Finite Group Databasebeta

← #13144006

Subgroups — #13144006

Order 1728.

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

normal subgroups only

This group has 891 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×#88748C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3C4⋊C4×#272C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4803—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10#88748—22no—
11#88748C414yescomplemented
12C2×C4⋊#272—21no—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272—21no—
15C2×C4×#272—21no—
16C2×C4×#272C2211yescharacteristic
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540—21no—
21C2×#6180C2211yescharacteristic
22#88527—22no—
23C2×C3×#239—34no—
24C2×C3×ΓL(1,𝔽25)—34no—
25C3×D3×C4⋊C4—34no—
26C32×#103D314yescomplemented
27C3×#835—34no—
28C3×#821—34no—
29C3×#809—34no—
30C22×#272—21no—
31C22×#272D411yescharacteristic
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)C2×C412yescomplemented
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—21no—
38C2×#1084D411yescharacteristic
39C4⋊#272—24no—
40C4⋊#272—21no—
41C4⋊#272C2×C412yescomplemented
42C4×#272—24no—
43C4×#272—42no—
44C4×#272C2×C412yes—
45C4×#272D412yes—
46C8×H(3,𝔽3)—21no—
47#6180—21no—
48C3×#417—91no—
49#4803—34no—
50C2×C32×Q8C2×D314yes—
51C3×#239—316no—
52C3×#239—68no—
53C2×C3×D12—34no—
54C2×C3×D12—64no—
55C2×C3×C4×D3—34no—
56C2×C3×C4×D3—64no—
57C2×C3×C4×D3—64no—
58C2×C32×C8C2×D314yes—
59C32×C4⋊C4C2×D314yes—
60C3×C4⋊Dic3—34no—
61C3×Dic3⋊C4—64no—
62C2×C3×C3⋊C8—34no—
63C3×ΓL(1,𝔽25)—68no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—21no—
66C2×#272—21no—
67C2×#272—22no—
68C2×#272—41no—
69C2×#272—41no—
70C2×#272C22⋊C412yes—
71C4×H(3,𝔽3)—22no—
72C4×H(3,𝔽3)—41no—
73C4×H(3,𝔽3)C22×C411yescharacteristic
74C4×H(3,𝔽3)C2×D411yescharacteristic
75#1084—41no—
76#1084C22⋊C412yes—
77C2×C3×#41—91no—
78C2×#239—34no—
79C2×C3×M16—91no—
80C2×C3×C4⋊C4—91no—
81C2×ΓL(1,𝔽25)—34no—
82D3×C4⋊C4—34no—
83C3×#103—34no—
84C3×#103—91no—
85C3×#103C3⋊D311yescharacteristic
86C3×#102—91no—
87C3×#102—91no—
88#835—34no—
89#821—34no—
90#809—34no—
91C22×C3×D3—34no—
92C22×C3×D3—64no—
93C32×Q8—24no—
94C32×Q8C4×D318yescomplemented
95C2×C32×C4—24no—
96C2×C32×C4—24no—
97C2×C32×C4C22×D314yes—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—64no—
100C3×D12—38no—
101C3×D12—616no—
102C3×D12—64no—
103C3×C4×D3—38no—
104C3×C4×D3—38no—
105C3×C4×D3—616no—
106C3×C4×D3—128no—
107C32×C8—24no—
108C3×C3⋊C8—64no—
109#417—91noSylow
110C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
111C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
112C2×H(3,𝔽3)#13711yescharacteristic
113#272—22no—
114#272—42no—
115C3×#41—94no—
116C3×#41—182no—
117C2×C3×Q8—34no—
118C2×C3×Q8C2×C3⋊D311yescharacteristic
119C2×C3×D4—91no—
120C2×C3×D4—181no—
121C22×C3×C4—91no—
122C22×C3×C4—181no—
123C22×C3×C4—181no—
124#239—316no—
125#239—68no—
126C2×D12—34no—
127C2×D12—64no—
128C2×C4×D3—34no—
129C2×C4×D3—64no—
130C2×C4×D3—64no—
131C3×M16—182no—
132C2×C3×C8—34no—
133C2×C3×C8—91no—
134C2×C3×C8C2×C3⋊D311yescharacteristic
135C3×C4⋊C4—34no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4—181no—
138C3×C4⋊C4C2×C3⋊D311yescharacteristic
139C4⋊Dic3—34no—
140Dic3⋊C4—64no—
141C2×C3⋊C8—34no—
142ΓL(1,𝔽25)—68no—
143C22×C32D3×D414yes—
144C2×C3×D3—38no—
145C2×C3×D3—64no—
146C2×C3×D3—64no—
147C2×C3×D3—68no—
148C2×C3×D3—124no—
149C2×C3×D3—124no—
150C32×C4—28no—
151C32×C4—44no—
152C32×C4C2×C4×D314yes—
153C32×C4D3×D414yes—
154C3×Dic3—38no—
155C3×Dic3—124no—
156C2×#41—91no—
157C2×M16—91no—
158C2×C4⋊C4—91no—
159#103—91no—
160#103#27211yescharacteristic, complemented
161#102—91no—
162#102—91no—
163H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
164C23×C3—91no—
165C23×C3—181no—
166C22×D3—34no—
167C22×D3—64no—
168C3×Q8—21no—
169C3×Q8—38no—
170C3×Q8—64no—
171C3×Q8C4×C3⋊D312yes—
172C3×D4—92no—
173C3×D4—184no—
174C3×D4—181no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—34no—
178C2×C3×C4—64no—
179C2×C3×C4—64no—
180C2×C3×C4—91no—
181C2×C3×C4—92no—
182C2×C3×C4—92no—
183C2×C3×C4—184no—
184C2×C3×C4—181no—
185C2×C3×C4—362no—
186C2×C3×C4C22×C3⋊D311yescharacteristic
187C2×Dic3—34no—
188C2×Dic3—64no—
189D12—38no—
190D12—64no—
191D12—616no—
192C4×D3—38no—
193C4×D3—38no—
194C4×D3—616no—
195C4×D3—128no—
196C3×C8—21no—
197C3×C8—64no—
198C3×C8—181no—
199C3⋊C8—64no—
200C2×C32D3×C22⋊C414yes—
201C2×C32#91414yes—
202C2×C32#91814yes—
203C3×D3—68no—
204C3×D3—128no—
205#41—94no—
206#41—182no—
207C2×Q8C2×#27211yescharacteristic
208C2×D4—91no—
209C2×D4—181no—
210C22×C4—91no—
211C22×C4—181no—
212C22×C4—181no—
213M16—182no—
214C2×C8—91no—
215C2×C8C2×#27211yescharacteristic
216C4⋊C4—91no—
217C4⋊C4—181no—
218C4⋊C4C2×#27211yescharacteristic
219C22×C3—34no—
220C22×C3—92no—
221C22×C3—181no—
222C22×C3—181no—
223C22×C3—182no—
224C22×C3—361no—
225C22×C3—361no—
226C22×C3D4×C3⋊D311yescharacteristic
227C2×D3—38no—
228C2×D3—64no—
229C2×D3—64no—
230C2×D3—68no—
231C2×D3—124no—
232C2×D3—124no—
233C3×C4—22no—
234C3×C4—34no—
235C3×C4—34no—
236C3×C4—41no—
237C3×C4—68no—
238C3×C4—92no—
239C3×C4—124no—
240C3×C4—361no—
241C3×C4C2×C4×C3⋊D311yescharacteristic
242C3×C4D4×C3⋊D311yescharacteristic
243Dic3—38no—
244Dic3—124no—
245C32#480314yescomplemented
246C23—91no—
247C23—181no—
248Q8—21no—
249Q8C4×#27212yescomplemented
250D4—92no—
251D4—184no—
252D4—181no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—91no—
256C2×C4—92no—
257C2×C4—92no—
258C2×C4—184no—
259C2×C4—181no—
260C2×C4—362no—
261C2×C4C22×#27211yescharacteristic
262C8—21no—
263C8—181no—
264C2×C3—34no—
265C2×C3—34no—
266C2×C3—34no—
267C2×C3—182no—
268C2×C3—362no—
269C2×C3C22⋊C4×C3⋊D311yescharacteristic
270C2×C3#6412311yescharacteristic
271C2×C3#6412711yescharacteristic
272D3—68no—
273D3—128no—
274C22—92no—
275C22—181no—
276C22—181no—
277C22—182no—
278C22—361no—
279C22—361no—
280C22D4×#27211yescharacteristic
281C4—22no—
282C4—41no—
283C4—92no—
284C4—361no—
285C4C2×C4×#27211yescharacteristic
286C4D4×#27211yescharacteristic
287C3—34no—
288C3#1059250211yescharacteristic
289C2—182no—
290C2—362no—
291C2C22⋊C4×#27211yescharacteristic
292C2#1172180011yescharacteristic
293C2#1172180711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC2×#88748C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#417
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
2. C2×#88527C2semidirect
3. C4⋊C4×#272C2semidirect
4. H(3,𝔽3)×#103C2semidirect
5. #11720583C2semidirect
6. #11720554C2semidirect
7. #11720531C2semidirect
11. #88748C4semidirect
26. C32×#103D3semidirect
33. Q8×H(3,𝔽3)C2×C4semidirect
41. C4⋊#272C2×C4semidirect
94. C32×Q8C4×D3semidirect
160. #103#272semidirect
163. H(3,𝔽3)#417semidirect
245. C32#4803semidirect
249. Q8C4×#272semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
C2×#885272
C4⋊C4×#2722
H(3,𝔽3)×#1032
#117205832
#117205542
#117205312
C3×#48033