Finite Group Databasebeta

← #13166016

Subgroups — #13166016

Order 1728.

502 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 72 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×C3×C3⋊(#41⋊C3)C211yesmaximal, characteristic
2C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593658—31nomaximal
5#10593191—33nomaximal
6#10591556—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊(#41⋊C3)C412yescomplemented
9Dic3×C3⋊Dic3—41nomaximal
10C2×C3×#239D311yescharacteristic
11C2×C3×#41⋊C3—31no—
12C2×C3⋊(#41⋊C3)D313yes—
13#63841—31no—
14C2×D3×C3⋊C8—31no—
15C3⋊#859—31no—
16C3⋊#859—33no—
17C3⋊#859—61no—
18C3⋊#859—63no—
19C3⋊#859D311yescharacteristic, complemented
20C3×#859—33no—
21#63662—31no—
22C3×#835—31no—
23#63563—31no—
24#63562—31no—
25C2×C3⋊(C3⋊Dic3)—41no—
26C2×C3×C3⋊Dic3—41no—
27C2×C32×Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic
29#5424—93no—
30#5233—91no—
31#4804—91no—
32C2×C32×Q8C2×D311yescharacteristic
33C3×#239—32no—
34C3×#239Dic312yescomplemented
35C2×C3×D12—31no—
36C2×C3×C4×D3—31no—
37C3×#41⋊C3—32no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—23no—
40C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
41C2×C3×SL(2,𝔽3)C2×D313yes—
42C3⋊(#41⋊C3)Dic316yescomplemented
43C4⋊(C3⋊Dic3)—31no—
44C4×C3⋊Dic3—121no—
45C2×C3⋊(C3⋊C8)—31no—
46C3×C4⋊Dic3—31no—
47C2×C3×C3⋊C8—31no—
48(C2×D3)⋊Dic3—61no—
49Dic32—61no—
50Dic32—123no—
51D3×C3⋊C8—62no—
52C22×C33—41no—
53C3⋊(C3⋊Dic3)—42no—
54C3×C3⋊Dic3—42no—
55C32×Dic3—42no—
56C2×C3×#41—31no—
57C2×#239C3⋊D311yescharacteristic
58C2×#41⋊C3—33no—
59#926—91no—
60C22×C3⋊C8—91no—
61C2×D3×C8—91no—
62#892—91no—
63#859—33no—
64#859—93no—
65#859—183no—
66C3×#103—91no—
67#835—31no—
68#835—91no—
69#835—91no—
70#835—181no—
71#832—91no—
72#832—91no—
73#821—91no—
74#809—91no—
75C22×C3×D3—31no—
76C32×Q8—31no—
77C32×Q8C2×Dic311yescharacteristic
78C2×C32×C4—31no—
79C2×C32×C4—121no—
80C2×C3⋊Dic3—41no—
81C2×C3⋊Dic3—61no—
82C2×C3⋊Dic3—121no—
83C2×C3⋊Dic3—123no—
84C2×C3⋊Dic3—241no—
85C2×C3⋊Dic3—243no—
86C2×C3×Dic3—43no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—123no—
89C2×C3×Dic3S411yescharacteristic
90C3×D12—32no—
91C3×D12—61no—
92C3×C4×D3—32no—
93C3×C4×D3—32no—
94C3×SL(2,𝔽3)—21no—
95C3×SL(2,𝔽3)—23no—
96C3×SL(2,𝔽3)C4×D311yescharacteristic
97C3×SL(2,𝔽3)C2×Dic313yes—
98C3⋊(C3⋊C8)—61no—
99C3×C3⋊C8—61no—
100#416—271noSylow
101C2×C33—41no—
102C2×C33—41no—
103C2×C33—41no—
104C3×#41—32no—
105C3×#41—92no—
106C2×C3×Q8—21no—
107C2×C3×Q8D3211yescharacteristic
108C2×C3×Q8C2×C3⋊D311yescharacteristic
109C2×C3×D4—91no—
110C22×C3×C4—91no—
111#239—32no—
112#239C3⋊Dic312yescomplemented
113C2×D12—31no—
114C2×C4×D3—31no—
115#41⋊C3—36no—
116C2×SL(2,𝔽3)—23no—
117C2×SL(2,𝔽3)D3213yes—
118C2×C3×C8—91no—
119C3×C4⋊C4—91no—
120#217—181no—
121(C2×D3)⋊C4—181no—
122C4⋊Dic3—31no—
123C4⋊Dic3—91no—
124C4⋊Dic3—91no—
125C4⋊Dic3—181no—
126C4×Dic3—181no—
127C4×Dic3—181no—
128C4×Dic3—363no—
129C2×C3⋊C8—31no—
130C2×C3⋊C8—91no—
131C2×C3⋊C8—91no—
132C2×C3⋊C8—181no—
133C2×C3⋊C8—182no—
134D3×C8—182no—
135C22×C32—41no—
136C22×C32—43no—
137C22×C32—83no—
138C22×C32—81no—
139C22×C32C2×S411yescharacteristic
140C2×C3×D3—32no—
141C2×C3×D3—61no—
142C2×C3×D3—61no—
143C32×C4—31no—
144C32×C4—31no—
145C32×C4—122no—
146C3⋊Dic3—42no—
147C3⋊Dic3—122no—
148C3⋊Dic3—126no—
149C3⋊Dic3—121no—
150C3⋊Dic3—246no—
151C3⋊Dic3—242no—
152C3×Dic3—46no—
153C3×Dic3—126no—
154C3×Dic3—121no—
155C3×Dic3SL(2,ℤ/4)12yes—
156C2×#41—31no—
157C22×C8—271no—
158#117—271no—
159#103—91no—
160#103—271no—
161#102—271no—
162#102—271no—
163C33—41noSylow
164C23×C3—91no—
165C22×D3—31no—
166C3×Q8—21no—
167C3×Q8—31no—
168C3×Q8—31no—
169C3×Q8—61no—
170C3×Q8D3×Dic311yescharacteristic
171C3×Q8C2×C3⋊Dic311yescharacteristic
172C3×D4—92no—
173C3×D4—181no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—61no—
178C2×C3×C4—92no—
179C2×C3×C4—92no—
180C2×C3×C4—123no—
181C2×C3×C4—181no—
182C2×Dic3—61no—
183C2×Dic3—123no—
184C2×Dic3—181no—
185C2×Dic3—181no—
186C2×Dic3—363no—
187C2×Dic3—361no—
188C2×Dic3—723no—
189C2×Dic3C3⋊S411yescharacteristic
190D12—32no—
191D12—61no—
192C4×D3—32no—
193C4×D3—32no—
194SL(2,𝔽3)—23no—
195SL(2,𝔽3)D3×Dic313yes—
196C3×C8—181no—
197C3⋊C8—61no—
198C3⋊C8—181no—
199C3⋊C8—181no—
200C3⋊C8—361no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—43no—
205C2×C32—43no—
206C2×C32—43no—
207C2×C32—83no—
208C2×C32—83no—
209C2×C32—83no—
210C2×C32—81no—
211C2×C32—81no—
212C2×C32—81no—
213C2×C32#98511yescharacteristic
214C2×C32#98511yescharacteristic
215C2×C32C2×SL(2,ℤ/4)11yescharacteristic
216C3×D3—62no—
217#41—32no—
218#41—92no—
219C2×Q8D3×C3⋊D311yescharacteristic
220C2×D4—91no—
221C22×C4—91no—
222C2×C8—91no—
223C2×C8—271no—
224C2×C8—542no—
225C4⋊C4—91no—
226C4⋊C4—271no—
227C22⋊C4—541no—
228C42—541no—
229C22×C3—21no—
230C22×C3—43no—
231C22×C3—83no—
232C22×C3—92no—
233C22×C3—181no—
234C22×C3—181no—
235C22×C3D3×S411yescharacteristic
236C22×C3C2×C3⋊S411yescharacteristic
237C2×D3—32no—
238C2×D3—61no—
239C2×D3—61no—
240C3×C4—31no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—32no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—126no—
248C3×C4—361no—
249Dic3—126no—
250Dic3—121no—
251Dic3—366no—
252Dic3—361no—
253Dic3—361no—
254Dic3—721no—
255Dic3—726no—
256Dic3C3⋊SL(2,ℤ/4)12yes—
257C32—41no—
258C32—43no—
259C32—83no—
260C32—81no—
261C32#542411yescharacteristic, complemented
262C23—91no—
263Q8—31no—
264Q8D3×C3⋊Dic311yescharacteristic
265D4—92no—
266D4—181no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—92no—
270C2×C4—92no—
271C2×C4—181no—
272C2×C4—541no—
273C8—181no—
274C8—541no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—43no—
279C2×C3—43no—
280C2×C3—43no—
281C2×C3—83no—
282C2×C3—83no—
283C2×C3—83no—
284C2×C3—182no—
285C2×C3#6419711yescharacteristic
286C2×C3#6419811yescharacteristic
287C2×C3D3×SL(2,ℤ/4)11yescharacteristic
288C2×C3#6426611yescharacteristic
289C2×C3#6426611yescharacteristic
290C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
291D3—62no—
292C22—92no—
293C22—181no—
294C22—181no—
295C22D3×C3⋊S411yescharacteristic
296C4—31no—
297C4—31no—
298C4—32no—
299C4—361no—
300C4—1081no—
301C3—21no—
302C3—43no—
303C3—83no—
304C3#1059319111yescharacteristic, complemented
305C3#1059365811yescharacteristic, complemented
306C2—182no—
307C2#1172419211yescharacteristic
308C2#1172419311yescharacteristic
309C2D3×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊(#41⋊C3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊#859)C2semidirect
3. C3×C3⋊#859C2semidirect
8. C3×C3⋊(#41⋊C3)C4semidirect
19. C3⋊#859D3semidirect
34. C3×#239Dic3semidirect
42. C3⋊(#41⋊C3)Dic3semidirect
112. #239C3⋊Dic3semidirect
261. C32#5424semidirect
304. C3#10593191semidirect
305. C3#10593658semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊(#41⋊C3)2
C3⋊(C3⋊#859)2
C3×C3⋊#8592
#105936583
#105931913
#105915563
Dic3×C3⋊Dic34