Finite Group Databasebeta

← #13166042

Subgroups — #13166042

Order 1728.

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

normal subgroups only

This group has 697 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593655—31nomaximal
5#10593218—33nomaximal
6#10591584—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6291—41nomaximal
9C3×D3×SL(2,𝔽3)—21no—
10C3⋊(C3⋊GL(2,𝔽3))—21no—
11C2×C3×D3×Q8D311yescharacteristic, complemented
12C22×C3×SL(2,𝔽3)—31no—
13C2×D3×SL(2,𝔽3)D313yescomplemented
14C2×C3⋊GL(2,𝔽3)—31no—
15C2×C3⋊GL(2,𝔽3)—33no—
16C2×C3⋊GL(2,𝔽3)—61no—
17C2×C3⋊GL(2,𝔽3)—63no—
18C2×#3755—31no—
19#63909—31no—
20C2×#3717—31no—
21C3⋊#859D311yescharacteristic, complemented
22C3×#859—33no—
23C3×#835—31no—
24#63565—31no—
25(C2×D3)⋊(C3⋊C8)—31no—
26C22×C3⋊(C3⋊D3)—41no—
27C22×C32×D3—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic, complemented
30#6291—44no—
31#5421—93no—
32#5227—91no—
33#4806—91no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×#638—121no—
36C2×C4⋊(C3⋊D3)—31no—
37C3×D3×Q8—21no—
38C3×D3×Q8—61no—
39C2×C3×C4×D3—31no—
40C2×C3×Dic6—31no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—23no—
43C2×C3×SL(2,𝔽3)—61no—
44C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
45C2×C3×SL(2,𝔽3)C2×D313yes—
46C2×#626—61no—
47C2×#626—123no—
48D3×SL(2,𝔽3)—23no—
49C3⋊GL(2,𝔽3)—62no—
50C3⋊GL(2,𝔽3)—66no—
51C3⋊GL(2,𝔽3)—61no—
52C3⋊GL(2,𝔽3)—63no—
53#3755—61no—
54#3755—61no—
55C2×C3⋊(C3⋊C8)—31no—
56C3×C4⋊Dic3—31no—
57C2×C3×C3⋊C8—31no—
58#3722—61no—
59#3717—62no—
60C22×C33—41no—
61C2×C3⋊(C3⋊D3)—42no—
62C2×C3⋊(C3⋊D3)—81no—
63C2×C3⋊(C3⋊D3)—81no—
64C2×C32×D3—42no—
65C2×C32×D3—81no—
66C2×C32×D3—81no—
67C3×C3⋊Dic3—42no—
68C22×C3×Q8—31no—
69C2×D3×Q8C3⋊D311yescharacteristic, complemented
70C22×SL(2,𝔽3)—33no—
71C2×GL(2,𝔽3)—93no—
72C2×GL(2,𝔽3)—183no—
73C2×#215—91no—
74C2×#215—91no—
75C2×#215—91no—
76C2×#215—181no—
77#930—91no—
78C2×#204—91no—
79#895—91no—
80#859—33no—
81C3×#103—91no—
82#835—31no—
83#835—91no—
84#830—91no—
85(C2×D3)⋊C8—91no—
86#810—91no—
87C23×C32—121no—
88C22×C3⋊D3—61no—
89C22×C3⋊D3—121no—
90C22×C3⋊D3—123no—
91C22×C3⋊D3—241no—
92C22×C3⋊D3—243no—
93C22×C3×D3—43no—
94C22×C3×D3S411yescharacteristic
95C32×Q8—31no—
96C32×Q8#6711yescharacteristic, complemented
97C2×C32×C4—31no—
98#638—124no—
99C2×C3⋊Dic3—41no—
100C4⋊(C3⋊D3)—61no—
101C4⋊(C3⋊D3)—61no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—123no—
105C3×C4×D3—61no—
106C3×C4×D3—61no—
107C3×Dic6—32no—
108C3×Dic6—61no—
109C3×SL(2,𝔽3)—21no—
110C3×SL(2,𝔽3)—23no—
111C3×SL(2,𝔽3)D1211yescharacteristic, complemented
112C3×SL(2,𝔽3)#6713yescomplemented
113#626—1212no—
114#626—122no—
115C3⋊(C3⋊C8)—61no—
116C3×C3⋊C8—61no—
117#448—271noSylow
118C2×C33—41no—
119C2×C33—41no—
120C2×C33—41no—
121C3⋊(C3⋊D3)—82no—
122C32×D3—82no—
123C2×C3×Q8—21no—
124C2×C3×Q8—61no—
125C2×C3×Q8—91no—
126C2×C3×Q8—181no—
127C2×C3×Q8D3211yescharacteristic
128C2×C3×Q8C2×C3⋊D311yescharacteristic
129C22×C3×C4—91no—
130C2×#67—181no—
131C2×#67—363no—
132D3×Q8—21no—
133D3×Q8—61no—
134C2×D12—91no—
135C2×D12—91no—
136C2×D12—181no—
137C2×D12—181no—
138C2×C4×D3—31no—
139C2×Dic6—31no—
140C2×SL(2,𝔽3)—23no—
141C2×SL(2,𝔽3)—63no—
142C2×SL(2,𝔽3)D3213yes—
143GL(2,𝔽3)—186no—
144GL(2,𝔽3)—183no—
145C2×C3×C8—91no—
146C3×C4⋊C4—91no—
147#215—182no—
148#215—182no—
149#215—181no—
150#215—181no—
151#215—182no—
152#215—181no—
153#215—181no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—181no—
156C4⋊Dic3—31no—
157C2×C3⋊C8—31no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—181no—
161#204—182no—
162C22×C32—41no—
163C22×C32—43no—
164C22×C32—83no—
165C22×C32—81no—
166C22×C32—122no—
167C22×C32—241no—
168C22×C32—241no—
169C22×C32C2×S411yescharacteristic
170C2×C3⋊D3—122no—
171C2×C3⋊D3—121no—
172C2×C3⋊D3—126no—
173C2×C3⋊D3—121no—
174C2×C3⋊D3—121no—
175C2×C3⋊D3—243no—
176C2×C3⋊D3—241no—
177C2×C3⋊D3—243no—
178C2×C3⋊D3—241no—
179C2×C3⋊D3—246no—
180C2×C3⋊D3—242no—
181C2×C3⋊D3—242no—
182C2×C3⋊D3—246no—
183C2×C3⋊D3—242no—
184C2×C3⋊D3—246no—
185C2×C3×D3—21no—
186C2×C3×D3—21no—
187C2×C3×D3—46no—
188C2×C3×D3—83no—
189C2×C3×D3—83no—
190C2×C3×D3GL(2,𝔽3)12yescomplemented
191C32×C4—31no—
192C32×C4—31no—
193C3⋊Dic3—42no—
194C3×Dic3—61no—
195C3×Dic3—126no—
196C3×Dic3—121no—
197C22×Q8—31no—
198C2×SD16—271no—
199C2×SD16—271no—
200#121—271no—
201#103—91no—
202#103—271no—
203C22⋊C8—271no—
204C33—41noSylow
205C23×C3—31no—
206C23×C3—123no—
207C22×D3—181no—
208C22×D3—181no—
209C22×D3—361no—
210C22×D3—363no—
211C22×D3—723no—
212C22×D3C3⋊S411yescharacteristic
213C3×Q8—21no—
214C3×Q8—31no—
215C3×Q8—31no—
216C3×Q8—61no—
217C3×Q8—92no—
218C3×Q8—181no—
219C3×Q8#62611yescharacteristic, complemented
220C3×Q8#63811yescharacteristic, complemented
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228#67—362no—
229#67—3612no—
230C2×Dic3—31no—
231C2×Dic3—61no—
232C2×Dic3—123no—
233D12—181no—
234D12—181no—
235D12—181no—
236D12—181no—
237D12—182no—
238D12—361no—
239D12—362no—
240C4×D3—61no—
241C4×D3—61no—
242Dic6—32no—
243Dic6—61no—
244SL(2,𝔽3)—23no—
245SL(2,𝔽3)#62613yescomplemented
246C3×C8—181no—
247C3⋊C8—61no—
248C3⋊C8—181no—
249C3⋊C8—181no—
250C3⋊C8—182no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—43no—
255C2×C32—43no—
256C2×C32—43no—
257C2×C32—83no—
258C2×C32—83no—
259C2×C32—83no—
260C2×C32—81no—
261C2×C32—81no—
262C2×C32—81no—
263C2×C32—242no—
264C2×C32C2×GL(2,𝔽3)11yescharacteristic
265C2×C32#98311yescharacteristic
266C2×C32GL(2,ℤ/4)11yescharacteristic
267C3⋊D3—242no—
268C3⋊D3—246no—
269C3⋊D3—2412no—
270C3⋊D3—244no—
271C3⋊D3—241no—
272C3×D3—22no—
273C3×D3—86no—
274C2×Q8—61no—
275C2×Q8—91no—
276C2×Q8—181no—
277C2×Q8D3×C3⋊D311yescharacteristic
278C2×D4—271no—
279C2×D4—541no—
280C22×C4—91no—
281SD16—542no—
282SD16—541no—
283SD16—541no—
284C2×C8—91no—
285C2×C8—271no—
286C4⋊C4—91no—
287C22⋊C4—541no—
288C22×C3—21no—
289C22×C3—32no—
290C22×C3—43no—
291C22×C3—61no—
292C22×C3—61no—
293C22×C3—83no—
294C22×C3—126no—
295C22×C3—243no—
296C22×C3—243no—
297C22×C3D3×S411yescharacteristic
298C22×C3C2×C3⋊S411yescharacteristic
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—362no—
302C2×D3—366no—
303C2×D3—361no—
304C2×D3—361no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—361no—
308C2×D3—361no—
309C2×D3—721no—
310C2×D3—721no—
311C2×D3—723no—
312C2×D3—723no—
313C2×D3—726no—
314C2×D3—726no—
315C2×D3—726no—
316C2×D3C3⋊GL(2,𝔽3)12yescomplemented
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—181no—
324C3×C4—361no—
325Dic3—61no—
326Dic3—126no—
327Dic3—121no—
328C32—41no—
329C32—43no—
330C32—83no—
331C32—81no—
332C32#542111yescharacteristic, complemented
333C23—31no—
334C23—541no—
335Q8—31no—
336Q8—92no—
337Q8—181no—
338Q8#629111yescharacteristic, complemented
339D4—541no—
340D4—541no—
341D4—1082no—
342C2×C4—31no—
343C2×C4—91no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—181no—
347C8—181no—
348C8—541no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—43no—
353C2×C3—43no—
354C2×C3—43no—
355C2×C3—62no—
356C2×C3—83no—
357C2×C3—83no—
358C2×C3—83no—
359C2×C3—246no—
360C2×C3#6420211yescharacteristic
361C2×C3D3×GL(2,𝔽3)11yescharacteristic
362C2×C3#6421011yescharacteristic
363C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
364C2×C3#6426411yescharacteristic
365C2×C3#6426911yescharacteristic
366D3—22no—
367D3—722no—
368D3—726no—
369D3—7212no—
370D3—721no—
371D3—721no—
372C22—32no—
373C22—61no—
374C22—61no—
375C22—1081no—
376C22—1081no—
377C22—1081no—
378C22D3×C3⋊S411yescharacteristic
379C4—31no—
380C4—31no—
381C4—181no—
382C4—361no—
383C3—21no—
384C3—43no—
385C3—83no—
386C3#1059321811yescharacteristic, complemented
387C3#1059365511yescharacteristic, complemented
388C2—62no—
389C2—2161no—
390C2#1172419711yescharacteristic
391C2D3×C3⋊GL(2,𝔽3)11yescharacteristic
392C2#1172420511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, 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
1. C2×C3×D3×SL(2,𝔽3)C2semidirect
2. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3×C3⋊#859C2semidirect
11. C2×C3×D3×Q8D3semidirect
13. C2×D3×SL(2,𝔽3)D3semidirect
21. C3⋊#859D3semidirect
29. C32×SL(2,𝔽3)D4semidirect
69. C2×D3×Q8C3⋊D3semidirect
96. C32×Q8#67semidirect
111. C3×SL(2,𝔽3)D12semidirect
112. C3×SL(2,𝔽3)#67semidirect
190. C2×C3×D3GL(2,𝔽3)semidirect
219. C3×Q8#626semidirect
220. C3×Q8#638semidirect
245. SL(2,𝔽3)#626semidirect
316. C2×D3C3⋊GL(2,𝔽3)semidirect
332. C32#5421semidirect
338. Q8#6291semidirect
386. C3#10593218semidirect
387. C3#10593655semidirect

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
C2×C3⋊(C3⋊GL(2,𝔽3))2
C3×C3⋊#8592
#105936553
#105932183
#105915843
C2×#62914