Finite Group Databasebeta

← C3×#10590284 #13157326

Subgroups — C3×#10590284 #13157326

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
1C3×C4×#634C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic
3C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic
4C3×#63575C212yesmaximal
5C3×#63567C212yesmaximal
6C3×#5091—31nomaximal
7C3×#4850—31nomaximal
8#10590284C311yesmaximal, characteristic, direct product factor
9C2×C3×#634C2211yescharacteristic
10C2×C32×Dic6—21no—
11C33×C42C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×#3750—21no—
14C3×C4×C3⋊Dic3—21no—
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2212yes—
17C2×C32×C3⋊C8C2212yes—
18C32×C4⋊Q8—31no—
19C4×#634C2×C311yescharacteristic
20C3×#869—21no—
21C3×#869D311yescharacteristic
22C3×#869C2×C311yescharacteristic
23C3×C4×Dic6—31no—
24C3×C4×Dic6—31no—
25C3×C4×Dic6—61no—
26C32×C4⋊C8—31no—
27C3×#818—32no—
28C3×#816—32no—
29C3×#810—32no—
30C3×#807—32no—
31C3×C4⋊(C3⋊C8)—21no—
32C3×C4⋊(C3⋊C8)D311yescharacteristic
33C3×C4⋊(C3⋊C8)C2×C311yescharacteristic
34#63575C2×C312yes—
35#63567C2×C312yes—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C3×#634—21no—
42C3×#634Q812yescomplemented
43C2×C32×Dic3—22no—
44C32×Dic6—42no—
45C32×C3⋊C8—22no—
46C3×#477—91no—
47#5091—31no—
48#4850—31no—
49C2×C32×Q8—61no—
50C2×#634C22×C311yescharacteristic
51C2×C3×Dic6—22no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—31no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—61no—
57C2×C32×C8—32no—
58C32×C4⋊C4—32no—
59C32×C4⋊C4—61no—
60C32×C42—21no—
61C32×C42—21no—
62C32×C42—21no—
63C32×C42—41no—
64C32×C42C2×D311yescharacteristic
65C32×C42C2×D311yescharacteristic
66C32×C42C22×C311yescharacteristic
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68#3750—21no—
69C4×C3⋊Dic3—21no—
70C3×C4⋊Dic3—22no—
71C3×C4⋊Dic3—22no—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—61no—
77C3×C4⋊Dic3C2×D312yes—
78C3×C4⋊Dic3C22×C312yes—
79C3×Dic3⋊C4—62no—
80C3×Dic3⋊C4—61no—
81C3×Dic3⋊C4—61no—
82C3×C4×Dic3—62no—
83C3×C4×Dic3—61no—
84C3×C4×Dic3—61no—
85C2×C3×C3⋊C8—22no—
86C2×C3×C3⋊C8C2×D312yes—
87C2×C3×C3⋊C8C22×C312yes—
88C22×C33C2×D411yescharacteristic
89C33×C4—21no—
90C33×C4Q1612yes—
91C33×C4C2×Q811yescharacteristic
92C33×C4#4111yescharacteristic
93C3×C3⋊Dic3—22no—
94C3×C3⋊Dic3—41no—
95C32×Dic3—42no—
96C3×C4⋊Q8—31no—
97C3×C4⋊Q8—31no—
98C3×C4⋊Q8—61no—
99C3×C4×Q8—91no—
100#869C3×D311yescharacteristic
101C4×Dic6—31no—
102C4×Dic6—31no—
103C4×Dic6—61no—
104C3×#107—92no—
105C3×C4⋊C8—31no—
106C3×C4⋊C8—31no—
107C3×C4⋊C8—61no—
108C3×#103—92no—
109#818—32no—
110#816—32no—
111#810—32no—
112#807—32no—
113C4⋊(C3⋊C8)C3×D311yescharacteristic
114C32×Q8—122no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
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—41no—
125C2×C32×C4—41no—
126C2×C32×C4—41no—
127C2×C32×C4—62no—
128C2×C32×C4D1211yescharacteristic
129C2×C32×C4D1211yescharacteristic
130C2×C32×C4#6711yescharacteristic
131C2×C32×C4#6711yescharacteristic
132C2×C32×C4C3×D411yescharacteristic
133C2×C32×C4C3×D411yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C23×C311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139#634—21no—
140#634C3×Q812yescomplemented
141C2×C3×Dic3—24no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C3×Dic6—32no—
151C3×Dic6—32no—
152C3×Dic6—44no—
153C3×Dic6—42no—
154C3×Dic6—42no—
155C3×Dic6—62no—
156C3×Dic6—62no—
157C3×Dic6—61no—
158C3×Dic6—61no—
159C32×C8—62no—
160C3×C3⋊C8—22no—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—42no—
163#477—91noSylow
164C2×C33C22⋊Q811yescharacteristic
165C2×C33C2×Q1611yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C2×C3×Q8—62no—
168C2×C3×Q8—61no—
169C2×C3×Q8—61no—
170C2×C3×Q8—91no—
171C2×Dic6—21no—
172C2×Dic6—31no—
173C2×Dic6—31no—
174C2×Dic6—61no—
175C2×C3×C8—32no—
176C2×C3×C8—32no—
177C2×C3×C8—62no—
178C3×C4⋊C4—32no—
179C3×C4⋊C4—32no—
180C3×C4⋊C4—62no—
181C3×C4⋊C4—62no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4—61no—
184C3×C4⋊C4—91no—
185C3×C4⋊C4—181no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42—21no—
189C3×C42—41no—
190C3×C42—181no—
191C3×C42D3211yescharacteristic
192C3×C42C2×C3×D311yescharacteristic
193C3×C42C2×C3×D311yescharacteristic
194C4⋊Dic3—21no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197C4⋊Dic3—61no—
198C4⋊Dic3C2×C3×D312yes—
199Dic3⋊C4—62no—
200Dic3⋊C4—61no—
201Dic3⋊C4—61no—
202C4×Dic3—62no—
203C4×Dic3—61no—
204C4×Dic3—61no—
205C2×C3⋊C8C2×C3×D312yes—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—41no—
210C22×C32C2×D1211yescharacteristic
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—122no—
233C32×C4Dic1212yes—
234C32×C4#21612yes—
235C32×C4C3×Q1612yes—
236C32×C4#23711yescharacteristic
237C32×C4D3×Q811yescharacteristic
238C32×C4D3×Q811yescharacteristic
239C32×C4#23911yescharacteristic
240C32×C4C2×C3×Q811yescharacteristic
241C32×C4C3×#4111yescharacteristic
242C3⋊Dic3—22no—
243C3⋊Dic3—41no—
244C3×Dic3—44no—
245C3×Dic3—42no—
246C3×Dic3—42no—
247C3×Dic3—64no—
248C3×Dic3—62no—
249C3×Dic3—62no—
250C3×Dic3—122no—
251C3×Dic3—121no—
252C3×Dic3—121no—
253C4⋊Q8—31no—
254C4×Q8—91no—
255#107—92no—
256C4⋊C8—31no—
257#103—92no—
258C33#47711yescharacteristic, complemented, Sylow
259C3×Q8—92no—
260C3×Q8—124no—
261C3×Q8—122no—
262C3×Q8—122no—
263C3×Q8—181no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—64no—
277C2×C3×C4—62no—
278C2×C3×C4—62no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4#62611yescharacteristic
282C2×C3×C4#62611yescharacteristic
283C2×C3×C4C3×D1211yescharacteristic
284C2×C3×C4C3×D1211yescharacteristic
285C2×C3×C4C3×#6711yescharacteristic
286C2×C3×C4C3×#6711yescharacteristic
287C2×C3×C4C2×D3211yescharacteristic
288C2×C3×C4C22×C3×D311yescharacteristic
289C2×C3×C4C22×C3×D311yescharacteristic
290C2×Dic3—22no—
291C2×Dic3—62no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—62no—
295C2×Dic3—61no—
296C2×Dic3—61no—
297Dic6—32no—
298Dic6—32no—
299Dic6—42no—
300Dic6—62no—
301Dic6—62no—
302Dic6—61no—
303Dic6—61no—
304C3×C8—62no—
305C3×C8—62no—
306C3×C8—122no—
307C3⋊C8—22no—
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#89711yescharacteristic
321C2×C32C2×Dic1211yescharacteristic
322C2×C32#90811yescharacteristic
323C2×C32#93211yescharacteristic
324C2×C32C2×#21611yescharacteristic
325C2×C32#94611yescharacteristic
326C2×C32C3×C22⋊Q811yescharacteristic
327C2×C32C2×C3×Q1611yescharacteristic
328C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
329C2×Q8—61no—
330C2×Q8—91no—
331C2×C8—32no—
332C4⋊C4—32no—
333C4⋊C4—61no—
334C4⋊C4—91no—
335C4⋊C4—181no—
336C42—181no—
337C42C3×D3211yescharacteristic
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—41no—
342C22×C3C2×#62611yescharacteristic
343C22×C3C2×C3×D1211yescharacteristic
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—124no—
365C3×C4—122no—
366C3×C4—122no—
367C3×C4—182no—
368C3×C4—361no—
369C3×C4#371912yes—
370C3×C4C3×Dic1212yes—
371C3×C4C3×#21612yes—
372C3×C4#379611yescharacteristic
373C3×C4#379711yescharacteristic
374C3×C4C3×#23711yescharacteristic
375C3×C4C3×D3×Q811yescharacteristic
376C3×C4C3×D3×Q811yescharacteristic
377C3×C4C3×#23911yescharacteristic
378Dic3—42no—
379Dic3—64no—
380Dic3—62no—
381Dic3—62no—
382Dic3—122no—
383Dic3—121no—
384Dic3—121no—
385C32—21no—
386C32—21no—
387C32—21no—
388C32—41no—
389C32#485011yescharacteristic, complemented
390C32#509111yescharacteristic, complemented
391C32C3×#47711yescharacteristic, complemented
392Q8—92no—
393Q8—122no—
394Q8—181no—
395C2×C4—62no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4C3×#62611yescharacteristic
399C2×C4C3×#62611yescharacteristic
400C2×C4C2×C3×D3211yescharacteristic
401C8—62no—
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#6383011yescharacteristic
415C2×C3C2×#371911yescharacteristic
416C2×C3#6387111yescharacteristic
417C2×C3C3×#89711yescharacteristic
418C2×C3C2×C3×Dic1211yescharacteristic
419C2×C3C3×#90811yescharacteristic
420C2×C3C3×#93211yescharacteristic
421C2×C3C2×C3×#21611yescharacteristic
422C2×C3C3×#94611yescharacteristic
423C22C2×C3×#62611yescharacteristic
424C4—21no—
425C4—122no—
426C4—182no—
427C4—361no—
428C4C3×#371912yes—
429C4C3×#379611yescharacteristic
430C4C3×#379711yescharacteristic
431C3—21no—
432C3—21no—
433C3—21no—
434C3—41no—
435C3#1059028411yescharacteristic, direct product factor
436C3C3×#485011yescharacteristic, complemented
437C3C3×#509111yescharacteristic, complemented
438C2C3×#6383011yescharacteristic
439C2C2×C3×#371911yescharacteristic
440C2C3×#6387111yescharacteristic
subgroupG/H
centerC22×C3C2×#626
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. #10590284C3direct (N × H)
42. C3×#634Q8semidirect
140. #634C3×Q8semidirect
258. C33#477semidirect
389. C32#4850semidirect
390. C32#5091semidirect
391. C32C3×#477semidirect
435. C3#10590284direct (N × H)
436. C3C3×#4850semidirect
437. C3C3×#5091semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6342
C32×#8692
C32×C4⋊(C3⋊C8)2
C3×#635752
C3×#635672
C3×#50913
C3×#48503
#105902843