Finite Group Databasebeta

← C3×#10591008 #13158025

Subgroups — C3×#10591008 #13158025

Order 1728.

856 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64306C211yesmaximal, characteristic
2C3×M16×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×C4×#624C211yesmaximal, characteristic, complemented
4C32×#825C212yesmaximal, complemented
5C3×#63569C212yesmaximal, complemented
6C3×D3×#104—32nomaximal
7#10591008C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C32×#235C2212yes—
10C3×#3811—21no—
11C2×C3×#624—21no—
12C3×#3805—21no—
13C3×#3802C411yescharacteristic, complemented
14C3×C4×D32—21no—
15C3×#3798—21no—
16C3×#3797C411yescharacteristic, complemented
17C3×#3796C412yescomplemented
18C33×M16C2211yescharacteristic, complemented
19C3×#3748C2211yescharacteristic, complemented
20C3×#3743—21no—
21C3×C8×C3⋊D3—21no—
22C32×C4×Dic3C2212yescomplemented
23C3×Dic32—21no—
24C3×D3×#41—32no—
25#64306C2×C311yescharacteristic
26M16×C3⋊D3C2×C311yescharacteristic, complemented
27C3×D3×M16—32no—
28C3×D3×M16—61no—
29C3×C42×D3—32no—
30C4×#624C2×C311yescharacteristic, complemented
31C32×#104—32no—
32C3×#837—32no—
33C3×#825—22no—
34C3×#825D312yescomplemented
35C3×#825C2×C312yescomplemented
36C3×#805—32no—
37#63569C2×C312yescomplemented
38C22×C3×C3⋊D3D411yescharacteristic
39C2×C3×D32—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4×C3⋊D3C2×C411yescharacteristic
43C3×C4×C3⋊D3C2×C411yescharacteristic
44C3×C4×C3⋊D3D411yescharacteristic
45C3×C4×C3⋊D3D411yescharacteristic
46C32×#67—22no—
47C2×C32×Dic3—22no—
48C32×D12C2×C412yescomplemented
49C32×C4×D3—22no—
50C32×Dic6C2×C412yescomplemented
51C3×#627—21no—
52C3×#626—22no—
53C3×#625—21no—
54C3×#624—24no—
55C3×#624—21no—
56C3×D3×Dic3—22no—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C2×C3×#104—91no—
60D3×#104—32no—
61C32×#41—32no—
62C2×C4×C3⋊D3C22×C311yescharacteristic
63C3×#239—32no—
64C3×D3×Q8—32no—
65C3×#237—32no—
66C3×#237—62no—
67C3×D3×D4—32no—
68C3×D3×D4—62no—
69C3×#235—22no—
70C3×#235—62no—
71C3×#235C2×D312yes—
72C3×#235C22×C312yes—
73C2×C3×C4×D3—32no—
74C2×C3×C4×D3—62no—
75C2×C3×C4×D3—62no—
76C2×C3×C4×D3—61no—
77#3811—21no—
78C2×#624—21no—
79#3805—21no—
80#3802C3×C411yescharacteristic, complemented
81C4×D32—21no—
82#3798—21no—
83#3797C3×C411yescharacteristic, complemented
84#3796C3×C412yescomplemented
85C32×M16—21no—
86C32×M16—22no—
87C32×M16—41no—
88C32×M16C2×D312yescomplemented
89C32×M16C22×C311yescharacteristic, complemented
90C32×C42—32no—
91#3748C22×C311yescharacteristic, complemented
92#3743—21no—
93C8×C3⋊D3—21no—
94C3×C4×Dic3—22no—
95C3×C4×Dic3—62no—
96C3×C4×Dic3C2×D312yescomplemented
97C3×C4×Dic3C22×C312yescomplemented
98C3×#208—32no—
99C3×#208—61no—
100C3×ΓL(1,𝔽25)—62no—
101C3×ΓL(1,𝔽25)—62no—
102C3×D3×C8—62no—
103C3×D3×C8—62no—
104Dic32—21no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3C22⋊C411yescharacteristic
108C2×C3×C3⋊D3C22⋊C411yescharacteristic
109C2×C32×D3—22no—
110C3×D32—42no—
111C33×C4C22×C411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3×C3⋊Dic3C22⋊C411yescharacteristic
114C3×C3⋊Dic3C22⋊C411yescharacteristic
115C32×Dic3—24no—
116C32×Dic3—22no—
117C2×C3×#41—91no—
118D3×#41—32no—
119C2×C3×M16—91no—
120C2×C3×C42—91no—
121D3×M16—32no—
122D3×M16—61no—
123C42×D3—32no—
124C3×#104—32no—
125C3×#104—32no—
126C3×#104—62no—
127C3×#104—92no—
128#837—32no—
129#825C3×D312yescomplemented
130#805—32no—
131C22×C3⋊D3C3×D411yescharacteristic
132C22×C3×D3—32no—
133C22×C3×D3—61no—
134C22×C3×D3—62no—
135C2×D32—21no—
136C32×Q8—32no—
137C32×D4—32no—
138C32×D4—62no—
139C2×C32×C4—21no—
140C2×C32×C4—22no—
141C2×C32×C4—41no—
142C2×C32×C4—62no—
143C2×C32×C4—62no—
144C2×C32×C4C22×D312yes—
145C2×C32×C4C23×C311yescharacteristic
146C2×C3⋊Dic3C3×D411yescharacteristic
147C4×C3⋊D3C2×C3×C411yescharacteristic
148C4×C3⋊D3C2×C3×C411yescharacteristic
149C4×C3⋊D3C3×D411yescharacteristic
150C4×C3⋊D3C3×D411yescharacteristic
151C3×#67—22no—
152C3×#67—22no—
153C3×#67—42no—
154C3×#67—62no—
155C3×#67—62no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—42no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—62no—
162C3×D12—22no—
163C3×D12—62no—
164C3×D12C4×D312yescomplemented
165C3×D12C2×C3×C412yescomplemented
166C3×C4×D3—22no—
167C3×C4×D3—22no—
168C3×C4×D3—32no—
169C3×C4×D3—32no—
170C3×C4×D3—32no—
171C3×C4×D3—32no—
172C3×C4×D3—42no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—62no—
178C3×C4×D3—68no—
179C3×C4×D3—62no—
180C3×Dic6—22no—
181C3×Dic6—62no—
182C3×Dic6C4×D312yescomplemented
183C3×Dic6C2×C3×C412yescomplemented
184#627—21no—
185#626—22no—
186#625—21no—
187#624—24no—
188#624—21no—
189D3×Dic3—22no—
190C32×C8—21no—
191C32×C8—22no—
192C32×C8—22no—
193C32×C8—42no—
194C32×C8—42no—
195C3⋊(C3⋊C8)—21no—
196C3×C3⋊C8—62no—
197C3×C3⋊C8—62no—
198C2×#104—91noSylow
199C2×C33—21no—
200C2×C33C2×C22⋊C411yescharacteristic
201C3×C3⋊D3—21no—
202C3×C3⋊D3#10412yescomplemented
203C32×D3—42no—
204C3×#41—32no—
205C3×#41—32no—
206C3×#41—62no—
207C3×#41—92no—
208C3×#41—181no—
209C3×#41—181no—
210C2×C3×Q8—91no—
211C2×C3×D4—91no—
212C2×C3×D4—181no—
213C22×C3×C4—91no—
214C22×C3×C4—181no—
215C22×C3×C4—181no—
216#239—32no—
217D3×Q8—32no—
218#237—32no—
219#237—62no—
220D3×D4—32no—
221D3×D4—62no—
222#235—62no—
223#235C2×C3×D312yes—
224C2×C4×D3—32no—
225C2×C4×D3—61no—
226C2×C4×D3—62no—
227C2×C4×D3—62no—
228C3×M16—21no—
229C3×M16—22no—
230C3×M16—41no—
231C3×M16—91no—
232C3×M16—181no—
233C3×M16D3211yescharacteristic, complemented
234C3×M16C2×C3×D312yescomplemented
235C2×C3×C8—181no—
236C3×C42—32no—
237C3×C42—32no—
238C3×C42—62no—
239C3×C42—181no—
240C4×Dic3—62no—
241C4×Dic3C2×C3×D312yescomplemented
242#208—32no—
243#208—61no—
244ΓL(1,𝔽25)—62no—
245ΓL(1,𝔽25)—62no—
246D3×C8—62no—
247D3×C8—62no—
248C22×C32—21no—
249C22×C32—22no—
250C22×C32—41no—
251C22×C32—62no—
252C22×C32D3×D412yes—
253C22×C32C2×C3×D411yescharacteristic
254C2×C3⋊D3—22no—
255C2×C3⋊D3C3×C22⋊C411yescharacteristic
256C2×C3⋊D3C3×C22⋊C411yescharacteristic
257C2×C3×D3—22no—
258C2×C3×D3—22no—
259C2×C3×D3—32no—
260C2×C3×D3—32no—
261C2×C3×D3—42no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—64no—
265C2×C3×D3—122no—
266C2×C3×D3—124no—
267D32—42no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—22no—
271C32×C4—22no—
272C32×C4—41no—
273C32×C4—41no—
274C32×C4—64no—
275C32×C4—62no—
276C32×C4C2×C4×D312yes—
277C32×C4D3×D412yes—
278C32×C4C22×C3×C411yescharacteristic
279C32×C4C2×C3×D411yescharacteristic
280C3⋊Dic3C3×C22⋊C411yescharacteristic
281C3⋊Dic3C3×C22⋊C411yescharacteristic
282C3×Dic3—24no—
283C3×Dic3—24no—
284C3×Dic3—22no—
285C3×Dic3—22no—
286C3×Dic3—32no—
287C3×Dic3—32no—
288C3×Dic3—42no—
289C3×Dic3—44no—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C2×#41—91no—
293C2×M16—91no—
294C2×C42—91no—
295#104—32no—
296#104—92no—
297C33C2×#10411yescharacteristic, complemented, Sylow
298C23×C3—91no—
299C23×C3—181no—
300C22×D3—32no—
301C22×D3—61no—
302C22×D3—62no—
303C3×Q8—32no—
304C3×Q8—32no—
305C3×Q8—62no—
306C3×Q8—181no—
307C3×D4—32no—
308C3×D4—32no—
309C3×D4—62no—
310C3×D4—62no—
311C3×D4—62no—
312C3×D4—122no—
313C3×D4—182no—
314C3×D4—181no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—91no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—122no—
328C2×C3×C4—122no—
329C2×C3×C4—181no—
330C2×C3×C4—184no—
331C2×C3×C4—182no—
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C22×C3×D312yes—
334#67—22no—
335#67—62no—
336#67—62no—
337C2×Dic3—22no—
338C2×Dic3—32no—
339C2×Dic3—61no—
340C2×Dic3—62no—
341D12—62no—
342D12C3×C4×D312yescomplemented
343C4×D3—22no—
344C4×D3—32no—
345C4×D3—32no—
346C4×D3—32no—
347C4×D3—32no—
348C4×D3—61no—
349C4×D3—61no—
350C4×D3—61no—
351C4×D3—61no—
352C4×D3—62no—
353C4×D3—68no—
354C4×D3—62no—
355Dic6—62no—
356Dic6C3×C4×D312yescomplemented
357C3×C8—22no—
358C3×C8—21no—
359C3×C8—22no—
360C3×C8—42no—
361C3×C8—42no—
362C3×C8—181no—
363C3⋊C8—62no—
364C3⋊C8—62no—
365C2×C32—21no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—21no—
369C2×C32—41no—
370C2×C32—42no—
371C2×C32—41no—
372C2×C32—81no—
373C2×C32—122no—
374C2×C32D3×C22⋊C412yes—
375C2×C32C2×C3×C22⋊C411yescharacteristic
376C3⋊D3—21no—
377C3⋊D3C3×#10412yescomplemented
378C3×D3—34no—
379C3×D3—42no—
380C3×D3—42no—
381C3×D3—62no—
382C3×D3—62no—
383C3×D3—82no—
384C3×D3—121no—
385#41—32no—
386#41—92no—
387#41—181no—
388#41—181no—
389C2×Q8—91no—
390C2×D4—91no—
391C2×D4—181no—
392C22×C4—91no—
393C22×C4—181no—
394C22×C4—181no—
395M16—91no—
396M16—181no—
397M16C3×D3211yescharacteristic, complemented
398C2×C8—181no—
399C42—32no—
400C42—181no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—91no—
407C22×C3—91no—
408C22×C3—122no—
409C22×C3—182no—
410C22×C3—362no—
411C22×C3#380211yescharacteristic
412C22×C3C3×D3×D412yes—
413C2×D3—22no—
414C2×D3—32no—
415C2×D3—32no—
416C2×D3—61no—
417C2×D3—61no—
418C2×D3—64no—
419C2×D3—122no—
420C2×D3—124no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—21no—
424C3×C4—22no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—64no—
428C3×C4—64no—
429C3×C4—62no—
430C3×C4—62no—
431C3×C4—91no—
432C3×C4—91no—
433C3×C4—122no—
434C3×C4—124no—
435C3×C4C4×D3211yescharacteristic
436C3×C4#381111yescharacteristic
437C3×C4C2×C3×C4×D312yes—
438C3×C4C3×D3×D412yes—
439Dic3—24no—
440Dic3—22no—
441Dic3—32no—
442Dic3—32no—
443Dic3—61no—
444Dic3—61no—
445C32—21no—
446C32—22no—
447C32—41no—
448C32D3×#10412yescomplemented
449C32C2×C3×#10411yescharacteristic, complemented
450C23—91no—
451C23—181no—
452Q8—32no—
453Q8—181no—
454D4—32no—
455D4—62no—
456D4—182no—
457D4—181no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—91no—
464C2×C4—91no—
465C2×C4—181no—
466C2×C4—184no—
467C2×C4—182no—
468C2×C4C2×C3×D3211yescharacteristic
469C8—21no—
470C8—181no—
471C2×C3—21no—
472C2×C3—22no—
473C2×C3—21no—
474C2×C3—22no—
475C2×C3—42no—
476C2×C3—41no—
477C2×C3—41no—
478C2×C3—81no—
479C2×C3—92no—
480C2×C3—122no—
481C2×C3—122no—
482C2×C3—181no—
483C2×C3—242no—
484C2×C3#6392111yescharacteristic
485C2×C3C3×D3×C22⋊C412yes—
486D3—34no—
487D3—42no—
488D3—62no—
489D3—62no—
490D3—121no—
491C22—62no—
492C22—91no—
493C22—91no—
494C22—182no—
495C22—362no—
496C22C3×#380211yescharacteristic
497C4—64no—
498C4—62no—
499C4—91no—
500C4—91no—
501C4C3×C4×D3211yescharacteristic
502C4C3×#381111yescharacteristic
503C3—21no—
504C3—22no—
505C3—41no—
506C3#1059100811yescharacteristic, direct product factor
507C3C3×D3×#10412yescomplemented
508C2—21no—
509C2—92no—
510C2—122no—
511C2—181no—
512C2C3×#6392111yescharacteristic
subgroupG/H
centerC3×C4#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33C2×C22⋊C4
prime radicalC3×#64306C2
cosocleC2×C32×C4C23×C3
hypercenterC3×M16D32
hypocenterC32C2×C3×#104
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. C3×C4×#624C2semidirect
4. C32×#825C2semidirect
5. C3×#63569C2semidirect
7. #10591008C3direct (N × H)
13. C3×#3802C4semidirect
16. C3×#3797C4semidirect
17. C3×#3796C4semidirect
18. C33×M16C22semidirect
19. C3×#3748C22semidirect
22. C32×C4×Dic3C22semidirect
26. M16×C3⋊D3C2×C3semidirect
30. C4×#624C2×C3semidirect
34. C3×#825D3semidirect
35. C3×#825C2×C3semidirect
37. #63569C2×C3semidirect
48. C32×D12C2×C4semidirect
50. C32×Dic6C2×C4semidirect
80. #3802C3×C4semidirect
83. #3797C3×C4semidirect
84. #3796C3×C4semidirect
88. C32×M16C2×D3semidirect
89. C32×M16C22×C3semidirect
91. #3748C22×C3semidirect
96. C3×C4×Dic3C2×D3semidirect
97. C3×C4×Dic3C22×C3semidirect
129. #825C3×D3semidirect
164. C3×D12C4×D3semidirect
165. C3×D12C2×C3×C4semidirect
182. C3×Dic6C4×D3semidirect
183. C3×Dic6C2×C3×C4semidirect
202. C3×C3⋊D3#104semidirect
233. C3×M16D32semidirect
234. C3×M16C2×C3×D3semidirect
241. C4×Dic3C2×C3×D3semidirect
297. C33C2×#104semidirect
342. D12C3×C4×D3semidirect
356. Dic6C3×C4×D3semidirect
377. C3⋊D3C3×#104semidirect
397. M16C3×D32semidirect
448. C32D3×#104semidirect
449. C32C2×C3×#104semidirect
506. C3#10591008direct (N × H)
507. C3C3×D3×#104semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
C3×M16×C3⋊D32
C3×C4×#6242
C32×#8252
C3×#635692
C3×D3×#1043
#105910083