Finite Group Databasebeta

← C3×#10590556 #13157540

Subgroups — C3×#10590556 #13157540

Order 1728.

552 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#887C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3C3×#63635C211yesmaximal, characteristic
4C32×#808C211yesmaximal, characteristic
5C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
6C3×#63566C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8C3×#5020—31nomaximal
9C3×#4854—31nomaximal
10#10590556C311yesmaximal, characteristic, direct product factor
11C2×C32×Dic6C2211yescharacteristic
12C2×C3×#627—21no—
13C33×C4⋊C4C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×C4×Q8—31no—
23C3×#888—31no—
24C3×#887—21no—
25C3×#887D311yescharacteristic
26C3×#887C2×C311yescharacteristic
27C3×#869—31no—
28#63918C2×C311yescharacteristic
29C32×#106—31no—
30#63635C2×C311yescharacteristic
31C3×#835—31no—
32C3×#835—31no—
33C3×#816—31no—
34C3×Dic3⋊C8—31no—
35C3×#810—31no—
36C3×#808—21no—
37C3×#808—31no—
38C3×#808—31no—
39C3×#808—61no—
40C3×#808D311yescharacteristic
41C3×#808C2×C311yescharacteristic
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)C2×C311yescharacteristic
44#63566C2×C311yescharacteristic
45#63565C2×C311yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×Dic6—21no—
54C32×Dic6Q812yescomplemented
55C3×#627—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58C3×#475—91no—
59#5020—31no—
60#4854—31no—
61C2×C32×Q8—31no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—61no—
64C2×C3×Dic6—61no—
65C2×C3×Dic6C2×D311yescharacteristic
66C2×C3×Dic6C22×C311yescharacteristic
67C2×#627—21no—
68C2×C32×C8—31no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—31no—
73C32×C4⋊C4—41no—
74C32×C4⋊C4—61no—
75C32×C4⋊C4C2×D311yescharacteristic
76C32×C4⋊C4C2×D311yescharacteristic
77C32×C4⋊C4C22×C311yescharacteristic
78C32×C42—31no—
79C32×C42—61no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C22×C311yescharacteristic
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4—21no—
91C3×Dic3⋊C4—21no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—22no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C22×C311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105Dic3⋊Dic3—21no—
106C22×C33C2×D411yescharacteristic
107C33×C4—41no—
108C33×C4C2×Q811yescharacteristic
109C33×C4#4111yescharacteristic
110C3×C3⋊Dic3—41no—
111C32×Dic3—22no—
112C32×Dic3—21no—
113C32×Dic3—41no—
114C32×Dic3SD1612yes—
115C3×C4⋊Q8—91no—
116C3×C4×Q8—31no—
117C3×C4×Q8—31no—
118C3×C4×Q8—61no—
119#888—31no—
120#887C3×D311yescharacteristic
121#869—31no—
122C3×#106—31no—
123C3×#106—31no—
124C3×#106—61no—
125C3×#106—91no—
126C3×C4⋊C8—91no—
127C3×#103—91no—
128C3×#103—91no—
129#835—31no—
130#835—31no—
131#816—31no—
132Dic3⋊C8—31no—
133#810—31no—
134#808—31no—
135#808—31no—
136#808—61no—
137#808C3×D311yescharacteristic
138C4⋊(C3⋊C8)—31no—
139C32×Q8—32no—
140C32×Q8—61no—
141C2×C32×C4—21no—
142C2×C32×C4—22no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—31no—
149C2×C32×C4—31no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4—41no—
153C2×C32×C4—42no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C23×C311yescharacteristic
159C2×C3⋊Dic3—21no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—121no—
171C2×C3×Dic3#6711yescharacteristic
172C2×C3×Dic3#6711yescharacteristic
173C2×C3×Dic3C3×D411yescharacteristic
174C2×C3×Dic3C3×D411yescharacteristic
175C3×Dic6—22no—
176C3×Dic6—21no—
177C3×Dic6—21no—
178C3×Dic6—22no—
179C3×Dic6—122no—
180C3×Dic6—122no—
181C3×Dic6Dic612yescomplemented
182C3×Dic6C3×Q812yescomplemented
183#627—42no—
184C32×C8—61no—
185C3⋊(C3⋊C8)—21no—
186C3×C3⋊C8—22no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—62no—
190C3×C3⋊C8—61no—
191C3×C3⋊C8—61no—
192#475—91noSylow
193C2×C33C22⋊Q811yescharacteristic
194C2×C33C2×SD1611yescharacteristic
195C2×C33#13711yescharacteristic
196C2×C3×Q8—31no—
197C2×C3×Q8—31no—
198C2×C3×Q8—61no—
199C2×C3×Q8—181no—
200C2×Dic6—61no—
201C2×Dic6—61no—
202C2×Dic6C2×C3×D311yescharacteristic
203C2×C3×C8—31no—
204C2×C3×C8—31no—
205C2×C3×C8—61no—
206C2×C3×C8—91no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—31no—
212C3×C4⋊C4—41no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4—61no—
215C3×C4⋊C4—61no—
216C3×C4⋊C4—62no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—181no—
219C3×C4⋊C4D3211yescharacteristic
220C3×C4⋊C4C2×C3×D311yescharacteristic
221C3×C4⋊C4C2×C3×D311yescharacteristic
222C3×C42—31no—
223C3×C42—31no—
224C3×C42—61no—
225C3×C42—61no—
226C3×C42—61no—
227C3×C42—62no—
228C4⋊Dic3—31no—
229C4⋊Dic3—31no—
230C4⋊Dic3—61no—
231C4⋊Dic3—61no—
232C4⋊Dic3C2×C3×D311yescharacteristic
233Dic3⋊C4—21no—
234Dic3⋊C4—61no—
235C4×Dic3—21no—
236C4×Dic3C2×C3×D311yescharacteristic
237C2×C3⋊C8—31no—
238C2×C3⋊C8—31no—
239C2×C3⋊C8—61no—
240C2×C3⋊C8C2×C3×D311yescharacteristic
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—41no—
245C22×C32D3×D411yescharacteristic
246C22×C32C2×#6711yescharacteristic
247C22×C32C2×C3×D411yescharacteristic
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—32no—
255C32×C4—41no—
256C32×C4—41no—
257C32×C4—41no—
258C32×C4—42no—
259C32×C4—41no—
260C32×C4—41no—
261C32×C4—62no—
262C32×C4—61no—
263C32×C4—81no—
264C32×C4—81no—
265C32×C4—82no—
266C32×C4—121no—
267C32×C4C2×Dic611yescharacteristic
268C32×C4#23511yescharacteristic
269C32×C4#23511yescharacteristic
270C32×C4D3×Q811yescharacteristic
271C32×C4C2×C3×Q811yescharacteristic
272C32×C4C3×#4111yescharacteristic
273C3⋊Dic3—41no—
274C3×Dic3—24no—
275C3×Dic3—22no—
276C3×Dic3—21no—
277C3×Dic3—22no—
278C3×Dic3—21no—
279C3×Dic3—22no—
280C3×Dic3—41no—
281C3×Dic3—41no—
282C3×Dic3—41no—
283C3×Dic3—81no—
284C3×Dic3—121no—
285C3×Dic3—121no—
286C3×Dic3—241no—
287C3×Dic3#21512yes—
288C3×Dic3C3×SD1612yes—
289C4⋊Q8—91no—
290C4×Q8—31no—
291#106—31no—
292#106—91no—
293C4⋊C8—91no—
294#103—91no—
295#103—91no—
296C33#47511yescharacteristic, complemented, Sylow
297C3×Q8—32no—
298C3×Q8—32no—
299C3×Q8—62no—
300C3×Q8—62no—
301C3×Q8—61no—
302C3×Q8—61no—
303C3×Q8—362no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—31no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—62no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—121no—
327C2×C3×C4—181no—
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C22×C3×D311yescharacteristic
330C2×C3×C4C22×C3×D311yescharacteristic
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—121no—
336C2×Dic3C3×#6711yescharacteristic
337C2×Dic3C3×#6711yescharacteristic
338Dic6—21no—
339Dic6—122no—
340Dic6—122no—
341Dic6C3×Dic612yescomplemented
342C3×C8—62no—
343C3×C8—61no—
344C3×C8—61no—
345C3×C8—181no—
346C3⋊C8—21no—
347C3⋊C8—62no—
348C3⋊C8—61no—
349C3⋊C8—61no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32#89611yescharacteristic
363C2×C32D3×SD1611yescharacteristic
364C2×C32#91511yescharacteristic
365C2×C32#92411yescharacteristic
366C2×C32C2×#21511yescharacteristic
367C2×C32#95111yescharacteristic
368C2×C32C3×C22⋊Q811yescharacteristic
369C2×C32C2×C3×SD1611yescharacteristic
370C2×C32C3×#13711yescharacteristic
371C2×Q8—31no—
372C2×Q8—181no—
373C2×C8—31no—
374C2×C8—91no—
375C4⋊C4—31no—
376C4⋊C4—61no—
377C4⋊C4—91no—
378C4⋊C4—181no—
379C4⋊C4C3×D3211yescharacteristic
380C42—31no—
381C42—61no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—41no—
386C22×C3D3×#6711yescharacteristic
387C22×C3C3×D3×D411yescharacteristic
388C22×C3C2×C3×#6711yescharacteristic
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—32no—
396C3×C4—32no—
397C3×C4—42no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—41no—
401C3×C4—41no—
402C3×C4—41no—
403C3×C4—62no—
404C3×C4—64no—
405C3×C4—62no—
406C3×C4—61no—
407C3×C4—62no—
408C3×C4—61no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4—82no—
412C3×C4—121no—
413C3×C4—121no—
414C3×C4—121no—
415C3×C4—241no—
416C3×C4—361no—
417C3×C4D3×Dic611yescharacteristic
418C3×C4#379811yescharacteristic
419C3×C4C2×C3×Dic611yescharacteristic
420C3×C4C3×#23511yescharacteristic
421C3×C4C3×#23511yescharacteristic
422C3×C4C3×D3×Q811yescharacteristic
423Dic3—22no—
424Dic3—21no—
425Dic3—41no—
426Dic3—121no—
427Dic3—121no—
428Dic3—241no—
429Dic3C3×#21512yes—
430C32—21no—
431C32—21no—
432C32—21no—
433C32—41no—
434C32#485411yescharacteristic, complemented
435C32#502011yescharacteristic, complemented
436C32C3×#47511yescharacteristic, complemented
437Q8—32no—
438Q8—61no—
439Q8—362no—
440C2×C4—21no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—181no—
446C2×C4C2×C3×D3211yescharacteristic
447C8—61no—
448C8—181no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3#6385111yescharacteristic
462C2×C3#6392911yescharacteristic
463C2×C3D3×#21511yescharacteristic
464C2×C3C3×#89611yescharacteristic
465C2×C3C3×D3×SD1611yescharacteristic
466C2×C3C3×#91511yescharacteristic
467C2×C3C3×#92411yescharacteristic
468C2×C3C2×C3×#21511yescharacteristic
469C2×C3C3×#95111yescharacteristic
470C22C3×D3×#6711yescharacteristic
471C4—32no—
472C4—41no—
473C4—62no—
474C4—61no—
475C4—121no—
476C4—361no—
477C4C3×D3×Dic611yescharacteristic
478C4C3×#379811yescharacteristic
479C3—21no—
480C3—21no—
481C3—21no—
482C3—41no—
483C3#1059055611yescharacteristic, direct product factor
484C3C3×#485411yescharacteristic, complemented
485C3C3×#502011yescharacteristic, complemented
486C2C3×#6385111yescharacteristic
487C2C3×#6392911yescharacteristic
488C2C3×D3×#21511yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#475
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
10. #10590556C3direct (N × H)
54. C32×Dic6Q8semidirect
181. C3×Dic6Dic6semidirect
182. C3×Dic6C3×Q8semidirect
296. C33#475semidirect
341. Dic6C3×Dic6semidirect
434. C32#4854semidirect
435. C32#5020semidirect
436. C32C3×#475semidirect
483. C3#10590556direct (N × H)
484. C3C3×#4854semidirect
485. C3C3×#5020semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
C3×#639182
C3×#636352
C32×#8082
C3×Dic3⋊(C3⋊C8)2
C3×#635662
C3×#635652
C3×#50203
C3×#48543
#105905563