Finite Group Databasebeta

← C3×#10590862 #13157825

Subgroups — C3×#10590862 #13157825

Order 1728.

722 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3744C211yesmaximal, characteristic, complemented
2C2×C32×#204C211yesmaximal, characteristic, complemented
3C3×#63864C211yesmaximal, characteristic, complemented
4C3×#63860C211yesmaximal, characteristic, complemented
5C2×C3×#3719C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic
7C32×C8×Dic3C211yesmaximal, characteristic, complemented
8C3×#5168—31nomaximal
9C3×#4696—31nomaximal
10#10590862C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C33×C8C2211yescharacteristic
16C3×#3744—22no—
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C32×#204—22no—
20C3×#3722—22no—
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3719—22no—
23C3×#3714—22no—
24C2×C32×SD16—31no—
25C2×#3744C2×C311yescharacteristic, complemented
26C3×#947—31no—
27C2×C3×#216—31no—
28C3×#936—31no—
29C2×C3×#213—31no—
30C2×C3×Dic12—31no—
31C2×C3×D24—31no—
32C2×C3×#204—21no—
33C2×C3×#204—31no—
34C2×C3×#204—31no—
35C2×C3×#204—61no—
36C2×C3×#204D311yescharacteristic, complemented
37C2×C3×#204C2×C311yescharacteristic, complemented
38C3×#875—31no—
39C3×#875—31no—
40#63864C2×C311yescharacteristic, complemented
41#63860C2×C311yescharacteristic, complemented
42C2×#3719C2×C311yescharacteristic, complemented
43C2×#3714C2×C311yescharacteristic
44C32×C4×C8—31no—
45C3×C8×Dic3—21no—
46C3×C8×Dic3D311yescharacteristic, complemented
47C3×C8×Dic3C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C33×C8D412yes—
63C32×C3⋊C8D412yes—
64C3×#495—91no—
65#5168—31no—
66#4696—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C22×C311yescharacteristic
77C2×C3×Dic6—21no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C22×C311yescharacteristic
83C32×SD16—62no—
84C2×C32×C8—21no—
85C2×C32×C8—21no—
86C2×C32×C8—21no—
87C2×C32×C8—31no—
88C2×C32×C8—41no—
89C2×C32×C8C2×D311yescharacteristic
90C2×C32×C8C2×D311yescharacteristic
91C2×C32×C8C22×C311yescharacteristic
92C32×C42—31no—
93#3744—22no—
94C3×#217—62no—
95C3×#216—62no—
96C3×#213—62no—
97C3×(C2×D3)⋊C4—62no—
98C3×(C2×D3)⋊C4—62no—
99C3×(C2×D3)⋊C4—62no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3C2×D311yescharacteristic
102C3×C4×Dic3C22×C311yescharacteristic
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106C3×Dic12—62no—
107C3×D24—62no—
108C3×#204—24no—
109C3×#204—22no—
110C3×#204—22no—
111C3×#204—64no—
112C3×#204—62no—
113C3×#204—62no—
114#3722—22no—
115(C2×D3)⋊Dic3—22no—
116#3719—22no—
117#3714—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—42no—
121C2×C32×D3—41no—
122C2×C32×D3—42no—
123C33×C4C2×D411yescharacteristic
124C33×C4C2×D411yescharacteristic
125C3×C3⋊Dic3—41no—
126C32×Dic3—22no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×SD16—31no—
130C2×C3×SD16—31no—
131C2×C3×SD16—61no—
132C2×C3×SD16—91no—
133C2×C3×D8—91no—
134C3×#124—91no—
135C3×#124—91no—
136#947—31no—
137C2×#216—31no—
138#936—31no—
139C2×#213—31no—
140C2×Dic12—31no—
141C2×D24—31no—
142C2×#204—31no—
143C2×#204—31no—
144C2×#204—61no—
145C2×#204C3×D311yescharacteristic, complemented
146#875—31no—
147#875—31no—
148C3×C4×C8—31no—
149C3×C4×C8—31no—
150C3×C4×C8—61no—
151C8×Dic3C3×D311yescharacteristic, complemented
152C23×C32—61no—
153C22×C3⋊D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—21no—
156C22×C3×D3—41no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—121no—
160C32×Q8—61no—
161C32×Q8—61no—
162C32×D4—61no—
163C32×D4—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—32no—
168C2×C32×C4—41no—
169C2×C32×C4—61no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C23×C311yescharacteristic
173C2×C3⋊Dic3—21no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—21no—
176#634—21no—
177#634—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—41no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—121no—
185C2×C3×Dic3D1212yes—
186C2×C3×Dic3C3×D412yes—
187C3×D12—22no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—41no—
193C3×D12—62no—
194C3×D12—61no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—121no—
199C3×Dic6—22no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—41no—
205C3×Dic6—62no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—121no—
211C32×C8—22no—
212C32×C8—22no—
213C32×C8—22no—
214C32×C8—32no—
215C32×C8—42no—
216C32×C8D1212yes—
217C32×C8#6712yes—
218C32×C8C3×D412yes—
219C3×C3⋊C8—22no—
220C3×C3⋊C8D1212yes—
221C3×C3⋊C8C3×D412yes—
222#495—91noSylow
223C2×C33#12711yescharacteristic
224C2×C33#13512yes—
225C3×C3⋊D3—81no—
226C32×D3—81no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—61no—
230C2×C3×Q8—91no—
231C2×C3×D4—31no—
232C2×C3×D4—31no—
233C2×C3×D4—61no—
234C2×C3×D4—91no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×D12C2×C3×D311yescharacteristic
239C2×Dic6—31no—
240C2×Dic6—31no—
241C2×Dic6—61no—
242C2×Dic6C2×C3×D311yescharacteristic
243C3×Q16—182no—
244C3×SD16—64no—
245C3×SD16—62no—
246C3×SD16—62no—
247C3×SD16—182no—
248C3×D8—182no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—31no—
253C2×C3×C8—31no—
254C2×C3×C8—41no—
255C2×C3×C8—61no—
256C2×C3×C8D3211yescharacteristic
257C2×C3×C8C2×C3×D311yescharacteristic
258C2×C3×C8C2×C3×D311yescharacteristic
259C3×C22⋊C4—182no—
260C3×C22⋊C4—182no—
261C3×C42—31no—
262C3×C42—31no—
263C3×C42—61no—
264#217—62no—
265#216—62no—
266#213—62no—
267(C2×D3)⋊C4—62no—
268(C2×D3)⋊C4—62no—
269(C2×D3)⋊C4—62no—
270C4×Dic3C2×C3×D311yescharacteristic
271C2×C3⋊C8C2×C3×D311yescharacteristic
272Dic12—62no—
273D24—62no—
274#204—22no—
275#204—64no—
276#204—62no—
277#204—62no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—41no—
282C22×C32—121no—
283C22×C32—122no—
284C22×C32C2×D1211yescharacteristic
285C22×C32D3×D411yescharacteristic
286C22×C32C2×C3×D411yescharacteristic
287C2×C3⋊D3—41no—
288C2×C3⋊D3—42no—
289C2×C3×D3—42no—
290C2×C3×D3—42no—
291C2×C3×D3—41no—
292C2×C3×D3—41no—
293C2×C3×D3—42no—
294C2×C3×D3—82no—
295C2×C3×D3—122no—
296C2×C3×D3—121no—
297C2×C3×D3—122no—
298C2×C3×D3—122no—
299C2×C3×D3—121no—
300C2×C3×D3—242no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—41no—
308C32×C4—41no—
309C32×C4—62no—
310C32×C4—121no—
311C32×C4C2×D1211yescharacteristic
312C32×C4C2×D1211yescharacteristic
313C32×C4D3×D411yescharacteristic
314C32×C4C2×#6711yescharacteristic
315C32×C4C2×C3×D411yescharacteristic
316C32×C4C2×C3×D411yescharacteristic
317C3⋊Dic3—41no—
318C3×Dic3—24no—
319C3×Dic3—22no—
320C3×Dic3—22no—
321C3×Dic3—42no—
322C3×Dic3—41no—
323C3×Dic3—41no—
324C3×Dic3—122no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C2×Q16—91no—
328C2×SD16—31no—
329C2×SD16—91no—
330C2×D8—91no—
331#124—91no—
332#124—91no—
333C4×C8—31no—
334C33#49511yescharacteristic, complemented, Sylow
335C23×C3—61no—
336C23×C3—61no—
337C23×C3—121no—
338C23×C3—181no—
339C22×D3—21no—
340C22×D3—61no—
341C22×D3—61no—
342C22×D3—121no—
343C3×Q8—62no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—121no—
349C3×Q8—181no—
350C3×Q8—181no—
351C3×D4—62no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—61no—
355C3×D4—61no—
356C3×D4—121no—
357C3×D4—181no—
358C3×D4—181no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—32no—
363C2×C3×C4—32no—
364C2×C3×C4—41no—
365C2×C3×C4—62no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—121no—
369C2×C3×C4—181no—
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C22×C3×D311yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×Dic3—21no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—121no—
377C2×Dic3C3×D1212yes—
378D12—21no—
379D12—21no—
380D12—62no—
381D12—61no—
382D12—61no—
383D12—61no—
384D12—61no—
385D12—121no—
386Dic6—21no—
387Dic6—21no—
388Dic6—62no—
389Dic6—61no—
390Dic6—61no—
391Dic6—61no—
392Dic6—61no—
393Dic6—121no—
394C3×C8—22no—
395C3×C8—22no—
396C3×C8—22no—
397C3×C8—32no—
398C3×C8—32no—
399C3×C8—42no—
400C3×C8—62no—
401C3×C8#62612yes—
402C3×C8C3×D1212yes—
403C3×C8C3×#6712yes—
404C3⋊C8C3×D1212yes—
405C2×C32—22no—
406C2×C32—22no—
407C2×C32—22no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—41no—
412C2×C32—42no—
413C2×C32—241no—
414C2×C32#87411yescharacteristic
415C2×C32#90412yes—
416C2×C32#91612yes—
417C2×C32#94011yescharacteristic
418C2×C32C3×#12711yescharacteristic
419C2×C32C3×#13512yes—
420C3⋊D3—81no—
421C3×D3—82no—
422C3×D3—81no—
423C3×D3—81no—
424C3×D3—242no—
425C3×D3—241no—
426C3×D3—241no—
427C2×Q8—31no—
428C2×Q8—91no—
429C2×D4—31no—
430C2×D4—91no—
431Q16—182no—
432SD16—62no—
433SD16—182no—
434D8—182no—
435C2×C8—31no—
436C2×C8C3×D3211yescharacteristic
437C22⋊C4—182no—
438C22⋊C4—182no—
439C42—31no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—41no—
444C22×C3—122no—
445C22×C3—122no—
446C22×C3—122no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—242no—
450C22×C3—362no—
451C22×C3—361no—
452C22×C3D3×D1211yescharacteristic
453C22×C3C2×C3×D1211yescharacteristic
454C22×C3C3×D3×D411yescharacteristic
455C2×D3—42no—
456C2×D3—41no—
457C2×D3—122no—
458C2×D3—122no—
459C2×D3—122no—
460C2×D3—121no—
461C2×D3—121no—
462C2×D3—242no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—64no—
472C3×C4—62no—
473C3×C4—62no—
474C3×C4—122no—
475C3×C4—121no—
476C3×C4—121no—
477C3×C4—361no—
478C3×C4D3×D1211yescharacteristic
479C3×C4C2×#62611yescharacteristic
480C3×C4C2×C3×D1211yescharacteristic
481C3×C4C2×C3×D1211yescharacteristic
482C3×C4C3×D3×D411yescharacteristic
483C3×C4C2×C3×#6711yescharacteristic
484Dic3—22no—
485Dic3—41no—
486Dic3—122no—
487Dic3—121no—
488Dic3—121no—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#469611yescharacteristic, complemented
494C32#516811yescharacteristic, complemented
495C32C3×#49511yescharacteristic, complemented
496C23—61no—
497C23—181no—
498Q8—61no—
499Q8—61no—
500Q8—181no—
501Q8—181no—
502D4—61no—
503D4—61no—
504D4—181no—
505D4—181no—
506C2×C4—32no—
507C2×C4—61no—
508C2×C4—181no—
509C2×C4C2×C3×D3211yescharacteristic
510C8—32no—
511C8C3×#62612yes—
512C2×C3—22no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—42no—
519C2×C3—41no—
520C2×C3—242no—
521C2×C3—241no—
522C2×C3—241no—
523C2×C3—721no—
524C2×C3#6380612yes—
525C2×C3#6391111yescharacteristic
526C2×C3C3×#87411yescharacteristic
527C2×C3C3×#90412yes—
528C2×C3C3×#91612yes—
529C2×C3C3×#94011yescharacteristic
530D3—81no—
531D3—242no—
532D3—241no—
533D3—241no—
534C22—122no—
535C22—121no—
536C22—362no—
537C22—361no—
538C22C3×D3×D1211yescharacteristic
539C4—62no—
540C4—121no—
541C4—361no—
542C4C3×D3×D1211yescharacteristic
543C4C2×C3×#62611yescharacteristic
544C3—21no—
545C3—21no—
546C3—21no—
547C3—41no—
548C3#1059086211yescharacteristic, direct product factor
549C3C3×#469611yescharacteristic, complemented
550C3C3×#516811yescharacteristic, complemented
551C2—241no—
552C2—721no—
553C2C3×#6380612yes—
554C2C3×#6391111yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypercenterC2×C3×C8D32
hypocenterC32C3×#495
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3744C2semidirect
2. C2×C32×#204C2semidirect
3. C3×#63864C2semidirect
4. C3×#63860C2semidirect
5. C2×C3×#3719C2semidirect
7. C32×C8×Dic3C2semidirect
10. #10590862C3direct (N × H)
25. C2×#3744C2×C3semidirect
36. C2×C3×#204D3semidirect
37. C2×C3×#204C2×C3semidirect
40. #63864C2×C3semidirect
41. #63860C2×C3semidirect
42. C2×#3719C2×C3semidirect
46. C3×C8×Dic3D3semidirect
47. C3×C8×Dic3C2×C3semidirect
145. C2×#204C3×D3semidirect
151. C8×Dic3C3×D3semidirect
334. C33#495semidirect
493. C32#4696semidirect
494. C32#5168semidirect
495. C32C3×#495semidirect
548. C3#10590862direct (N × H)
549. C3C3×#4696semidirect
550. C3C3×#5168semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
C2×C32×#2042
C3×#638642
C3×#638602
C2×C3×#37192
C2×C3×#37142
C32×C8×Dic32
C3×#51683
C3×#46963
#105908623