Finite Group Databasebeta

← C3×#10590321 #13157336

Subgroups — C3×#10590321 #13157336

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64081C211yesmaximal, characteristic
2C2×C3×#3712C211yesmaximal, characteristic
3C2×C3×#3710C211yesmaximal, characteristic
4C32×#803C211yesmaximal, characteristic, complemented
5C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
6C3×#63555C211yesmaximal, characteristic, complemented
7C3×#63553C211yesmaximal, characteristic, complemented
8C3×#4708—31nomaximal
9C3×#4689—31nomaximal
10#10590321C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C3×#3752C412yes—
14C3×#3750C412yes—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3712C414yescomplemented
21C3×#3710C414yescomplemented
22#64081C2×C311yescharacteristic
23C2×C3×ΓL(1,𝔽25)—31no—
24C2×C3×ΓL(1,𝔽25)—31no—
25C2×C3×D3×C8—31no—
26C2×C3×D3×C8—31no—
27C3×#872—31no—
28C3×#872—31no—
29C3×#872—61no—
30C2×#3712C2×C311yescharacteristic
31C2×#3710C2×C311yescharacteristic
32C32×#97—31no—
33C32×C4×C8—31no—
34C3×#815—31no—
35C3×#815—31no—
36C3×C8×Dic3—31no—
37C3×C8×Dic3—31no—
38C3×#803—21no—
39C3×#803D311yescharacteristic, complemented
40C3×#803C2×C311yescharacteristic, complemented
41C3×C4×C3⋊C8—21no—
42C3×C4×C3⋊C8D311yescharacteristic, complemented
43C3×C4×C3⋊C8C2×C311yescharacteristic, complemented
44#63555C2×C311yescharacteristic, complemented
45#63553C2×C311yescharacteristic, complemented
46C22×C3×C3⋊D3C2×C411yescharacteristic
47C2×C33×C4C2×C412yes—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3C2×C412yes—
50C2×C3×C3⋊Dic3C2×C411yescharacteristic
51C3×C4×C3⋊D3C2×C412yes—
52C3×C4×C3⋊D3C2×C412yes—
53C32×C3⋊C8C2×C412yes—
54C32×C3⋊C8C2×C412yes—
55C32×C3⋊C8C2×C412yes—
56C32×C3⋊C8C2×C412yes—
57C3×#405—91no—
58#4708—31no—
59#4689—31no—
60C2×C4×C3⋊D3C22×C311yescharacteristic
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—61no—
64C2×C32×C8—31no—
65C2×C32×C8—31no—
66C2×C32×C8—31no—
67C2×C32×C8—31no—
68C32×C42—21no—
69C32×C42—21no—
70C32×C42—21no—
71C32×C42—41no—
72C32×C42C2×D311yescharacteristic
73C32×C42C2×D311yescharacteristic
74C32×C42C22×C311yescharacteristic
75#3752C3×C412yes—
76#3750C3×C412yes—
77C4×C3⋊Dic3C22×C311yescharacteristic
78C3×(C2×D3)⋊C4—32no—
79C3×(C2×D3)⋊C4—32no—
80C3×(C2×D3)⋊C4—62no—
81C3×Dic3⋊C4—32no—
82C3×Dic3⋊C4—32no—
83C3×Dic3⋊C4—62no—
84C3×C4×Dic3—31no—
85C3×C4×Dic3—31no—
86C3×C4×Dic3—61no—
87C2×C3×C3⋊C8—21no—
88C2×C3×C3⋊C8—21no—
89C2×C3×C3⋊C8—21no—
90C2×C3×C3⋊C8—21no—
91C2×C3×C3⋊C8C2×D311yescharacteristic
92C2×C3×C3⋊C8C2×D311yescharacteristic
93C2×C3×C3⋊C8C2×D311yescharacteristic
94C2×C3×C3⋊C8C2×D311yescharacteristic
95C2×C3×C3⋊C8C22×C311yescharacteristic
96C2×C3×C3⋊C8C22×C311yescharacteristic
97C2×C3×C3⋊C8C22×C311yescharacteristic
98C2×C3×C3⋊C8C22×C311yescharacteristic
99C3×ΓL(1,𝔽25)—34no—
100C3×ΓL(1,𝔽25)—34no—
101C3×D3×C8—34no—
102C3×D3×C8—34no—
103#3712C3×C414yescomplemented
104#3710C3×C414yescomplemented
105C22×C33C22×C411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3C4212yes—
108C33×C4—22no—
109C33×C4C22×C411yescharacteristic
110C33×C4C22×C411yescharacteristic
111C3×C3⋊Dic3—22no—
112C3×C3⋊Dic3C4212yes—
113C2×C3×M16—91no—
114C22×C3×C8—91no—
115C3×#117—91no—
116C2×ΓL(1,𝔽25)—31no—
117C2×ΓL(1,𝔽25)—31no—
118C2×D3×C8—31no—
119C2×D3×C8—31no—
120#872—31no—
121#872—31no—
122#872—61no—
123C3×#97—31no—
124C3×#97—31no—
125C3×#97—61no—
126C3×#97—91no—
127C3×C4×C8—31no—
128C3×C4×C8—31no—
129C3×C4×C8—61no—
130C3×C4×C8—91no—
131#815—31no—
132#815—31no—
133C8×Dic3—31no—
134C8×Dic3—31no—
135#803C3×D311yescharacteristic, complemented
136C4×C3⋊C8C3×D311yescharacteristic, complemented
137C22×C3⋊D3C2×C3×C411yescharacteristic
138C22×C3×D3—31no—
139C22×C3×D3—31no—
140C22×C3×D3—61no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—42no—
148C2×C32×C4—41no—
149C2×C32×C4C4×D312yes—
150C2×C32×C4C4×D312yes—
151C2×C32×C4C2×C3×C412yes—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C23×C311yescharacteristic
155C2×C3⋊Dic3C2×C3×C412yes—
156C2×C3⋊Dic3C2×C3×C411yescharacteristic
157C4×C3⋊D3C2×C3×C412yes—
158C4×C3⋊D3C2×C3×C412yes—
159C2×C3×Dic3—31no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—62no—
165C3×C4×D3—32no—
166C3×C4×D3—32no—
167C3×C4×D3—32no—
168C3×C4×D3—32no—
169C3×C4×D3—62no—
170C3×C4×D3—62no—
171C32×C8—32no—
172C32×C8—32no—
173C32×C8—32no—
174C32×C8—32no—
175C3×C3⋊C8—22no—
176C3×C3⋊C8—22no—
177C3×C3⋊C8—22no—
178C3×C3⋊C8—22no—
179C3×C3⋊C8C4×D312yes—
180C3×C3⋊C8C4×D312yes—
181C3×C3⋊C8C4×D312yes—
182C3×C3⋊C8C4×D312yes—
183C3×C3⋊C8C2×C3×C412yes—
184C3×C3⋊C8C2×C3×C412yes—
185C3×C3⋊C8C2×C3×C412yes—
186C3×C3⋊C8C2×C3×C412yes—
187#405—91noSylow
188C2×C33C2×C4211yescharacteristic
189C2×C33#13112yes—
190C3×C3⋊D3—22no—
191C22×C3×C4—91no—
192C2×C4×D3—31no—
193C2×C4×D3—31no—
194C2×C4×D3—61no—
195C3×M16—94no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—31no—
199C2×C3×C8—31no—
200C2×C3×C8—31no—
201C2×C3×C8—31no—
202C2×C3×C8—31no—
203C2×C3×C8—31no—
204C2×C3×C8—61no—
205C2×C3×C8—61no—
206C2×C3×C8—61no—
207C2×C3×C8—61no—
208C2×C3×C8—94no—
209C3×C4⋊C4—92no—
210C3×C22⋊C4—92no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—41no—
215C3×C42—91no—
216C3×C42D3211yescharacteristic
217C3×C42C2×C3×D311yescharacteristic
218C3×C42C2×C3×D311yescharacteristic
219(C2×D3)⋊C4—32no—
220(C2×D3)⋊C4—32no—
221(C2×D3)⋊C4—62no—
222Dic3⋊C4—32no—
223Dic3⋊C4—32no—
224Dic3⋊C4—62no—
225C4×Dic3—31no—
226C4×Dic3—31no—
227C4×Dic3—61no—
228C2×C3⋊C8C2×C3×D311yescharacteristic
229C2×C3⋊C8C2×C3×D311yescharacteristic
230C2×C3⋊C8C2×C3×D311yescharacteristic
231C2×C3⋊C8C2×C3×D311yescharacteristic
232ΓL(1,𝔽25)—34no—
233ΓL(1,𝔽25)—34no—
234D3×C8—34no—
235D3×C8—34no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—41no—
240C22×C32C2×C4×D311yescharacteristic
241C22×C32C2×C4×D311yescharacteristic
242C22×C32C22×C3×C411yescharacteristic
243C2×C3⋊D3—22no—
244C2×C3⋊D3C3×C4212yes—
245C2×C3×D3—32no—
246C2×C3×D3—32no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—122no—
251C32×C4—22no—
252C32×C4—24no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—41no—
262C32×C4—41no—
263C32×C4—42no—
264C32×C4—42no—
265C32×C4—44no—
266C32×C4C2×C4×D311yescharacteristic
267C32×C4C2×C4×D311yescharacteristic
268C32×C4C2×C4×D311yescharacteristic
269C32×C4C2×C4×D311yescharacteristic
270C32×C4C22×C3×C411yescharacteristic
271C32×C4C22×C3×C411yescharacteristic
272C3⋊Dic3—22no—
273C3⋊Dic3C3×C4212yes—
274C3×Dic3—32no—
275C3×Dic3—32no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C3×Dic3—122no—
280C2×M16—91no—
281C22×C8—91no—
282#117—91no—
283#97—31no—
284#97—91no—
285C4×C8—31no—
286C4×C8—91no—
287C33#40511yescharacteristic, complemented, Sylow
288C23×C3—91no—
289C22×D3—31no—
290C22×D3—31no—
291C22×D3—61no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—91no—
301C2×C3×C4—92no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4#62412yes—
305C2×C3×C4C3×C4×D312yes—
306C2×C3×C4C3×C4×D312yes—
307C2×C3×C4C2×D3211yescharacteristic
308C2×C3×C4C22×C3×D311yescharacteristic
309C2×C3×C4C22×C3×D311yescharacteristic
310C2×Dic3—31no—
311C2×Dic3—31no—
312C2×Dic3—32no—
313C2×Dic3—32no—
314C2×Dic3—61no—
315C2×Dic3—62no—
316C4×D3—32no—
317C4×D3—32no—
318C4×D3—32no—
319C4×D3—32no—
320C4×D3—62no—
321C4×D3—62no—
322C3×C8—32no—
323C3×C8—32no—
324C3×C8—32no—
325C3×C8—32no—
326C3×C8—32no—
327C3×C8—32no—
328C3×C8—32no—
329C3×C8—32no—
330C3×C8—62no—
331C3×C8—62no—
332C3×C8—62no—
333C3×C8—62no—
334C3⋊C8C3×C4×D312yes—
335C3⋊C8C3×C4×D312yes—
336C3⋊C8C3×C4×D312yes—
337C3⋊C8C3×C4×D312yes—
338C2×C32—22no—
339C2×C32—22no—
340C2×C32—22no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—41no—
345C2×C32—42no—
346C2×C32C42×D311yescharacteristic
347C2×C32C42×D311yescharacteristic
348C2×C32#90112yes—
349C2×C32#90712yes—
350C2×C32C2×C3×C4211yescharacteristic
351C2×C32C3×#13112yes—
352C3⋊D3—22no—
353C3×D3—62no—
354C3×D3—62no—
355C3×D3—122no—
356C22×C4—91no—
357M16—94no—
358C2×C8—31no—
359C2×C8—31no—
360C2×C8—31no—
361C2×C8—31no—
362C2×C8—94no—
363C4⋊C4—92no—
364C22⋊C4—92no—
365C42—91no—
366C42C3×D3211yescharacteristic
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—41no—
371C22×C3—92no—
372C22×C3—182no—
373C22×C3C2×#62411yescharacteristic
374C22×C3C2×C3×C4×D311yescharacteristic
375C22×C3C2×C3×C4×D311yescharacteristic
376C2×D3—32no—
377C2×D3—32no—
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—122no—
382C3×C4—24no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—44no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—92no—
398C3×C4—182no—
399C3×C4C4×D3211yescharacteristic
400C3×C4C4×D3211yescharacteristic
401C3×C4C2×C3×C4×D311yescharacteristic
402C3×C4C2×C3×C4×D311yescharacteristic
403C3×C4C2×C3×C4×D311yescharacteristic
404C3×C4C2×C3×C4×D311yescharacteristic
405Dic3—32no—
406Dic3—32no—
407Dic3—62no—
408Dic3—62no—
409Dic3—62no—
410Dic3—122no—
411C32—21no—
412C32—21no—
413C32—21no—
414C32—41no—
415C32#468911yescharacteristic, complemented
416C32#470811yescharacteristic, complemented
417C32C3×#40511yescharacteristic, complemented
418C23—91no—
419C2×C4—91no—
420C2×C4—92no—
421C2×C4—92no—
422C2×C4—92no—
423C2×C4C3×#62412yes—
424C2×C4C2×C3×D3211yescharacteristic
425C8—32no—
426C8—32no—
427C8—32no—
428C8—32no—
429C2×C3—22no—
430C2×C3—22no—
431C2×C3—22no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—42no—
436C2×C3—41no—
437C2×C3—182no—
438C2×C3#6381712yes—
439C2×C3C4×#62411yescharacteristic
440C2×C3C3×C42×D311yescharacteristic
441C2×C3C3×C42×D311yescharacteristic
442C2×C3C3×#90112yes—
443C2×C3C3×#90712yes—
444D3—62no—
445D3—62no—
446D3—122no—
447C22—92no—
448C22—182no—
449C22C2×C3×#62411yescharacteristic
450C4—22no—
451C4—92no—
452C4—182no—
453C4C3×C4×D3211yescharacteristic
454C4C3×C4×D3211yescharacteristic
455C3—21no—
456C3—21no—
457C3—21no—
458C3—41no—
459C3#1059032111yescharacteristic, direct product factor
460C3C3×#468911yescharacteristic, complemented
461C3C3×#470811yescharacteristic, complemented
462C2—182no—
463C2C3×#6381712yes—
464C2C3×C4×#62411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#405
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
4. C32×#803C2semidirect
5. C32×C4×C3⋊C8C2semidirect
6. C3×#63555C2semidirect
7. C3×#63553C2semidirect
10. #10590321C3direct (N × H)
20. C3×#3712C4semidirect
21. C3×#3710C4semidirect
39. C3×#803D3semidirect
40. C3×#803C2×C3semidirect
42. C3×C4×C3⋊C8D3semidirect
43. C3×C4×C3⋊C8C2×C3semidirect
44. #63555C2×C3semidirect
45. #63553C2×C3semidirect
103. #3712C3×C4semidirect
104. #3710C3×C4semidirect
135. #803C3×D3semidirect
136. C4×C3⋊C8C3×D3semidirect
287. C33#405semidirect
415. C32#4689semidirect
416. C32#4708semidirect
417. C32C3×#405semidirect
459. C3#10590321direct (N × H)
460. C3C3×#4689semidirect
461. C3C3×#4708semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
C2×C3×#37122
C2×C3×#37102
C32×#8032
C32×C4×C3⋊C82
C3×#635552
C3×#635532
C3×#47083
C3×#46893
#105903213