Finite Group Databasebeta

← C3×#10591009 #13158026

Subgroups — C3×#10591009 #13158026

Order 1728.

792 conjugacy classes of subgroups.

The subgroup lattice has 508 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×#63874C211yesmaximal, characteristic, complemented
4C32×#825C212yesmaximal, complemented
5C3×#63569C212yesmaximal, complemented
6C3×#4821—32nomaximal
7#10591009C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C32×#235C2212yes—
10C3×#3811—21no—
11C3×#3805—21no—
12C3×#3802C411yescharacteristic, complemented
13C3×C4×D32—21no—
14C3×#3798—21no—
15C3×#3797C411yescharacteristic, complemented
16C3×#3796C412yescomplemented
17C33×M16C2211yescharacteristic, complemented
18C3×#3748C2211yescharacteristic, complemented
19C3×#3743—21no—
20C3×C8×C3⋊D3—21no—
21C32×C4×Dic3C2212yescomplemented
22C3×#3724—21no—
23C3×#3722—21no—
24C3×D3×#41—32no—
25#64306C2×C311yescharacteristic
26M16×C3⋊D3C2×C311yescharacteristic, complemented
27C3×D3×M16—32no—
28C3×D3×M16—61no—
29C3×#872—32no—
30#63874C2×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×C412yes—
49C32×C4×D3—22no—
50C32×Dic6C2×C412yes—
51C3×#627—21no—
52C3×#626—22no—
53C3×#625—21no—
54C3×#624—21no—
55C3×D3×Dic3—22no—
56C33×C8—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C3×#421—91no—
59#4821—32no—
60C32×#41—32no—
61C2×C4×C3⋊D3C22×C311yescharacteristic
62C3×#239—32no—
63C3×D3×Q8—32no—
64C3×#237—32no—
65C3×#237—62no—
66C3×D3×D4—32no—
67C3×D3×D4—62no—
68C3×#235—22no—
69C3×#235—62no—
70C3×#235C2×D312yes—
71C3×#235C22×C312yes—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—62no—
74C2×C3×C4×D3—61no—
75#3811—21no—
76#3805—21no—
77#3802C3×C411yescharacteristic, complemented
78C4×D32—21no—
79#3798—21no—
80#3797C3×C411yescharacteristic, complemented
81#3796C3×C412yescomplemented
82C32×M16—21no—
83C32×M16—22no—
84C32×M16—41no—
85C32×M16C2×D312yescomplemented
86C32×M16C22×C311yescharacteristic, complemented
87C32×C42—32no—
88#3748C22×C311yescharacteristic, complemented
89#3743—21no—
90C8×C3⋊D3—21no—
91C3×(C2×D3)⋊C4—62no—
92C3×Dic3⋊C4—62no—
93C3×C4×Dic3—22no—
94C3×C4×Dic3C2×D312yescomplemented
95C3×C4×Dic3C22×C312yescomplemented
96C3×#208—32no—
97C3×#208—61no—
98C3×ΓL(1,𝔽25)—62no—
99C3×ΓL(1,𝔽25)—62no—
100C3×D3×C8—62no—
101C3×D3×C8—62no—
102#3724—21no—
103#3722—21no—
104C22×C33C2×D411yescharacteristic
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3C22⋊C411yescharacteristic
107C2×C3×C3⋊D3C22⋊C411yescharacteristic
108C2×C32×D3—22no—
109C3×D32—42no—
110C33×C4C22×C411yescharacteristic
111C33×C4C2×D411yescharacteristic
112C3×C3⋊Dic3C22⋊C411yescharacteristic
113C3×C3⋊Dic3C22⋊C411yescharacteristic
114C32×Dic3—22no—
115C32×Dic3—42no—
116C2×C3×#41—91no—
117D3×#41—32no—
118C2×C3×M16—91no—
119C3×#117—91no—
120D3×M16—32no—
121D3×M16—61no—
122#872—32no—
123C3×#104—32no—
124C3×#104—32no—
125C3×#104—62no—
126C3×#104—92no—
127#837—32no—
128#825C3×D312yescomplemented
129#805—32no—
130C22×C3⋊D3C3×D411yescharacteristic
131C22×C3×D3—32no—
132C22×C3×D3—61no—
133C22×C3×D3—62no—
134C2×D32—21no—
135C32×Q8—32no—
136C32×D4—32no—
137C32×D4—62no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—41no—
141C2×C32×C4—62no—
142C2×C32×C4—62no—
143C2×C32×C4C22×D312yes—
144C2×C32×C4C23×C311yescharacteristic
145C2×C3⋊Dic3C3×D411yescharacteristic
146C4×C3⋊D3C2×C3×C411yescharacteristic
147C4×C3⋊D3C2×C3×C411yescharacteristic
148C4×C3⋊D3C3×D411yescharacteristic
149C4×C3⋊D3C3×D411yescharacteristic
150C3×#67—22no—
151C3×#67—22no—
152C3×#67—42no—
153C3×#67—62no—
154C3×#67—62no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—62no—
161C3×D12—22no—
162C3×D12—62no—
163C3×D12C4×D312yes—
164C3×D12C2×C3×C412yes—
165C3×C4×D3—22no—
166C3×C4×D3—22no—
167C3×C4×D3—32no—
168C3×C4×D3—32no—
169C3×C4×D3—32no—
170C3×C4×D3—32no—
171C3×C4×D3—42no—
172C3×C4×D3—61no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—62no—
177C3×C4×D3—62no—
178C3×Dic6—22no—
179C3×Dic6—62no—
180C3×Dic6C4×D312yes—
181C3×Dic6C2×C3×C412yes—
182#627—21no—
183#626—22no—
184#625—21no—
185#624—21no—
186D3×Dic3—22no—
187C32×C8—21no—
188C32×C8—22no—
189C32×C8—22no—
190C32×C8—42no—
191C32×C8—42no—
192C3⋊(C3⋊C8)—21no—
193C3×C3⋊C8—62no—
194C3×C3⋊C8—62no—
195#421—91noSylow
196C2×C33—21no—
197C2×C33C2×C22⋊C411yescharacteristic
198C3×C3⋊D3—21no—
199C3×C3⋊D3—21no—
200C32×D3—42no—
201C3×#41—32no—
202C3×#41—32no—
203C3×#41—62no—
204C3×#41—92no—
205C3×#41—181no—
206C3×#41—181no—
207C2×C3×Q8—91no—
208C2×C3×D4—91no—
209C2×C3×D4—181no—
210C22×C3×C4—91no—
211C22×C3×C4—181no—
212#239—32no—
213D3×Q8—32no—
214#237—32no—
215#237—62no—
216D3×D4—32no—
217D3×D4—62no—
218#235—62no—
219#235C2×C3×D312yes—
220C2×C4×D3—32no—
221C2×C4×D3—61no—
222C2×C4×D3—62no—
223C3×M16—21no—
224C3×M16—22no—
225C3×M16—41no—
226C3×M16—91no—
227C3×M16—181no—
228C3×M16D3211yescharacteristic, complemented
229C3×M16C2×C3×D312yescomplemented
230C2×C3×C8—181no—
231C3×C4⋊C4—181no—
232C3×C22⋊C4—181no—
233C3×C42—32no—
234C3×C42—32no—
235C3×C42—62no—
236(C2×D3)⋊C4—62no—
237Dic3⋊C4—62no—
238C4×Dic3C2×C3×D312yescomplemented
239#208—32no—
240#208—61no—
241ΓL(1,𝔽25)—62no—
242ΓL(1,𝔽25)—62no—
243D3×C8—62no—
244D3×C8—62no—
245C22×C32—21no—
246C22×C32—22no—
247C22×C32—41no—
248C22×C32—62no—
249C22×C32D3×D412yes—
250C22×C32C2×C3×D411yescharacteristic
251C2×C3⋊D3—41no—
252C2×C3⋊D3C3×C22⋊C411yescharacteristic
253C2×C3⋊D3C3×C22⋊C411yescharacteristic
254C2×C3×D3—22no—
255C2×C3×D3—22no—
256C2×C3×D3—32no—
257C2×C3×D3—32no—
258C2×C3×D3—42no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—124no—
262C2×C3×D3—122no—
263C2×C3×D3—122no—
264D32—42no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—22no—
268C32×C4—22no—
269C32×C4—41no—
270C32×C4—41no—
271C32×C4—62no—
272C32×C4—122no—
273C32×C4C2×C4×D312yes—
274C32×C4D3×D412yes—
275C32×C4C22×C3×C411yescharacteristic
276C32×C4C2×C3×D411yescharacteristic
277C3⋊Dic3C3×C22⋊C411yescharacteristic
278C3⋊Dic3C3×C22⋊C411yescharacteristic
279C3×Dic3—22no—
280C3×Dic3—22no—
281C3×Dic3—32no—
282C3×Dic3—32no—
283C3×Dic3—42no—
284C3×Dic3—42no—
285C3×Dic3—42no—
286C3×Dic3—44no—
287C3×Dic3—61no—
288C3×Dic3—61no—
289C2×#41—91no—
290C2×M16—91no—
291#117—91no—
292#104—32no—
293#104—92no—
294C33#42111yescharacteristic, complemented, Sylow
295C23×C3—91no—
296C23×C3—181no—
297C22×D3—32no—
298C22×D3—61no—
299C22×D3—62no—
300C3×Q8—32no—
301C3×Q8—32no—
302C3×Q8—62no—
303C3×Q8—181no—
304C3×D4—32no—
305C3×D4—32no—
306C3×D4—62no—
307C3×D4—62no—
308C3×D4—62no—
309C3×D4—122no—
310C3×D4—182no—
311C3×D4—181no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—91no—
320C2×C3×C4—91no—
321C2×C3×C4—91no—
322C2×C3×C4—91no—
323C2×C3×C4—91no—
324C2×C3×C4—122no—
325C2×C3×C4—122no—
326C2×C3×C4—181no—
327C2×C3×C4—182no—
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C22×C3×D312yes—
330#67—22no—
331#67—62no—
332#67—62no—
333C2×Dic3—22no—
334C2×Dic3—32no—
335C2×Dic3—61no—
336C2×Dic3—62no—
337D12—62no—
338D12C3×C4×D312yes—
339C4×D3—22no—
340C4×D3—32no—
341C4×D3—32no—
342C4×D3—32no—
343C4×D3—32no—
344C4×D3—61no—
345C4×D3—61no—
346C4×D3—61no—
347C4×D3—61no—
348C4×D3—62no—
349C4×D3—62no—
350Dic6—62no—
351Dic6C3×C4×D312yes—
352C3×C8—22no—
353C3×C8—21no—
354C3×C8—22no—
355C3×C8—42no—
356C3×C8—42no—
357C3×C8—181no—
358C3⋊C8—62no—
359C3⋊C8—62no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32—41no—
365C2×C32—42no—
366C2×C32—41no—
367C2×C32—81no—
368C2×C32—122no—
369C2×C32D3×C22⋊C412yes—
370C2×C32C2×C3×C22⋊C411yescharacteristic
371C3⋊D3—21no—
372C3⋊D3—21no—
373C3×D3—42no—
374C3×D3—42no—
375C3×D3—62no—
376C3×D3—62no—
377C3×D3—62no—
378C3×D3—82no—
379C3×D3—121no—
380#41—32no—
381#41—92no—
382#41—181no—
383#41—181no—
384C2×Q8—91no—
385C2×D4—91no—
386C2×D4—181no—
387C22×C4—91no—
388C22×C4—181no—
389M16—91no—
390M16—181no—
391M16C3×D3211yescharacteristic, complemented
392C2×C8—181no—
393C4⋊C4—181no—
394C22⋊C4—181no—
395C42—32no—
396C22×C3—21no—
397C22×C3—22no—
398C22×C3—41no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—91no—
402C22×C3—91no—
403C22×C3—122no—
404C22×C3—361no—
405C22×C3—362no—
406C22×C3#380211yescharacteristic
407C22×C3C3×D3×D412yes—
408C2×D3—22no—
409C2×D3—32no—
410C2×D3—32no—
411C2×D3—61no—
412C2×D3—61no—
413C2×D3—124no—
414C2×D3—122no—
415C2×D3—122no—
416C3×C4—21no—
417C3×C4—22no—
418C3×C4—21no—
419C3×C4—22no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—62no—
423C3×C4—62no—
424C3×C4—91no—
425C3×C4—91no—
426C3×C4—122no—
427C3×C4—122no—
428C3×C4—122no—
429C3×C4—124no—
430C3×C4C4×D3211yescharacteristic
431C3×C4#381111yescharacteristic
432C3×C4C2×C3×C4×D312yes—
433C3×C4C3×D3×D412yes—
434Dic3—22no—
435Dic3—32no—
436Dic3—32no—
437Dic3—42no—
438Dic3—61no—
439Dic3—61no—
440C32—21no—
441C32—22no—
442C32—41no—
443C32#482112yescomplemented
444C32C3×#42111yescharacteristic, complemented
445C23—91no—
446C23—181no—
447Q8—32no—
448Q8—181no—
449D4—32no—
450D4—62no—
451D4—182no—
452D4—181no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—91no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—181no—
461C2×C4—182no—
462C2×C4C2×C3×D3211yescharacteristic
463C8—21no—
464C8—181no—
465C2×C3—21no—
466C2×C3—22no—
467C2×C3—21no—
468C2×C3—22no—
469C2×C3—42no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—81no—
473C2×C3—122no—
474C2×C3—122no—
475C2×C3—181no—
476C2×C3—181no—
477C2×C3—242no—
478C2×C3#6392111yescharacteristic
479C2×C3C3×D3×C22⋊C412yes—
480D3—42no—
481D3—62no—
482D3—62no—
483D3—62no—
484D3—121no—
485C22—62no—
486C22—91no—
487C22—91no—
488C22—361no—
489C22—362no—
490C22C3×#380211yescharacteristic
491C4—62no—
492C4—91no—
493C4—91no—
494C4—122no—
495C4C3×C4×D3211yescharacteristic
496C4C3×#381111yescharacteristic
497C3—21no—
498C3—22no—
499C3—41no—
500C3#1059100911yescharacteristic, direct product factor
501C3C3×#482112yescomplemented
502C2—21no—
503C2—122no—
504C2—181no—
505C2—181no—
506C2C3×#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
hypocenterC32C3×#421
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×#63874C2semidirect
4. C32×#825C2semidirect
5. C3×#63569C2semidirect
7. #10591009C3direct (N × H)
12. C3×#3802C4semidirect
15. C3×#3797C4semidirect
16. C3×#3796C4semidirect
17. C33×M16C22semidirect
18. C3×#3748C22semidirect
21. C32×C4×Dic3C22semidirect
26. M16×C3⋊D3C2×C3semidirect
30. #63874C2×C3semidirect
34. C3×#825D3semidirect
35. C3×#825C2×C3semidirect
37. #63569C2×C3semidirect
77. #3802C3×C4semidirect
80. #3797C3×C4semidirect
81. #3796C3×C4semidirect
85. C32×M16C2×D3semidirect
86. C32×M16C22×C3semidirect
88. #3748C22×C3semidirect
94. C3×C4×Dic3C2×D3semidirect
95. C3×C4×Dic3C22×C3semidirect
128. #825C3×D3semidirect
228. C3×M16D32semidirect
229. C3×M16C2×C3×D3semidirect
238. C4×Dic3C2×C3×D3semidirect
294. C33#421semidirect
391. M16C3×D32semidirect
443. C32#4821semidirect
444. C32C3×#421semidirect
500. C3#10591009direct (N × H)
501. C3C3×#4821semidirect

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×#638742
C32×#8252
C3×#635692
C3×#48213
#105910093