Finite Group Databasebeta

← C2×C3×D3×#41⋊C3 #13181447

Subgroups — C2×C3×D3×#41⋊C3 #13181447

Order 1728.

1179 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1179 conjugacy classes of subgroups — too many to draw. Only its 196 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×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C3×D3×#41⋊C3C214yesmaximal, direct product factor
3C2×C3×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C3×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
5C2×C3×D3×#41C311yesmaximal, characteristic, complemented
6C22×C3×#41⋊C3—31nomaximal
7C2×D3×#41⋊C3C313yesmaximal, direct product factor
8C2×C32×C4×D3—41nomaximal
9C32×#41⋊C3C2212yescomplemented
10C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
11C3×D3×SL(2,𝔽3)C2212yescomplemented
12C3×C3⋊(#41⋊C3)C2212yescomplemented
13C2×C32×#41C2×C311yescharacteristic, complemented
14C3×D3×#41—34no—
15C3×D3×#41C2×C314yescomplemented
16C2×C3×#239C2×C311yescharacteristic, complemented
17C2×C3×D3×Q8C2×C311yescharacteristic
18C2×C3×#237—31no—
19C2×C3×D3×D4—31no—
20C2×C3×#235—31no—
21C22×C3×C4×D3—31no—
22C2×C3×#41⋊C3—21no—
23C2×C3×#41⋊C3—23no—
24C2×C3×#41⋊C3—34no—
25C2×C3×#41⋊C3—31no—
26C2×C3×#41⋊C3D311yescharacteristic, direct product factor
27C2×C3×#41⋊C3C2×C313yescomplemented
28C22×C3×SL(2,𝔽3)—31no—
29D3×#41⋊C3C2×C3112yesdirect product factor
30C2×D3×SL(2,𝔽3)C2×C313yescomplemented
31C2×C3⋊(#41⋊C3)C2×C313yescomplemented
32C22×C32×D3—41no—
33C2×C33×C4—41no—
34C2×C32×Dic3—41no—
35C32×C4×D3—44no—
36C32×SL(2,𝔽3)C2311yescharacteristic, complemented
37C22×C3×#41—31no—
38C2×D3×#41C3211yescharacteristic, complemented
39C22×#41⋊C3—33no—
40C23×C3×D3—31no—
41C32×#41—32no—
42C32×#41C22×C312yescomplemented
43C2×C32×Q8C22×C311yescharacteristic
44C2×C32×D4—31no—
45C22×C32×C4—31no—
46C22×C32×C4—121no—
47C2×C3×#67—32no—
48C22×C3×Dic3—31no—
49C3×#239—32no—
50C3×#239C22×C312yescomplemented
51C3×D3×Q8—32no—
52C3×D3×Q8C22×C312yes—
53C3×#237—34no—
54C3×#237—34no—
55C3×D3×D4—34no—
56C3×D3×D4—34no—
57C3×#235—34no—
58C3×#235—34no—
59C2×C3×D12—31no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—34no—
64C2×C3×C4×D3—34no—
65C2×C3×C4×D3—43no—
66C2×C3×C4×D3A411yescharacteristic
67C2×C3×Dic6—31no—
68C3×#41⋊C3—22no—
69C3×#41⋊C3—26no—
70C3×#41⋊C3—32no—
71C3×#41⋊C3C2×D312yesdirect product factor
72C3×#41⋊C3C22×C316yescomplemented
73C2×C3×SL(2,𝔽3)—21no—
74C2×C3×SL(2,𝔽3)—23no—
75C2×C3×SL(2,𝔽3)—32no—
76C2×C3×SL(2,𝔽3)C2×D311yescharacteristic, complemented
77C2×C3×SL(2,𝔽3)C22×C313yescomplemented
78D3×SL(2,𝔽3)C22×C316yescomplemented
79C3⋊(#41⋊C3)C22×C316yescomplemented
80C22×C33—41no—
81C2×C32×D3—44no—
82C2×C32×D3—42no—
83C33×C4—42no—
84C32×Dic3—42no—
85C2×C3×#41—21no—
86C2×C3×#41—34no—
87C2×C3×#41—31no—
88C2×C3×#41—91no—
89C2×C3×#41—91no—
90C2×C3×#41—94no—
91C2×C3×#41C3×D311yescharacteristic, complemented
92C2×C3×#41C2×C3211yescharacteristic, complemented
93C22×C3×Q8—31no—
94C22×C3×D4—91no—
95C23×C3×C4—91no—
96D3×#41—34no—
97D3×#41C2×C3214yescomplemented
98C2×#239C2×C3211yescharacteristic, complemented
99C2×D3×Q8C2×C3211yescharacteristic
100C2×#237—31no—
101C2×D3×D4—31no—
102C2×#235—31no—
103C22×C4×D3—31no—
104C2×#41⋊C3—23no—
105C2×#41⋊C3—312no—
106C2×#41⋊C3—33no—
107C2×#41⋊C3C3×D313yesdirect product factor
108C22×SL(2,𝔽3)—33no—
109C23×C32—31no—
110C23×C32—121no—
111C22×C3×D3—31no—
112C22×C3×D3—34no—
113C22×C3×D3—43no—
114C22×C3×D3—64no—
115C22×C3×D3C2×A411yescharacteristic
116C32×Q8—31no—
117C32×Q8C23×C311yescharacteristic
118C32×D4—32no—
119C32×D4—32no—
120C2×C32×C4—31no—
121C2×C32×C4—32no—
122C2×C32×C4—32no—
123C2×C32×C4—41no—
124C2×C32×C4—43no—
125C2×C32×C4—83no—
126C2×C32×C4—81no—
127C2×C32×C4—124no—
128C2×C32×C4—121no—
129C2×C32×C4C2×A411yescharacteristic
130C3×#67—38no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—43no—
134C2×C3×Dic3C2×A411yescharacteristic
135C3×D12—32no—
136C3×D12—32no—
137C3×C4×D3—32no—
138C3×C4×D3—32no—
139C3×C4×D3—34no—
140C3×C4×D3—32no—
141C3×C4×D3—32no—
142C3×C4×D3—412no—
143C3×C4×D3C2×A414yes—
144C3×Dic6—32no—
145C3×Dic6—32no—
146C3×SL(2,𝔽3)—21no—
147C3×SL(2,𝔽3)—23no—
148C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
149C3×SL(2,𝔽3)C23×C313yescomplemented
150C22×#41—31noSylow
151C2×C33—42no—
152C2×C33—41no—
153C32×D3—44no—
154C24×C3—91no—
155C23×D3—31no—
156C3×#41—22no—
157C3×#41—32no—
158C3×#41—32no—
159C3×#41—32no—
160C3×#41—62no—
161C3×#41—94no—
162C3×#41—94no—
163C3×#41—94no—
164C3×#41—94no—
165C3×#41—92no—
166C3×#41C2×C3×D312yescomplemented
167C3×#41C22×C3212yescomplemented
168C2×C3×Q8—21no—
169C2×C3×Q8—32no—
170C2×C3×Q8—91no—
171C2×C3×Q8—92no—
172C2×C3×Q8C2×C3×D311yescharacteristic
173C2×C3×Q8C22×C3211yescharacteristic
174C2×C3×D4—31no—
175C2×C3×D4—31no—
176C2×C3×D4—61no—
177C2×C3×D4—91no—
178C2×C3×D4—94no—
179C2×C3×D4—94no—
180C2×C3×D4—92no—
181C22×C3×C4—31no—
182C22×C3×C4—31no—
183C22×C3×C4—31no—
184C22×C3×C4—61no—
185C22×C3×C4—91no—
186C22×C3×C4—94no—
187C22×C3×C4—94no—
188C22×C3×C4—91no—
189C22×C3×C4—91no—
190C22×C3×C4—91no—
191C22×C3×C4—123no—
192C2×#67—32no—
193C22×Dic3—31no—
194#239—32no—
195#239C22×C3212yescomplemented
196D3×Q8—32no—
197D3×Q8C22×C3212yes—
198#237—34no—
199#237—34no—
200D3×D4—34no—
201D3×D4—34no—
202#235—34no—
203#235—34no—
204C2×D12—31no—
205C2×C4×D3—31no—
206C2×C4×D3—31no—
207C2×C4×D3—31no—
208C2×C4×D3—34no—
209C2×C4×D3—34no—
210C2×C4×D3C3×A411yescharacteristic
211C2×Dic6—31no—
212#41⋊C3—26no—
213#41⋊C3—36no—
214#41⋊C3C2×C3×D316yesdirect product factor
215C2×SL(2,𝔽3)—23no—
216C2×SL(2,𝔽3)—36no—
217C2×SL(2,𝔽3)C2×C3×D313yescomplemented
218C22×C32—32no—
219C22×C32—41no—
220C22×C32—43no—
221C22×C32—62no—
222C22×C32—83no—
223C22×C32—81no—
224C22×C32—122no—
225C22×C32—124no—
226C22×C32C22×A411yescharacteristic
227C2×C3×D3—32no—
228C2×C3×D3—46no—
229C2×C3×D3—412no—
230C2×C3×D3—62no—
231C2×C3×D3—68no—
232C2×C3×D3#41⋊C314yesdirect product factor
233C2×C3×D3C22×A412yes—
234C32×C4—31no—
235C32×C4—31no—
236C32×C4—42no—
237C32×C4—46no—
238C32×C4—86no—
239C32×C4—82no—
240C32×C4—122no—
241C32×C4C22×A412yes—
242C3×Dic3—32no—
243C3×Dic3—46no—
244C3×Dic3C22×A412yes—
245C2×#41—34no—
246C2×#41—31no—
247C2×#41—91no—
248C2×#41—91no—
249C2×#41—94no—
250C2×#41C32×D311yescharacteristic, complemented
251C22×Q8—31no—
252C22×D4—91no—
253C23×C4—91no—
254C33—41noSylow
255C23×C3—31no—
256C23×C3—31no—
257C23×C3—31no—
258C23×C3—61no—
259C23×C3—94no—
260C23×C3—91no—
261C23×C3—123no—
262C23×C3—184no—
263C22×D3—31no—
264C22×D3—34no—
265C22×D3—64no—
266C22×D3C2×C3×A411yescharacteristic
267C3×Q8—21no—
268C3×Q8—31no—
269C3×Q8—31no—
270C3×Q8—61no—
271C3×Q8—92no—
272C3×Q8—92no—
273C3×Q8C22×C3×D311yescharacteristic
274C3×Q8C23×C3211yescharacteristic
275C3×D4—32no—
276C3×D4—32no—
277C3×D4—32no—
278C3×D4—32no—
279C3×D4—62no—
280C3×D4—62no—
281C3×D4—92no—
282C3×D4—92no—
283C3×D4—98no—
284C2×C3×C4—21no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—34no—
289C2×C3×C4—32no—
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—32no—
293C2×C3×C4—43no—
294C2×C3×C4—62no—
295C2×C3×C4—62no—
296C2×C3×C4—61no—
297C2×C3×C4—83no—
298C2×C3×C4—92no—
299C2×C3×C4—92no—
300C2×C3×C4—92no—
301C2×C3×C4—92no—
302C2×C3×C4—94no—
303C2×C3×C4—91no—
304C2×C3×C4—94no—
305C2×C3×C4—1212no—
306C2×C3×C4—123no—
307C2×C3×C4D3×A411yescharacteristic
308C2×C3×C4C2×C3×A411yescharacteristic
309#67—38no—
310C2×Dic3—31no—
311C2×Dic3—34no—
312C2×Dic3C2×C3×A411yescharacteristic
313D12—32no—
314D12—32no—
315C4×D3—32no—
316C4×D3—32no—
317C4×D3—34no—
318C4×D3—32no—
319C4×D3—32no—
320C4×D3C2×C3×A414yes—
321Dic6—32no—
322Dic6—32no—
323SL(2,𝔽3)—23no—
324SL(2,𝔽3)C22×C3×D313yescomplemented
325C2×C32—42no—
326C2×C32—41no—
327C2×C32—46no—
328C2×C32—43no—
329C2×C32—62no—
330C2×C32—83no—
331C2×C32—86no—
332C2×C32—82no—
333C2×C32—81no—
334C2×C32—124no—
335C2×C32C2×#41⋊C312yescomplemented
336C2×C32C23×A411yescharacteristic
337C3×D3—412no—
338C3×D3—62no—
339C3×D3C2×#41⋊C314yesdirect product factor
340C24—91no—
341#41—32no—
342#41—32no—
343#41—94no—
344#41—94no—
345#41—94no—
346#41—94no—
347#41—92no—
348#41C2×C32×D312yescomplemented
349C2×Q8—32no—
350C2×Q8—91no—
351C2×Q8—92no—
352C2×Q8C2×C32×D311yescharacteristic
353C2×D4—31no—
354C2×D4—91no—
355C2×D4—94no—
356C2×D4—94no—
357C2×D4—92no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—91no—
361C22×C4—94no—
362C22×C4—94no—
363C22×C4—91no—
364C22×C4—91no—
365C22×C4—91no—
366C22×C3—21no—
367C22×C3—34no—
368C22×C3—32no—
369C22×C3—32no—
370C22×C3—32no—
371C22×C3—43no—
372C22×C3—62no—
373C22×C3—62no—
374C22×C3—62no—
375C22×C3—83no—
376C22×C3—92no—
377C22×C3—126no—
378C22×C3—1212no—
379C22×C3—122no—
380C22×C3—188no—
381C22×C3—182no—
382C22×C3C2×D3×A411yescharacteristic
383C22×C3C22×C3×A411yescharacteristic
384C2×D3—32no—
385C2×D3—62no—
386C2×D3—68no—
387C2×D3C3×#41⋊C314yesdirect product factor
388C2×D3C22×C3×A412yes—
389C3×C4—22no—
390C3×C4—31no—
391C3×C4—31no—
392C3×C4—32no—
393C3×C4—31no—
394C3×C4—31no—
395C3×C4—46no—
396C3×C4—61no—
397C3×C4—61no—
398C3×C4—86no—
399C3×C4—92no—
400C3×C4—126no—
401C3×C4C2×D3×A412yes—
402C3×C4C22×C3×A412yes—
403Dic3—32no—
404Dic3C22×C3×A412yes—
405C32—41no—
406C32—43no—
407C32—83no—
408C32—81no—
409C32C22×#41⋊C311yescharacteristic, complemented
410C23—31no—
411C23—31no—
412C23—94no—
413C23—91no—
414C23—184no—
415Q8—31no—
416Q8—92no—
417Q8—92no—
418Q8C22×C32×D311yescharacteristic
419D4—32no—
420D4—32no—
421D4—92no—
422D4—92no—
423D4—98no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—32no—
427C2×C4—32no—
428C2×C4—34no—
429C2×C4—92no—
430C2×C4—92no—
431C2×C4—92no—
432C2×C4—92no—
433C2×C4—94no—
434C2×C4—91no—
435C2×C4—94no—
436C2×C4C3×D3×A411yescharacteristic
437C2×C3—22no—
438C2×C3—21no—
439C2×C3—34no—
440C2×C3—46no—
441C2×C3—43no—
442C2×C3—62no—
443C2×C3—62no—
444C2×C3—86no—
445C2×C3—83no—
446C2×C3—1212no—
447C2×C3—122no—
448C2×C3—182no—
449C2×C3D3×#41⋊C312yesdirect product factor
450C2×C3C2×C3×#41⋊C312yescomplemented
451C2×C3C22×D3×A411yescharacteristic
452C2×C3C23×C3×A411yescharacteristic
453D3—62no—
454D3C2×C3×#41⋊C314yesdirect product factor
455C22—34no—
456C22—32no—
457C22—32no—
458C22—62no—
459C22—92no—
460C22—188no—
461C22—182no—
462C22C2×C3×D3×A411yescharacteristic
463C4—31no—
464C4—31no—
465C4—32no—
466C4—92no—
467C4C2×C3×D3×A412yes—
468C3—21no—
469C3—43no—
470C3—83no—
471C3C2×D3×#41⋊C311yescharacteristic, direct product factor
472C3C22×C3×#41⋊C311yescharacteristic, complemented
473C2—34no—
474C2—62no—
475C2—182no—
476C2C3×D3×#41⋊C312yesdirect product factor
477C2C22×C3×D3×A411yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C32
FrattiniC2C22×C3×D3×A4
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C3×D3×#41⋊C3C2direct (N × H)
3. C2×C3×D3×SL(2,𝔽3)C2semidirect
4. C2×C3×C3⋊(#41⋊C3)C2semidirect
5. C2×C3×D3×#41C3semidirect
7. C2×D3×#41⋊C3C3direct (N × H)
9. C32×#41⋊C3C22semidirect
10. C2×C32×SL(2,𝔽3)C22semidirect
11. C3×D3×SL(2,𝔽3)C22semidirect
12. C3×C3⋊(#41⋊C3)C22semidirect
13. C2×C32×#41C2×C3semidirect
15. C3×D3×#41C2×C3semidirect
16. C2×C3×#239C2×C3semidirect
26. C2×C3×#41⋊C3D3direct (N × H)
27. C2×C3×#41⋊C3C2×C3semidirect
29. D3×#41⋊C3C2×C3direct (N × H)
30. C2×D3×SL(2,𝔽3)C2×C3semidirect
31. C2×C3⋊(#41⋊C3)C2×C3semidirect
36. C32×SL(2,𝔽3)C23semidirect
38. C2×D3×#41C32semidirect
42. C32×#41C22×C3semidirect
50. C3×#239C22×C3semidirect
71. C3×#41⋊C3C2×D3direct (N × H)
72. C3×#41⋊C3C22×C3semidirect
76. C2×C3×SL(2,𝔽3)C2×D3semidirect
77. C2×C3×SL(2,𝔽3)C22×C3semidirect
78. D3×SL(2,𝔽3)C22×C3semidirect
79. C3⋊(#41⋊C3)C22×C3semidirect
91. C2×C3×#41C3×D3semidirect
92. C2×C3×#41C2×C32semidirect
97. D3×#41C2×C32semidirect
98. C2×#239C2×C32semidirect
107. C2×#41⋊C3C3×D3direct (N × H)
148. C3×SL(2,𝔽3)C22×D3semidirect
149. C3×SL(2,𝔽3)C23×C3semidirect
166. C3×#41C2×C3×D3semidirect
167. C3×#41C22×C32semidirect
195. #239C22×C32semidirect
214. #41⋊C3C2×C3×D3direct (N × H)
217. C2×SL(2,𝔽3)C2×C3×D3semidirect
232. C2×C3×D3#41⋊C3direct (N × H)
250. C2×#41C32×D3semidirect
324. SL(2,𝔽3)C22×C3×D3semidirect
335. C2×C32C2×#41⋊C3semidirect
339. C3×D3C2×#41⋊C3direct (N × H)
348. #41C2×C32×D3semidirect
387. C2×D3C3×#41⋊C3direct (N × H)
409. C32C22×#41⋊C3semidirect
449. C2×C3D3×#41⋊C3direct (N × H)
450. C2×C3C2×C3×#41⋊C3semidirect
454. D3C2×C3×#41⋊C3direct (N × H)
471. C3C2×D3×#41⋊C3direct (N × H)
472. C3C22×C3×#41⋊C3semidirect
476. C2C3×D3×#41⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3×D3×#41⋊C32
C2×C3×D3×SL(2,𝔽3)2
C2×C3×C3⋊(#41⋊C3)2
C2×C3×D3×#413
C22×C3×#41⋊C33
C2×D3×#41⋊C33
C2×C32×C4×D34