Finite Group Databasebeta

← #13138963

Subgroups — #13138963

Order 1728.

538 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88529C211yesmaximal, characteristic
2#11721149C212yesmaximal, complemented
3#63644⋊C3C211yesmaximal, characteristic, complemented
4#63642⋊C3C211yesmaximal, characteristic, complemented
5#11720324C212yesmaximal, complemented
6#10590781—31nomaximal
7#10590771—31nomaximal
8C2×C4⋊#272C2212yes—
9C2×#6199—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88529C414yescomplemented
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#88451—22no—
16C2×C3×D24—31no—
17C2×C3×D24—31no—
18C2×C3×D24—61no—
19#63898—32no—
20Dic3×D12—32no—
21#63644D311yescharacteristic, complemented
22#63642D311yescharacteristic, complemented
23C3×#817—31no—
24C3×#815—31no—
25#63563—32no—
26#63562—32no—
27C22×#272—22no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C4⋊#272—22no—
30C4⋊#272C2×C414yescomplemented
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3D412yes—
33#6199—44no—
34C8×H(3,𝔽3)C2×C412yescomplemented
35(C3⋊(C3⋊C8))⋊C3—21no—
36#5152—91no—
37#4869—91no—
38#4711—91no—
39C2×C3×D12—32no—
40C2×C3×D12—32no—
41C2×C3×D12—62no—
42C2×D3×Dic3—62no—
43C2×D3×Dic3—62no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C2×D311yescharacteristic
47C4⋊(C3⋊Dic3)C2×D312yes—
48C4×C3⋊Dic3C2×D311yescharacteristic
49C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
50C3×C4⋊Dic3—32no—
51C3×C4×Dic3—31no—
52C2×C3×C3⋊C8—31no—
53C3×D24—34no—
54C3×D24—34no—
55C3×D24—64no—
56(C2×D3)⋊Dic3—62no—
57(C2×D3)⋊Dic3—62no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×#272—24no—
60C2×#272—42no—
61C2×#272—42no—
62C4×H(3,𝔽3)C22×C411yescharacteristic
63C4×H(3,𝔽3)#4111yescharacteristic
64(C3⋊Dic3)⋊C3—22no—
65(C3⋊Dic3)⋊C3—42no—
66C2×C3×D8—91no—
67D4×Dic3—92no—
68C2×D24—31no—
69C2×D24—31no—
70C2×D24—61no—
71#893—92no—
72C4×D12—92no—
73C3×#106—91no—
74C3×#97—91no—
75#832—92no—
76#821—92no—
77#817—31no—
78#817—91no—
79#815—31no—
80#815—91no—
81#809—92no—
82C22×C3×D3—62no—
83C22×C3×D3—62no—
84C22×C3×D3—122no—
85C2×C32×C4—21no—
86C2×C32×C4C22×D311yescharacteristic
87C2×C32×C4C22×D311yescharacteristic
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3D1212yes—
90C2×C3×Dic3—32no—
91C2×C3×Dic3—62no—
92C3×D12—34no—
93C3×D12—34no—
94C3×D12—64no—
95C3×D12—62no—
96C3×D12—62no—
97C3×D12—122no—
98D3×Dic3—124no—
99D3×Dic3—124no—
100C32×C8—22no—
101C32×C8C4×D312yescomplemented
102C32×C8C4×D312yescomplemented
103C3⋊(C3⋊C8)—21no—
104C3×C3⋊C8—61no—
105#442—271noSylow
106C2×H(3,𝔽3)C4×D411yescharacteristic
107C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
108#272—44no—
109C2×C3×D4—92no—
110C22×Dic3—182no—
111C2×D12—32no—
112C2×D12—32no—
113C2×D12—62no—
114C2×C4×D3—182no—
115C2×C4×D3—182no—
116C3×D8—94no—
117C2×C3×C8—31no—
118C2×C3×C8—31no—
119C2×C3×C8—61no—
120C2×C3×C8—91no—
121C2×C3×C8D3211yescharacteristic
122C3×C4⋊C4—92no—
123C3×C42—91no—
124#217—182no—
125(C2×D3)⋊C4—182no—
126(C2×D3)⋊C4—182no—
127C4⋊Dic3—32no—
128C4⋊Dic3—92no—
129C4×Dic3—31no—
130C4×Dic3—91no—
131C2×C3⋊C8—31no—
132C2×C3⋊C8—91no—
133D24—34no—
134D24—34no—
135D24—64no—
136C22×C32—21no—
137C22×C32C2×D1211yescharacteristic
138C22×C32D3×D411yescharacteristic
139C2×C3×D3—64no—
140C2×C3×D3—64no—
141C2×C3×D3—124no—
142C2×C3×D3—122no—
143C2×C3×D3—122no—
144C2×C3×D3—122no—
145C2×C3×D3—122no—
146C2×C3×D3—242no—
147C2×C3×D3—242no—
148C32×C4—21no—
149C32×C4—21no—
150C32×C4C2×C4×D311yescharacteristic
151C32×C4C2×C4×D311yescharacteristic
152C32×C4#23511yescharacteristic
153C32×C4#23911yescharacteristic
154C3⋊Dic3—22no—
155C3⋊Dic3—42no—
156C3×Dic3—62no—
157C3×Dic3—122no—
158C2×D8—91no—
159C4×D4—272no—
160#106—91no—
161#102—272no—
162#97—91no—
163H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
164C23×C3—182no—
165C22×D3—62no—
166C22×D3—62no—
167C22×D3—122no—
168C3×D4—94no—
169C3×D4—182no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—61no—
173C2×C3×C4—92no—
174C2×C3×C4—182no—
175C2×C3×C4C2×D3211yescharacteristic
176C2×Dic3—32no—
177C2×Dic3—62no—
178C2×Dic3—92no—
179C2×Dic3—182no—
180C2×Dic3—364no—
181D12—34no—
182D12—34no—
183D12—64no—
184D12—62no—
185D12—62no—
186D12—122no—
187C4×D3—364no—
188C4×D3—364no—
189C3×C8—32no—
190C3×C8—32no—
191C3×C8—62no—
192C3×C8—181no—
193C3×C8#62412yes—
194C3⋊C8—61no—
195C3⋊C8—181no—
196C2×C32—22no—
197C2×C32—21no—
198C2×C32C4×D1211yescharacteristic
199C2×C32#89311yescharacteristic
200C2×C32#90812yes—
201C2×C32#91412yes—
202C3×D3—124no—
203C3×D3—124no—
204C3×D3—244no—
205C2×D4—92no—
206C22×C4—542no—
207D8—94no—
208C2×C8—91no—
209C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
210C4⋊C4—92no—
211C22⋊C4—542no—
212C42—91no—
213C22×C3—31no—
214C22×C3—31no—
215C22×C3—61no—
216C22×C3—184no—
217C22×C3—362no—
218C22×C3—362no—
219C22×C3D3×D1211yescharacteristic
220C2×D3—64no—
221C2×D3—64no—
222C2×D3—124no—
223C2×D3—122no—
224C2×D3—122no—
225C2×D3—122no—
226C2×D3—122no—
227C2×D3—242no—
228C2×D3—242no—
229C3×C4—31no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—182no—
236C3×C4—362no—
237C3×C4#379911yescharacteristic
238C3×C4C2×#62411yescharacteristic
239Dic3—62no—
240Dic3—122no—
241Dic3—182no—
242Dic3—362no—
243C32—21no—
244C32#471111yescharacteristic, complemented
245C32#486911yescharacteristic, complemented
246C23—182no—
247D4—94no—
248D4—182no—
249C2×C4—92no—
250C2×C4—182no—
251C2×C4—1084no—
252C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
253C8—181no—
254C8#619912yescomplemented
255C2×C3—32no—
256C2×C3—32no—
257C2×C3—31no—
258C2×C3—31no—
259C2×C3—62no—
260C2×C3—61no—
261C2×C3—364no—
262C2×C3#6379412yes—
263C2×C3#6389411yescharacteristic
264D3—124no—
265D3—124no—
266D3—244no—
267C22—184no—
268C22—362no—
269C22—362no—
270C22#8865411yescharacteristic
271C4—182no—
272C4—362no—
273C4#8865211yescharacteristic
274C4C2×#619911yescharacteristic
275C3—31no—
276C3—31no—
277C3—61no—
278C3#1059076811yescharacteristic
279C2—364no—
280C2#1172084812yes—
281C2#1172114911yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#88529C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721149C2semidirect
3. #63644⋊C3C2semidirect
4. #63642⋊C3C2semidirect
5. #11720324C2semidirect
11. #88529C4semidirect
21. #63644D3semidirect
22. #63642D3semidirect
30. C4⋊#272C2×C4semidirect
34. C8×H(3,𝔽3)C2×C4semidirect
101. C32×C8C4×D3semidirect
102. C32×C8C4×D3semidirect
163. H(3,𝔽3)#442semidirect
244. C32#4711semidirect
245. C32#4869semidirect
254. C8#6199semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
#117211492
#63644⋊C32
#63642⋊C32
#117203242
#105907813
#105907713