Finite Group Databasebeta

← D3×Dic3×SL(2,𝔽3) #13166148

Subgroups — D3×Dic3×SL(2,𝔽3) #13166148

Order 1728.

540 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 92 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
2SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4D3×Q8×Dic3C311yesmaximal, characteristic, complemented
5C2×Dic3×SL(2,𝔽3)—31nomaximal
6C4×D3×SL(2,𝔽3)—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×D3×Dic3—41nomaximal
9C3×D3×SL(2,𝔽3)C412yescomplemented
10C2×C3×D3×Q8C2×C311yescharacteristic
11C22×C3×SL(2,𝔽3)—31no—
12C2×D3×SL(2,𝔽3)—21no—
13C2×D3×SL(2,𝔽3)D311yescharacteristic
14Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
15C3×Q8×Dic3C2×C311yescharacteristic, complemented
16C3×C4×SL(2,𝔽3)—31no—
17D3×C4⋊Dic3—31no—
18C4×D3×Dic3—31no—
19#63843—31no—
20Dic3×Dic6—31no—
21Dic3×SL(2,𝔽3)—21no—
22Dic3×SL(2,𝔽3)—31no—
23Dic3×SL(2,𝔽3)—31no—
24Dic3×SL(2,𝔽3)—61no—
25Dic3×SL(2,𝔽3)D311yescharacteristic, direct product factor
26C22×C32×D3—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
30C3×D3×Dic3—44no—
31C2×Q8×Dic3—31no—
32C4×D3×Q8—31no—
33C2×C4×SL(2,𝔽3)—91no—
34C2×C32×Q8C22×C311yescharacteristic
35C22×C3×Dic3—121no—
36C3×D3×Q8—32no—
37C3×D3×Q8C3×C412yescomplemented
38C2×C3×C4×D3—31no—
39C2×C3×C4×D3—121no—
40C2×C3×Dic6—31no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)—32no—
45C2×C3×SL(2,𝔽3)—41no—
46C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
47C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
48C2×D3×Dic3—31no—
49C2×D3×Dic3A411yescharacteristic
50D3×SL(2,𝔽3)—22no—
51D3×SL(2,𝔽3)Dic312yesdirect product factor
52C4⋊(C3⋊Dic3)—31no—
53C4×C3⋊Dic3—31no—
54C3×C4⋊Dic3—31no—
55C3×C4×Dic3—31no—
56Dic3⋊Dic3—32no—
57Dic32—31no—
58Dic32—31no—
59C22×C33—41no—
60C2×C32×D3—42no—
61C2×C32×D3—42no—
62C2×C32×D3—42no—
63C3×C3⋊Dic3—42no—
64C32×Dic3—42no—
65C22×C3×Q8—31no—
66C2×D3×Q8C3×D311yescharacteristic
67C22×SL(2,𝔽3)—31no—
68C22×SL(2,𝔽3)—61no—
69C3×C4×Q8—31no—
70Q8×Dic3—31no—
71Q8×Dic3—31no—
72Q8×Dic3—61no—
73Q8×Dic3—91no—
74Q8×Dic3—91no—
75Q8×Dic3C3×D311yescharacteristic, complemented
76C2×C4⋊Dic3—91no—
77C2×C4×Dic3—91no—
78D3×C4⋊C4—91no—
79#887—91no—
80C42×D3—91no—
81C4×Dic6—91no—
82C4×SL(2,𝔽3)—31no—
83C4×SL(2,𝔽3)—61no—
84C4×SL(2,𝔽3)—91no—
85C23×C32—121no—
86C22×C3×D3—41no—
87C22×C3×D3—81no—
88C22×C3×D3C2×A411yescharacteristic
89C32×Q8—31no—
90C32×Q8C2×C3×C411yescharacteristic, complemented
91C2×C32×C4—31no—
92C2×C32×C4—121no—
93C2×C3⋊Dic3—31no—
94C2×C3⋊Dic3C2×A411yescharacteristic
95C2×C3×Dic3—31no—
96C2×C3×Dic3—31no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—81no—
99C2×C3×Dic3—124no—
100C2×C3×Dic3—121no—
101C2×C3×Dic3—121no—
102C2×C3×Dic3—241no—
103C2×C3×Dic3C2×A411yescharacteristic
104C3×C4×D3—32no—
105C3×C4×D3—32no—
106C3×C4×D3—124no—
107C3×Dic6—32no—
108C3×Dic6—32no—
109C3×SL(2,𝔽3)—21no—
110C3×SL(2,𝔽3)—21no—
111C3×SL(2,𝔽3)—21no—
112C3×SL(2,𝔽3)—41no—
113C3×SL(2,𝔽3)C4×D311yescharacteristic, complemented
114C3×SL(2,𝔽3)C2×Dic311yescharacteristic, complemented
115D3×Dic3—62no—
116D3×Dic3SL(2,𝔽3)14yesdirect product factor
117C2×C4×Q8—91noSylow
118C2×C33—41no—
119C2×C33—41no—
120C2×C33—41no—
121C32×D3—44no—
122C2×C3×Q8—21no—
123C2×C3×Q8—32no—
124C2×C3×Q8—91no—
125C2×C3×Q8—92no—
126C2×C3×Q8C2×C3×D311yescharacteristic
127C2×C3×Q8C2×C3×D311yescharacteristic
128C22×C3×C4—91no—
129C22×C3×C4—361no—
130C22×Dic3—31no—
131C22×Dic3—91no—
132D3×Q8—32no—
133D3×Q8C3×Dic312yescomplemented
134C2×C4×D3—31no—
135C2×C4×D3—31no—
136C2×C4×D3—91no—
137C2×Dic6—31no—
138C2×SL(2,𝔽3)—21no—
139C2×SL(2,𝔽3)—21no—
140C2×SL(2,𝔽3)—32no—
141C2×SL(2,𝔽3)—41no—
142C2×SL(2,𝔽3)—62no—
143C2×SL(2,𝔽3)D3211yescharacteristic
144C3×C4⋊C4—91no—
145C3×C42—91no—
146C4⋊Dic3—31no—
147C4⋊Dic3—92no—
148C4⋊Dic3—91no—
149C4⋊Dic3—91no—
150C4⋊Dic3—181no—
151Dic3⋊C4—92no—
152C4×Dic3—31no—
153C4×Dic3—91no—
154C4×Dic3—91no—
155C4×Dic3—91no—
156C4×Dic3—91no—
157C4×Dic3—91no—
158C4×Dic3—91no—
159C4×Dic3—181no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32—81no—
163C22×C32—81no—
164C22×C32—81no—
165C22×C32—122no—
166C22×C32—122no—
167C22×C32—122no—
168C22×C32—161no—
169C22×C32C22×A411yescharacteristic
170C2×C3×D3—42no—
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—82no—
174C2×C3×D3—82no—
175C2×C3×D3—82no—
176C2×C3×D3C4×A412yes—
177C2×C3×D3C2×SL(2,𝔽3)12yes—
178C2×C3×D3#41⋊C312yes—
179C32×C4—31no—
180C32×C4—31no—
181C32×C4—122no—
182C3⋊Dic3—61no—
183C3⋊Dic3C2×SL(2,𝔽3)12yescomplemented
184C3×Dic3—32no—
185C3×Dic3—42no—
186C3×Dic3—61no—
187C3×Dic3—82no—
188C3×Dic3—122no—
189C3×Dic3—122no—
190C3×Dic3—242no—
191C3×Dic3C2×SL(2,𝔽3)12yescomplemented
192C22×Q8—31no—
193C4×Q8—31no—
194C4×Q8—91no—
195C4×Q8—271no—
196C4×Q8—271no—
197C2×C4⋊C4—271no—
198C2×C42—271no—
199C33—41noSylow
200C23×C3—31no—
201C23×C3—121no—
202C23×C3—241no—
203C22×D3D3×A411yescharacteristic
204C3×Q8—21no—
205C3×Q8—31no—
206C3×Q8—31no—
207C3×Q8—61no—
208C3×Q8—92no—
209C3×Q8—92no—
210C3×Q8C3×C4×D311yescharacteristic, complemented
211C3×Q8C2×C3×Dic311yescharacteristic, complemented
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—61no—
216C2×C3×C4—92no—
217C2×C3×C4—92no—
218C2×C3×C4—91no—
219C2×C3×C4—91no—
220C2×C3×C4—121no—
221C2×C3×C4—241no—
222C2×C3×C4—364no—
223C2×C3×C4—361no—
224C2×Dic3—31no—
225C2×Dic3—31no—
226C2×Dic3—31no—
227C2×Dic3—34no—
228C2×Dic3—31no—
229C2×Dic3—61no—
230C2×Dic3—91no—
231C2×Dic3—91no—
232C2×Dic3—181no—
233C2×Dic3—182no—
234C2×Dic3D3×A411yescharacteristic
235C4×D3—34no—
236C4×D3—32no—
237C4×D3—32no—
238C4×D3—182no—
239Dic6—32no—
240Dic6—32no—
241SL(2,𝔽3)—21no—
242SL(2,𝔽3)—21no—
243SL(2,𝔽3)—41no—
244SL(2,𝔽3)D3×Dic311yescharacteristic, direct product factor
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32—81no—
252C2×C32—81no—
253C2×C32—81no—
254C2×C32—81no—
255C2×C32—81no—
256C2×C32—81no—
257C2×C32—81no—
258C2×C32—81no—
259C2×C32—81no—
260C2×C32—124no—
261C2×C32—161no—
262C2×C32—161no—
263C2×C32—161no—
264C2×C32C2×C4×A411yescharacteristic
265C2×C32C22×SL(2,𝔽3)11yescharacteristic
266C2×C32C2×#41⋊C311yescharacteristic
267C3×D3—44no—
268C3×D3—84no—
269C3×D3C4×SL(2,𝔽3)14yescomplemented
270C2×Q8—32no—
271C2×Q8—91no—
272C2×Q8—92no—
273C2×Q8C3×D3211yescharacteristic
274C22×C4—91no—
275C22×C4—91no—
276C22×C4—271no—
277C4⋊C4—91no—
278C4⋊C4—271no—
279C4⋊C4—272no—
280C42—91no—
281C42—271no—
282C42—271no—
283C42—271no—
284C22×C3—21no—
285C22×C3—32no—
286C22×C3—32no—
287C22×C3—32no—
288C22×C3—41no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3—122no—
292C22×C3—122no—
293C22×C3—122no—
294C22×C3—161no—
295C22×C3—242no—
296C22×C3—242no—
297C22×C3—242no—
298C22×C3C2×D3×A411yescharacteristic
299C22×C3C2×D3×A411yescharacteristic
300C2×D3C3⋊(#41⋊C3)12yes—
301C2×D3D3×SL(2,𝔽3)12yes—
302C2×D3Dic3×A412yes—
303C3×C4—31no—
304C3×C4—31no—
305C3×C4—32no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—61no—
309C3×C4—61no—
310C3×C4—92no—
311C3×C4—122no—
312C3×C4—181no—
313C3×C4—242no—
314C3×C4—362no—
315Dic3—32no—
316Dic3—32no—
317Dic3—32no—
318Dic3—62no—
319Dic3—61no—
320Dic3—181no—
321Dic3—181no—
322Dic3—361no—
323Dic3D3×SL(2,𝔽3)12yesdirect product factor
324C32—41no—
325C32—41no—
326C32—81no—
327C32—81no—
328C32—81no—
329C32—161no—
330C32C2×C4×SL(2,𝔽3)11yescharacteristic, complemented
331C23—31no—
332Q8—31no—
333Q8—92no—
334Q8—92no—
335Q8C3×D3×Dic311yescharacteristic, complemented
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—92no—
339C2×C4—92no—
340C2×C4—94no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—91no—
344C2×C4—271no—
345C2×C4—542no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—34no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—81no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3—81no—
359C2×C3—124no—
360C2×C3—161no—
361C2×C3—161no—
362C2×C3—161no—
363C2×C3—244no—
364C2×C3C4×D3×A411yescharacteristic
365C2×C3C2×C3⋊(#41⋊C3)11yescharacteristic
366C2×C3C2×D3×SL(2,𝔽3)11yescharacteristic
367C2×C3C2×D3×SL(2,𝔽3)11yescharacteristic
368C2×C3D3×#41⋊C311yescharacteristic
369C2×C3C2×Dic3×A411yescharacteristic
370D3Dic3×SL(2,𝔽3)14yesdirect product factor
371C22—32no—
372C22—32no—
373C22—32no—
374C22D32×A411yescharacteristic
375C4—32no—
376C4—31no—
377C4—31no—
378C4—92no—
379C4—92no—
380C4—181no—
381C4—541no—
382C3—21no—
383C3—41no—
384C3—81no—
385C3—81no—
386C3—161no—
387C3C4×D3×SL(2,𝔽3)11yescharacteristic, complemented
388C3C2×Dic3×SL(2,𝔽3)11yescharacteristic, complemented
389C2—34no—
390C2D3×C3⋊(#41⋊C3)11yescharacteristic
391C2D32×SL(2,𝔽3)11yescharacteristic
392C2D3×Dic3×A411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, hypocenterC32×Q8C2×C3×C4
Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
prime radicalC2×C3×D3×SL(2,𝔽3)C2
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×C3⋊Dic3C2semidirect
3. C3×Dic3×SL(2,𝔽3)C2semidirect
4. D3×Q8×Dic3C3semidirect
9. C3×D3×SL(2,𝔽3)C4semidirect
14. Q8×C3⋊Dic3C2×C3semidirect
15. C3×Q8×Dic3C2×C3semidirect
25. Dic3×SL(2,𝔽3)D3direct (N × H)
29. C32×SL(2,𝔽3)C2×C4semidirect
37. C3×D3×Q8C3×C4semidirect
51. D3×SL(2,𝔽3)Dic3direct (N × H)
75. Q8×Dic3C3×D3semidirect
90. C32×Q8C2×C3×C4semidirect
113. C3×SL(2,𝔽3)C4×D3semidirect
114. C3×SL(2,𝔽3)C2×Dic3semidirect
116. D3×Dic3SL(2,𝔽3)direct (N × H)
133. D3×Q8C3×Dic3semidirect
183. C3⋊Dic3C2×SL(2,𝔽3)semidirect
191. C3×Dic3C2×SL(2,𝔽3)semidirect
210. C3×Q8C3×C4×D3semidirect
211. C3×Q8C2×C3×Dic3semidirect
244. SL(2,𝔽3)D3×Dic3direct (N × H)
269. C3×D3C4×SL(2,𝔽3)semidirect
323. Dic3D3×SL(2,𝔽3)direct (N × H)
330. C32C2×C4×SL(2,𝔽3)semidirect
335. Q8C3×D3×Dic3semidirect
370. D3Dic3×SL(2,𝔽3)direct (N × H)
387. C3C4×D3×SL(2,𝔽3)semidirect
388. C3C2×Dic3×SL(2,𝔽3)semidirect

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
SL(2,𝔽3)×C3⋊Dic32
C3×Dic3×SL(2,𝔽3)2
D3×Q8×Dic33
C2×Dic3×SL(2,𝔽3)3
C4×D3×SL(2,𝔽3)3
C2×C3×D3×Dic34