Finite Group Databasebeta

← SL(2,𝔽3)×#626 #13166151

Subgroups — SL(2,𝔽3)×#626 #13166151

Order 1728.

559 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 68 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4Q8×#626C311yesmaximal, characteristic, complemented
5SL(2,𝔽3)×#67—31nomaximal
6SL(2,𝔽3)×D12—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×#626—41nomaximal
9C3⋊D3×SL(2,𝔽3)—21no—
10C3×D3×SL(2,𝔽3)—21no—
11C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
12C2×C3×D3×Q8C2×C311yescharacteristic, complemented
13C22×C3×SL(2,𝔽3)—31no—
14C2×D3×SL(2,𝔽3)—21no—
15C2×D3×SL(2,𝔽3)—31no—
16C2×D3×SL(2,𝔽3)—31no—
17C2×D3×SL(2,𝔽3)—61no—
18C2×D3×SL(2,𝔽3)D311yescharacteristic, complemented
19C3×Q8×Dic3C2×C311yescharacteristic, complemented
20C3×C4×SL(2,𝔽3)—31no—
21C4×#626—31no—
22#63871—31no—
23#63866—31no—
24#63857—31no—
25Dic3×SL(2,𝔽3)—21no—
26Dic3×SL(2,𝔽3)D311yescharacteristic, complemented
27C22×C3×C3⋊D3—41no—
28C22×C32×D3—41no—
29C2×C32×Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic, complemented
31C3×#626—44no—
32Q8×#67—31no—
33Q8×D12—31no—
34D4×SL(2,𝔽3)—91no—
35C2×C32×Q8C22×C311yescharacteristic
36Q8×C3⋊D3—21no—
37Q8×C3⋊D3—61no—
38C2×C4×C3⋊D3—31no—
39C2×#634—31no—
40C2×C3×#67—121no—
41C3×D3×Q8—21no—
42C3×D3×Q8—61no—
43C2×C3×D12—121no—
44C2×C3×C4×D3—31no—
45C2×C3×Dic6—31no—
46C2×C3×SL(2,𝔽3)—21no—
47C2×C3×SL(2,𝔽3)—21no—
48C2×C3×SL(2,𝔽3)—21no—
49C2×C3×SL(2,𝔽3)—41no—
50C2×C3×SL(2,𝔽3)—61no—
51C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
52C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
53C2×#626A411yescharacteristic
54D3×SL(2,𝔽3)—22no—
55D3×SL(2,𝔽3)—21no—
56D3×SL(2,𝔽3)—62no—
57D3×SL(2,𝔽3)—61no—
58D3×SL(2,𝔽3)—61no—
59C3×C4⋊Dic3—31no—
60C3×C4×Dic3—31no—
61Dic3⋊Dic3—32no—
62Dic3⋊Dic3—31no—
63#3722—31no—
64(C2×D3)⋊Dic3—31no—
65C22×C33—41no—
66C2×C3×C3⋊D3—42no—
67C2×C3×C3⋊D3—81no—
68C2×C3×C3⋊D3—81no—
69C2×C32×D3—42no—
70C2×C32×D3—81no—
71C2×C32×D3—81no—
72C32×Dic3—42no—
73C22×C3×Q8—31no—
74C2×D3×Q8—31no—
75C2×D3×Q8—31no—
76C2×D3×Q8—61no—
77C2×D3×Q8C3×D311yescharacteristic, complemented
78C22×SL(2,𝔽3)—31no—
79C22×SL(2,𝔽3)—61no—
80C22×SL(2,𝔽3)—91no—
81C3×C4×Q8—31no—
82#946—91no—
83Q8×Dic3C3×D311yescharacteristic, complemented
84#944—91no—
85C4×#67—91no—
86#924—91no—
87#897—91no—
88#897—91no—
89C4×D12—91no—
90#869—91no—
91C4×SL(2,𝔽3)—31no—
92C4×SL(2,𝔽3)—61no—
93C23×C32—121no—
94C22×C3⋊D3C2×A411yescharacteristic
95C22×C3×D3—41no—
96C22×C3×D3—81no—
97C22×C3×D3—121no—
98C22×C3×D3—121no—
99C22×C3×D3—241no—
100C22×C3×D3C2×A411yescharacteristic
101C32×Q8—31no—
102C32×Q8C3×D411yescharacteristic, complemented
103C2×C32×C4—31no—
104C2×C32×C4—121no—
105C2×C3⋊Dic3—31no—
106C4×C3⋊D3—61no—
107C4×C3⋊D3—61no—
108#634—61no—
109#634—61no—
110C3×#67—124no—
111C2×C3×Dic3—31no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—81no—
115C2×C3×Dic3C2×A411yescharacteristic
116C3×D12—124no—
117C3×C4×D3—61no—
118C3×C4×D3—61no—
119C3×Dic6—61no—
120C3×Dic6—61no—
121C3×SL(2,𝔽3)—21no—
122C3×SL(2,𝔽3)—21no—
123C3×SL(2,𝔽3)—21no—
124C3×SL(2,𝔽3)—41no—
125C3×SL(2,𝔽3)D1211yescharacteristic, complemented
126C3×SL(2,𝔽3)#6711yescharacteristic, complemented
127#626SL(2,𝔽3)14yesdirect product factor
128D4×Q8—91noSylow
129C2×C33—41no—
130C2×C33—41no—
131C2×C33—41no—
132C3×C3⋊D3—82no—
133C32×D3—82no—
134C2×C3×Q8—21no—
135C2×C3×Q8—61no—
136C2×C3×Q8—91no—
137C2×C3×Q8—181no—
138C2×C3×Q8C2×C3×D311yescharacteristic
139C2×C3×Q8C2×C3×D311yescharacteristic
140C2×C3×D4—361no—
141C22×C3×C4—91no—
142C2×#67—31no—
143D3×Q8—21no—
144D3×Q8—61no—
145D3×Q8—62no—
146D3×Q8—61no—
147D3×Q8—61no—
148D3×Q8—182no—
149D3×Q8—181no—
150D3×Q8—181no—
151C2×D12—31no—
152C2×C4×D3—31no—
153C2×C4×D3—91no—
154C2×C4×D3—91no—
155C2×C4×D3—181no—
156C2×Dic6—31no—
157C2×Dic6—91no—
158C2×Dic6—91no—
159C2×Dic6—181no—
160C2×SL(2,𝔽3)—21no—
161C2×SL(2,𝔽3)—21no—
162C2×SL(2,𝔽3)—41no—
163C2×SL(2,𝔽3)—62no—
164C2×SL(2,𝔽3)—61no—
165C2×SL(2,𝔽3)—181no—
166C2×SL(2,𝔽3)D3211yescharacteristic
167C3×C4⋊C4—91no—
168C3×C42—91no—
169#217—91no—
170(C2×D3)⋊C4—91no—
171(C2×D3)⋊C4—91no—
172(C2×D3)⋊C4—91no—
173C4⋊Dic3—31no—
174C4⋊Dic3—91no—
175C4⋊Dic3—92no—
176Dic3⋊C4—92no—
177Dic3⋊C4—91no—
178C4×Dic3—31no—
179C22×C32—41no—
180C22×C32—41no—
181C22×C32—81no—
182C22×C32—81no—
183C22×C32—81no—
184C22×C32—122no—
185C22×C32—161no—
186C22×C32—241no—
187C22×C32—241no—
188C22×C32C22×A411yescharacteristic
189C2×C3⋊D3—21no—
190C2×C3⋊D3—21no—
191C2×C3⋊D3C2×SL(2,𝔽3)12yescomplemented
192C2×C3×D3—21no—
193C2×C3×D3—21no—
194C2×C3×D3—42no—
195C2×C3×D3—81no—
196C2×C3×D3—81no—
197C2×C3×D3—82no—
198C2×C3×D3—82no—
199C2×C3×D3—82no—
200C2×C3×D3—122no—
201C2×C3×D3—122no—
202C2×C3×D3—242no—
203C2×C3×D3—241no—
204C2×C3×D3—241no—
205C2×C3×D3—241no—
206C2×C3×D3—241no—
207C2×C3×D3—242no—
208C2×C3×D3—242no—
209C2×C3×D3C2×SL(2,𝔽3)12yescomplemented
210C32×C4—31no—
211C32×C4—31no—
212C32×C4—122no—
213C3⋊Dic3—61no—
214C3×Dic3—42no—
215C3×Dic3—61no—
216C3×Dic3—61no—
217C3×Dic3—82no—
218C3×Dic3C2×SL(2,𝔽3)12yescomplemented
219C22×Q8—31no—
220C22×Q8—91no—
221C4⋊Q8—271no—
222C22⋊Q8—271no—
223C22⋊Q8—271no—
224C4×Q8—31no—
225C4×D4—271no—
226C33—41noSylow
227C23×C3—31no—
228C23×C3—121no—
229C23×C3—241no—
230C23×C3—361no—
231C22×D3—31no—
232C22×D3—31no—
233C22×D3—61no—
234C22×D3D3×A411yescharacteristic
235C3×Q8—21no—
236C3×Q8—31no—
237C3×Q8—31no—
238C3×Q8—61no—
239C3×Q8—181no—
240C3×Q8—181no—
241C3×Q8C3×D1211yescharacteristic, complemented
242C3×Q8C3×#6711yescharacteristic, complemented
243C3×D4—364no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—61no—
248C2×C3×C4—91no—
249C2×C3×C4—91no—
250C2×C3×C4—121no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—241no—
254#67—34no—
255C2×Dic3—31no—
256C2×Dic3—31no—
257C2×Dic3—91no—
258C2×Dic3—91no—
259C2×Dic3—181no—
260C2×Dic3D3×A411yescharacteristic
261D12—34no—
262C4×D3—61no—
263C4×D3—61no—
264C4×D3—181no—
265C4×D3—181no—
266C4×D3—181no—
267C4×D3—181no—
268C4×D3—182no—
269C4×D3—182no—
270Dic6—61no—
271Dic6—61no—
272Dic6—182no—
273Dic6—181no—
274Dic6—181no—
275Dic6—181no—
276Dic6—181no—
277Dic6—182no—
278SL(2,𝔽3)—21no—
279SL(2,𝔽3)—21no—
280SL(2,𝔽3)—41no—
281SL(2,𝔽3)#62611yescharacteristic, direct product factor
282C2×C32—41no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—81no—
289C2×C32—81no—
290C2×C32—81no—
291C2×C32—81no—
292C2×C32—81no—
293C2×C32—81no—
294C2×C32—81no—
295C2×C32—81no—
296C2×C32—81no—
297C2×C32—161no—
298C2×C32—161no—
299C2×C32—161no—
300C2×C32—242no—
301C2×C32D4×A411yescharacteristic
302C2×C32C22×SL(2,𝔽3)11yescharacteristic
303C2×C322−1+4⋊C311yescharacteristic
304C3⋊D3—22no—
305C3×D3—22no—
306C3×D3—82no—
307C3×D3—84no—
308C3×D3—242no—
309C3×D3—242no—
310C3×D3—244no—
311C2×Q8—61no—
312C2×Q8—91no—
313C2×Q8—181no—
314C2×Q8—181no—
315C2×Q8—271no—
316C2×Q8—541no—
317C2×Q8C3×D3211yescharacteristic
318C2×D4—91no—
319C22×C4—91no—
320C22×C4—271no—
321C4⋊C4—91no—
322C4⋊C4—271no—
323C4⋊C4—272no—
324C22⋊C4—271no—
325C22⋊C4—271no—
326C42—91no—
327C22×C3—21no—
328C22×C3—32no—
329C22×C3—41no—
330C22×C3—61no—
331C22×C3—61no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—122no—
335C22×C3—161no—
336C22×C3—242no—
337C22×C3—241no—
338C22×C3—241no—
339C22×C3—242no—
340C22×C3—242no—
341C22×C3—362no—
342C22×C3—721no—
343C22×C3—721no—
344C22×C3C2×D3×A411yescharacteristic
345C22×C3C2×D3×A411yescharacteristic
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—32no—
349C2×D3—32no—
350C2×D3—62no—
351C2×D3—62no—
352C2×D3—62no—
353C2×D3—61no—
354C2×D3—61no—
355C2×D3—61no—
356C2×D3—61no—
357C2×D3D3×SL(2,𝔽3)12yescomplemented
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—32no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—61no—
364C3×C4—61no—
365C3×C4—122no—
366C3×C4—181no—
367C3×C4—181no—
368C3×C4—242no—
369Dic3—61no—
370Dic3—61no—
371Dic3—182no—
372Dic3—181no—
373Dic3—181no—
374Dic3D3×SL(2,𝔽3)12yescomplemented
375C32—41no—
376C32—41no—
377C32—81no—
378C32—81no—
379C32—81no—
380C32—161no—
381C32D4×SL(2,𝔽3)11yescharacteristic, complemented
382C23—31no—
383C23—91no—
384Q8—31no—
385Q8—181no—
386Q8—181no—
387Q8—541no—
388Q8—541no—
389Q8C3×#62611yescharacteristic, complemented
390D4—94no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4—271no—
398C2×C4—541no—
399C2×C4—541no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3—41no—
406C2×C3—62no—
407C2×C3—81no—
408C2×C3—81no—
409C2×C3—81no—
410C2×C3—81no—
411C2×C3—81no—
412C2×C3—81no—
413C2×C3—161no—
414C2×C3—161no—
415C2×C3—161no—
416C2×C3—242no—
417C2×C3—244no—
418C2×C3—722no—
419C2×C3A4×D1211yescharacteristic
420C2×C3C2×D3×SL(2,𝔽3)11yescharacteristic
421C2×C3C2×D3×SL(2,𝔽3)11yescharacteristic
422C2×C3#1007⋊C311yescharacteristic
423C2×C3#1010⋊C311yescharacteristic
424C2×C3A4×#6711yescharacteristic
425D3—22no—
426D3—64no—
427D3—62no—
428D3—62no—
429C22—32no—
430C22—61no—
431C22—61no—
432C22—92no—
433C22—181no—
434C22—181no—
435C22D32×A411yescharacteristic
436C4—32no—
437C4—31no—
438C4—31no—
439C4—181no—
440C4—181no—
441C4—541no—
442C3—21no—
443C3—41no—
444C3—81no—
445C3—81no—
446C3—161no—
447C3SL(2,𝔽3)×D1211yescharacteristic, complemented
448C3SL(2,𝔽3)×#6711yescharacteristic, complemented
449C2—62no—
450C2—182no—
451C2#64315⋊C311yescharacteristic
452C2D32×SL(2,𝔽3)11yescharacteristic
453C2A4×#62611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
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×D3×SL(2,𝔽3)C2semidirect
3. C3×Dic3×SL(2,𝔽3)C2semidirect
4. Q8×#626C3semidirect
11. C2×Q8×C3⋊D3C2×C3semidirect
12. C2×C3×D3×Q8C2×C3semidirect
18. C2×D3×SL(2,𝔽3)D3semidirect
19. C3×Q8×Dic3C2×C3semidirect
26. Dic3×SL(2,𝔽3)D3semidirect
30. C32×SL(2,𝔽3)D4semidirect
77. C2×D3×Q8C3×D3semidirect
83. Q8×Dic3C3×D3semidirect
102. C32×Q8C3×D4semidirect
125. C3×SL(2,𝔽3)D12semidirect
126. C3×SL(2,𝔽3)#67semidirect
127. #626SL(2,𝔽3)direct (N × H)
191. C2×C3⋊D3C2×SL(2,𝔽3)semidirect
209. C2×C3×D3C2×SL(2,𝔽3)semidirect
218. C3×Dic3C2×SL(2,𝔽3)semidirect
241. C3×Q8C3×D12semidirect
242. C3×Q8C3×#67semidirect
281. SL(2,𝔽3)#626direct (N × H)
357. C2×D3D3×SL(2,𝔽3)semidirect
374. Dic3D3×SL(2,𝔽3)semidirect
381. C32D4×SL(2,𝔽3)semidirect
389. Q8C3×#626semidirect
447. C3SL(2,𝔽3)×D12semidirect
448. C3SL(2,𝔽3)×#67semidirect

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×D3×SL(2,𝔽3)2
C3×Dic3×SL(2,𝔽3)2
Q8×#6263
SL(2,𝔽3)×#673
SL(2,𝔽3)×D123
C2×C3×#6264