Finite Group Databasebeta

← C3×#10594765 #13174362

Subgroups — C3×#10594765 #13174362

Order 1728.

1192 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64316C212yesmaximal, complemented
2C3×#64128C211yesmaximal, characteristic, complemented
3C32×D3×Q16C212yesmaximal, complemented
4C3×#63948C214yesmaximal, complemented
5C3×#63945C212yesmaximal, complemented
6C3×#63808C212yesmaximal, complemented
7C3×#63803C211yesmaximal, characteristic, complemented
8C3×#63802C211yesmaximal, characteristic, complemented
9C3×#5766—32nomaximal
10#10594765C311yesmaximal, characteristic, direct product factor
11C3×#3832C2212yes—
12C32×D3×Q8C2214yes—
13C3×#3799—24no—
14C3×#3799C2214yes—
15C3×#3798—22no—
16C3×#3798C2211yescharacteristic
17C3×#3797—22no—
18C3×#3797C2212yes—
19C33×Q16C2211yescharacteristic
20C3×#3755C2212yes—
21C3×#3745C2211yescharacteristic
22C3×C8×C3⋊D3C2211yescharacteristic
23C32×#216C2214yes—
24C32×Dic12C2212yes—
25C32×D3×C8C2212yes—
26C3×#3718C2212yes—
27C3×#3717C2214yes—
28C3×#3712C2211yescharacteristic
29C3×#3711C2212yes—
30C3×#1007—34no—
31#64316C2×C312yescomplemented
32C2×C32×Q16—32no—
33#64128C2×C311yescharacteristic, complemented
34C3×#942—34no—
35C3×#919—32no—
36C3×#919—32no—
37C3×#919—61no—
38C3×#918—34no—
39C3×#918—34no—
40C3×D3×Q16—22no—
41C3×D3×Q16—32no—
42C3×D3×Q16D312yescomplemented
43C3×D3×Q16C2×C312yescomplemented
44C3×#904—32no—
45C3×#901—32no—
46#63948C2×C314yescomplemented
47#63945C2×C312yescomplemented
48#63808C2×C312yescomplemented
49#63803C2×C311yescharacteristic, complemented
50#63802C2×C311yescharacteristic, complemented
51C33×Q8C2312yes—
52C3×C4⋊(C3⋊D3)—22no—
53C3×C4⋊(C3⋊D3)C2312yes—
54C3×C4×C3⋊D3—22no—
55C3×C4×C3⋊D3C2311yescharacteristic
56C32×C4×D3—24no—
57C32×C4×D3C2312yes—
58C32×Dic6—24no—
59C32×Dic6C2314yes—
60C3×#627—22no—
61C3×#627D411yescharacteristic
62C3×#626—24no—
63C3×#626D412yes—
64C3×#625D411yescharacteristic
65C3×#624—22no—
66C3×#624—24no—
67C33×C8C2311yescharacteristic
68C3×C3⋊(C3⋊C8)C2311yescharacteristic
69C32×C3⋊C8C2312yes—
70C3×#578—91no—
71#5766—32no—
72C2×C32×Q8—34no—
73#3832C22×C312yes—
74C3×#239—34no—
75C3×#239—34no—
76C3×#239—64no—
77C3×#239—62no—
78C3×D3×Q8—24no—
79C3×D3×Q8—34no—
80C3×D3×Q8—64no—
81C3×D3×Q8C2×D314yes—
82C3×D3×Q8C22×C314yes—
83C3×#235—32no—
84C3×#235—34no—
85C3×#235—64no—
86C3×#235—64no—
87#3799—24no—
88#3799C22×C314yes—
89#3798—22no—
90#3798C22×C311yescharacteristic
91#3797—22no—
92#3797C22×C312yes—
93C32×Q16—21no—
94C32×Q16—22no—
95C32×Q16—34no—
96C32×Q16—32no—
97C32×Q16—41no—
98C32×Q16C2×D312yes—
99C32×Q16C22×C311yescharacteristic
100C2×C32×C8—32no—
101#3755C22×C312yes—
102#3745C22×C311yescharacteristic
103C8×C3⋊D3C22×C311yescharacteristic
104C3×#216—24no—
105C3×#216—34no—
106C3×#216C2×D314yes—
107C3×#216C22×C314yes—
108C3×#215—34no—
109C3×#215—34no—
110C3×#215—62no—
111C3×#208—32no—
112C3×Dic12—22no—
113C3×Dic12C2×D312yes—
114C3×Dic12C22×C312yes—
115C3×D24—32no—
116C3×D24—61no—
117C3×#204—34no—
118C3×ΓL(1,𝔽25)—32no—
119C3×ΓL(1,𝔽25)—32no—
120C3×D3×C8—22no—
121C3×D3×C8—32no—
122C3×D3×C8—61no—
123C3×D3×C8C2×D312yes—
124C3×D3×C8C22×C312yes—
125#3718C22×C312yes—
126#3717C22×C314yes—
127#3712C22×C311yescharacteristic
128#3711C22×C312yes—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3C2×D411yescharacteristic
131C2×C32×D3C2×D412yes—
132C33×C4—22no—
133C33×C4C2411yescharacteristic
134C3×C3⋊Dic3C2×D411yescharacteristic
135C32×Dic3—24no—
136C32×Dic3C2×D412yes—
137C3×2−1+4—92no—
138#1007—34no—
139C3×#137—92no—
140C3×#137—94no—
141C3×#135—92no—
142C3×#135—91no—
143C2×C3×Q16—32no—
144C2×C3×Q16—32no—
145C2×C3×Q16—62no—
146C2×C3×Q16—91no—
147C3×#131—91no—
148#942—34no—
149#919—32no—
150#919—32no—
151#919—61no—
152#918—34no—
153#918—34no—
154D3×Q16—32no—
155D3×Q16C3×D312yescomplemented
156#904—32no—
157#901—32no—
158C32×Q8—22no—
159C32×Q8—24no—
160C32×Q8—34no—
161C32×Q8—42no—
162C32×Q8—64no—
163C32×Q8C22×D314yes—
164C32×Q8C23×C312yes—
165C2×C32×C4—32no—
166C2×C32×C4—64no—
167C4⋊(C3⋊D3)—22no—
168C4⋊(C3⋊D3)C23×C312yes—
169C4×C3⋊D3—22no—
170C4×C3⋊D3C23×C311yescharacteristic
171C3×#67—32no—
172C3×#67—32no—
173C3×#67—64no—
174C3×D12—32no—
175C3×D12—34no—
176C3×D12—62no—
177C3×D12—64no—
178C3×D12—64no—
179C3×D12—122no—
180C3×C4×D3—24no—
181C3×C4×D3—24no—
182C3×C4×D3—22no—
183C3×C4×D3—32no—
184C3×C4×D3—44no—
185C3×C4×D3—61no—
186C3×C4×D3—64no—
187C3×C4×D3—64no—
188C3×C4×D3—64no—
189C3×C4×D3—64no—
190C3×C4×D3—122no—
191C3×C4×D3C22×D312yes—
192C3×C4×D3C23×C312yes—
193C3×Dic6—24no—
194C3×Dic6—24no—
195C3×Dic6—24no—
196C3×Dic6—32no—
197C3×Dic6—44no—
198C3×Dic6—64no—
199C3×Dic6C22×D314yes—
200C3×Dic6C23×C314yes—
201#627—22no—
202#627C3×D411yescharacteristic
203#626—24no—
204#626C3×D412yes—
205#625C3×D411yescharacteristic
206#624—22no—
207#624—24no—
208C32×C8—21no—
209C32×C8—22no—
210C32×C8—32no—
211C32×C8—41no—
212C32×C8C22×D312yes—
213C32×C8C23×C311yescharacteristic
214C3⋊(C3⋊C8)C23×C311yescharacteristic
215C3×C3⋊C8—22no—
216C3×C3⋊C8—32no—
217C3×C3⋊C8—61no—
218C3×C3⋊C8C22×D312yes—
219C3×C3⋊C8C23×C312yes—
220#578—91noSylow
221C2×C33C22×D411yescharacteristic
222C3×C3⋊D3—21no—
223C3×C3⋊D3—42no—
224C32×D3—22no—
225C3×#41—91no—
226C3×#41—94no—
227C3×#41—92no—
228C3×#41—182no—
229C3×#41—184no—
230C2×C3×Q8—34no—
231C2×C3×Q8—34no—
232C2×C3×Q8—64no—
233C2×C3×Q8—92no—
234C2×C3×Q8—182no—
235#239—34no—
236#239—34no—
237#239—62no—
238#239—64no—
239D3×Q8—34no—
240D3×Q8—64no—
241D3×Q8C2×C3×D314yes—
242#235—32no—
243#235—34no—
244#235—64no—
245#235—64no—
246C3×Q16—21no—
247C3×Q16—22no—
248C3×Q16—34no—
249C3×Q16—34no—
250C3×Q16—32no—
251C3×Q16—32no—
252C3×Q16—41no—
253C3×Q16—62no—
254C3×Q16—64no—
255C3×Q16—92no—
256C3×Q16D3211yescharacteristic
257C3×Q16C2×C3×D312yes—
258C3×SD16—94no—
259C3×SD16—92no—
260C3×D8—91no—
261C3×M16—91no—
262C3×M16—92no—
263C2×C3×C8—32no—
264C2×C3×C8—32no—
265C2×C3×C8—62no—
266C2×C3×C8—91no—
267#216—34no—
268#216C2×C3×D314yes—
269#215—34no—
270#215—34no—
271#215—62no—
272#208—32no—
273Dic12C2×C3×D312yes—
274D24—32no—
275D24—61no—
276#204—34no—
277ΓL(1,𝔽25)—32no—
278ΓL(1,𝔽25)—32no—
279D3×C8—32no—
280D3×C8—61no—
281D3×C8C2×C3×D312yes—
282C22×C32—32no—
283C2×C3⋊D3—22no—
284C2×C3⋊D3C2×C3×D411yescharacteristic
285C2×C3×D3—22no—
286C2×C3×D3—32no—
287C2×C3×D3—61no—
288C2×C3×D3—64no—
289C2×C3×D3—122no—
290C2×C3×D3D3×D412yes—
291C2×C3×D3C2×C3×D412yes—
292C32×C4—24no—
293C32×C4—22no—
294C32×C4—21no—
295C32×C4—22no—
296C32×C4—32no—
297C32×C4—41no—
298C32×C4—44no—
299C32×C4—42no—
300C32×C4—64no—
301C32×C4—82no—
302C32×C4C23×D312yes—
303C32×C4C24×C311yescharacteristic
304C3⋊Dic3C2×C3×D411yescharacteristic
305C3×Dic3—24no—
306C3×Dic3—24no—
307C3×Dic3—22no—
308C3×Dic3—32no—
309C3×Dic3—44no—
310C3×Dic3—61no—
311C3×Dic3D3×D412yes—
312C3×Dic3C2×C3×D412yes—
3132−1+4—92no—
314#137—92no—
315#137—94no—
316#135—92no—
317#135—91no—
318C2×Q16—32no—
319C2×Q16—91no—
320#131—91no—
321C33#57811yescharacteristic, complemented, Sylow
322C3×Q8—24no—
323C3×Q8—22no—
324C3×Q8—34no—
325C3×Q8—34no—
326C3×Q8—42no—
327C3×Q8—64no—
328C3×Q8—64no—
329C3×Q8—64no—
330C3×Q8—91no—
331C3×Q8—124no—
332C3×Q8—182no—
333C3×Q8C2×D3212yes—
334C3×Q8C22×C3×D314yes—
335C3×D4—92no—
336C3×D4—91no—
337C3×D4—92no—
338C3×D4—182no—
339C3×D4—184no—
340C2×C3×C4—32no—
341C2×C3×C4—32no—
342C2×C3×C4—62no—
343C2×C3×C4—64no—
344C2×C3×C4—64no—
345C2×C3×C4—91no—
346C2×C3×C4—124no—
347C2×C3×C4—182no—
348C2×C3×C4—184no—
349C2×C3×C4—182no—
350#67—32no—
351#67—32no—
352#67—64no—
353D12—32no—
354D12—34no—
355D12—62no—
356D12—64no—
357D12—64no—
358D12—122no—
359C4×D3—24no—
360C4×D3—32no—
361C4×D3—61no—
362C4×D3—64no—
363C4×D3—64no—
364C4×D3—64no—
365C4×D3—64no—
366C4×D3—122no—
367C4×D3C22×C3×D312yes—
368Dic6—24no—
369Dic6—32no—
370Dic6—64no—
371Dic6C22×C3×D314yes—
372C3×C8—22no—
373C3×C8—21no—
374C3×C8—32no—
375C3×C8—32no—
376C3×C8—41no—
377C3×C8—62no—
378C3×C8—91no—
379C3×C8C2×D3211yescharacteristic
380C3×C8C22×C3×D312yes—
381C3⋊C8—32no—
382C3⋊C8—61no—
383C3⋊C8C22×C3×D312yes—
384C2×C32—21no—
385C2×C32—22no—
386C2×C32—41no—
387C2×C32—62no—
388C2×C32C2×D3×D412yes—
389C2×C32C22×C3×D411yescharacteristic
390C3⋊D3—21no—
391C3⋊D3—42no—
392C3×D3—24no—
393C3×D3—22no—
394C3×D3—22no—
395C3×D3—62no—
396C3×D3—121no—
397C3×D3—124no—
398C3×D3—242no—
399#41—91no—
400#41—94no—
401#41—92no—
402#41—182no—
403#41—184no—
404C2×Q8—34no—
405C2×Q8—92no—
406C2×Q8—182no—
407Q16—34no—
408Q16—32no—
409Q16—92no—
410Q16C3×D3211yescharacteristic
411SD16—94no—
412SD16—92no—
413D8—91no—
414M16—91no—
415M16—92no—
416C2×C8—32no—
417C2×C8—91no—
418C22×C3—32no—
419C22×C3—32no—
420C22×C3—62no—
421C22×C3—91no—
422C22×C3—182no—
423C2×D3—32no—
424C2×D3—61no—
425C2×D3—64no—
426C2×D3—122no—
427C2×D3C3×D3×D412yes—
428C3×C4—22no—
429C3×C4—24no—
430C3×C4—21no—
431C3×C4—22no—
432C3×C4—32no—
433C3×C4—32no—
434C3×C4—42no—
435C3×C4—44no—
436C3×C4—41no—
437C3×C4—62no—
438C3×C4—64no—
439C3×C4—64no—
440C3×C4—82no—
441C3×C4—91no—
442C3×C4—124no—
443C3×C4C22×D3211yescharacteristic
444C3×C4C23×C3×D312yes—
445Dic3—24no—
446Dic3—32no—
447Dic3—61no—
448Dic3C3×D3×D412yes—
449C32—22no—
450C32—21no—
451C32—41no—
452C32#576612yescomplemented
453C32C3×#57811yescharacteristic, complemented
454Q8—34no—
455Q8—64no—
456Q8—91no—
457Q8—182no—
458Q8C2×C3×D3212yes—
459D4—92no—
460D4—91no—
461D4—92no—
462D4—182no—
463D4—184no—
464C2×C4—32no—
465C2×C4—64no—
466C2×C4—91no—
467C2×C4—182no—
468C2×C4—184no—
469C2×C4—182no—
470C8—32no—
471C8—91no—
472C8C2×C3×D3211yescharacteristic
473C2×C3—21no—
474C2×C3—22no—
475C2×C3—41no—
476C2×C3—64no—
477C2×C3—62no—
478C2×C3—62no—
479C2×C3—181no—
480C2×C3—362no—
481C2×C3D32×D411yescharacteristic
482C2×C3C2×C3×D3×D412yes—
483D3—22no—
484D3—62no—
485D3—121no—
486D3—124no—
487D3—242no—
488C22—32no—
489C22—91no—
490C22—182no—
491C4—22no—
492C4—32no—
493C4—64no—
494C4—91no—
495C4C22×C3×D3211yescharacteristic
496C3—21no—
497C3—22no—
498C3—41no—
499C3#1059476511yescharacteristic, direct product factor
500C3C3×#576612yescomplemented
501C2—62no—
502C2—181no—
503C2—362no—
504C2C3×D32×D411yescharacteristic
subgroupG/H
centerC2×C3D32×D4
commutator, cosocleC32×C4C24×C3
FrattiniC4C22×C3×D32
FittingC33×Q16C22
socleC2×C33C22×D4
hypercenterC3×Q16D32
hypocenterC32C3×#578
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q16

Complements

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

NH ≅ G/Ntype
1. C3×#64316C2semidirect
2. C3×#64128C2semidirect
3. C32×D3×Q16C2semidirect
4. C3×#63948C2semidirect
5. C3×#63945C2semidirect
6. C3×#63808C2semidirect
7. C3×#63803C2semidirect
8. C3×#63802C2semidirect
10. #10594765C3direct (N × H)
31. #64316C2×C3semidirect
33. #64128C2×C3semidirect
42. C3×D3×Q16D3semidirect
43. C3×D3×Q16C2×C3semidirect
46. #63948C2×C3semidirect
47. #63945C2×C3semidirect
48. #63808C2×C3semidirect
49. #63803C2×C3semidirect
50. #63802C2×C3semidirect
155. D3×Q16C3×D3semidirect
321. C33#578semidirect
452. C32#5766semidirect
453. C32C3×#578semidirect
499. C3#10594765direct (N × H)
500. C3C3×#5766semidirect

Maximal subgroups

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

subgroupindex
C3×#643162
C3×#641282
C32×D3×Q162
C3×#639482
C3×#639452
C3×#638082
C3×#638032
C3×#638022
C3×#57663
#105947653