Finite Group Databasebeta

← C3×#68778 #11788560

Subgroups — C3×#68778 #11788560

Order 1152.

719 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5748C211yesmaximal, characteristic
2C3×C4×#235C211yesmaximal, characteristic, complemented
3C3×#5006C211yesmaximal, characteristic, complemented
4C3×#5005C211yesmaximal, characteristic, complemented
5C2×C3×#805C211yesmaximal, characteristic, complemented
6C32×#339C211yesmaximal, characteristic, complemented
7C3×#4559C211yesmaximal, characteristic, complemented
8C3×#1906—31nomaximal
9#68778C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C2×C32×C42C2211yescharacteristic
15C3×C4×#67—22no—
16C3×#926C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18C3×#909—21no—
19C3×#909C412yescomplemented
20C3×#908—21no—
21C3×#908C412yescomplemented
22C3×#904—22no—
23C3×C4×D12—21no—
24C3×#872—21no—
25C3×C42×D3—21no—
26C3×C4×Dic6—21no—
27C3×#806—22no—
28C3×#805—22no—
29C3×#805—21no—
30C3×#805—21no—
31C3×#575—31no—
32C3×C4×#41—31no—
33#5748C2×C311yescharacteristic
34C4×#235C2×C311yescharacteristic, complemented
35C3×#421—31no—
36C2×C3×#104—31no—
37C3×#410—31no—
38#5006C2×C311yescharacteristic, complemented
39#5005C2×C311yescharacteristic, complemented
40C2×#805C2×C311yescharacteristic, complemented
41C3×#344—31no—
42C3×#339—21no—
43C3×#339D311yescharacteristic, complemented
44C3×#339C2×C311yescharacteristic, complemented
45#4559C2×C311yescharacteristic, complemented
46C22×C32×C4—21no—
47C22×C32×C4C2311yescharacteristic
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C3×#235—21no—
51C3×#235—21no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C3×#235—21no—
56C3×#235C2×C412yes—
57C3×#235D412yes—
58C2×C3×D12C2×C411yescharacteristic, complemented
59C2×C3×D12D411yescharacteristic
60C2×C3×C4×D3—21no—
61C2×C3×C4×D3—22no—
62C2×C3×C4×D3—21no—
63C2×C3×Dic6C2×C411yescharacteristic, complemented
64C2×C3×Dic6D411yescharacteristic
65C32×M16—21no—
66C32×M16C2×C412yescomplemented
67C2×C32×C8—21no—
68C32×C4⋊C4—21no—
69C32×C22⋊C4—21no—
70C32×C42—21no—
71C32×C42—21no—
72C32×C42D411yescharacteristic
73C32×C42D411yescharacteristic
74C3×#217D412yes—
75C3×(C2×D3)⋊C4—22no—
76C3×C4⋊Dic3—21no—
77C3×Dic3⋊C4—22no—
78C3×C4×Dic3—22no—
79C3×C4×Dic3—21no—
80C3×#208—21no—
81C3×#208—21no—
82C2×C3×C3⋊C8—21no—
83C3×Dic12—22no—
84C3×D24—22no—
85C3×#204—22no—
86C3×#204—22no—
87#1906—31noSylow
88C2×C3×#41—31no—
89C2×C3×#41—31no—
90C2×#235C22×C311yescharacteristic
91C2×#235C22×C311yescharacteristic
92C3×#137—32no—
93C3×#137—61no—
94C3×AGL(1,ℤ/8)—32no—
95C3×AGL(1,ℤ/8)—61no—
96C3×#135—62no—
97C2×C3×M16—21no—
98C2×C3×M16—31no—
99C2×C3×M16C2×D311yescharacteristic
100C2×C3×M16C22×C311yescharacteristic
101C3×C4×Q8—61no—
102C3×C4×D4—61no—
103C3×C4×D4—62no—
104C3×#117—21no—
105C3×#117—31no—
106C3×#117—61no—
107C3×#117C2×D311yescharacteristic
108C3×#117C22×C311yescharacteristic
109C2×C3×C42—21no—
110C2×C3×C42—61no—
111C2×C3×C42C2×D311yescharacteristic
112C2×C3×C42C22×C311yescharacteristic
113C4×#67—22no—
114#926C22×C311yescharacteristic
115C2×#208C22×C311yescharacteristic
116#909—21no—
117#909C3×C412yescomplemented
118#908—21no—
119#908C3×C412yescomplemented
120#904—22no—
121C4×D12—21no—
122#872—21no—
123C42×D3—21no—
124C4×Dic6—21no—
125C3×#104—62no—
126C3×#104—61no—
127C3×#104—61no—
128C3×#99—62no—
129#806—22no—
130#805—22no—
131#805—21no—
132#805—21no—
133C23×C32C2×D411yescharacteristic
134C22×C3×D3—21no—
135C22×C3×D3—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—42no—
139C2×C32×C4—41no—
140C2×C32×C4C22×C411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4#4111yescharacteristic
145C2×C32×C4#4111yescharacteristic
146C3×#67—22no—
147C3×#67—22no—
148C3×#67—41no—
149C3×#67—41no—
150C2×C3×Dic3—21no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—22no—
153C3×D12—21no—
154C3×D12—21no—
155C3×D12—21no—
156C3×D12C22⋊C412yes—
157C3×C4×D3—22no—
158C3×C4×D3—22no—
159C3×C4×D3—44no—
160C3×C4×D3—41no—
161C3×C4×D3—41no—
162C3×Dic6—21no—
163C3×Dic6—21no—
164C3×Dic6—21no—
165C3×Dic6C22⋊C412yes—
166C32×C8—22no—
167C3×C3⋊C8—41no—
168#575—31no—
169C4×#41—31no—
170#421—31no—
171C2×#104—31no—
172#410—31no—
173#344—31no—
174#339C3×D311yescharacteristic, complemented
175C3×#41—32no—
176C3×#41—32no—
177C3×#41—61no—
178C3×#41—61no—
179C3×#41—61no—
180C3×#41—61no—
181C3×#41—61no—
182C3×#41—61no—
183C2×C3×Q8—31no—
184C2×C3×Q8—31no—
185C2×C3×D4—31no—
186C2×C3×D4—31no—
187C2×C3×D4—61no—
188C2×C3×D4—61no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—41no—
193C22×C3×C4—62no—
194C22×C3×C4—61no—
195C22×C3×C4—61no—
196C22×C3×C4C22×D311yescharacteristic
197C22×C3×C4C23×C311yescharacteristic
198C2×#67—21no—
199C2×#67—21no—
200#235—21no—
201#235—21no—
202#235—21no—
203#235—21no—
204#235—21no—
205#235—21no—
206#235C2×C3×C412yes—
207#235C3×D412yes—
208C2×D12C2×C3×C411yescharacteristic, complemented
209C2×D12C3×D411yescharacteristic
210C2×C4×D3—21no—
211C2×C4×D3—22no—
212C2×C4×D3—21no—
213C2×Dic6C2×C3×C411yescharacteristic, complemented
214C2×Dic6C3×D411yescharacteristic
215C3×Q16—62no—
216C3×SD16—62no—
217C3×SD16—62no—
218C3×D8—62no—
219C3×M16—21no—
220C3×M16—21no—
221C3×M16—22no—
222C3×M16—41no—
223C3×M16—61no—
224C3×M16—61no—
225C3×M16C4×D312yescomplemented
226C3×M16C2×C3×C412yescomplemented
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—41no—
230C2×C3×C8—61no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—62no—
235C3×C4⋊C4—61no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—21no—
238C3×C22⋊C4—32no—
239C3×C22⋊C4—41no—
240C3×C22⋊C4—62no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—41no—
248C3×C42—41no—
249C3×C42—62no—
250C3×C42—61no—
251C3×C42D1211yescharacteristic
252C3×C42D1211yescharacteristic
253C3×C42C3×D411yescharacteristic
254C3×C42C3×D411yescharacteristic
255#217C3×D412yes—
256(C2×D3)⋊C4—22no—
257C4⋊Dic3—21no—
258Dic3⋊C4—22no—
259C4×Dic3—22no—
260C4×Dic3—21no—
261#208—21no—
262#208—21no—
263C2×C3⋊C8—21no—
264Dic12—22no—
265D24—22no—
266#204—22no—
267#204—22no—
268C22×C32—41no—
269C22×C32C4×D411yescharacteristic
270C22×C32#12011yescharacteristic
271C22×C32#12111yescharacteristic
272C2×C3×D3—22no—
273C2×C3×D3—42no—
274C2×C3×D3—41no—
275C2×C3×D3—81no—
276C32×C4—22no—
277C32×C4—41no—
278C32×C4—81no—
279C32×C4C2×C22⋊C411yescharacteristic
280C32×C4C4×D411yescharacteristic
281C32×C4#12111yescharacteristic
282C32×C4#12311yescharacteristic
283C3×Dic3—22no—
284C3×Dic3—42no—
285C3×Dic3—41no—
286C2×#41—31no—
287C2×#41—31no—
288#137—32no—
289#137—61no—
290AGL(1,ℤ/8)—32no—
291AGL(1,ℤ/8)—61no—
292#135—62no—
293C2×M16—31no—
294C2×M16C2×C3×D311yescharacteristic
295C4×Q8—61no—
296C4×D4—61no—
297C4×D4—62no—
298#117—31no—
299#117—61no—
300#117C2×C3×D311yescharacteristic
301C2×C42—61no—
302C2×C42C2×C3×D311yescharacteristic
303#104—62no—
304#104—61no—
305#104—61no—
306#99—62no—
307C23×C3—21no—
308C23×C3—61no—
309C23×C3—61no—
310C23×C3D3×D411yescharacteristic
311C23×C3C2×C3×D411yescharacteristic
312C22×D3—21no—
313C22×D3—21no—
314C3×Q8—32no—
315C3×Q8—61no—
316C3×Q8—61no—
317C3×Q8—61no—
318C3×D4—32no—
319C3×D4—62no—
320C3×D4—62no—
321C3×D4—61no—
322C3×D4—61no—
323C3×D4—61no—
324C3×D4—121no—
325C3×D4—121no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—44no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—61no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—61no—
349C2×C3×C4—124no—
350C2×C3×C4—121no—
351C2×C3×C4—121no—
352C2×C3×C4C2×C4×D311yescharacteristic
353C2×C3×C4C2×D1211yescharacteristic
354C2×C3×C4#23511yescharacteristic
355C2×C3×C4D3×D411yescharacteristic
356C2×C3×C4D3×D411yescharacteristic
357C2×C3×C4#23911yescharacteristic
358C2×C3×C4C22×C3×C411yescharacteristic
359C2×C3×C4C2×C3×D411yescharacteristic
360C2×C3×C4C2×C3×D411yescharacteristic
361C2×C3×C4C2×C3×D411yescharacteristic
362C2×C3×C4C3×#4111yescharacteristic
363C2×C3×C4C3×#4111yescharacteristic
364#67—22no—
365#67—22no—
366#67—41no—
367#67—41no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371D12—21no—
372D12—21no—
373D12—21no—
374D12C3×C22⋊C412yes—
375C4×D3—22no—
376C4×D3—22no—
377C4×D3—44no—
378C4×D3—41no—
379C4×D3—41no—
380Dic6—21no—
381Dic6—21no—
382Dic6—21no—
383Dic6C3×C22⋊C412yes—
384C3×C8—22no—
385C3×C8—22no—
386C3×C8—42no—
387C3×C8—121no—
388C3⋊C8—41no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32#38611yescharacteristic
393C3×D3—42no—
394C3×D3—81no—
395#41—32no—
396#41—32no—
397#41—61no—
398#41—61no—
399#41—61no—
400#41—61no—
401#41—61no—
402#41—61no—
403C2×Q8—31no—
404C2×Q8—31no—
405C2×D4—31no—
406C2×D4—31no—
407C2×D4—61no—
408C2×D4—61no—
409C22×C4—21no—
410C22×C4—62no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4C22×C3×D311yescharacteristic
414Q16—62no—
415SD16—62no—
416SD16—62no—
417D8—62no—
418M16—21no—
419M16—61no—
420M16—61no—
421M16C3×C4×D312yescomplemented
422C2×C8—21no—
423C2×C8—61no—
424C4⋊C4—21no—
425C4⋊C4—62no—
426C4⋊C4—61no—
427C22⋊C4—21no—
428C22⋊C4—32no—
429C22⋊C4—62no—
430C42—21no—
431C42—21no—
432C42—62no—
433C42—61no—
434C42C3×D1211yescharacteristic
435C42C3×D1211yescharacteristic
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—62no—
443C22×C3—122no—
444C22×C3—121no—
445C22×C3—241no—
446C22×C3#88211yescharacteristic
447C22×C3#88411yescharacteristic
448C22×C3#89311yescharacteristic
449C22×C3C3×C4×D411yescharacteristic
450C22×C3C3×#12011yescharacteristic
451C22×C3C3×#12111yescharacteristic
452C2×D3—22no—
453C2×D3—42no—
454C2×D3—41no—
455C2×D3—81no—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—42no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—42no—
466C3×C4—62no—
467C3×C4—82no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4—122no—
471C3×C4—121no—
472C3×C4C4×D1211yescharacteristic
473C3×C4D3×C22⋊C411yescharacteristic
474C3×C4#89411yescharacteristic
475C3×C4#89511yescharacteristic
476C3×C4C2×C3×C22⋊C411yescharacteristic
477C3×C4C3×C4×D411yescharacteristic
478C3×C4C3×#12111yescharacteristic
479C3×C4C3×#12311yescharacteristic
480Dic3—22no—
481Dic3—42no—
482Dic3—41no—
483C32#190611yescharacteristic, complemented, Sylow
484C23—61no—
485C23—61no—
486C23C3×D3×D411yescharacteristic
487Q8—32no—
488Q8—61no—
489Q8—61no—
490Q8—61no—
491D4—32no—
492D4—62no—
493D4—62no—
494D4—61no—
495D4—61no—
496D4—61no—
497D4—121no—
498D4—121no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—61no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—61no—
508C2×C4—124no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4C2×C3×C4×D311yescharacteristic
512C2×C4C2×C3×D1211yescharacteristic
513C2×C4C3×#23511yescharacteristic
514C2×C4C3×D3×D411yescharacteristic
515C2×C4C3×D3×D411yescharacteristic
516C2×C4C3×#23911yescharacteristic
517C8—22no—
518C8—121no—
519C2×C3—22no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—122no—
530C2×C3—241no—
531C2×C3#466511yescharacteristic
532C2×C3C3×#38611yescharacteristic
533D3—42no—
534D3—81no—
535C22—41no—
536C22—62no—
537C22—122no—
538C22—121no—
539C22—241no—
540C22C3×#88211yescharacteristic
541C22C3×#88411yescharacteristic
542C22C3×#89311yescharacteristic
543C4—22no—
544C4—41no—
545C4—62no—
546C4—81no—
547C4—122no—
548C4—121no—
549C4C3×C4×D1211yescharacteristic
550C4C3×D3×C22⋊C411yescharacteristic
551C4C3×#89411yescharacteristic
552C4C3×#89511yescharacteristic
553C3—21no—
554C3#6877811yescharacteristic, direct product factor
555C3C3×#190611yescharacteristic, complemented
556C2—21no—
557C2—21no—
558C2—21no—
559C2—122no—
560C2—241no—
561C2C3×#466511yescharacteristic
subgroupG/H
centerC3×C4#895
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#339C2
socleC2×C32#386
prime radicalC3×#5748C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#339D3
hypocenterC3C3×#1906
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#339

Complements

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

NH ≅ G/Ntype
2. C3×C4×#235C2semidirect
3. C3×#5006C2semidirect
4. C3×#5005C2semidirect
5. C2×C3×#805C2semidirect
6. C32×#339C2semidirect
7. C3×#4559C2semidirect
9. #68778C3direct (N × H)
19. C3×#909C4semidirect
21. C3×#908C4semidirect
34. C4×#235C2×C3semidirect
38. #5006C2×C3semidirect
39. #5005C2×C3semidirect
40. C2×#805C2×C3semidirect
43. C3×#339D3semidirect
44. C3×#339C2×C3semidirect
45. #4559C2×C3semidirect
58. C2×C3×D12C2×C4semidirect
63. C2×C3×Dic6C2×C4semidirect
66. C32×M16C2×C4semidirect
117. #909C3×C4semidirect
119. #908C3×C4semidirect
174. #339C3×D3semidirect
208. C2×D12C2×C3×C4semidirect
213. C2×Dic6C2×C3×C4semidirect
225. C3×M16C4×D3semidirect
226. C3×M16C2×C3×C4semidirect
421. M16C3×C4×D3semidirect
483. C32#1906semidirect
554. C3#68778direct (N × H)
555. C3C3×#1906semidirect

Maximal subgroups

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

subgroupindex
C3×#57482
C3×C4×#2352
C3×#50062
C3×#50052
C2×C3×#8052
C32×#3392
C3×#45592
C3×#19063
#687783