Finite Group Databasebeta

← D3×#64267 #13180828

Subgroups — D3×#64267 #13180828

Order 1728.

968 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C3×D3×#41⋊C3C211yesmaximal, characteristic, complemented
2#11724471C211yesmaximal, characteristic, complemented
3C3×#64267C211yesmaximal, characteristic, complemented
4D3×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
5#11724194C212yesmaximal, complemented
6C2×#64267—31nomaximal
7D3×#986—33nomaximal
8D3×#949—31nomaximal
9C32×#41⋊C3C2211yescharacteristic, complemented
10D3×C4⋊(C3⋊D3)—41nomaximal
11C3×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C3×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
13C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
14C3×C3⋊GL(2,𝔽3)C2212yescomplemented
15C3×D3×#41D311yescharacteristic, complemented
16C2×C3×#41⋊C3—31no—
17C2×D3×D12—31no—
18D3×#41⋊C3D313yescomplemented
19#64267—31no—
20#64267—33no—
21#64267—32no—
22#64267—61no—
23#64267—63no—
24#64267D311yescharacteristic, direct product factor
25C2×C3⋊GL(2,𝔽3)—32no—
26C3×#986—33no—
27D3×GL(2,𝔽3)—36no—
28#64199—36no—
29#64158—31no—
30C3×#949—31no—
31#63939—32no—
32D3×#215—32no—
33#63925—32no—
34D3×#213—32no—
35#63825—31no—
36#63823—31no—
37D3×#208—31no—
38C2×D3×C3⋊D3—42no—
39C4⋊(C3⋊(C3⋊D3))—41no—
40C3×C4⋊(C3⋊D3)—41no—
41C32×C4×D3—41no—
42C32×SL(2,𝔽3)C2311yescharacteristic, complemented
43#6292—42no—
44C2×#986—93no—
45C2×#949—91no—
46D3×AGL(1,ℤ/8)—91no—
47C22×D32—61no—
48C32×#41C2×D311yescharacteristic, complemented
49C2×C4⋊(C3⋊D3)—31no—
50C2×C4⋊(C3⋊D3)—121no—
51C3×#239C2×D311yescharacteristic, complemented
52C3×D3×Q8C2×D311yescharacteristic
53C3×#237—31no—
54C3×D3×D4—31no—
55C3×#235—31no—
56C2×C3×D12—31no—
57C2×C3×C4×D3—31no—
58C3×#41⋊C3—21no—
59C3×#41⋊C3—23no—
60C3×#41⋊C3—31no—
61C3×#41⋊C3C2×D313yescomplemented
62C3×#41⋊C3C2×D311yescharacteristic, complemented
63C2×C3×SL(2,𝔽3)—31no—
64C2×#626—61no—
65D3×D12—32no—
66D3×D12—32no—
67D3×D12—61no—
68D3×D12—61no—
69D3×D12—123no—
70D3×SL(2,𝔽3)C2×D313yescomplemented
71C3⋊(#41⋊C3)C2×D313yescomplemented
72C3⋊GL(2,𝔽3)—32no—
73C3⋊GL(2,𝔽3)—36no—
74C3⋊GL(2,𝔽3)—62no—
75C3⋊GL(2,𝔽3)—66no—
76C3⋊GL(2,𝔽3)C2×D312yescomplemented
77C3×GL(2,𝔽3)—36no—
78#3755—32no—
79#3753—32no—
80#3748—31no—
81C3×#215—32no—
82C3×#213—32no—
83C3×#208—31no—
84#3717—32no—
85#3715—32no—
86#3714—32no—
87#3713—32no—
88#3711—32no—
89D3×C3⋊C8—32no—
90C2×C3⋊(C3⋊D3)—42no—
91C2×C3×C3⋊D3—42no—
92C2×C32×D3—41no—
93D3×C3⋊D3—84no—
94C33×C4—41no—
95C32×Dic3—41no—
96C2×C3×#41—31no—
97D3×#41C3⋊D311yescharacteristic, complemented
98C2×D3×D4—91no—
99C22×D12—91no—
100C2×#41⋊C3—33no—
101#986—33no—
102#986—96no—
103#986—93no—
104#986—183no—
105C2×GL(2,𝔽3)—96no—
106C3×AGL(1,ℤ/8)—91no—
107#949—31no—
108#949—91no—
109#949—92no—
110#949—92no—
111#949—91no—
112#949—91no—
113#949—91no—
114#949—181no—
115C2×#215—92no—
116#932—91no—
117C2×#213—92no—
118C2×#208—91no—
119#914—92no—
120D3×SD16—92no—
121#911—92no—
122D3×D8—92no—
123#908—91no—
124D3×M16—91no—
125C22×C3⋊D3—61no—
126C22×C3⋊D3—122no—
127C22×C3×D3—31no—
128C22×C3×D3—61no—
129C2×D32—62no—
130C2×D32—62no—
131C2×D32—126no—
132C2×D32—124no—
133C32×Q8C22×D311yescharacteristic
134C32×D4—31no—
135C2×C32×C4—31no—
136C2×C32×C4—121no—
137C4⋊(C3⋊D3)—32no—
138C4⋊(C3⋊D3)—41no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—121no—
141C4⋊(C3⋊D3)—122no—
142C4⋊(C3⋊D3)—123no—
143C4⋊(C3⋊D3)—241no—
144C4⋊(C3⋊D3)—243no—
145C3×#67—61no—
146C2×C3×Dic3—31no—
147C3×D12—31no—
148C3×D12—32no—
149C3×D12—61no—
150C3×D12—123no—
151C3×C4×D3—31no—
152C3×C4×D3—31no—
153C3×C4×D3—31no—
154C3×C4×D3—43no—
155C3×C4×D3S411yescharacteristic
156C3×Dic6—31no—
157C3×SL(2,𝔽3)—21no—
158C3×SL(2,𝔽3)—23no—
159C3×SL(2,𝔽3)C22×D313yescomplemented
160C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
161#626—62no—
162#626—62no—
163#626—126no—
164C3⋊(C3⋊C8)—32no—
165C3×C3⋊C8—32no—
166C2×AGL(1,ℤ/8)—271noSylow
167C2×C33—41no—
168C3⋊(C3⋊D3)—82no—
169C3×C3⋊D3—82no—
170C32×D3—42no—
171C23×D3—181no—
172C23×D3—181no—
173C3×#41—21no—
174C3×#41—31no—
175C3×#41—91no—
176C3×#41—91no—
177C3×#41D3211yescharacteristic, complemented
178C3×#41C2×C3⋊D311yescharacteristic, complemented
179C2×C3×Q8—31no—
180C2×C3×D4—91no—
181C2×C3×D4—91no—
182C22×C3×C4—91no—
183C2×#67—181no—
184#239C2×C3⋊D311yescharacteristic, complemented
185D3×Q8C2×C3⋊D311yescharacteristic
186#237—31no—
187D3×D4—31no—
188D3×D4—92no—
189D3×D4—92no—
190D3×D4—181no—
191D3×D4—181no—
192#235—31no—
193C2×D12—31no—
194C2×D12—92no—
195C2×D12—92no—
196C2×D12—91no—
197C2×D12—91no—
198C2×D12—181no—
199C2×D12—181no—
200C2×D12—181no—
201C2×D12—181no—
202C2×D12—363no—
203C2×C4×D3—31no—
204#41⋊C3—23no—
205#41⋊C3—33no—
206#41⋊C3D3213yescomplemented
207C2×SL(2,𝔽3)—33no—
208GL(2,𝔽3)—36no—
209GL(2,𝔽3)—96no—
210GL(2,𝔽3)—186no—
211C3×SD16—92no—
212C3×D8—92no—
213C3×M16—91no—
214#215—32no—
215#215—92no—
216#215—92no—
217#215—92no—
218#215—92no—
219#215—182no—
220#214—92no—
221#213—32no—
222#213—92no—
223#213—92no—
224#213—92no—
225#213—92no—
226#213—92no—
227#213—182no—
228#208—31no—
229#208—92no—
230#208—91no—
231#208—91no—
232#208—181no—
233C2×C3⋊C8—92no—
234D24—92no—
235#204—92no—
236ΓL(1,𝔽25)—92no—
237D3×C8—92no—
238C22×C32—31no—
239C22×C32—121no—
240C2×C3⋊D3—42no—
241C2×C3⋊D3—62no—
242C2×C3⋊D3—122no—
243C2×C3⋊D3—126no—
244C2×C3⋊D3—122no—
245C2×C3⋊D3—242no—
246C2×C3⋊D3—246no—
247C2×C3⋊D3—244no—
248C2×C3×D3—31no—
249C2×C3×D3—43no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—126no—
253C2×C3×D3—122no—
254C2×C3×D3C2×S411yescharacteristic
255D32—124no—
256D32—124no—
257D32—2412no—
258C32×C4—31no—
259C32×C4—41no—
260C32×C4—43no—
261C32×C4—83no—
262C32×C4—81no—
263C32×C4—121no—
264C32×C4C2×S411yescharacteristic
265C3×Dic3—31no—
266C3×Dic3—43no—
267C3×Dic3C2×S411yescharacteristic
268C2×#41—31no—
269C22×D4—271no—
270AGL(1,ℤ/8)—91no—
271AGL(1,ℤ/8)—271no—
272AGL(1,ℤ/8)—272no—
273AGL(1,ℤ/8)—272no—
274AGL(1,ℤ/8)—271no—
275AGL(1,ℤ/8)—271no—
276C2×SD16—272no—
277C2×D8—272no—
278C2×M16—271no—
279C33—41noSylow
280C23×C3—91no—
281C23×C3—181no—
282C22×D3—31no—
283C22×D3—61no—
284C22×D3—182no—
285C22×D3—182no—
286C22×D3—182no—
287C22×D3—181no—
288C22×D3—181no—
289C22×D3—182no—
290C22×D3—366no—
291C22×D3—361no—
292C22×D3—364no—
293C22×D3—364no—
294C3×Q8—21no—
295C3×Q8—91no—
296C3×Q8C2×D3211yescharacteristic
297C3×Q8C22×C3⋊D311yescharacteristic
298C3×D4—31no—
299C3×D4—31no—
300C3×D4—61no—
301C3×D4—91no—
302C3×D4—92no—
303C3×D4—181no—
304C3×D4—181no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—61no—
309C2×C3×C4—91no—
310C2×C3×C4—91no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—123no—
314#67—61no—
315#67—182no—
316#67—182no—
317C2×Dic3—31no—
318D12—31no—
319D12—32no—
320D12—61no—
321D12—92no—
322D12—92no—
323D12—123no—
324D12—182no—
325D12—182no—
326D12—182no—
327D12—181no—
328D12—181no—
329D12—361no—
330D12—363no—
331D12—366no—
332D12—723no—
333C4×D3—31no—
334C4×D3—31no—
335C4×D3—31no—
336C4×D3C3⋊S411yescharacteristic
337Dic6—31no—
338SL(2,𝔽3)—23no—
339SL(2,𝔽3)C2×D3213yescomplemented
340C3×C8—92no—
341C3⋊C8—32no—
342C3⋊C8—92no—
343C3⋊C8—92no—
344C3⋊C8—182no—
345C2×C32—41no—
346C2×C32—43no—
347C2×C32—61no—
348C2×C32—83no—
349C2×C32—81no—
350C2×C32—122no—
351C2×C32C22×S411yescharacteristic
352C3⋊D3—82no—
353C3⋊D3—122no—
354C3⋊D3—242no—
355C3⋊D3—246no—
356C3⋊D3—482no—
357C3⋊D3—486no—
358C3×D3—46no—
359C3×D3—61no—
360C3×D3—122no—
361C3×D3—246no—
362C3×D3#98612yescomplemented
363C24—541no—
364#41—31no—
365#41—91no—
366#41—91no—
367#41D3×C3⋊D311yescharacteristic, complemented
368C2×Q8—31no—
369C2×D4—91no—
370C2×D4—91no—
371C2×D4—271no—
372C2×D4—272no—
373C2×D4—272no—
374C2×D4—541no—
375C2×D4—541no—
376C2×D4—541no—
377C22×C4—91no—
378SD16—92no—
379SD16—272no—
380SD16—272no—
381SD16—272no—
382D8—92no—
383D8—272no—
384D8—272no—
385D8—272no—
386M16—91no—
387M16—272no—
388M16—271no—
389C2×C8—272no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—61no—
394C22×C3—91no—
395C22×C3—123no—
396C22×C3—182no—
397C22×C3—182no—
398C22×C3—362no—
399C2×D3—31no—
400C2×D3—62no—
401C2×D3—62no—
402C2×D3—126no—
403C2×D3—122no—
404C2×D3—182no—
405C2×D3—182no—
406C2×D3—362no—
407C2×D3—366no—
408C2×D3—364no—
409C2×D3—364no—
410C2×D3—364no—
411C2×D3—362no—
412C2×D3—362no—
413C2×D3—364no—
414C2×D3—726no—
415C2×D3—7212no—
416C2×D3—722no—
417C2×D3C2×C3⋊S411yescharacteristic
418C3×C4—21no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—31no—
422C3×C4—43no—
423C3×C4—61no—
424C3×C4—83no—
425C3×C4—91no—
426C3×C4—123no—
427C3×C4D3×S411yescharacteristic
428C3×C4C2×C3⋊S411yescharacteristic
429Dic3—31no—
430Dic3C2×C3⋊S411yescharacteristic
431C32—41no—
432C32—43no—
433C32—83no—
434C32—81no—
435C32C2×#98611yescharacteristic, complemented
436C23—91no—
437C23—181no—
438C23—542no—
439C23—541no—
440C23—542no—
441C23—1084no—
442Q8—91no—
443Q8C2×D3×C3⋊D311yescharacteristic
444D4—31no—
445D4—91no—
446D4—92no—
447D4—181no—
448D4—181no—
449D4—272no—
450D4—542no—
451D4—542no—
452D4—541no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—91no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—91no—
459C8—92no—
460C8—272no—
461C2×C3—21no—
462C2×C3—32no—
463C2×C3—43no—
464C2×C3—61no—
465C2×C3—61no—
466C2×C3—83no—
467C2×C3—126no—
468C2×C3—121no—
469C2×C3—181no—
470C2×C3—362no—
471C2×C3C2×D3×S411yescharacteristic
472C2×C3C22×C3⋊S411yescharacteristic
473D3—61no—
474D3—122no—
475D3—246no—
476D3—362no—
477D3—362no—
478D3—722no—
479D3—726no—
480D3—1446no—
481D3#6426712yesdirect product factor
482C22—31no—
483C22—31no—
484C22—91no—
485C22—182no—
486C22—182no—
487C22—362no—
488C22—542no—
489C22—1084no—
490C22—1082no—
491C22—1084no—
492C4—31no—
493C4—31no—
494C4—91no—
495C4D3×C3⋊S411yescharacteristic
496C3—21no—
497C3—43no—
498C3—83no—
499C3D3×#98611yescharacteristic, complemented
500C3C2×#6426711yescharacteristic, complemented
501C2—32no—
502C2—61no—
503C2—181no—
504C2—362no—
505C2—1082no—
506C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×C3⋊S4
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FittingC32×#41C2×D3
socleC2×C32C22×S4
hypercenterC4D3×C3⋊S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×D3×#41⋊C3C2semidirect
2. #11724471C2semidirect
3. C3×#64267C2semidirect
4. D3×C3⋊GL(2,𝔽3)C2semidirect
5. #11724194C2semidirect
9. C32×#41⋊C3C22semidirect
11. C3×D3×SL(2,𝔽3)C22semidirect
12. C3×C3⋊(#41⋊C3)C22semidirect
13. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
14. C3×C3⋊GL(2,𝔽3)C22semidirect
15. C3×D3×#41D3semidirect
18. D3×#41⋊C3D3semidirect
24. #64267D3direct (N × H)
42. C32×SL(2,𝔽3)C23semidirect
48. C32×#41C2×D3semidirect
51. C3×#239C2×D3semidirect
61. C3×#41⋊C3C2×D3semidirect
62. C3×#41⋊C3C2×D3semidirect
70. D3×SL(2,𝔽3)C2×D3semidirect
71. C3⋊(#41⋊C3)C2×D3semidirect
76. C3⋊GL(2,𝔽3)C2×D3semidirect
97. D3×#41C3⋊D3semidirect
159. C3×SL(2,𝔽3)C22×D3semidirect
160. C3×SL(2,𝔽3)C22×D3semidirect
177. C3×#41D32semidirect
178. C3×#41C2×C3⋊D3semidirect
184. #239C2×C3⋊D3semidirect
206. #41⋊C3D32semidirect
339. SL(2,𝔽3)C2×D32semidirect
362. C3×D3#986semidirect
367. #41D3×C3⋊D3semidirect
435. C32C2×#986semidirect
481. D3#64267direct (N × H)
499. C3D3×#986semidirect
500. C3C2×#64267semidirect

Maximal subgroups

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

subgroupindex
C3×D3×#41⋊C32
#117244712
C3×#642672
D3×C3⋊GL(2,𝔽3)2
#117241942
C2×#642673
D3×#9863
D3×#9493
D3×C4⋊(C3⋊D3)4