Finite Group Databasebeta

← C3×#10590394 #13157411

Subgroups — C3×#10590394 #13157411

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64306C211yesmaximal, characteristic
2C3×#64081C211yesmaximal, characteristic, complemented
3C3×#63818C211yesmaximal, characteristic, complemented
4C32×#805C212yesmaximal, complemented
5C3×#63568C212yesmaximal, complemented
6C3×#4821—32nomaximal
7#10590394C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C32×#235C2212yes—
10C3×#3811—21no—
11C3×#3805—21no—
12C3×#3802C411yescharacteristic, complemented
13C3×C4×D32—21no—
14C3×#3798—21no—
15C3×#3797C411yescharacteristic, complemented
16C3×#3796C412yescomplemented
17C33×C42C2211yescharacteristic, complemented
18C3×#3752—21no—
19C3×#3750—21no—
20C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
21C32×#208C2212yescomplemented
22C3×#3712—21no—
23C3×#3710—21no—
24C3×D3×#41—32no—
25#64306C2×C311yescharacteristic
26#64081C2×C311yescharacteristic, complemented
27C3×D3×M16—32no—
28C3×#872—32no—
29C3×#872—61no—
30#63818C2×C311yescharacteristic, complemented
31C32×#104—32no—
32C3×#837—32no—
33C3×#825—32no—
34C3×#805—22no—
35C3×#805D312yescomplemented
36C3×#805C2×C312yescomplemented
37#63568C2×C312yescomplemented
38C22×C3×C3⋊D3D411yescharacteristic
39C2×C3×D32—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4×C3⋊D3C2×C411yescharacteristic
45C3×C4×C3⋊D3C2×C411yescharacteristic
46C3×C4×C3⋊D3D411yescharacteristic
47C3×C4×C3⋊D3D411yescharacteristic
48C32×#67—22no—
49C32×D12C2×C412yes—
50C32×C4×D3—22no—
51C32×Dic6C2×C412yes—
52C3×#627—21no—
53C3×#626—22no—
54C3×#625—21no—
55C3×#624—21no—
56C3×D3×Dic3—22no—
57C32×C3⋊C8—22no—
58C3×#421—91no—
59#4821—32no—
60C32×#41—32no—
61C2×C4×C3⋊D3C22×C311yescharacteristic
62C3×#239—32no—
63C3×D3×Q8—32no—
64C3×#237—32no—
65C3×#237—62no—
66C3×D3×D4—32no—
67C3×D3×D4—62no—
68C3×#235—22no—
69C3×#235—62no—
70C3×#235C2×D312yes—
71C3×#235C22×C312yes—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—62no—
74C2×C3×C4×D3—61no—
75#3811—21no—
76#3805—21no—
77#3802C3×C411yescharacteristic, complemented
78C4×D32—21no—
79#3798—21no—
80#3797C3×C411yescharacteristic, complemented
81#3796C3×C412yescomplemented
82C32×M16—32no—
83C32×C42—21no—
84C32×C42—22no—
85C32×C42—41no—
86C32×C42C2×D312yescomplemented
87C32×C42C22×C311yescharacteristic, complemented
88#3752—21no—
89#3750—21no—
90C4×C3⋊Dic3C22×C311yescharacteristic, complemented
91C3×(C2×D3)⋊C4—62no—
92C3×(C2×D3)⋊C4—62no—
93C3×Dic3⋊C4—62no—
94C3×Dic3⋊C4—62no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—61no—
97C3×#208—22no—
98C3×#208C2×D312yescomplemented
99C3×#208C22×C312yescomplemented
100C3×ΓL(1,𝔽25)—62no—
101C3×D3×C8—62no—
102#3712—21no—
103#3710—21no—
104C22×C33C2×D411yescharacteristic
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3C22⋊C411yescharacteristic
107C2×C3×C3⋊D3C22⋊C411yescharacteristic
108C2×C32×D3—22no—
109C3×D32—42no—
110C33×C4—41no—
111C33×C4C22×C411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3×C3⋊Dic3—41no—
114C3×C3⋊Dic3C22⋊C411yescharacteristic
115C3×C3⋊Dic3C22⋊C411yescharacteristic
116C32×Dic3—22no—
117C2×C3×#41—91no—
118D3×#41—32no—
119C2×C3×M16—91no—
120C3×#117—91no—
121D3×M16—32no—
122#872—32no—
123#872—61no—
124C3×#104—32no—
125C3×#104—32no—
126C3×#104—62no—
127C3×#104—92no—
128#837—32no—
129#825—32no—
130#805C3×D312yescomplemented
131C22×C3⋊D3C3×D411yescharacteristic
132C22×C3×D3—32no—
133C22×C3×D3—61no—
134C22×C3×D3—62no—
135C2×D32—21no—
136C32×Q8—32no—
137C32×D4—32no—
138C32×D4—62no—
139C2×C32×C4—21no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—21no—
143C2×C32×C4—22no—
144C2×C32×C4—41no—
145C2×C32×C4—42no—
146C2×C32×C4—42no—
147C2×C32×C4—62no—
148C2×C32×C4C22×D312yes—
149C2×C32×C4C23×C311yescharacteristic
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3C3×D411yescharacteristic
152C4×C3⋊D3C2×C3×C411yescharacteristic
153C4×C3⋊D3C2×C3×C411yescharacteristic
154C4×C3⋊D3C3×D411yescharacteristic
155C4×C3⋊D3C3×D411yescharacteristic
156C3×#67—22no—
157C3×#67—22no—
158C3×#67—42no—
159C3×#67—62no—
160C3×#67—62no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C3×D12—22no—
167C3×D12—62no—
168C3×D12C4×D312yes—
169C3×D12C2×C3×C412yes—
170C3×C4×D3—22no—
171C3×C4×D3—22no—
172C3×C4×D3—32no—
173C3×C4×D3—32no—
174C3×C4×D3—32no—
175C3×C4×D3—32no—
176C3×C4×D3—42no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179C3×C4×D3—61no—
180C3×C4×D3—61no—
181C3×C4×D3—62no—
182C3×C4×D3—62no—
183C3×Dic6—22no—
184C3×Dic6—62no—
185C3×Dic6C4×D312yes—
186C3×Dic6C2×C3×C412yes—
187#627—21no—
188#626—22no—
189#625—21no—
190#624—21no—
191D3×Dic3—22no—
192C32×C8—62no—
193C3×C3⋊C8—22no—
194C3×C3⋊C8—22no—
195C3×C3⋊C8—42no—
196#421—91noSylow
197C2×C33—21no—
198C2×C33C2×C22⋊C411yescharacteristic
199C3×C3⋊D3—21no—
200C3×C3⋊D3—21no—
201C32×D3—42no—
202C3×#41—32no—
203C3×#41—32no—
204C3×#41—62no—
205C3×#41—92no—
206C3×#41—181no—
207C3×#41—181no—
208C2×C3×Q8—91no—
209C2×C3×D4—91no—
210C2×C3×D4—181no—
211C22×C3×C4—91no—
212C22×C3×C4—181no—
213#239—32no—
214D3×Q8—32no—
215#237—32no—
216#237—62no—
217D3×D4—32no—
218D3×D4—62no—
219#235—62no—
220#235C2×C3×D312yes—
221C2×C4×D3—32no—
222C2×C4×D3—61no—
223C2×C4×D3—62no—
224C3×M16—32no—
225C3×M16—32no—
226C3×M16—62no—
227C3×M16—181no—
228C2×C3×C8—181no—
229C3×C4⋊C4—181no—
230C3×C22⋊C4—181no—
231C3×C42—21no—
232C3×C42—22no—
233C3×C42—41no—
234C3×C42—91no—
235C3×C42D3211yescharacteristic, complemented
236C3×C42C2×C3×D312yescomplemented
237(C2×D3)⋊C4—62no—
238(C2×D3)⋊C4—62no—
239Dic3⋊C4—62no—
240Dic3⋊C4—62no—
241C4×Dic3—32no—
242C4×Dic3—61no—
243#208C2×C3×D312yescomplemented
244ΓL(1,𝔽25)—62no—
245D3×C8—62no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—41no—
249C22×C32—62no—
250C22×C32D3×D412yes—
251C22×C32C2×C3×D411yescharacteristic
252C2×C3⋊D3—41no—
253C2×C3⋊D3C3×C22⋊C411yescharacteristic
254C2×C3⋊D3C3×C22⋊C411yescharacteristic
255C2×C3×D3—22no—
256C2×C3×D3—22no—
257C2×C3×D3—32no—
258C2×C3×D3—32no—
259C2×C3×D3—42no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—124no—
263C2×C3×D3—122no—
264C2×C3×D3—122no—
265D32—42no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—22no—
269C32×C4—22no—
270C32×C4—41no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—42no—
274C32×C4—42no—
275C32×C4—44no—
276C32×C4—44no—
277C32×C4—62no—
278C32×C4C2×C4×D312yes—
279C32×C4D3×D412yes—
280C32×C4C22×C3×C411yescharacteristic
281C32×C4C2×C3×D411yescharacteristic
282C3⋊Dic3—41no—
283C3⋊Dic3C3×C22⋊C411yescharacteristic
284C3⋊Dic3C3×C22⋊C411yescharacteristic
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—32no—
288C3×Dic3—32no—
289C3×Dic3—42no—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3—122no—
293C3×Dic3—122no—
294C2×#41—91no—
295C2×M16—91no—
296#117—91no—
297#104—32no—
298#104—92no—
299C33#42111yescharacteristic, complemented, Sylow
300C23×C3—91no—
301C23×C3—181no—
302C22×D3—32no—
303C22×D3—61no—
304C22×D3—62no—
305C3×Q8—32no—
306C3×Q8—32no—
307C3×Q8—62no—
308C3×Q8—181no—
309C3×D4—32no—
310C3×D4—32no—
311C3×D4—62no—
312C3×D4—62no—
313C3×D4—62no—
314C3×D4—122no—
315C3×D4—182no—
316C3×D4—181no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—122no—
333C2×C3×C4—181no—
334C2×C3×C4—182no—
335C2×C3×C4—181no—
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C22×C3×D312yes—
338#67—22no—
339#67—62no—
340#67—62no—
341C2×Dic3—32no—
342C2×Dic3—61no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—62no—
346D12—62no—
347D12C3×C4×D312yes—
348C4×D3—22no—
349C4×D3—32no—
350C4×D3—32no—
351C4×D3—32no—
352C4×D3—32no—
353C4×D3—61no—
354C4×D3—61no—
355C4×D3—61no—
356C4×D3—61no—
357C4×D3—62no—
358C4×D3—62no—
359Dic6—62no—
360Dic6C3×C4×D312yes—
361C3×C8—62no—
362C3×C8—62no—
363C3×C8—122no—
364C3⋊C8—22no—
365C2×C32—21no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—22no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—42no—
372C2×C32—42no—
373C2×C32—122no—
374C2×C32D3×C22⋊C412yes—
375C2×C32C2×C3×C22⋊C411yescharacteristic
376C3⋊D3—21no—
377C3⋊D3—21no—
378C3×D3—42no—
379C3×D3—42no—
380C3×D3—62no—
381C3×D3—62no—
382C3×D3—82no—
383C3×D3—121no—
384C3×D3—121no—
385#41—32no—
386#41—92no—
387#41—181no—
388#41—181no—
389C2×Q8—91no—
390C2×D4—91no—
391C2×D4—181no—
392C22×C4—91no—
393C22×C4—181no—
394M16—32no—
395M16—181no—
396C2×C8—181no—
397C4⋊C4—181no—
398C22⋊C4—181no—
399C42—91no—
400C42C3×D3211yescharacteristic, complemented
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—91no—
407C22×C3—91no—
408C22×C3—122no—
409C22×C3—361no—
410C22×C3—362no—
411C22×C3#381111yescharacteristic
412C22×C3C3×D3×D412yes—
413C2×D3—22no—
414C2×D3—32no—
415C2×D3—32no—
416C2×D3—61no—
417C2×D3—61no—
418C2×D3—124no—
419C2×D3—122no—
420C2×D3—122no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—41no—
428C3×C4—44no—
429C3×C4—44no—
430C3×C4—41no—
431C3×C4—41no—
432C3×C4—62no—
433C3×C4—62no—
434C3×C4—91no—
435C3×C4—91no—
436C3×C4—122no—
437C3×C4—361no—
438C3×C4C4×D3211yescharacteristic
439C3×C4#380211yescharacteristic
440C3×C4C2×C3×C4×D312yes—
441C3×C4C3×D3×D412yes—
442Dic3—22no—
443Dic3—32no—
444Dic3—32no—
445Dic3—61no—
446Dic3—61no—
447Dic3—122no—
448Dic3—122no—
449C32—22no—
450C32—21no—
451C32—41no—
452C32#482112yescomplemented
453C32C3×#42111yescharacteristic, complemented
454C23—91no—
455C23—181no—
456Q8—32no—
457Q8—181no—
458D4—32no—
459D4—62no—
460D4—182no—
461D4—181no—
462C2×C4—21no—
463C2×C4—62no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—91no—
468C2×C4—91no—
469C2×C4—181no—
470C2×C4—182no—
471C2×C4—181no—
472C2×C4C2×C3×D3211yescharacteristic
473C8—62no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—22no—
479C2×C3—42no—
480C2×C3—42no—
481C2×C3—41no—
482C2×C3—122no—
483C2×C3—122no—
484C2×C3—181no—
485C2×C3—181no—
486C2×C3—242no—
487C2×C3#6392111yescharacteristic
488C2×C3C3×D3×C22⋊C412yes—
489D3—42no—
490D3—62no—
491D3—62no—
492D3—121no—
493D3—121no—
494C22—62no—
495C22—91no—
496C22—91no—
497C22—361no—
498C22—362no—
499C22C3×#381111yescharacteristic
500C4—41no—
501C4—62no—
502C4—91no—
503C4—91no—
504C4—361no—
505C4C3×C4×D3211yescharacteristic
506C4C3×#380211yescharacteristic
507C3—21no—
508C3—22no—
509C3—41no—
510C3#1059039411yescharacteristic, direct product factor
511C3C3×#482112yescomplemented
512C2—21no—
513C2—122no—
514C2—181no—
515C2—181no—
516C2C3×#6392111yescharacteristic
subgroupG/H
centerC3×C4#3802
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC2×C33C2×C22⋊C4
prime radicalC3×#64306C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#421
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C3×#64081C2semidirect
3. C3×#63818C2semidirect
4. C32×#805C2semidirect
5. C3×#63568C2semidirect
7. #10590394C3direct (N × H)
12. C3×#3802C4semidirect
15. C3×#3797C4semidirect
16. C3×#3796C4semidirect
17. C33×C42C22semidirect
20. C3×C4×C3⋊Dic3C22semidirect
21. C32×#208C22semidirect
26. #64081C2×C3semidirect
30. #63818C2×C3semidirect
35. C3×#805D3semidirect
36. C3×#805C2×C3semidirect
37. #63568C2×C3semidirect
77. #3802C3×C4semidirect
80. #3797C3×C4semidirect
81. #3796C3×C4semidirect
86. C32×C42C2×D3semidirect
87. C32×C42C22×C3semidirect
90. C4×C3⋊Dic3C22×C3semidirect
98. C3×#208C2×D3semidirect
99. C3×#208C22×C3semidirect
130. #805C3×D3semidirect
235. C3×C42D32semidirect
236. C3×C42C2×C3×D3semidirect
243. #208C2×C3×D3semidirect
299. C33#421semidirect
400. C42C3×D32semidirect
452. C32#4821semidirect
453. C32C3×#421semidirect
510. C3#10590394direct (N × H)
511. C3C3×#4821semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
C3×#640812
C3×#638182
C32×#8052
C3×#635682
C3×#48213
#105903943