Finite Group Databasebeta

← #13166024

Subgroups — #13166024

Order 1728.

469 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 60 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⋊#226)C211yesmaximal, characteristic
2C2×C3×C3⋊#226C211yesmaximal, characteristic
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
4C4⋊(C3⋊#226)—31nomaximal
5#10593201—33nomaximal
6#10591625—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊#226)—21no—
9C3×C3⋊#226—21no—
10Dic3⋊(C3⋊Dic3)—41nomaximal
11C2×C3⋊#226—31no—
12C2×C3⋊#226—33no—
13C2×C3⋊#226—61no—
14C2×C3⋊#226—63no—
15C2×C3⋊#226D311yescharacteristic
16C2×C3×#226—33no—
17C2×#3756—31no—
18C3×Q8×Dic3D311yescharacteristic
19C2×C3×#216—31no—
20C3×C4×SL(2,𝔽3)—31no—
21#63866—31no—
22Dic3×SL(2,𝔽3)D313yes—
23Dic3⋊(C3⋊C8)—31no—
24#63567—31no—
25#63565—31no—
26C2×C3⋊(C3⋊Dic3)—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C4⋊#226—93no—
31#5182—91no—
32#5031—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×#634—31no—
35C2×C3×Dic6—31no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)—23no—
38C2×C3×SL(2,𝔽3)C2×D313yes—
39C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
40C3⋊#226—21no—
41C3⋊#226—62no—
42C3⋊#226—66no—
43C3⋊#226—63no—
44C3⋊#226—61no—
45C3×#226—63no—
46#3756—61no—
47#3756—61no—
48C4⋊(C3⋊Dic3)—121no—
49C2×C3⋊(C3⋊C8)—31no—
50C3×#216—61no—
51C3×#216—61no—
52C3×C4⋊Dic3—31no—
53C3×C4×Dic3—31no—
54C2×C3×C3⋊C8—31no—
55Dic3⋊Dic3—61no—
56Dic3⋊Dic3—61no—
57Dic3⋊Dic3—123no—
58C22×C33—41no—
59C3⋊(C3⋊Dic3)—81no—
60C3×C3⋊Dic3—81no—
61C32×Dic3—42no—
62C2×#226—33no—
63C2×#226—93no—
64C2×#226—183no—
65C2×C3×Q16—91no—
66C3×C4×Q8—31no—
67Q8×Dic3C3⋊D311yescharacteristic
68#944—91no—
69C2×#216—31no—
70C2×#216—91no—
71C2×#216—91no—
72C2×#216—181no—
73#869—91no—
74C4×SL(2,𝔽3)—33no—
75#835—91no—
76#816—91no—
77Dic3⋊C8—91no—
78#810—91no—
79#810—91no—
80C4⋊(C3⋊C8)—91no—
81C32×Q8—31no—
82C32×Q8D1211yescharacteristic
83C2×C32×C4—31no—
84C2×C32×C4—121no—
85C2×C3⋊Dic3—41no—
86C2×C3⋊Dic3—61no—
87C2×C3⋊Dic3—121no—
88C2×C3⋊Dic3—123no—
89C2×C3⋊Dic3—241no—
90C2×C3⋊Dic3—243no—
91#634—61no—
92#634—61no—
93C2×C3×Dic3—31no—
94C2×C3×Dic3—43no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—123no—
97C2×C3×Dic3S411yescharacteristic
98C3×Dic6—61no—
99C3×Dic6—61no—
100C3×SL(2,𝔽3)—21no—
101C3×SL(2,𝔽3)—23no—
102C3×SL(2,𝔽3)D1213yes—
103C3×SL(2,𝔽3)#6711yescharacteristic
104C3⋊(C3⋊C8)—61no—
105C3×C3⋊C8—61no—
106#462—271noSylow
107C2×C33—41no—
108C2×C33—41no—
109C2×C33—41no—
110C2×C3×Q8—21no—
111C2×C3×Q8—91no—
112C2×C3×Q8D3211yescharacteristic
113C2×C3×Q8C2×C3⋊D311yescharacteristic
114C2×Dic6—31no—
115C2×Dic6—91no—
116C2×Dic6—91no—
117C2×Dic6—181no—
118C2×SL(2,𝔽3)—23no—
119C2×SL(2,𝔽3)D3213yes—
120#226—63no—
121#226—186no—
122#226—183no—
123C3×Q16—181no—
124C3×Q16—181no—
125C2×C3×C8—91no—
126C3×C4⋊C4—91no—
127C3×C42—91no—
128#216—61no—
129#216—61no—
130#216—182no—
131#216—181no—
132#216—181no—
133#216—182no—
134#216—181no—
135#216—181no—
136C4⋊Dic3—31no—
137C4⋊Dic3—181no—
138C4⋊Dic3—181no—
139C4⋊Dic3—363no—
140Dic3⋊C4—181no—
141Dic3⋊C4—181no—
142C4×Dic3—31no—
143C2×C3⋊C8—31no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—181no—
147C22×C32—41no—
148C22×C32—43no—
149C22×C32—83no—
150C22×C32—81no—
151C22×C32C2×S411yescharacteristic
152C32×C4—31no—
153C32×C4—31no—
154C32×C4—122no—
155C3⋊Dic3—81no—
156C3⋊Dic3—121no—
157C3⋊Dic3—243no—
158C3⋊Dic3—241no—
159C3⋊Dic3—246no—
160C3⋊Dic3—242no—
161C3×Dic3—46no—
162C3×Dic3—61no—
163C3×Dic3—121no—
164C3×Dic3—243no—
165C3×Dic3#22612yescomplemented
166C2×Q16—91no—
167C2×Q16—271no—
168C4⋊Q8—271no—
169C4×Q8—31no—
170C4⋊C8—271no—
171#103—271no—
172#103—271no—
173C33—41noSylow
174C3×Q8—21no—
175C3×Q8—31no—
176C3×Q8—31no—
177C3×Q8—61no—
178C3×Q8—181no—
179C3×Q8—181no—
180C3×Q8#62611yescharacteristic
181C3×Q8C4⋊(C3⋊D3)11yescharacteristic
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—61no—
186C2×C3×C4—91no—
187C2×C3×C4—123no—
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—
197C2×Dic3C3⋊S411yescharacteristic
198Dic6—61no—
199Dic6—61no—
200Dic6—181no—
201Dic6—181no—
202Dic6—181no—
203Dic6—181no—
204Dic6—182no—
205Dic6—361no—
206SL(2,𝔽3)—23no—
207SL(2,𝔽3)#62613yes—
208C3×C8—181no—
209C3⋊C8—61no—
210C3⋊C8—181no—
211C3⋊C8—181no—
212C3⋊C8—182no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32—43no—
217C2×C32—43no—
218C2×C32—43no—
219C2×C32—83no—
220C2×C32—83no—
221C2×C32—83no—
222C2×C32—81no—
223C2×C32—81no—
224C2×C32—81no—
225C2×C32C4⋊S411yescharacteristic
226C2×C32C2×#22611yescharacteristic
227C2×C32#98411yescharacteristic
228C2×Q8—91no—
229C2×Q8—271no—
230C2×Q8D3×C3⋊D311yescharacteristic
231Q16—181no—
232Q16—181no—
233Q16—541no—
234Q16—541no—
235C2×C8—91no—
236C2×C8—271no—
237C4⋊C4—91no—
238C4⋊C4—541no—
239C4⋊C4—541no—
240C42—91no—
241C22×C3—21no—
242C22×C3—43no—
243C22×C3—83no—
244C22×C3D3×S411yescharacteristic
245C22×C3C2×C3⋊S411yescharacteristic
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—32no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—126no—
254C3×C4—181no—
255C3×C4—361no—
256Dic3—61no—
257Dic3—121no—
258Dic3—243no—
259Dic3—362no—
260Dic3—361no—
261Dic3—361no—
262Dic3—723no—
263Dic3—726no—
264Dic3C3⋊#22612yescomplemented
265C32—41no—
266C32—43no—
267C32—83no—
268C32—81no—
269C32C4⋊#22611yescharacteristic, complemented
270Q8—31no—
271Q8—181no—
272Q8—181no—
273Q8—541no—
274Q8—541no—
275Q8#629211yescharacteristic
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—91no—
279C2×C4—181no—
280C2×C4—541no—
281C8—181no—
282C8—541no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—43no—
287C2×C3—43no—
288C2×C3—43no—
289C2×C3—83no—
290C2×C3—83no—
291C2×C3—83no—
292C2×C3#6419611yescharacteristic
293C2×C3D3×#22611yescharacteristic
294C2×C3#6420711yescharacteristic
295C2×C3C4⋊(C3⋊S4)11yescharacteristic
296C2×C3C2×C3⋊#22611yescharacteristic
297C2×C3#6426511yescharacteristic
298C22D3×C3⋊S411yescharacteristic
299C4—32no—
300C4—31no—
301C4—31no—
302C4—181no—
303C4—361no—
304C4—1081no—
305C3—21no—
306C3—43no—
307C3—83no—
308C3#1059320111yescharacteristic, complemented
309C3C4⋊(C3⋊#226)11yescharacteristic, complemented
310C2#1172419111yescharacteristic
311C2D3×C3⋊#22611yescharacteristic
312C2#1172420211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, prime radical, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
165. C3×Dic3#226semidirect
264. Dic3C3⋊#226semidirect
269. C32C4⋊#226semidirect
308. C3#10593201semidirect
309. C3C4⋊(C3⋊#226)semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊#226)2
C2×C3×C3⋊#2262
C3×Dic3×SL(2,𝔽3)2
C4⋊(C3⋊#226)3
#105932013
#105916253
Dic3⋊(C3⋊Dic3)4