Finite Group Databasebeta

← #13144005

Subgroups — #13144005

Order 1728.

825 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 825 conjugacy classes of subgroups — too many to draw. Only its 125 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×Q8×#272C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic
3#11721760C211yesmaximal, characteristic
4H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720549C211yesmaximal, characteristic, complemented
7#11720532C211yesmaximal, characteristic, complemented
8C3×#4802—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10Q8×#272—22no—
11Q8×#272C414yescomplemented
12C2×C4×#272—21no—
13C2×C4×#272C2211yescharacteristic
14C2×#6229—21no—
15C2×#6229C2211yescharacteristic
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18#88542—21no—
19C4⋊#1084C2211yescharacteristic
20C4×#1084—21no—
21C2×#6180C2211yescharacteristic
22#88527—22no—
23C2×C3×D3×Q8—34no—
24C2×C3×ΓL(1,𝔽25)—34no—
25C3×#892—34no—
26C32×#103D314yescomplemented
27C3×#835—34no—
28C3×#816—34no—
29C3×#810—34no—
30C22×#272D411yescharacteristic
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)C2×C412yes—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×#1084—21no—
37C2×#1084—21no—
38C2×#1084D411yescharacteristic
39C4×#272—24no—
40C4×#272C2×C412yes—
41C4×#272D412yes—
42#6229—21no—
43#6229—24no—
44#6229C2×C412yes—
45C8×H(3,𝔽3)—21no—
46#6180—21no—
47C3×#419—91no—
48#4802—34no—
49C2×C32×Q8C2×D314yes—
50C3×D3×Q8—316no—
51C3×D3×Q8—68no—
52C2×C3×C4×D3—34no—
53C2×C3×C4×D3—64no—
54C2×C3×Dic6—34no—
55C2×C3×Dic6—64no—
56C2×C32×C8C2×D314yes—
57C32×C4⋊C4C2×D314yes—
58C3×(C2×D3)⋊C4—64no—
59C3×C4⋊Dic3—34no—
60C3×C4×Dic3—64no—
61C2×C3×C3⋊C8—34no—
62C3×ΓL(1,𝔽25)—68no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—21no—
65C2×#272—21no—
66C2×#272C22⋊C412yes—
67C4×H(3,𝔽3)—22no—
68C4×H(3,𝔽3)—41no—
69C4×H(3,𝔽3)C22×C411yescharacteristic
70C4×H(3,𝔽3)C2×D411yescharacteristic
71#1084—22no—
72#1084—41no—
73#1084C22⋊C412yes—
74C22×C3×Q8—91no—
75C2×D3×Q8—34no—
76C2×C3×M16—91no—
77C3×#117—91no—
78C2×ΓL(1,𝔽25)—34no—
79#892—34no—
80C3×#103—34no—
81C3×#103—91no—
82C3×#103—91no—
83C3×#103—91no—
84C3×#103C3⋊D311yescharacteristic
85#835—34no—
86#816—34no—
87#810—34no—
88C22×C3×D3—34no—
89C32×Q8—24no—
90C32×Q8C4×D318yes—
91C2×C32×C4—24no—
92C2×C32×C4—24no—
93C2×C32×C4C22×D314yes—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—64no—
97C3×C4×D3—38no—
98C3×C4×D3—38no—
99C3×C4×D3—616no—
100C3×Dic6—38no—
101C3×Dic6—616no—
102C3×Dic6—64no—
103C32×C8—24no—
104C3×C3⋊C8—64no—
105#419—91noSylow
106C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
107C2×H(3,𝔽3)#13711yescharacteristic
108C2×H(3,𝔽3)#13711yescharacteristic
109#272—22no—
110C2×C3×Q8—34no—
111C2×C3×Q8—94no—
112C2×C3×Q8—91no—
113C2×C3×Q8—181no—
114C2×C3×Q8—182no—
115C2×C3×Q8C2×C3⋊D311yescharacteristic
116C22×C3×C4—91no—
117C22×C3×C4—181no—
118D3×Q8—316no—
119D3×Q8—68no—
120C2×C4×D3—34no—
121C2×C4×D3—64no—
122C2×Dic6—34no—
123C2×Dic6—64no—
124C3×M16—182no—
125C2×C3×C8—34no—
126C2×C3×C8—91no—
127C2×C3×C8C2×C3⋊D311yescharacteristic
128C3×C4⋊C4—34no—
129C3×C4⋊C4—91no—
130C3×C4⋊C4C2×C3⋊D311yescharacteristic
131C3×C22⋊C4—181no—
132C3×C42—181no—
133(C2×D3)⋊C4—64no—
134C4⋊Dic3—34no—
135C4×Dic3—64no—
136C2×C3⋊C8—34no—
137ΓL(1,𝔽25)—68no—
138C22×C32D3×D414yes—
139C2×C3×D3—38no—
140C2×C3×D3—64no—
141C2×C3×D3—64no—
142C32×C4—28no—
143C32×C4—44no—
144C32×C4C2×C4×D314yes—
145C32×C4D3×D414yes—
146C3×Dic3—38no—
147C3×Dic3—68no—
148C3×Dic3—124no—
149C22×Q8—91no—
150C2×M16—91no—
151#117—91no—
152#103—91no—
153#103—91no—
154#103—91no—
155#103#27211yescharacteristic, complemented
156H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
157C23×C3—91no—
158C22×D3—34no—
159C3×Q8—21no—
160C3×Q8—38no—
161C3×Q8—64no—
162C3×Q8—92no—
163C3×Q8—184no—
164C3×Q8—181no—
165C3×Q8C4×C3⋊D312yes—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—34no—
169C2×C3×C4—64no—
170C2×C3×C4—64no—
171C2×C3×C4—91no—
172C2×C3×C4—92no—
173C2×C3×C4—92no—
174C2×C3×C4—184no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177C2×C3×C4C22×C3⋊D311yescharacteristic
178C2×Dic3—34no—
179C2×Dic3—64no—
180C2×Dic3—64no—
181C4×D3—38no—
182C4×D3—38no—
183C4×D3—616no—
184Dic6—38no—
185Dic6—616no—
186Dic6—64no—
187C3×C8—21no—
188C3×C8—64no—
189C3×C8—181no—
190C3⋊C8—64no—
191C2×C32D3×C22⋊C414yes—
192C2×C32#91514yes—
193C2×C32#91814yes—
194C3×D3—68no—
195C2×Q8—94no—
196C2×Q8—91no—
197C2×Q8—181no—
198C2×Q8—182no—
199C2×Q8C2×#27211yescharacteristic
200C22×C4—91no—
201C22×C4—181no—
202M16—182no—
203C2×C8—91no—
204C2×C8C2×#27211yescharacteristic
205C4⋊C4—91no—
206C4⋊C4C2×#27211yescharacteristic
207C22⋊C4—181no—
208C42—181no—
209C22×C3—34no—
210C22×C3—92no—
211C22×C3—181no—
212C22×C3—181no—
213C22×C3D4×C3⋊D311yescharacteristic
214C2×D3—38no—
215C2×D3—64no—
216C2×D3—64no—
217C3×C4—22no—
218C3×C4—34no—
219C3×C4—34no—
220C3×C4—41no—
221C3×C4—68no—
222C3×C4—92no—
223C3×C4—124no—
224C3×C4—182no—
225C3×C4—361no—
226C3×C4C2×C4×C3⋊D311yescharacteristic
227C3×C4D4×C3⋊D311yescharacteristic
228Dic3—38no—
229Dic3—68no—
230Dic3—124no—
231C32#480214yescomplemented
232C23—91no—
233Q8—21no—
234Q8—92no—
235Q8—184no—
236Q8—181no—
237Q8C4×#27212yes—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—91no—
241C2×C4—92no—
242C2×C4—92no—
243C2×C4—184no—
244C2×C4—181no—
245C2×C4—181no—
246C2×C4C22×#27211yescharacteristic
247C8—21no—
248C8—181no—
249C2×C3—34no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—182no—
253C2×C3C22⋊C4×C3⋊D311yescharacteristic
254C2×C3#6412411yescharacteristic
255C2×C3#6412711yescharacteristic
256D3—68no—
257C22—92no—
258C22—181no—
259C22—181no—
260C22D4×#27211yescharacteristic
261C4—22no—
262C4—41no—
263C4—92no—
264C4—182no—
265C4—361no—
266C4C2×C4×#27211yescharacteristic
267C4D4×#27211yescharacteristic
268C3—34no—
269C3#1059250111yescharacteristic
270C2—182no—
271C2C22⋊C4×#27211yescharacteristic
272C2#1172180111yescharacteristic
273C2#1172180711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#419
LCSG ▷ 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
4. H(3,𝔽3)×#103C2semidirect
5. #11720583C2semidirect
6. #11720549C2semidirect
7. #11720532C2semidirect
11. Q8×#272C4semidirect
26. C32×#103D3semidirect
155. #103#272semidirect
156. H(3,𝔽3)#419semidirect
231. C32#4802semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×#885272
#117217602
H(3,𝔽3)×#1032
#117205832
#117205492
#117205322
C3×#48023