no—
194C3×C4—361no—
195C3×C4#382811yescharacteristic
196C3×C4#383011yescharacteristic
197Dic3—64no—
198Dic3—64no—
199Dic3—124no—
200C32#480014yescomplemented
201C23—181no—
202Q8—21no—
203Q8—21no—
204D4—181no—
205D4—181no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—181no—
211C2×C4C22×#27211yescharacteristic
212C8—21no—
213C8—181no—
214C2×C3—34no—
215C2×C3—34no—
216C2×C3—34no—
217C2×C3—721no—
218C2×C3#6409411yescharacteristic
219C2×C3#6412511yescharacteristic
220C2×C3#6412811yescharacteristic
221D3—244no—
222C22—361no—
223C22—361no—
224C22—361no—
225C22D4×#27211yescharacteristic
226C4—41no—
227C4—41no—
228C4—181no—
229C4—181no—
230C4—361no—
231C4#8874011yescharacteristic
232C4#8874411yescharacteristic
233C3—34no—
234C3#1059249911yescharacteristic
235C2—721no—
236C2#1172174111yescharacteristic
237C2#1172180211yescharacteristic
238C2#1172180811yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC2×C4⋊#272C22
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. #11721758C2semidirect
3. H(3,𝔽3)×#103C2semidirect
4. #11720583C2semidirect
6. C8×#1084C2semidirect
22. C32×#103D3semidirect
132. #103#272semidirect
136. H(3,𝔽3)#488semidirect
200. C32#4800semidirect

Maximal subgroups

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

subgroupindex
#117218642
#117217582
H(3,𝔽3)×#1032
#117205832
#117205542
C8×#10842
#117205312
C3×#48003