Finite Group Databasebeta

← C2×D3×C3⋊(#41⋊C3) #13180889

Subgroups — C2×D3×C3⋊(#41⋊C3) #13180889

Order 1728.

1011 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 112 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⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
2C2×C3×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
4D3×C3⋊(#41⋊C3)C214yesmaximal, direct product factor
5C2×D3×#239C311yesmaximal, characteristic, complemented
6C2×D3×#41⋊C3—31nomaximal
7C22×C3⋊(#41⋊C3)—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
9C2×C3×D3×Dic3—41nomaximal
10C3⋊(C3⋊(#41⋊C3))C2212yescomplemented
11C3×D3×SL(2,𝔽3)C2212yescomplemented
12C3×C3⋊(#41⋊C3)C2212yescomplemented
13C2×#3832C2×C311yescharacteristic, complemented
14C2×C3×#239C2×C311yescharacteristic, complemented
15C2×C3×D3×Q8C2×C311yescharacteristic
16C2×C3×#41⋊C3—31no—
17C22×C3×SL(2,𝔽3)—31no—
18D3×#239—34no—
19D3×#239C2×C314yescomplemented
20C2×D3×D12—31no—
21C2×C4×D32—31no—
22C2×#3799—31no—
23C2×#3796—31no—
24D3×#41⋊C3—34no—
25C2×D3×SL(2,𝔽3)—21no—
26C2×D3×SL(2,𝔽3)D311yescharacteristic, complemented
27C2×C3⋊(#41⋊C3)—21no—
28C2×C3⋊(#41⋊C3)—31no—
29C2×C3⋊(#41⋊C3)—31no—
30C2×C3⋊(#41⋊C3)—34no—
31C2×C3⋊(#41⋊C3)—61no—
32C2×C3⋊(#41⋊C3)D311yescharacteristic, direct product factor
33C22×C32×D3—41no—
34C2×C3×C3⋊Dic3—41no—
35C2×C32×Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic, complemented
37C3×D3×Dic3—44no—
38C2×D3×#41—31no—
39C22×#239—31no—
40C22×#41⋊C3—91no—
41C22×D32—31no—
42C2×C32×Q8C22×C311yescharacteristic
43#3832—32no—
44#3832C22×C312yescomplemented
45C2×C4⋊(C3⋊D3)—31no—
46C2×C4×C3⋊D3—31no—
47C22×C3×Dic3—121no—
48C3×#239—32no—
49C3×#239C22×C312yescomplemented
50C3×D3×Q8—32no—
51C3×D3×Q8C22×C312yes—
52C2×C3×D12—31no—
53C2×C3×C4×D3—31no—
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—121no—
56C2×C3×Dic6—31no—
57C3×#41⋊C3—32no—
58C2×C3×SL(2,𝔽3)—21no—
59C2×C3×SL(2,𝔽3)—21no—
60C2×C3×SL(2,𝔽3)—21no—
61C2×C3×SL(2,𝔽3)—32no—
62C2×C3×SL(2,𝔽3)—41no—
63C2×C3×SL(2,𝔽3)C2×D311yescharacteristic, complemented
64C2×C3×SL(2,𝔽3)C2×D311yescharacteristic, complemented
65C2×#626—32no—
66C2×#624—31no—
67C2×D3×Dic3—31no—
68C2×D3×Dic3A411yescharacteristic
69D3×D12—34no—
70D3×D12—34no—
71C4×D32—34no—
72C4×D32—34no—
73#3799—34no—
74#3799—34no—
75#3796—34no—
76#3796—34no—
77D3×SL(2,𝔽3)—22no—
78D3×SL(2,𝔽3)C2×D312yescomplemented
79C3⋊(#41⋊C3)—22no—
80C3⋊(#41⋊C3)—32no—
81C3⋊(#41⋊C3)—32no—
82C3⋊(#41⋊C3)—62no—
83C3⋊(#41⋊C3)C2×D312yesdirect product factor
84C22×C33—41no—
85C2×C32×D3—44no—
86C2×C32×D3—42no—
87C3×C3⋊Dic3—42no—
88C32×Dic3—42no—
89C2×C3×#41—31no—
90C22×C3×Q8—31no—
91D3×#41—34no—
92D3×#41—94no—
93C2×#239—31no—
94C2×#239—31no—
95C2×#239—34no—
96C2×#239—61no—
97C2×#239—91no—
98C2×#239—91no—
99C2×#239—94no—
100C2×#239C3×D311yescharacteristic, complemented
101C2×D3×Q8C3×D311yescharacteristic
102C2×#237—91no—
103C2×D3×D4—91no—
104C2×#235—91no—
105C22×D12—91no—
106C22×C4×D3—91no—
107C22×C4×D3—91no—
108C2×#41⋊C3—31no—
109C2×#41⋊C3—61no—
110C2×#41⋊C3—94no—
111C2×#41⋊C3—91no—
112C22×SL(2,𝔽3)—31no—
113C22×SL(2,𝔽3)—61no—
114C23×C32—121no—
115C22×C3⋊D3—31no—
116C22×C3×D3—31no—
117C22×C3×D3—41no—
118C22×C3×D3—81no—
119C22×C3×D3C2×A411yescharacteristic
120C2×D32—34no—
121C2×D32—64no—
122C32×Q8—31no—
123C32×Q8C23×C311yescharacteristic
124C2×C32×C4—31no—
125C2×C32×C4—121no—
126C2×C3⋊Dic3C2×A411yescharacteristic
127C4⋊(C3⋊D3)—32no—
128C4⋊(C3⋊D3)—32no—
129C4×C3⋊D3—32no—
130C4×C3⋊D3—32no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—81no—
134C2×C3×Dic3—124no—
135C2×C3×Dic3—121no—
136C2×C3×Dic3—121no—
137C2×C3×Dic3—241no—
138C2×C3×Dic3C2×A411yescharacteristic
139C3×D12—32no—
140C3×D12—32no—
141C3×C4×D3—32no—
142C3×C4×D3—32no—
143C3×C4×D3—32no—
144C3×C4×D3—32no—
145C3×C4×D3—124no—
146C3×Dic6—32no—
147C3×Dic6—32no—
148C3×SL(2,𝔽3)—21no—
149C3×SL(2,𝔽3)—21no—
150C3×SL(2,𝔽3)—21no—
151C3×SL(2,𝔽3)—41no—
152C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
153C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
154#626—38no—
155#624—34no—
156D3×Dic3—34no—
157D3×Dic3C2×A414yes—
158C22×#41—91noSylow
159C2×C33—42no—
160C2×C33—41no—
161C32×D3—44no—
162C23×D3—91no—
163C23×D3—91no—
164C3×#41—32no—
165C3×#41—92no—
166C2×C3×Q8—21no—
167C2×C3×Q8—32no—
168C2×C3×Q8—91no—
169C2×C3×Q8—92no—
170C2×C3×Q8C2×C3×D311yescharacteristic
171C2×C3×Q8C2×C3×D311yescharacteristic
172C2×C3×D4—91no—
173C22×C3×C4—91no—
174C22×C3×C4—91no—
175C22×C3×C4—361no—
176C2×#67—92no—
177C22×Dic3—31no—
178C22×Dic3—91no—
179#239—32no—
180#239—32no—
181#239—32no—
182#239—62no—
183#239—94no—
184#239—94no—
185#239—94no—
186#239—94no—
187#239—92no—
188#239—92no—
189#239—182no—
190#239C2×C3×D312yescomplemented
191D3×Q8—32no—
192D3×Q8C2×C3×D312yes—
193#237—94no—
194#237—94no—
195D3×D4—94no—
196D3×D4—94no—
197#235—94no—
198#235—94no—
199C2×D12—31no—
200C2×D12—94no—
201C2×D12—94no—
202C2×D12—91no—
203C2×D12—91no—
204C2×D12—92no—
205C2×D12—181no—
206C2×C4×D3—31no—
207C2×C4×D3—31no—
208C2×C4×D3—31no—
209C2×C4×D3—94no—
210C2×C4×D3—94no—
211C2×C4×D3—94no—
212C2×C4×D3—94no—
213C2×C4×D3—91no—
214C2×C4×D3—91no—
215C2×C4×D3—91no—
216C2×C4×D3—91no—
217C2×C4×D3—91no—
218C2×C4×D3—181no—
219C2×Dic6—31no—
220#41⋊C3—32no—
221#41⋊C3—62no—
222#41⋊C3—92no—
223C2×SL(2,𝔽3)—21no—
224C2×SL(2,𝔽3)—21no—
225C2×SL(2,𝔽3)—32no—
226C2×SL(2,𝔽3)—41no—
227C2×SL(2,𝔽3)—62no—
228C2×SL(2,𝔽3)D3211yescharacteristic, complemented
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—81no—
232C22×C32—81no—
233C22×C32—81no—
234C22×C32—122no—
235C22×C32—124no—
236C22×C32—161no—
237C22×C32C22×A411yescharacteristic
238C2×C3⋊D3—32no—
239C2×C3⋊D3—62no—
240C2×C3×D3—32no—
241C2×C3×D3—42no—
242C2×C3×D3—44no—
243C2×C3×D3—62no—
244C2×C3×D3—82no—
245C2×C3×D3—84no—
246C2×C3×D3#41⋊C314yescomplemented
247C2×C3×D3C22×A412yes—
248D32—68no—
249C32×C4—31no—
250C32×C4—31no—
251C32×C4—122no—
252C3⋊Dic3C22×A412yes—
253C3×Dic3—32no—
254C3×Dic3—42no—
255C3×Dic3—82no—
256C3×Dic3—122no—
257C3×Dic3—122no—
258C3×Dic3—242no—
259C3×Dic3C22×A412yes—
260C2×#41—31no—
261C2×#41—94no—
262C2×#41—91no—
263C2×#41—271no—
264C2×#41—271no—
265C2×#41—274no—
266C22×Q8—31no—
267C22×D4—271no—
268C23×C4—271no—
269C33—41noSylow
270C23×C3—31no—
271C23×C3—91no—
272C23×C3—121no—
273C23×C3—241no—
274C22×D3—31no—
275C22×D3—94no—
276C22×D3—91no—
277C22×D3—91no—
278C22×D3—94no—
279C22×D3—181no—
280C22×D3—184no—
281C22×D3—184no—
282C22×D3D3×A411yescharacteristic
283C3×Q8—21no—
284C3×Q8—31no—
285C3×Q8—31no—
286C3×Q8—61no—
287C3×Q8—92no—
288C3×Q8—92no—
289C3×Q8C22×C3×D311yescharacteristic
290C3×Q8C22×C3×D311yescharacteristic
291C3×D4—92no—
292C3×D4—92no—
293C2×C3×C4—31no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—61no—
297C2×C3×C4—92no—
298C2×C3×C4—92no—
299C2×C3×C4—92no—
300C2×C3×C4—92no—
301C2×C3×C4—91no—
302C2×C3×C4—121no—
303C2×C3×C4—241no—
304C2×C3×C4—364no—
305C2×C3×C4—361no—
306#67—98no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—31no—
310C2×Dic3—34no—
311C2×Dic3—61no—
312C2×Dic3—94no—
313C2×Dic3D3×A411yescharacteristic
314D12—32no—
315D12—32no—
316D12—92no—
317D12—92no—
318D12—92no—
319D12—92no—
320D12—98no—
321D12—182no—
322D12—182no—
323C4×D3—34no—
324C4×D3—32no—
325C4×D3—32no—
326C4×D3—32no—
327C4×D3—32no—
328C4×D3—94no—
329C4×D3—92no—
330C4×D3—92no—
331C4×D3—92no—
332C4×D3—92no—
333C4×D3—94no—
334C4×D3—94no—
335C4×D3—182no—
336C4×D3—182no—
337Dic6—32no—
338Dic6—32no—
339SL(2,𝔽3)—21no—
340SL(2,𝔽3)—21no—
341SL(2,𝔽3)—41no—
342SL(2,𝔽3)C2×D3211yescharacteristic, complemented
343C2×C32—42no—
344C2×C32—42no—
345C2×C32—41no—
346C2×C32—41no—
347C2×C32—81no—
348C2×C32—82no—
349C2×C32—82no—
350C2×C32—82no—
351C2×C32—81no—
352C2×C32—81no—
353C2×C32—124no—
354C2×C32—161no—
355C2×C32—162no—
356C2×C32C2×#41⋊C312yescomplemented
357C2×C32C23×A411yescharacteristic
358C3⋊D3—62no—
359C3×D3—44no—
360C3×D3—62no—
361C3×D3—84no—
362C3×D3C2×#41⋊C314yescomplemented
363C24—271no—
364#41—32no—
365#41—92no—
366#41—92no—
367#41—274no—
368#41—274no—
369#41—274no—
370#41—274no—
371#41—272no—
372C2×Q8—32no—
373C2×Q8—91no—
374C2×Q8—92no—
375C2×Q8C3×D3211yescharacteristic
376C2×D4—91no—
377C2×D4—271no—
378C2×D4—274no—
379C2×D4—274no—
380C2×D4—272no—
381C22×C4—91no—
382C22×C4—91no—
383C22×C4—91no—
384C22×C4—271no—
385C22×C4—271no—
386C22×C4—274no—
387C22×C4—274no—
388C22×C4—271no—
389C22×C3—21no—
390C22×C3—34no—
391C22×C3—32no—
392C22×C3—41no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—92no—
396C22×C3—122no—
397C22×C3—124no—
398C22×C3—161no—
399C22×C3—182no—
400C22×C3—244no—
401C22×C3—242no—
402C22×C3C2×D3×A411yescharacteristic
403C22×C3C2×D3×A411yescharacteristic
404C2×D3—32no—
405C2×D3—62no—
406C2×D3—92no—
407C2×D3—92no—
408C2×D3—182no—
409C2×D3—188no—
410C2×D3—182no—
411C2×D3—182no—
412C2×D3—188no—
413C2×D3—362no—
414C2×D3C3⋊(#41⋊C3)14yesdirect product factor
415C2×D3C2×D3×A412yes—
416C3×C4—31no—
417C3×C4—31no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—32no—
421C3×C4—61no—
422C3×C4—61no—
423C3×C4—92no—
424C3×C4—122no—
425C3×C4—242no—
426C3×C4—362no—
427Dic3—32no—
428Dic3—32no—
429Dic3—32no—
430Dic3—62no—
431Dic3C2×D3×A412yes—
432C32—41no—
433C32—41no—
434C32—81no—
435C32—81no—
436C32—81no—
437C32—161no—
438C32C22×#41⋊C311yescharacteristic, complemented
439C23—31no—
440C23—91no—
441C23—274no—
442C23—271no—
443C23—544no—
444Q8—31no—
445Q8—92no—
446Q8—92no—
447Q8C2×C3×D3211yescharacteristic
448D4—92no—
449D4—92no—
450D4—272no—
451D4—272no—
452D4—278no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—92no—
456C2×C4—92no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—94no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—272no—
463C2×C4—274no—
464C2×C4—272no—
465C2×C4—274no—
466C2×C3—21no—
467C2×C3—22no—
468C2×C3—34no—
469C2×C3—41no—
470C2×C3—42no—
471C2×C3—81no—
472C2×C3—81no—
473C2×C3—82no—
474C2×C3—82no—
475C2×C3—124no—
476C2×C3—161no—
477C2×C3—162no—
478C2×C3—182no—
479C2×C3—244no—
480C2×C3C2×C3⋊(#41⋊C3)12yescomplemented
481C2×C3D3×#41⋊C312yescomplemented
482C2×C3C22×D3×A411yescharacteristic
483C2×C3C22×D3×A411yescharacteristic
484D3—62no—
485D3—182no—
486D3—182no—
487D3—362no—
488D3C2×C3⋊(#41⋊C3)14yesdirect product factor
489C22—32no—
490C22—34no—
491C22—92no—
492C22—182no—
493C22—272no—
494C22—548no—
495C22—542no—
496C22D32×A411yescharacteristic
497C4—31no—
498C4—31no—
499C4—32no—
500C4—92no—
501C4—92no—
502C3—21no—
503C3—41no—
504C3—81no—
505C3—81no—
506C3—161no—
507C3C22×C3⋊(#41⋊C3)11yescharacteristic, complemented
508C3C2×D3×#41⋊C311yescharacteristic, complemented
509C2—34no—
510C2—182no—
511C2—542no—
512C2D3×C3⋊(#41⋊C3)12yesdirect product factor
513C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC22D32×A4
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C2×D32×A4
FittingC2×C32×Q8C22×C3
socleC22×C32C22×A4
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊(C3⋊(#41⋊C3))C2semidirect
2. C2×C3×D3×SL(2,𝔽3)C2semidirect
3. C2×C3×C3⋊(#41⋊C3)C2semidirect
4. D3×C3⋊(#41⋊C3)C2direct (N × H)
5. C2×D3×#239C3semidirect
8. C2×C32×SL(2,𝔽3)C22semidirect
10. C3⋊(C3⋊(#41⋊C3))C22semidirect
11. C3×D3×SL(2,𝔽3)C22semidirect
12. C3×C3⋊(#41⋊C3)C22semidirect
13. C2×#3832C2×C3semidirect
14. C2×C3×#239C2×C3semidirect
19. D3×#239C2×C3semidirect
26. C2×D3×SL(2,𝔽3)D3semidirect
32. C2×C3⋊(#41⋊C3)D3direct (N × H)
36. C32×SL(2,𝔽3)C23semidirect
44. #3832C22×C3semidirect
49. C3×#239C22×C3semidirect
63. C2×C3×SL(2,𝔽3)C2×D3semidirect
64. C2×C3×SL(2,𝔽3)C2×D3semidirect
78. D3×SL(2,𝔽3)C2×D3semidirect
83. C3⋊(#41⋊C3)C2×D3direct (N × H)
100. C2×#239C3×D3semidirect
152. C3×SL(2,𝔽3)C22×D3semidirect
153. C3×SL(2,𝔽3)C22×D3semidirect
190. #239C2×C3×D3semidirect
228. C2×SL(2,𝔽3)D32semidirect
246. C2×C3×D3#41⋊C3semidirect
342. SL(2,𝔽3)C2×D32semidirect
356. C2×C32C2×#41⋊C3semidirect
362. C3×D3C2×#41⋊C3semidirect
414. C2×D3C3⋊(#41⋊C3)direct (N × H)
438. C32C22×#41⋊C3semidirect
480. C2×C3C2×C3⋊(#41⋊C3)semidirect
481. C2×C3D3×#41⋊C3semidirect
488. D3C2×C3⋊(#41⋊C3)direct (N × H)
507. C3C22×C3⋊(#41⋊C3)semidirect
508. C3C2×D3×#41⋊C3semidirect
512. C2D3×C3⋊(#41⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊(#41⋊C3))2
C2×C3×D3×SL(2,𝔽3)2
C2×C3×C3⋊(#41⋊C3)2
D3×C3⋊(#41⋊C3)2
C2×D3×#2393
C2×D3×#41⋊C33
C22×C3⋊(#41⋊C3)3
C2×C3×D3×Dic34