Finite Group Databasebeta

← C3×#10590816 #13157868

Subgroups — C3×#10590816 #13157868

Order 1728.

522 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63919C211yesmaximal, characteristic
2C3×#63843C211yesmaximal, characteristic
3C3×#63641C211yesmaximal, characteristic
4C32×#817C211yesmaximal, characteristic
5C32×#816C211yesmaximal, characteristic
6C3×#63574C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8C3×#4854—31nomaximal
9C3×#4791—31nomaximal
10#10590816C311yesmaximal, characteristic, direct product factor
11C2×C32×Dic6C2211yescharacteristic
12C2×C3×#627—21no—
13C2×C33×C8C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C3×#3724—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic32—21no—
22C3×Q8×Dic3—31no—
23C3×#888—31no—
24C3×#888—31no—
25C3×#887—31no—
26#63919C2×C311yescharacteristic
27#63843C2×C311yescharacteristic
28C32×#106—31no—
29C32×#103—31no—
30#63641C2×C311yescharacteristic
31C3×#835—31no—
32C3×#817—21no—
33C3×#817D311yescharacteristic
34C3×#817C2×C311yescharacteristic
35C3×#816—21no—
36C3×#816D311yescharacteristic
37C3×#816C2×C311yescharacteristic
38C3×Dic3⋊C8—31no—
39C3×Dic3⋊C8—31no—
40C3×Dic3⋊C8—61no—
41C3×#810—31no—
42C3×#808—31no—
43C3×#808—31no—
44#63574C2×C311yescharacteristic
45#63565C2×C311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3D411yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6Q812yescomplemented
55C3×#627—42no—
56C33×C8—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C3×#475—91no—
59#4854—31no—
60#4791—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—21no—
69C2×C32×C8—21no—
70C2×C32×C8—21no—
71C2×C32×C8—41no—
72C2×C32×C8C2×D311yescharacteristic
73C2×C32×C8C2×D311yescharacteristic
74C2×C32×C8C22×C311yescharacteristic
75C32×C4⋊C4—31no—
76C32×C4⋊C4—31no—
77C32×C4⋊C4—31no—
78C4×C3⋊Dic3C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—61no—
84C3×C4⋊Dic3C2×D311yescharacteristic
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C22×C311yescharacteristic
88C3×C4⋊Dic3C22×C311yescharacteristic
89C3×C4⋊Dic3C22×C311yescharacteristic
90C3×Dic3⋊C4—61no—
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3—61no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101#3724—21no—
102Dic3⋊Dic3—21no—
103Dic32—21no—
104C22×C33C2×D411yescharacteristic
105C33×C4C2×Q811yescharacteristic
106C33×C4#4111yescharacteristic
107C3×C3⋊Dic3—21no—
108C3×C3⋊Dic3SD1612yes—
109C32×Dic3—22no—
110C32×Dic3—41no—
111C32×Dic3—41no—
112C32×Dic3—41no—
113C3×C4⋊Q8—91no—
114C3×C4×Q8—91no—
115Q8×Dic3—31no—
116#888—31no—
117#888—31no—
118#887—31no—
119C3×#106—31no—
120C3×#106—31no—
121C3×#106—61no—
122C3×#106—91no—
123C3×C4⋊C8—91no—
124C3×#103—31no—
125C3×#103—31no—
126C3×#103—61no—
127C3×#103—91no—
128#835—31no—
129#817C3×D311yescharacteristic
130#816C3×D311yescharacteristic
131Dic3⋊C8—31no—
132Dic3⋊C8—31no—
133Dic3⋊C8—61no—
134#810—31no—
135#808—31no—
136#808—31no—
137C32×Q8—32no—
138C32×Q8—61no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4—61no—
144C2×C32×C4—61no—
145C2×C32×C4—61no—
146C2×C32×C4—61no—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C23×C311yescharacteristic
150C2×C3⋊Dic3C3×D411yescharacteristic
151C2×C3⋊Dic3C3×D411yescharacteristic
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—41no—
165C2×C3×Dic3—41no—
166C2×C3×Dic3—41no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C3×Dic6—21no—
171C3×Dic6—21no—
172C3×Dic6—22no—
173C3×Dic6—41no—
174C3×Dic6—122no—
175C3×Dic6—122no—
176C3×Dic6Dic612yescomplemented
177C3×Dic6C3×Q812yescomplemented
178#627—42no—
179C32×C8—21no—
180C32×C8—22no—
181C32×C8—21no—
182C32×C8—21no—
183C32×C8—41no—
184C32×C8—41no—
185C32×C8—42no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—61no—
188C3×C3⋊C8—61no—
189C3×C3⋊C8—121no—
190#475—91noSylow
191C2×C33C22⋊Q811yescharacteristic
192C2×C33C2×SD1611yescharacteristic
193C2×C33#13711yescharacteristic
194C2×C3×Q8—31no—
195C2×C3×Q8—31no—
196C2×C3×Q8—61no—
197C2×C3×Q8—181no—
198C2×Dic6—61no—
199C2×Dic6—61no—
200C2×Dic6C2×C3×D311yescharacteristic
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—41no—
205C2×C3×C8—91no—
206C2×C3×C8D3211yescharacteristic
207C2×C3×C8C2×C3×D311yescharacteristic
208C2×C3×C8C2×C3×D311yescharacteristic
209C3×C4⋊C4—31no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—31no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—61no—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—61no—
218C3×C4⋊C4—181no—
219C3×C4⋊C4—181no—
220C3×C42—91no—
221C3×C42—181no—
222C4⋊Dic3—61no—
223C4⋊Dic3C2×C3×D311yescharacteristic
224C4⋊Dic3C2×C3×D311yescharacteristic
225C4⋊Dic3C2×C3×D311yescharacteristic
226Dic3⋊C4—61no—
227Dic3⋊C4—61no—
228Dic3⋊C4—61no—
229C4×Dic3—31no—
230C4×Dic3—31no—
231C4×Dic3—61no—
232C4×Dic3—61no—
233C4×Dic3—61no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—31no—
236C2×C3⋊C8—61no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—41no—
241C22×C32D3×D411yescharacteristic
242C22×C32D3×D411yescharacteristic
243C22×C32C2×C3×D411yescharacteristic
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—41no—
251C32×C4—41no—
252C32×C4—62no—
253C32×C4—121no—
254C32×C4—121no—
255C32×C4—121no—
256C32×C4C2×Dic611yescharacteristic
257C32×C4#23511yescharacteristic
258C32×C4#23711yescharacteristic
259C32×C4D3×Q811yescharacteristic
260C32×C4C2×C3×Q811yescharacteristic
261C32×C4C3×#4111yescharacteristic
262C3⋊Dic3—21no—
263C3⋊Dic3C3×SD1612yes—
264C3×Dic3—22no—
265C3×Dic3—22no—
266C3×Dic3—32no—
267C3×Dic3—32no—
268C3×Dic3—42no—
269C3×Dic3—41no—
270C3×Dic3—41no—
271C3×Dic3—41no—
272C3×Dic3—41no—
273C3×Dic3—41no—
274C3×Dic3—41no—
275C3×Dic3—62no—
276C3×Dic3—61no—
277C3×Dic3—61no—
278C3×Dic3—81no—
279C3×Dic3—81no—
280C3×Dic3—81no—
281C3×Dic3—121no—
282C4⋊Q8—91no—
283C4×Q8—91no—
284#106—31no—
285#106—91no—
286C4⋊C8—91no—
287#103—31no—
288#103—91no—
289C33#47511yescharacteristic, complemented, Sylow
290C3×Q8—32no—
291C3×Q8—32no—
292C3×Q8—62no—
293C3×Q8—61no—
294C3×Q8—61no—
295C3×Q8—121no—
296C3×Q8—362no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—41no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—91no—
310C2×C3×C4—91no—
311C2×C3×C4—121no—
312C2×C3×C4—121no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4C2×D3211yescharacteristic
316C2×C3×C4C22×C3×D311yescharacteristic
317C2×C3×C4C22×C3×D311yescharacteristic
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—31no—
323C2×Dic3—31no—
324C2×Dic3—31no—
325C2×Dic3—31no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328Dic6—21no—
329Dic6—122no—
330Dic6—122no—
331Dic6C3×Dic612yescomplemented
332C3×C8—22no—
333C3×C8—21no—
334C3×C8—21no—
335C3×C8—21no—
336C3×C8—41no—
337C3×C8—41no—
338C3×C8—42no—
339C3×C8—181no—
340C3⋊C8—61no—
341C3⋊C8—61no—
342C3⋊C8—121no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—41no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32#87811yescharacteristic
356C2×C32#89611yescharacteristic
357C2×C32D3×SD1611yescharacteristic
358C2×C32D3×SD1611yescharacteristic
359C2×C32#91511yescharacteristic
360C2×C32#91811yescharacteristic
361C2×C32C3×C22⋊Q811yescharacteristic
362C2×C32C2×C3×SD1611yescharacteristic
363C2×C32C3×#13711yescharacteristic
364C2×Q8—31no—
365C2×Q8—181no—
366C2×C8—91no—
367C2×C8C3×D3211yescharacteristic
368C4⋊C4—31no—
369C4⋊C4—31no—
370C4⋊C4—31no—
371C4⋊C4—181no—
372C4⋊C4—181no—
373C42—91no—
374C42—181no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—41no—
379C22×C3#380211yescharacteristic
380C22×C3C3×D3×D411yescharacteristic
381C22×C3C3×D3×D411yescharacteristic
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—62no—
391C3×C4—62no—
392C3×C4—92no—
393C3×C4—122no—
394C3×C4—121no—
395C3×C4—121no—
396C3×C4—121no—
397C3×C4—121no—
398C3×C4—121no—
399C3×C4—121no—
400C3×C4—181no—
401C3×C4—241no—
402C3×C4—241no—
403C3×C4—241no—
404C3×C4D3×Dic611yescharacteristic
405C3×C4#380411yescharacteristic
406C3×C4C2×C3×Dic611yescharacteristic
407C3×C4C3×#23511yescharacteristic
408C3×C4C3×#23711yescharacteristic
409C3×C4C3×D3×Q811yescharacteristic
410Dic3—22no—
411Dic3—32no—
412Dic3—32no—
413Dic3—41no—
414Dic3—41no—
415Dic3—41no—
416Dic3—62no—
417Dic3—61no—
418Dic3—61no—
419Dic3—121no—
420C32—21no—
421C32—21no—
422C32—21no—
423C32—41no—
424C32#479111yescharacteristic, complemented
425C32#485411yescharacteristic, complemented
426C32C3×#47511yescharacteristic, complemented
427Q8—32no—
428Q8—61no—
429Q8—362no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—91no—
435C2×C4—91no—
436C2×C4C2×C3×D3211yescharacteristic
437C8—21no—
438C8—181no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3#6379611yescharacteristic
452C2×C3#6381111yescharacteristic
453C2×C3#6389911yescharacteristic
454C2×C3C3×#87811yescharacteristic
455C2×C3C3×#89611yescharacteristic
456C2×C3C3×D3×SD1611yescharacteristic
457C2×C3C3×D3×SD1611yescharacteristic
458C2×C3C3×#91511yescharacteristic
459C2×C3C3×#91811yescharacteristic
460C22C3×#380211yescharacteristic
461C4—62no—
462C4—92no—
463C4—121no—
464C4—121no—
465C4—121no—
466C4—181no—
467C4C3×D3×Dic611yescharacteristic
468C4C3×#380411yescharacteristic
469C3—21no—
470C3—21no—
471C3—21no—
472C3—41no—
473C3#1059081611yescharacteristic, direct product factor
474C3C3×#479111yescharacteristic, complemented
475C3C3×#485411yescharacteristic, complemented
476C2C3×#6379611yescharacteristic
477C2C3×#6381111yescharacteristic
478C2C3×#6389911yescharacteristic
subgroupG/H
centerC22×C3#3802
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#475
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
10. #10590816C3direct (N × H)
54. C32×Dic6Q8semidirect
176. C3×Dic6Dic6semidirect
177. C3×Dic6C3×Q8semidirect
289. C33#475semidirect
331. Dic6C3×Dic6semidirect
424. C32#4791semidirect
425. C32#4854semidirect
426. C32C3×#475semidirect
473. C3#10590816direct (N × H)
474. C3C3×#4791semidirect
475. C3C3×#4854semidirect

Maximal subgroups

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

subgroupindex
C3×#639192
C3×#638432
C3×#636412
C32×#8172
C32×#8162
C3×#635742
C3×#635652
C3×#48543
C3×#47913
#105908163