Finite Group Databasebeta

← C3×#10591541 #13158582

Subgroups — C3×#10591541 #13158582

Order 1728.

642 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3756C211yesmaximal, characteristic
2C32×Q8×Dic3C211yesmaximal, characteristic
3C3×#63844C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5C3×#63574C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic
8C3×#5030—31nomaximal
9C3×#4789—31nomaximal
10#10591541C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C3×#3756C414yescomplemented
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×#3724—21no—
22C3×Dic32—21no—
23C32×C4×Q8—31no—
24C2×#3756C2×C311yescharacteristic
25C3×Q8×Dic3—21no—
26C3×Q8×Dic3D311yescharacteristic
27C3×Q8×Dic3C2×C311yescharacteristic
28C2×C3×#216—31no—
29C2×C3×#216—31no—
30C2×C3×#216—61no—
31C3×#887—31no—
32C3×C4×Dic6—31no—
33#63844C2×C311yescharacteristic
34C32×#103—31no—
35C3×#835—21no—
36C3×#835D311yescharacteristic
37C3×#835C2×C311yescharacteristic
38C3×#816—31no—
39C3×#815—31no—
40C3×#810—31no—
41C3×#808—31no—
42C3×#808—31no—
43C3×#803—31no—
44#63574C2×C311yescharacteristic
45#63567C2×C311yescharacteristic
46#63554C2×C311yescharacteristic
47C33×Q8—21no—
48C33×Q8C2×C412yes—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634C2×C412yes—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58C3×C3⋊(C3⋊C8)C2×C412yes—
59C32×C3⋊C8—21no—
60C3×#441—91no—
61#5030—31no—
62#4789—31no—
63C2×C32×Q8—21no—
64C2×C32×Q8—21no—
65C2×C32×Q8—21no—
66C2×C32×Q8—41no—
67C2×C32×Q8C2×D311yescharacteristic
68C2×C32×Q8C2×D311yescharacteristic
69C2×C32×Q8C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×C32×C8—31no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—31no—
77C32×C4⋊C4—61no—
78C32×C42—31no—
79C32×C42—61no—
80#3756C3×C414yescomplemented
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×#216—34no—
83C3×#216—34no—
84C3×#216—64no—
85C3×C4⋊Dic3—22no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×C4⋊Dic3C22×C311yescharacteristic
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×C4×Dic3—22no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3—21no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3C2×D311yescharacteristic
103C3×C4×Dic3C22×C311yescharacteristic
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C22×C311yescharacteristic
110#3724—21no—
111Dic32—21no—
112C22×C33C2×D411yescharacteristic
113C33×C4—22no—
114C33×C4C22×C411yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—22no—
117C32×Dic3—21no—
118C32×Dic3—21no—
119C32×Dic3—41no—
120C32×Dic3—41no—
121C2×C3×Q16—91no—
122C3×C4×Q8—31no—
123C3×C4×Q8—31no—
124C3×C4×Q8—61no—
125C3×C4×Q8—91no—
126Q8×Dic3C3×D311yescharacteristic
127C2×#216—31no—
128C2×#216—31no—
129C2×#216—61no—
130#887—31no—
131C4×Dic6—31no—
132C3×#106—91no—
133C3×#103—31no—
134C3×#103—31no—
135C3×#103—61no—
136C3×#103—91no—
137C3×#97—91no—
138#835C3×D311yescharacteristic
139#816—31no—
140#815—31no—
141#810—31no—
142#808—31no—
143#808—31no—
144#803—31no—
145C32×Q8—21no—
146C32×Q8—22no—
147C32×Q8—21no—
148C32×Q8—21no—
149C32×Q8—22no—
150C32×Q8—22no—
151C32×Q8—22no—
152C32×Q8—42no—
153C32×Q8—41no—
154C32×Q8—41no—
155C32×Q8—42no—
156C32×Q8C4×D312yes—
157C32×Q8C4×D312yes—
158C32×Q8C2×C3×C412yes—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—31no—
167C2×C32×C4—31no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—42no—
172C2×C32×C4—61no—
173C2×C32×C4—61no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C23×C311yescharacteristic
177C2×C3⋊Dic3—21no—
178#634—21no—
179#634C2×C3×C412yes—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—41no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—121no—
191C2×C3×Dic3#6711yescharacteristic
192C2×C3×Dic3#6711yescharacteristic
193C2×C3×Dic3C3×D411yescharacteristic
194C2×C3×Dic3C3×D411yescharacteristic
195C3×Dic6—32no—
196C3×Dic6—32no—
197C3×Dic6—62no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—121no—
201C32×C8—61no—
202C3⋊(C3⋊C8)C2×C3×C412yes—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—32no—
206C3×C3⋊C8—32no—
207C3×C3⋊C8—41no—
208C3×C3⋊C8—62no—
209#441—91noSylow
210C2×C33C4×D411yescharacteristic
211C2×C33#13711yescharacteristic
212C2×C33#13711yescharacteristic
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—41no—
217C2×C3×Q8—91no—
218C2×C3×Q8D3211yescharacteristic
219C2×C3×Q8C2×C3×D311yescharacteristic
220C2×C3×Q8C2×C3×D311yescharacteristic
221C2×Dic6—31no—
222C2×Dic6—31no—
223C2×Dic6—61no—
224C3×Q16—94no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—61no—
228C2×C3×C8—91no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—62no—
238C3×C4⋊C4—181no—
239C3×C42—31no—
240C3×C42—31no—
241C3×C42—61no—
242C3×C42—61no—
243C3×C42—61no—
244C3×C42—62no—
245C3×C42—181no—
246#216—34no—
247#216—34no—
248#216—64no—
249C4⋊Dic3—21no—
250C4⋊Dic3C2×C3×D311yescharacteristic
251C4⋊Dic3C2×C3×D311yescharacteristic
252Dic3⋊C4—61no—
253Dic3⋊C4—61no—
254C4×Dic3—21no—
255C4×Dic3—61no—
256C4×Dic3—61no—
257C4×Dic3C2×C3×D311yescharacteristic
258C2×C3⋊C8—31no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8C2×C3×D311yescharacteristic
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32D3×D411yescharacteristic
267C22×C32C2×#6711yescharacteristic
268C22×C32C2×C3×D411yescharacteristic
269C32×C4—22no—
270C32×C4—24no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—42no—
282C32×C4—42no—
283C32×C4—44no—
284C32×C4—61no—
285C32×C4—61no—
286C32×C4—121no—
287C32×C4—121no—
288C32×C4C2×C4×D311yescharacteristic
289C32×C4C2×C4×D311yescharacteristic
290C32×C4#23511yescharacteristic
291C32×C4#23711yescharacteristic
292C32×C4C22×C3×C411yescharacteristic
293C32×C4C3×#4111yescharacteristic
294C3⋊Dic3—22no—
295C3×Dic3—22no—
296C3×Dic3—21no—
297C3×Dic3—21no—
298C3×Dic3—21no—
299C3×Dic3—21no—
300C3×Dic3—41no—
301C3×Dic3—42no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—81no—
309C3×Dic3—122no—
310C2×Q16—91no—
311C4×Q8—31no—
312C4×Q8—91no—
313#106—91no—
314#103—31no—
315#103—91no—
316#97—91no—
317C33#44111yescharacteristic, complemented, Sylow
318C3×Q8—22no—
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—21no—
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—92no—
330C3×Q8—181no—
331C3×Q8#62412yes—
332C3×Q8C3×C4×D312yes—
333C3×Q8C3×C4×D312yes—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—62no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—121no—
357C2×C3×C4—181no—
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×C3×C4C22×C3×D311yescharacteristic
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—121no—
366C2×Dic3C3×#6711yescharacteristic
367C2×Dic3C3×#6711yescharacteristic
368Dic6—32no—
369Dic6—32no—
370Dic6—62no—
371Dic6—61no—
372Dic6—61no—
373Dic6—121no—
374C3×C8—61no—
375C3×C8—61no—
376C3×C8—92no—
377C3×C8—121no—
378C3⋊C8—21no—
379C3⋊C8—32no—
380C3⋊C8—32no—
381C3⋊C8—62no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32#67⋊C411yescharacteristic
395C2×C32#91511yescharacteristic
396C2×C32#91811yescharacteristic
397C2×C32C4×#6711yescharacteristic
398C2×C32#94211yescharacteristic
399C2×C32#95111yescharacteristic
400C2×C32C3×C4×D411yescharacteristic
401C2×C32C3×#13711yescharacteristic
402C2×C32C3×#13711yescharacteristic
403C2×Q8—91no—
404C2×Q8C3×D3211yescharacteristic
405Q16—94no—
406C2×C8—31no—
407C2×C8—91no—
408C4⋊C4—31no—
409C4⋊C4—31no—
410C4⋊C4—61no—
411C4⋊C4—181no—
412C42—31no—
413C42—61no—
414C42—181no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—41no—
419C22×C3D3×#6711yescharacteristic
420C22×C3C3×D3×D411yescharacteristic
421C22×C3C2×C3×#6711yescharacteristic
422C3×C4—24no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—44no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—62no—
438C3×C4—61no—
439C3×C4—61no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—121no—
443C3×C4—122no—
444C3×C4—121no—
445C3×C4—121no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—182no—
449C3×C4—241no—
450C3×C4#380411yescharacteristic
451C3×C4C2×#62411yescharacteristic
452C3×C4C2×C3×C4×D311yescharacteristic
453C3×C4C2×C3×C4×D311yescharacteristic
454C3×C4C3×#23511yescharacteristic
455C3×C4C3×#23711yescharacteristic
456Dic3—21no—
457Dic3—21no—
458Dic3—41no—
459Dic3—41no—
460Dic3—62no—
461Dic3—62no—
462Dic3—122no—
463C32—21no—
464C32—21no—
465C32—21no—
466C32—41no—
467C32#478911yescharacteristic, complemented
468C32#503011yescharacteristic, complemented
469C32C3×#44111yescharacteristic, complemented
470Q8—21no—
471Q8—92no—
472Q8—181no—
473Q8C3×#62412yes—
474C2×C4—21no—
475C2×C4—31no—
476C2×C4—31no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—181no—
480C2×C4C2×C3×D3211yescharacteristic
481C8—61no—
482C8—92no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3#6394311yescharacteristic
496C2×C3#6394611yescharacteristic
497C2×C3#6395211yescharacteristic
498C2×C3C3×#67⋊C411yescharacteristic
499C2×C3C3×#91511yescharacteristic
500C2×C3C3×#91811yescharacteristic
501C2×C3C3×C4×#6711yescharacteristic
502C2×C3C3×#94211yescharacteristic
503C2×C3C3×#95111yescharacteristic
504C22C3×D3×#6711yescharacteristic
505C4—22no—
506C4—61no—
507C4—61no—
508C4—121no—
509C4—121no—
510C4—182no—
511C4C3×#380411yescharacteristic
512C4C2×C3×#62411yescharacteristic
513C3—21no—
514C3—21no—
515C3—21no—
516C3—41no—
517C3#1059154111yescharacteristic, direct product factor
518C3C3×#478911yescharacteristic, complemented
519C3C3×#503011yescharacteristic, complemented
520C2C3×#6394311yescharacteristic
521C2C3×#6394611yescharacteristic
522C2C3×#6395211yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×Q8D32
hypocenterC32C3×#441
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
10. #10591541C3direct (N × H)
13. C3×#3756C4semidirect
80. #3756C3×C4semidirect
317. C33#441semidirect
467. C32#4789semidirect
468. C32#5030semidirect
469. C32C3×#441semidirect
517. C3#10591541direct (N × H)
518. C3C3×#4789semidirect
519. C3C3×#5030semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37562
C32×Q8×Dic32
C3×#638442
C32×#8352
C3×#635742
C3×#635672
C3×#635542
C3×#50303
C3×#47893
#105915413