Finite Group Databasebeta

← #13166034

Subgroups — #13166034

Order 1728.

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

normal subgroups only

This group has 518 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593654—31nomaximal
5#10593209—33nomaximal
6#10591557—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×D3×SL(2,𝔽3)C412yescomplemented
9(C2×D3)⋊(C3⋊Dic3)—41nomaximal
10C2×C3×D3×Q8D311yescharacteristic
11C22×C3×SL(2,𝔽3)—31no—
12C2×D3×SL(2,𝔽3)D313yes—
13#63841—31no—
14C2×#3711—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#63566—31no—
24#63565—31no—
25C22×C32×D3—41no—
26C2×C3⋊(C3⋊Dic3)—41no—
27C2×C3×C3⋊Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic
29#5420—93no—
30#5225—91no—
31#4802—91no—
32C2×C32×Q8C2×D311yescharacteristic
33C3×D3×Q8—32no—
34C3×D3×Q8Dic312yescomplemented
35C2×C3×C4×D3—31no—
36C2×C3×Dic6—31no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)—23no—
39C2×C3×SL(2,𝔽3)—32no—
40C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
41C2×C3×SL(2,𝔽3)C2×D313yes—
42D3×SL(2,𝔽3)Dic316yescomplemented
43#3757—121no—
44C4⋊(C3⋊Dic3)—31no—
45C2×C3⋊(C3⋊C8)—31no—
46C3×C4⋊Dic3—31no—
47C2×C3×C3⋊C8—31no—
48(C2×D3)⋊Dic3—61no—
49(C2×D3)⋊Dic3—123no—
50Dic32—61no—
51#3711—62no—
52C22×C33—41no—
53C2×C32×D3—42no—
54C2×C32×D3—81no—
55C2×C32×D3—81no—
56C3⋊(C3⋊Dic3)—81no—
57C3×C3⋊Dic3—81no—
58C22×C3×Q8—31no—
59C2×D3×Q8C3⋊D311yescharacteristic
60C22×SL(2,𝔽3)—33no—
61#926—91no—
62C2×#208—91no—
63C2×ΓL(1,𝔽25)—91no—
64#892—91no—
65#859—33no—
66#859—93no—
67#859—183no—
68C3×#103—91no—
69#835—31no—
70#835—91no—
71#835—91no—
72#835—91no—
73#835—91no—
74#835—181no—
75#816—91no—
76#810—91no—
77C23×C32—121no—
78C22×C3×D3—43no—
79C22×C3×D3S411yescharacteristic
80C32×Q8—31no—
81C32×Q8C2×Dic311yescharacteristic
82C2×C32×C4—31no—
83C2×C3⋊Dic3—41no—
84C2×C3⋊Dic3—61no—
85C2×C3⋊Dic3—121no—
86C2×C3⋊Dic3—123no—
87C2×C3⋊Dic3—241no—
88C2×C3⋊Dic3—243no—
89C2×C3×Dic3—31no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—123no—
92C3×C4×D3—32no—
93C3×C4×D3—32no—
94C3×Dic6—32no—
95C3×Dic6—61no—
96C3×SL(2,𝔽3)—21no—
97C3×SL(2,𝔽3)—23no—
98C3×SL(2,𝔽3)C4×D311yescharacteristic
99C3×SL(2,𝔽3)C2×Dic313yes—
100C3⋊(C3⋊C8)—61no—
101C3×C3⋊C8—61no—
102#419—271noSylow
103C2×C33—41no—
104C2×C33—41no—
105C2×C33—41no—
106C32×D3—82no—
107C2×C3×Q8—21no—
108C2×C3×Q8—32no—
109C2×C3×Q8—91no—
110C2×C3×Q8—92no—
111C2×C3×Q8D3211yescharacteristic
112C2×C3×Q8C2×C3⋊D311yescharacteristic
113C22×C3×C4—91no—
114D3×Q8—32no—
115D3×Q8C3⋊Dic312yescomplemented
116C2×C4×D3—31no—
117C2×Dic6—31no—
118C2×SL(2,𝔽3)—23no—
119C2×SL(2,𝔽3)—36no—
120C2×SL(2,𝔽3)D3213yes—
121C2×C3×C8—91no—
122C3×C4⋊C4—91no—
123#217—181no—
124#217—363no—
125(C2×D3)⋊C4—181no—
126C4⋊Dic3—31no—
127C4⋊Dic3—91no—
128C4⋊Dic3—91no—
129C4⋊Dic3—181no—
130C4×Dic3—181no—
131C4×Dic3—181no—
132#208—182no—
133C2×C3⋊C8—31no—
134C2×C3⋊C8—91no—
135C2×C3⋊C8—91no—
136C2×C3⋊C8—181no—
137ΓL(1,𝔽25)—182no—
138C22×C32—41no—
139C22×C32—43no—
140C22×C32—83no—
141C22×C32—81no—
142C22×C32—122no—
143C22×C32—241no—
144C22×C32—241no—
145C22×C32C2×S411yescharacteristic
146C2×C3×D3—21no—
147C2×C3×D3—21no—
148C2×C3×D3—46no—
149C2×C3×D3—83no—
150C2×C3×D3—83no—
151C2×C3×D3SL(2,ℤ/4)12yes—
152C32×C4—31no—
153C32×C4—31no—
154C3⋊Dic3—81no—
155C3⋊Dic3—121no—
156C3⋊Dic3—243no—
157C3⋊Dic3—241no—
158C3⋊Dic3—242no—
159C3⋊Dic3—246no—
160C3×Dic3—32no—
161C3×Dic3—121no—
162C3×Dic3—243no—
163C22×Q8—31no—
164C2×M16—271no—
165#117—271no—
166#103—91no—
167#103—271no—
168#103—271no—
169#103—271no—
170C33—41noSylow
171C23×C3—31no—
172C23×C3—123no—
173C22×D3C3⋊S411yescharacteristic
174C3×Q8—21no—
175C3×Q8—31no—
176C3×Q8—31no—
177C3×Q8—61no—
178C3×Q8—92no—
179C3×Q8—181no—
180C3×Q8D3×Dic311yescharacteristic
181C3×Q8C2×C3⋊Dic311yescharacteristic
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—61no—
185C2×C3×C4—92no—
186C2×C3×C4—92no—
187C2×C3×C4—91no—
188C2×C3×C4—181no—
189C2×Dic3—31no—
190C2×Dic3—61no—
191C2×Dic3—123no—
192C2×Dic3—181no—
193C2×Dic3—181no—
194C2×Dic3—361no—
195C2×Dic3—363no—
196C2×Dic3—723no—
197C4×D3—32no—
198C4×D3—32no—
199Dic6—32no—
200Dic6—61no—
201SL(2,𝔽3)—23no—
202SL(2,𝔽3)D3×Dic313yes—
203C3×C8—181no—
204C3⋊C8—61no—
205C3⋊C8—181no—
206C3⋊C8—181no—
207C3⋊C8—361no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—43no—
212C2×C32—43no—
213C2×C32—43no—
214C2×C32—83no—
215C2×C32—83no—
216C2×C32—83no—
217C2×C32—81no—
218C2×C32—81no—
219C2×C32—81no—
220C2×C32—242no—
221C2×C32#98311yescharacteristic
222C2×C32#98311yescharacteristic
223C2×C32C2×SL(2,ℤ/4)11yescharacteristic
224C3×D3—22no—
225C3×D3—86no—
226C2×Q8—32no—
227C2×Q8—91no—
228C2×Q8—92no—
229C2×Q8D3×C3⋊D311yescharacteristic
230C22×C4—91no—
231M16—542no—
232C2×C8—91no—
233C2×C8—271no—
234C4⋊C4—91no—
235C4⋊C4—271no—
236C22⋊C4—541no—
237C42—541no—
238C22×C3—21no—
239C22×C3—32no—
240C22×C3—43no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—83no—
244C22×C3—126no—
245C22×C3—243no—
246C22×C3—243no—
247C22×C3D3×S411yescharacteristic
248C22×C3C2×C3⋊S411yescharacteristic
249C2×D3—21no—
250C2×D3—21no—
251C2×D3C3⋊SL(2,ℤ/4)12yes—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—61no—
257C3×C4—61no—
258C3×C4—92no—
259C3×C4—361no—
260Dic3—32no—
261Dic3—121no—
262Dic3—243no—
263Dic3—361no—
264Dic3—361no—
265Dic3—721no—
266Dic3—723no—
267Dic3—726no—
268C32—41no—
269C32—43no—
270C32—83no—
271C32—81no—
272C32#542011yescharacteristic, complemented
273C23—31no—
274Q8—31no—
275Q8—92no—
276Q8—181no—
277Q8D3×C3⋊Dic311yescharacteristic
278C2×C4—31no—
279C2×C4—92no—
280C2×C4—92no—
281C2×C4—91no—
282C2×C4—181no—
283C2×C4—541no—
284C8—181no—
285C8—541no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—43no—
290C2×C3—43no—
291C2×C3—43no—
292C2×C3—62no—
293C2×C3—83no—
294C2×C3—83no—
295C2×C3—83no—
296C2×C3—246no—
297C2×C3#6420111yescharacteristic
298C2×C3#6420211yescharacteristic
299C2×C3D3×SL(2,ℤ/4)11yescharacteristic
300C2×C3#6426411yescharacteristic
301C2×C3#6426411yescharacteristic
302C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
303D3—22no—
304C22—32no—
305C22—61no—
306C22—61no—
307C22D3×C3⋊S411yescharacteristic
308C4—31no—
309C4—31no—
310C4—92no—
311C4—361no—
312C4—1081no—
313C3—21no—
314C3—43no—
315C3—83no—
316C3#1059320911yescharacteristic, complemented
317C3#1059365411yescharacteristic, complemented
318C2—62no—
319C2#1172419611yescharacteristic
320C2#1172419711yescharacteristic
321C2D3×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×D3×SL(2,𝔽3)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×D3×SL(2,𝔽3)C4semidirect
19. C3⋊#859D3semidirect
34. C3×D3×Q8Dic3semidirect
42. D3×SL(2,𝔽3)Dic3semidirect
115. D3×Q8C3⋊Dic3semidirect
272. C32#5420semidirect
316. C3#10593209semidirect
317. C3#10593654semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
C3×C3⋊#8592
#105936543
#105932093
#105915573
(C2×D3)⋊(C3⋊Dic3)4