Finite Group Databasebeta

← C3×#10590870 #13157831

Subgroups — C3×#10590870 #13157831

Order 1728.

686 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×Dic12C211yesmaximal, characteristic, complemented
2C2×C32×#204C211yesmaximal, characteristic
3C3×#63873C211yesmaximal, characteristic, complemented
4C3×#63863C211yesmaximal, characteristic
5C2×C3×#3718C211yesmaximal, characteristic, complemented
6C2×C3×#3715C211yesmaximal, characteristic
7C3×#63642C211yesmaximal, characteristic, complemented
8C3×#5190—31nomaximal
9C3×#5167—31nomaximal
10#10590870C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C33×C8C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×Dic12—22no—
19C32×#204—22no—
20C3×#3724—22no—
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3718—22no—
23C3×#3715—22no—
24C2×C32×Q16—31no—
25C2×C32×SD16—31no—
26C3×#947—31no—
27C3×#944—31no—
28C3×#944—31no—
29C2×C3×#216—31no—
30C2×C3×#216—31no—
31C2×C3×#215—31no—
32C3×#936—31no—
33C2×C3×#214—31no—
34C2×C3×Dic12—21no—
35C2×C3×Dic12D311yescharacteristic, complemented
36C2×C3×Dic12C2×C311yescharacteristic, complemented
37C2×C3×#204—21no—
38C2×C3×#204D311yescharacteristic
39C2×C3×#204C2×C311yescharacteristic
40#63873C2×C311yescharacteristic, complemented
41#63863C2×C311yescharacteristic
42C2×#3718C2×C311yescharacteristic, complemented
43C2×#3715C2×C311yescharacteristic
44#63642C2×C311yescharacteristic, complemented
45C3×#815—31no—
46C3×#815—31no—
47C3×#815—61no—
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3D412yes—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C32×D12—21no—
55C32×D12—21no—
56C32×Dic6—21no—
57C32×Dic6—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C33×C8D412yes—
63C3×C3⋊(C3⋊C8)D412yes—
64C3×#497—91no—
65#5190—31no—
66#5167—31no—
67C2×C32×Q8—31no—
68C2×C32×Q8—31no—
69C2×C32×Q8—31no—
70C2×C32×D4—31no—
71C2×C3×D12—21no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×D12C22×C311yescharacteristic
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6C2×D311yescharacteristic
78C2×C3×Dic6C2×D311yescharacteristic
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C2×C3×Dic6C22×C311yescharacteristic
82C2×C3×Dic6C22×C311yescharacteristic
83C32×Q16—62no—
84C32×SD16—62no—
85C2×C32×C8—21no—
86C2×C32×C8—21no—
87C2×C32×C8—21no—
88C2×C32×C8—41no—
89C2×C32×C8C2×D311yescharacteristic
90C2×C32×C8C2×D311yescharacteristic
91C2×C32×C8C22×C311yescharacteristic
92C4×C3⋊Dic3C22×C311yescharacteristic
93C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
94C3×#217—62no—
95C3×#216—62no—
96C3×#216—62no—
97C3×#215—62no—
98C3×#214—62no—
99C3×(C2×D3)⋊C4—62no—
100C3×Dic3⋊C4—62no—
101C3×Dic3⋊C4—62no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—61no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C3×Dic12—24no—
109C3×Dic12—22no—
110C3×Dic12—22no—
111C3×#204—24no—
112C3×#204—22no—
113C3×#204—22no—
114#3724—22no—
115(C2×D3)⋊Dic3—22no—
116#3718—22no—
117#3715—22no—
118C22×C33C2×D411yescharacteristic
119C2×C32×D3—41no—
120C2×C32×D3—42no—
121C33×C4C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C3×C3⋊Dic3—22no—
124C32×Dic3—41no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C2×C3×Q16—31no—
128C2×C3×Q16—31no—
129C2×C3×Q16—61no—
130C2×C3×Q16—91no—
131C2×C3×SD16—31no—
132C2×C3×SD16—31no—
133C2×C3×SD16—61no—
134C2×C3×SD16—91no—
135C3×C4⋊Q8—91no—
136C3×#124—91no—
137#947—31no—
138#944—31no—
139#944—31no—
140C2×#216—31no—
141C2×#216—31no—
142C2×#215—31no—
143#936—31no—
144C2×#214—31no—
145C2×Dic12C3×D311yescharacteristic, complemented
146C2×#204C3×D311yescharacteristic
147C3×#97—91no—
148#815—31no—
149#815—31no—
150#815—61no—
151C23×C32—61no—
152C22×C3×D3—21no—
153C22×C3×D3—21no—
154C22×C3×D3—41no—
155C32×Q8—61no—
156C32×Q8—61no—
157C32×Q8—61no—
158C32×Q8—61no—
159C32×Q8—61no—
160C32×Q8—61no—
161C32×D4—61no—
162C32×D4—61no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4—61no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C23×C311yescharacteristic
173C2×C3⋊Dic3C3×D412yes—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3—62no—
186C3×D12—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—41no—
192C3×Dic6—22no—
193C3×Dic6—22no—
194C3×Dic6—22no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—41no—
208C3×Dic6—41no—
209C3×Dic6—41no—
210C32×C8—22no—
211C32×C8—22no—
212C32×C8—22no—
213C32×C8—42no—
214C32×C8#6712yes—
215C32×C8#6712yes—
216C32×C8C3×D412yes—
217C3⋊(C3⋊C8)C3×D412yes—
218C3×C3⋊C8—32no—
219C3×C3⋊C8—32no—
220C3×C3⋊C8—62no—
221#497—91noSylow
222C2×C33#12711yescharacteristic
223C2×C33#13712yes—
224C32×D3—81no—
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—31no—
230C2×C3×Q8—31no—
231C2×C3×Q8—61no—
232C2×C3×Q8—61no—
233C2×C3×Q8—61no—
234C2×C3×D4—31no—
235C2×C3×D4—31no—
236C2×C3×D4—61no—
237C2×D12C2×C3×D311yescharacteristic
238C2×Dic6C2×C3×D311yescharacteristic
239C2×Dic6C2×C3×D311yescharacteristic
240C2×Dic6C2×C3×D311yescharacteristic
241C3×Q16—64no—
242C3×Q16—62no—
243C3×Q16—62no—
244C3×Q16—182no—
245C3×SD16—64no—
246C3×SD16—62no—
247C3×SD16—62no—
248C3×SD16—182no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—41no—
253C2×C3×C8—91no—
254C2×C3×C8D3211yescharacteristic
255C2×C3×C8C2×C3×D311yescharacteristic
256C2×C3×C8C2×C3×D311yescharacteristic
257C3×C4⋊C4—182no—
258C3×C22⋊C4—182no—
259C3×C42—91no—
260#217—62no—
261#216—62no—
262#216—62no—
263#215—62no—
264#214—62no—
265(C2×D3)⋊C4—62no—
266Dic3⋊C4—62no—
267Dic3⋊C4—62no—
268C4×Dic3—31no—
269C4×Dic3—31no—
270C4×Dic3—61no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—61no—
274Dic12—22no—
275#204—22no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—41no—
280C22×C32—121no—
281C22×C32—122no—
282C22×C32D3×D411yescharacteristic
283C22×C32D3×D411yescharacteristic
284C22×C32C2×C3×D411yescharacteristic
285C2×C3×D3—42no—
286C2×C3×D3—42no—
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C2×C3×D3—42no—
290C2×C3×D3—82no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—121no—
300C32×C4—121no—
301C32×C4—121no—
302C32×C4D3×D411yescharacteristic
303C32×C4D3×D411yescharacteristic
304C32×C4C2×#6711yescharacteristic
305C32×C4C2×#6711yescharacteristic
306C32×C4C2×C3×D411yescharacteristic
307C32×C4C2×C3×D411yescharacteristic
308C3⋊Dic3—22no—
309C3×Dic3—42no—
310C3×Dic3—42no—
311C3×Dic3—42no—
312C3×Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—41no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C3×Dic3—62no—
319C3×Dic3—62no—
320C3×Dic3—122no—
321C2×Q16—31no—
322C2×Q16—91no—
323C2×SD16—31no—
324C2×SD16—91no—
325C4⋊Q8—91no—
326#124—91no—
327#97—91no—
328C33#49711yescharacteristic, complemented, Sylow
329C23×C3—61no—
330C23×C3—61no—
331C23×C3—121no—
332C22×D3—21no—
333C3×Q8—62no—
334C3×Q8—62no—
335C3×Q8—62no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—61no—
342C3×Q8—61no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—121no—
349C3×Q8—121no—
350C3×Q8—121no—
351C3×D4—62no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—61no—
355C3×D4—61no—
356C3×D4—121no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—92no—
368C2×C3×C4—121no—
369C2×C3×C4—121no—
370C2×C3×C4—121no—
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×C3×C4C22×C3×D311yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—32no—
378C2×Dic3—32no—
379C2×Dic3—62no—
380D12—21no—
381D12—21no—
382Dic6—21no—
383Dic6—21no—
384Dic6—21no—
385Dic6—21no—
386Dic6—21no—
387Dic6—21no—
388C3×C8—22no—
389C3×C8—22no—
390C3×C8—22no—
391C3×C8—42no—
392C3×C8—92no—
393C3×C8#62512yes—
394C3×C8C3×#6712yes—
395C3×C8C3×#6712yes—
396C3⋊C8—32no—
397C3⋊C8—32no—
398C3⋊C8—62no—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—22no—
404C2×C32—21no—
405C2×C32—41no—
406C2×C32—42no—
407C2×C32—241no—
408C2×C32#91512yes—
409C2×C32#91812yes—
410C2×C32#94011yescharacteristic
411C2×C32#94011yescharacteristic
412C2×C32C3×#12711yescharacteristic
413C2×C32C3×#13712yes—
414C3×D3—82no—
415C3×D3—81no—
416C3×D3—81no—
417C2×Q8—31no—
418C2×Q8—31no—
419C2×Q8—31no—
420C2×D4—31no—
421Q16—62no—
422Q16—182no—
423SD16—62no—
424SD16—182no—
425C2×C8—91no—
426C2×C8C3×D3211yescharacteristic
427C4⋊C4—182no—
428C22⋊C4—182no—
429C42—91no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—122no—
435C22×C3—122no—
436C22×C3—121no—
437C22×C3—122no—
438C22×C3—121no—
439C22×C3—242no—
440C22×C3#380211yescharacteristic
441C22×C3C3×D3×D411yescharacteristic
442C22×C3C3×D3×D411yescharacteristic
443C2×D3—42no—
444C2×D3—41no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—122no—
454C3×C4—122no—
455C3×C4—122no—
456C3×C4—121no—
457C3×C4—121no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—182no—
463C3×C4#380211yescharacteristic
464C3×C4C2×#62511yescharacteristic
465C3×C4C3×D3×D411yescharacteristic
466C3×C4C3×D3×D411yescharacteristic
467C3×C4C2×C3×#6711yescharacteristic
468C3×C4C2×C3×#6711yescharacteristic
469Dic3—41no—
470Dic3—41no—
471Dic3—41no—
472Dic3—62no—
473Dic3—62no—
474Dic3—122no—
475C32—21no—
476C32—21no—
477C32—21no—
478C32—41no—
479C32#516711yescharacteristic, complemented
480C32#519011yescharacteristic, complemented
481C32C3×#49711yescharacteristic, complemented
482C23—61no—
483Q8—61no—
484Q8—61no—
485Q8—61no—
486Q8—61no—
487Q8—61no—
488Q8—61no—
489D4—61no—
490D4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—92no—
495C2×C4C2×C3×D3211yescharacteristic
496C8—92no—
497C8C3×#62512yes—
498C2×C3—22no—
499C2×C3—22no—
500C2×C3—22no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—42no—
505C2×C3—41no—
506C2×C3—242no—
507C2×C3—241no—
508C2×C3—241no—
509C2×C3#6381112yes—
510C2×C3#6391411yescharacteristic
511C2×C3C3×#91512yes—
512C2×C3C3×#91812yes—
513C2×C3C3×#94011yescharacteristic
514C2×C3C3×#94011yescharacteristic
515D3—81no—
516C22—122no—
517C22—121no—
518C22C3×#380211yescharacteristic
519C4—121no—
520C4—121no—
521C4—121no—
522C4—182no—
523C4C3×#380211yescharacteristic
524C4C2×C3×#62511yescharacteristic
525C3—21no—
526C3—21no—
527C3—21no—
528C3—41no—
529C3#1059087011yescharacteristic, direct product factor
530C3C3×#516711yescharacteristic, complemented
531C3C3×#519011yescharacteristic, complemented
532C2—241no—
533C2C3×#6381112yes—
534C2C3×#6391411yescharacteristic
subgroupG/H
centerC22×C3#3802
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×C3×C8D32
hypocenterC32C3×#497
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×C32×Dic12C2semidirect
3. C3×#63873C2semidirect
5. C2×C3×#3718C2semidirect
7. C3×#63642C2semidirect
10. #10590870C3direct (N × H)
35. C2×C3×Dic12D3semidirect
36. C2×C3×Dic12C2×C3semidirect
40. #63873C2×C3semidirect
42. C2×#3718C2×C3semidirect
44. #63642C2×C3semidirect
145. C2×Dic12C3×D3semidirect
328. C33#497semidirect
479. C32#5167semidirect
480. C32#5190semidirect
481. C32C3×#497semidirect
529. C3#10590870direct (N × H)
530. C3C3×#5167semidirect
531. C3C3×#5190semidirect

Maximal subgroups

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

subgroupindex
C2×C32×Dic122
C2×C32×#2042
C3×#638732
C3×#638632
C2×C3×#37182
C2×C3×#37152
C3×#636422
C3×#51903
C3×#51673
#105908703