Finite Group Databasebeta

← #13166025

Subgroups — #13166025

Order 1728.

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

normal subgroups only

This group has 494 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#10593660—31nomaximal
5#10593200—33nomaximal
6#10591550—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—
13D3×C4⋊Dic3—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#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, complemented
29#5426—93no—
30#5232—91no—
31#4803—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—
42C2×D3×Dic3—61no—
43C3⋊(#41⋊C3)Dic316yescomplemented
44C4⋊(C3⋊Dic3)—31no—
45C4⋊(C3⋊Dic3)—121no—
46C2×C3⋊(C3⋊C8)—31no—
47C3×C4⋊Dic3—31no—
48C2×C3×C3⋊C8—31no—
49Dic3⋊Dic3—61no—
50Dic3⋊Dic3—123no—
51#3711—62no—
52C22×C33—41no—
53C3⋊(C3⋊Dic3)—81no—
54C3×C3⋊Dic3—81no—
55C32×Dic3—42no—
56C2×C3×#41—31no—
57C2×#239C3⋊D311yescharacteristic
58C2×#41⋊C3—33no—
59C2×C4⋊Dic3—91no—
60C2×#208—91no—
61C2×ΓL(1,𝔽25)—91no—
62D3×C4⋊C4—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, complemented
97C3×SL(2,𝔽3)C2×Dic313yescomplemented
98D3×Dic3—122no—
99C3⋊(C3⋊C8)—61no—
100C3×C3⋊C8—61no—
101#417—271noSylow
102C2×C33—41no—
103C2×C33—41no—
104C2×C33—41no—
105C3×#41—32no—
106C3×#41—92no—
107C2×C3×Q8—21no—
108C2×C3×Q8D3211yescharacteristic
109C2×C3×Q8C2×C3⋊D311yescharacteristic
110C2×C3×D4—91no—
111C22×C3×C4—91no—
112C22×Dic3—181no—
113#239—32no—
114#239C3⋊Dic312yescomplemented
115C2×D12—31no—
116C2×C4×D3—31no—
117C2×C4×D3—181no—
118#41⋊C3—36no—
119C2×SL(2,𝔽3)—23no—
120C2×SL(2,𝔽3)D3213yes—
121C2×C3×C8—91no—
122C3×C4⋊C4—91no—
123C4⋊Dic3—31no—
124C4⋊Dic3—91no—
125C4⋊Dic3—91no—
126C4⋊Dic3—181no—
127C4⋊Dic3—181no—
128C4⋊Dic3—363no—
129Dic3⋊C4—181no—
130#208—182no—
131C2×C3⋊C8—31no—
132C2×C3⋊C8—91no—
133C2×C3⋊C8—91no—
134C2×C3⋊C8—181no—
135ΓL(1,𝔽25)—182no—
136C22×C32—41no—
137C22×C32—43no—
138C22×C32—83no—
139C22×C32—81no—
140C22×C32C2×S411yescharacteristic
141C2×C3×D3—32no—
142C2×C3×D3—61no—
143C2×C3×D3—61no—
144C32×C4—31no—
145C32×C4—31no—
146C32×C4—122no—
147C3⋊Dic3—81no—
148C3⋊Dic3—121no—
149C3⋊Dic3—243no—
150C3⋊Dic3—241no—
151C3⋊Dic3—242no—
152C3⋊Dic3—246no—
153C3×Dic3—46no—
154C3×Dic3—121no—
155C3×Dic3—243no—
156C3×Dic3SL(2,ℤ/4)12yes—
157C2×#41—31no—
158C2×M16—271no—
159C2×C4⋊C4—271no—
160#103—91no—
161#103—271no—
162#102—271no—
163#102—271no—
164C33—41noSylow
165C23×C3—91no—
166C22×D3—31no—
167C3×Q8—21no—
168C3×Q8—31no—
169C3×Q8—31no—
170C3×Q8—61no—
171C3×Q8D3×Dic311yescharacteristic
172C3×Q8C2×C3⋊Dic311yescharacteristic
173C3×D4—92no—
174C3×D4—181no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—61no—
179C2×C3×C4—92no—
180C2×C3×C4—92no—
181C2×C3×C4—123no—
182C2×C3×C4—181no—
183C2×Dic3—61no—
184C2×Dic3—123no—
185C2×Dic3—181no—
186C2×Dic3—181no—
187C2×Dic3—361no—
188C2×Dic3—363no—
189C2×Dic3—362no—
190C2×Dic3—723no—
191C2×Dic3C3⋊S411yescharacteristic
192D12—32no—
193D12—61no—
194C4×D3—32no—
195C4×D3—32no—
196C4×D3—362no—
197SL(2,𝔽3)—23no—
198SL(2,𝔽3)D3×Dic313yescomplemented
199C3×C8—181no—
200C3⋊C8—61no—
201C3⋊C8—181no—
202C3⋊C8—181no—
203C3⋊C8—361no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32—43no—
208C2×C32—43no—
209C2×C32—43no—
210C2×C32—83no—
211C2×C32—83no—
212C2×C32—83no—
213C2×C32—81no—
214C2×C32—81no—
215C2×C32—81no—
216C2×C32#98411yescharacteristic
217C2×C32#98611yescharacteristic
218C2×C32C2×SL(2,ℤ/4)11yescharacteristic
219C3×D3—62no—
220#41—32no—
221#41—92no—
222C2×Q8D3×C3⋊D311yescharacteristic
223C2×D4—91no—
224C22×C4—91no—
225C22×C4—541no—
226M16—542no—
227C2×C8—91no—
228C2×C8—271no—
229C4⋊C4—91no—
230C4⋊C4—271no—
231C4⋊C4—541no—
232C22×C3—21no—
233C22×C3—43no—
234C22×C3—83no—
235C22×C3—92no—
236C22×C3—181no—
237C22×C3—181no—
238C22×C3D3×S411yescharacteristic
239C22×C3C2×C3⋊S411yescharacteristic
240C2×D3—32no—
241C2×D3—61no—
242C2×D3—61no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—32no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—126no—
251C3×C4—361no—
252Dic3—121no—
253Dic3—243no—
254Dic3—361no—
255Dic3—361no—
256Dic3—721no—
257Dic3—723no—
258Dic3—726no—
259Dic3C3⋊SL(2,ℤ/4)12yes—
260C32—41no—
261C32—43no—
262C32—83no—
263C32—81no—
264C32#542611yescharacteristic, complemented
265C23—91no—
266Q8—31no—
267Q8D3×C3⋊Dic311yescharacteristic
268D4—92no—
269D4—181no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—92no—
273C2×C4—92no—
274C2×C4—181no—
275C2×C4—541no—
276C2×C4—1082no—
277C8—181no—
278C8—541no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—43no—
283C2×C3—43no—
284C2×C3—43no—
285C2×C3—83no—
286C2×C3—83no—
287C2×C3—83no—
288C2×C3—182no—
289C2×C3#6419611yescharacteristic
290C2×C3#6419911yescharacteristic
291C2×C3D3×SL(2,ℤ/4)11yescharacteristic
292C2×C3#6426511yescharacteristic
293C2×C3#6426711yescharacteristic
294C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
295D3—62no—
296C22—92no—
297C22—181no—
298C22—181no—
299C22D3×C3⋊S411yescharacteristic
300C4—31no—
301C4—31no—
302C4—32no—
303C4—361no—
304C4—1081no—
305C3—21no—
306C3—43no—
307C3—83no—
308C3#1059320011yescharacteristic, complemented
309C3#1059366011yescharacteristic, complemented
310C2—182no—
311C2#1172419111yescharacteristic
312C2#1172419411yescharacteristic
313C2D3×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
28. C32×SL(2,𝔽3)C2×C4semidirect
34. C3×#239Dic3semidirect
43. C3⋊(#41⋊C3)Dic3semidirect
96. C3×SL(2,𝔽3)C4×D3semidirect
97. C3×SL(2,𝔽3)C2×Dic3semidirect
114. #239C3⋊Dic3semidirect
198. SL(2,𝔽3)D3×Dic3semidirect
264. C32#5426semidirect
308. C3#10593200semidirect
309. C3#10593660semidirect

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
#105936603
#105932003
#105915503
Dic3⋊(C3⋊Dic3)4