Finite Group Databasebeta

← C3×#10590310 #13157299

Subgroups — C3×#10590310 #13157299

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#869C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
4C3×#63576C212yesmaximal
5C3×#63565C212yesmaximal
6C3×#5091—31nomaximal
7C3×#5021—31nomaximal
8#10590310C311yesmaximal, characteristic, direct product factor
9C2×C32×Dic6—21no—
10C2×C32×Dic6C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3—21no—
14C32×C4⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C2212yes—
16C32×Dic3⋊C4—21no—
17C32×C4×Dic3—21no—
18C32×C4⋊Q8—31no—
19C32×C4×Q8—31no—
20C3×#869—21no—
21C3×#869D311yescharacteristic
22C3×#869C2×C311yescharacteristic
23C3×C4×Dic6—21no—
24C3×C4×Dic6D311yescharacteristic
25C3×C4×Dic6C2×C311yescharacteristic
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
27C3×#835—32no—
28C3×#810—32no—
29C3×#807—32no—
30C3×#807—32no—
31C3×C4⋊(C3⋊C8)—31no—
32C3×C4⋊(C3⋊C8)—31no—
33C3×C4⋊(C3⋊C8)—61no—
34#63576C2×C312yes—
35#63565C2×C312yes—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C32×Dic3—21no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—21no—
42C32×Dic6—21no—
43C32×Dic6—42no—
44C32×Dic6Q812yescomplemented
45C3×C3⋊(C3⋊C8)—22no—
46C3×#477—91no—
47#5091—31no—
48#5021—31no—
49C2×C32×Q8—31no—
50C2×C32×Q8—61no—
51C2×C3×Dic6—21no—
52C2×C3×Dic6—22no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6C2×D311yescharacteristic
56C2×C3×Dic6C22×C311yescharacteristic
57C32×C4⋊C4—31no—
58C32×C4⋊C4—32no—
59C32×C4⋊C4—61no—
60C32×C4⋊C4—61no—
61C32×C42—21no—
62C32×C42—21no—
63C32×C42—21no—
64C32×C42—41no—
65C32×C42—61no—
66C32×C42C2×D311yescharacteristic
67C32×C42C2×D311yescharacteristic
68C32×C42C22×C311yescharacteristic
69C2×C3⋊(C3⋊C8)C22×C312yes—
70C3×C4⋊Dic3—22no—
71C3×C4⋊Dic3—21no—
72C3×C4⋊Dic3—22no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3C2×D311yescharacteristic
76C3×C4⋊Dic3C2×D312yes—
77C3×C4⋊Dic3C22×C312yes—
78C3×C4⋊Dic3C22×C311yescharacteristic
79C3×Dic3⋊C4—22no—
80C3×Dic3⋊C4—21no—
81C3×Dic3⋊C4—21no—
82C3×C4×Dic3—22no—
83C3×C4×Dic3—21no—
84C3×C4×Dic3—21no—
85C2×C3×C3⋊C8—32no—
86C2×C3×C3⋊C8—32no—
87C2×C3×C3⋊C8—62no—
88C22×C33C2×D411yescharacteristic
89C33×C4—21no—
90C33×C4Q1612yes—
91C33×C4C2×Q811yescharacteristic
92C33×C4#4111yescharacteristic
93C32×Dic3—22no—
94C32×Dic3—42no—
95C32×Dic3—41no—
96C3×C4⋊Q8—31no—
97C3×C4⋊Q8—31no—
98C3×C4⋊Q8—61no—
99C3×C4×Q8—31no—
100C3×C4×Q8—31no—
101C3×C4×Q8—61no—
102#869C3×D311yescharacteristic
103C4×Dic6C3×D311yescharacteristic
104C3×#107—92no—
105C3×C4⋊C8—91no—
106C3×#103—92no—
107#835—32no—
108#810—32no—
109#807—32no—
110#807—32no—
111C4⋊(C3⋊C8)—31no—
112C4⋊(C3⋊C8)—31no—
113C4⋊(C3⋊C8)—61no—
114C32×Q8—32no—
115C32×Q8—61no—
116C32×Q8—122no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—41no—
127C2×C32×C4—41no—
128C2×C32×C4—41no—
129C2×C32×C4—61no—
130C2×C32×C4—61no—
131C2×C32×C4—62no—
132C2×C32×C4#6711yescharacteristic
133C2×C32×C4#6711yescharacteristic
134C2×C32×C4#6711yescharacteristic
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C3×D411yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C23×C311yescharacteristic
141C2×C3×Dic3—22no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—22no—
150C3×Dic6—22no—
151C3×Dic6—21no—
152C3×Dic6—21no—
153C3×Dic6—22no—
154C3×Dic6—44no—
155C3×Dic6—42no—
156C3×Dic6—42no—
157C3×Dic6Dic612yescomplemented
158C3×Dic6C3×Q812yescomplemented
159C3⋊(C3⋊C8)—22no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—62no—
162C3×C3⋊C8—122no—
163#477—91noSylow
164C2×C33C22⋊Q811yescharacteristic
165C2×C33C2×Q1611yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C2×C3×Q8—31no—
168C2×C3×Q8—31no—
169C2×C3×Q8—61no—
170C2×C3×Q8—62no—
171C2×C3×Q8—61no—
172C2×C3×Q8—61no—
173C2×Dic6—21no—
174C2×Dic6C2×C3×D311yescharacteristic
175C2×C3×C8—92no—
176C3×C4⋊C4—32no—
177C3×C4⋊C4—31no—
178C3×C4⋊C4—31no—
179C3×C4⋊C4—32no—
180C3×C4⋊C4—61no—
181C3×C4⋊C4—62no—
182C3×C4⋊C4—62no—
183C3×C4⋊C4—62no—
184C3×C4⋊C4—61no—
185C3×C4⋊C4—61no—
186C3×C4⋊C4—61no—
187C3×C4⋊C4—61no—
188C3×C42—21no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42—41no—
192C3×C42—62no—
193C3×C42—61no—
194C3×C42—61no—
195C3×C42D3211yescharacteristic
196C3×C42C2×C3×D311yescharacteristic
197C3×C42C2×C3×D311yescharacteristic
198C4⋊Dic3—21no—
199C4⋊Dic3C2×C3×D311yescharacteristic
200C4⋊Dic3C2×C3×D312yes—
201Dic3⋊C4—21no—
202C4×Dic3—21no—
203C2×C3⋊C8—32no—
204C2×C3⋊C8—32no—
205C2×C3⋊C8—62no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—41no—
210C22×C32C2×#6711yescharacteristic
211C22×C32C2×#6711yescharacteristic
212C22×C32C2×C3×D411yescharacteristic
213C32×C4—21no—
214C32×C4—22no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—22no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—42no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4—41no—
230C32×C4—41no—
231C32×C4—42no—
232C32×C4—62no—
233C32×C4—121no—
234C32×C4—122no—
235C32×C4#21612yes—
236C32×C4#21612yes—
237C32×C4C3×Q1612yes—
238C32×C4C2×Dic611yescharacteristic
239C32×C4#23511yescharacteristic
240C32×C4D3×Q811yescharacteristic
241C32×C4#23911yescharacteristic
242C32×C4C2×C3×Q811yescharacteristic
243C32×C4C3×#4111yescharacteristic
244C3×Dic3—24no—
245C3×Dic3—22no—
246C3×Dic3—22no—
247C3×Dic3—44no—
248C3×Dic3—42no—
249C3×Dic3—42no—
250C3×Dic3—41no—
251C3×Dic3—42no—
252C3×Dic3—41no—
253C4⋊Q8—31no—
254C4×Q8—31no—
255#107—92no—
256C4⋊C8—91no—
257#103—92no—
258C33#47711yescharacteristic, complemented, Sylow
259C3×Q8—32no—
260C3×Q8—32no—
261C3×Q8—62no—
262C3×Q8—62no—
263C3×Q8—61no—
264C3×Q8—61no—
265C3×Q8—124no—
266C3×Q8—122no—
267C3×Q8—122no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—62no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—62no—
284C2×C3×C4—64no—
285C2×C3×C4—61no—
286C2×C3×C4—62no—
287C2×C3×C4—61no—
288C2×C3×C4—62no—
289C2×C3×C4#62511yescharacteristic
290C2×C3×C4#62511yescharacteristic
291C2×C3×C4C3×#6711yescharacteristic
292C2×C3×C4C3×#6711yescharacteristic
293C2×C3×C4C3×#6711yescharacteristic
294C2×C3×C4C3×#6711yescharacteristic
295C2×C3×C4C2×D3211yescharacteristic
296C2×C3×C4C22×C3×D311yescharacteristic
297C2×C3×C4C22×C3×D311yescharacteristic
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—22no—
301Dic6—21no—
302Dic6—42no—
303Dic6C3×Dic612yescomplemented
304C3×C8—182no—
305C3⋊C8—62no—
306C3⋊C8—62no—
307C3⋊C8—122no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32#92411yescharacteristic
321C2×C32#93211yescharacteristic
322C2×C32C2×#21611yescharacteristic
323C2×C32C2×#21611yescharacteristic
324C2×C32#94611yescharacteristic
325C2×C32#94911yescharacteristic
326C2×C32C3×C22⋊Q811yescharacteristic
327C2×C32C2×C3×Q1611yescharacteristic
328C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
329C2×Q8—31no—
330C2×Q8—61no—
331C2×C8—92no—
332C4⋊C4—32no—
333C4⋊C4—31no—
334C4⋊C4—61no—
335C4⋊C4—61no—
336C42—61no—
337C42C3×D3211yescharacteristic
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—41no—
342C22×C3C2×#62511yescharacteristic
343C22×C3C2×C3×#6711yescharacteristic
344C22×C3C2×C3×#6711yescharacteristic
345C3×C4—22no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—42no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4—42no—
364C3×C4—64no—
365C3×C4—62no—
366C3×C4—62no—
367C3×C4—124no—
368C3×C4—122no—
369C3×C4—121no—
370C3×C4—122no—
371C3×C4—122no—
372C3×C4—121no—
373C3×C4#371812yes—
374C3×C4C3×#21612yes—
375C3×C4C3×#21612yes—
376C3×C4D3×Dic611yescharacteristic
377C3×C4#379911yescharacteristic
378C3×C4C2×C3×Dic611yescharacteristic
379C3×C4C3×#23511yescharacteristic
380C3×C4C3×D3×Q811yescharacteristic
381C3×C4C3×#23911yescharacteristic
382Dic3—22no—
383Dic3—42no—
384Dic3—41no—
385C32—21no—
386C32—21no—
387C32—21no—
388C32—41no—
389C32#502111yescharacteristic, complemented
390C32#509111yescharacteristic, complemented
391C32C3×#47711yescharacteristic, complemented
392Q8—32no—
393Q8—61no—
394Q8—122no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—62no—
398C2×C4C3×#62511yescharacteristic
399C2×C4C3×#62511yescharacteristic
400C2×C4C2×C3×D3211yescharacteristic
401C8—182no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3#6382311yescharacteristic
415C2×C3C2×#371811yescharacteristic
416C2×C3#6385611yescharacteristic
417C2×C3C3×#92411yescharacteristic
418C2×C3C3×#93211yescharacteristic
419C2×C3C2×C3×#21611yescharacteristic
420C2×C3C2×C3×#21611yescharacteristic
421C2×C3C3×#94611yescharacteristic
422C2×C3C3×#94911yescharacteristic
423C22C2×C3×#62511yescharacteristic
424C4—21no—
425C4—62no—
426C4—122no—
427C4—121no—
428C4C3×#371812yes—
429C4C3×D3×Dic611yescharacteristic
430C4C3×#379911yescharacteristic
431C3—21no—
432C3—21no—
433C3—21no—
434C3—41no—
435C3#1059031011yescharacteristic, direct product factor
436C3C3×#502111yescharacteristic, complemented
437C3C3×#509111yescharacteristic, complemented
438C2C3×#6382311yescharacteristic
439C2C2×C3×#371811yescharacteristic
440C2C3×#6385611yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#477
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
8. #10590310C3direct (N × H)
44. C32×Dic6Q8semidirect
157. C3×Dic6Dic6semidirect
158. C3×Dic6C3×Q8semidirect
258. C33#477semidirect
303. Dic6C3×Dic6semidirect
389. C32#5021semidirect
390. C32#5091semidirect
391. C32C3×#477semidirect
435. C3#10590310direct (N × H)
436. C3C3×#5021semidirect
437. C3C3×#5091semidirect

Maximal subgroups

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

subgroupindex
C32×#8692
C32×C4×Dic62
C3×C4⋊(C3⋊(C3⋊C8))2
C3×#635762
C3×#635652
C3×#50913
C3×#50213
#105903103