Finite Group Databasebeta

← C3×#10594786 #13174421

Subgroups — C3×#10594786 #13174421

Order 1728.

1498 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C212yesmaximal, complemented
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C3×#63827C214yesmaximal, complemented
4C3×#63823C214yesmaximal, complemented
5C3×#63822C214yesmaximal, complemented
6C3×#5822—32nomaximal
7#10594786C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C2×C32×#67—22no—
10C32×#235—22no—
11C32×#235—22no—
12C32×#235C2214yes—
13C32×#235C2214yescomplemented
14C2×C32×D12C2212yescomplemented
15C2×C32×C4×D3—22no—
16C2×C32×Dic6C2212yescomplemented
17C3×#3748C2214yescomplemented
18C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
19C3×#3718C2214yescomplemented
20C3×#3715C2218yescomplemented
21C3×#3713C2214yescomplemented
22C2×C32×#41—32no—
23C2×C3×#235—22no—
24C2×C3×#235D312yescomplemented
25C2×C3×#235C2×C312yescomplemented
26C2×#3748C2×C311yescharacteristic, complemented
27C3×#951—34no—
28C3×#950—38no—
29C3×#949—34no—
30C3×#942—34no—
31C3×#932—34no—
32C2×C3×#208—32no—
33C2×C3×#208—61no—
34#63827C2×C314yescomplemented
35#63823C2×C314yescomplemented
36#63822C2×C314yescomplemented
37C23×C33D411yescharacteristic
38C22×C32×D3—22no—
39C2×C33×C4D412yes—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C33×C4C2312yes—
43C32×#67—24no—
44C32×#67—24no—
45C2×C32×Dic3—22no—
46C32×D12—22no—
47C32×D12C2314yes—
48C32×C4×D3—24no—
49C32×C4×D3—24no—
50C32×Dic6—22no—
51C32×Dic6C2314yes—
52C3×C3⋊(C3⋊C8)C2314yes—
53C3×#575—91no—
54#5822—32no—
55C32×#41—34no—
56C32×#41—34no—
57C32×#41—62no—
58C32×#41—62no—
59C2×C32×Q8—32no—
60C2×C32×D4—32no—
61C2×C32×D4—62no—
62C22×C32×C4—21no—
63C22×C32×C4—22no—
64C22×C32×C4—41no—
65C22×C32×C4—62no—
66C22×C32×C4C2×D312yes—
67C22×C32×C4C22×C311yescharacteristic
68C2×C3×#67—24no—
69C2×C3×#67—22no—
70C2×C3×#67—22no—
71C3×#235—24no—
72C3×#235—24no—
73C3×#235—24no—
74C3×#235—24no—
75C3×#235—22no—
76C3×#235—22no—
77C3×#235—22no—
78C3×#235—22no—
79C3×#235C2×D314yes—
80C3×#235C2×D314yescomplemented
81C3×#235C22×C314yes—
82C3×#235C22×C314yescomplemented
83C2×C3×D12—22no—
84C2×C3×D12C2×D312yescomplemented
85C2×C3×D12C22×C312yescomplemented
86C2×C3×C4×D3—24no—
87C2×C3×C4×D3—22no—
88C2×C3×C4×D3—22no—
89C2×C3×Dic6—22no—
90C2×C3×Dic6C2×D312yescomplemented
91C2×C3×Dic6C22×C312yescomplemented
92#3748C22×C314yescomplemented
93C2×C3⋊(C3⋊C8)C22×C312yescomplemented
94C3×#216—38no—
95C3×#215—38no—
96C3×#214—38no—
97C3×#213—38no—
98C3×#208—34no—
99C3×#208—34no—
100C3×#208—64no—
101C2×C3×C3⋊C8—34no—
102C2×C3×C3⋊C8—62no—
103#3718C22×C314yescomplemented
104#3715C22×C318yescomplemented
105#3713C22×C314yescomplemented
106C22×C33—41no—
107C22×C33C2×D412yes—
108C22×C33C2×D411yescharacteristic
109C2×C32×D3—24no—
110C2×C32×D3—44no—
111C33×C4C2×D412yes—
112C33×C4C2×D411yescharacteristic
113C33×C4C2411yescharacteristic
114C32×Dic3—24no—
115C2×C3×#41—32no—
116C2×C3×#41—32no—
117C2×C3×#41—62no—
118C2×#235C3×D312yescomplemented
119C3×#137—94no—
120C3×AGL(1,ℤ/8)—94no—
121C3×#135—94no—
122C2×C3×M16—91no—
123#951—34no—
124#950—38no—
125#949—34no—
126#942—34no—
127#932—34no—
128C2×#208—32no—
129C2×#208—61no—
130C23×C32—21no—
131C23×C32—22no—
132C23×C32—41no—
133C23×C32—62no—
134C23×C32#6712yes—
135C23×C32C3×D411yescharacteristic
136C22×C3×D3—24no—
137C22×C3×D3—22no—
138C22×C3×D3—22no—
139C32×Q8—34no—
140C32×Q8—62no—
141C32×D4—34no—
142C32×D4—64no—
143C32×D4—64no—
144C32×D4—62no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—42no—
156C2×C32×C4—41no—
157C2×C32×C4—42no—
158C2×C32×C4—41no—
159C2×C32×C4—62no—
160C2×C32×C4—64no—
161C2×C32×C4—64no—
162C2×C32×C4#6712yes—
163C2×C32×C4#6712yes—
164C2×C32×C4#6712yes—
165C2×C32×C4C3×D412yes—
166C2×C32×C4C3×D411yescharacteristic
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D312yes—
170C2×C32×C4C23×C311yescharacteristic
171C2×C32×C4C23×C312yes—
172C3×#67—28no—
173C3×#67—28no—
174C3×#67—24no—
175C3×#67—24no—
176C3×#67—24no—
177C3×#67—24no—
178C2×C3×Dic3—24no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C3×D12—24no—
182C3×D12—22no—
183C3×D12—22no—
184C3×D12—24no—
185C3×D12C22×D314yes—
186C3×D12C23×C314yes—
187C3×C4×D3—28no—
188C3×C4×D3—28no—
189C3×C4×D3—24no—
190C3×C4×D3—24no—
191C3×C4×D3—24no—
192C3×C4×D3—24no—
193C3×Dic6—24no—
194C3×Dic6—22no—
195C3×Dic6—22no—
196C3×Dic6—24no—
197C3×Dic6C22×D314yes—
198C3×Dic6C23×C314yes—
199C3⋊(C3⋊C8)C23×C314yes—
200C3×C3⋊C8—38no—
201C3×C3⋊C8—64no—
202#575—91noSylow
203C2×C33—22no—
204C2×C33—21no—
205C2×C33C22×D411yescharacteristic
206C32×D3—44no—
207C3×#41—34no—
208C3×#41—34no—
209C3×#41—34no—
210C3×#41—34no—
211C3×#41—64no—
212C3×#41—64no—
213C3×#41—64no—
214C3×#41—64no—
215C3×#41—62no—
216C3×#41—62no—
217C3×#41—62no—
218C3×#41—62no—
219C2×C3×Q8—32no—
220C2×C3×Q8—32no—
221C2×C3×Q8—62no—
222C2×C3×D4—32no—
223C2×C3×D4—32no—
224C2×C3×D4—62no—
225C2×C3×D4—64no—
226C2×C3×D4—62no—
227C2×C3×D4—62no—
228C22×C3×C4—21no—
229C22×C3×C4—22no—
230C22×C3×C4—41no—
231C22×C3×C4—64no—
232C22×C3×C4—62no—
233C22×C3×C4—62no—
234C22×C3×C4D3211yescharacteristic
235C22×C3×C4C2×C3×D312yes—
236C2×#67—22no—
237#235—22no—
238#235—22no—
239#235C2×C3×D314yes—
240#235C2×C3×D314yescomplemented
241C2×D12C2×C3×D312yescomplemented
242C2×C4×D3—22no—
243C2×Dic6C2×C3×D312yescomplemented
244C3×Q16—94no—
245C3×SD16—98no—
246C3×D8—94no—
247C3×M16—94no—
248C2×C3×C8—92no—
249#216—38no—
250#215—38no—
251#214—38no—
252#213—38no—
253#208—34no—
254#208—34no—
255#208—64no—
256C2×C3⋊C8—34no—
257C2×C3⋊C8—62no—
258C22×C32—21no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—42no—
264C22×C32—41no—
265C22×C32—42no—
266C22×C32—41no—
267C22×C32—42no—
268C22×C32—44no—
269C22×C32—44no—
270C22×C32—64no—
271C22×C32—124no—
272C22×C32C2×#6712yes—
273C22×C32C2×#6712yes—
274C22×C32C2×#6712yes—
275C22×C32C2×C3×D412yes—
276C22×C32C2×C3×D411yescharacteristic
277C2×C3×D3—28no—
278C2×C3×D3—24no—
279C2×C3×D3—24no—
280C2×C3×D3—48no—
281C2×C3×D3—44no—
282C2×C3×D3—44no—
283C32×C4—21no—
284C32×C4—22no—
285C32×C4—22no—
286C32×C4—21no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4—22no—
290C32×C4—41no—
291C32×C4—42no—
292C32×C4—41no—
293C32×C4—64no—
294C32×C4C2×#6712yes—
295C32×C4C2×#6712yes—
296C32×C4C2×#6712yes—
297C32×C4C2×C3×D412yes—
298C32×C4C2×C3×D411yescharacteristic
299C32×C4C23×D312yes—
300C32×C4C24×C311yescharacteristic
301C3×Dic3—28no—
302C3×Dic3—24no—
303C3×Dic3—24no—
304C2×#41—32no—
305#137—94no—
306AGL(1,ℤ/8)—94no—
307#135—94no—
308C2×M16—91no—
309C33#57511yescharacteristic, complemented, Sylow
310C23×C3—22no—
311C23×C3—21no—
312C23×C3—41no—
313C23×C3—64no—
314C23×C3—62no—
315C23×C3—62no—
316C23×C3#62511yescharacteristic
317C23×C3C3×#6712yes—
318C22×D3—22no—
319C3×Q8—34no—
320C3×Q8—34no—
321C3×Q8—64no—
322C3×Q8—64no—
323C3×Q8—62no—
324C3×Q8—62no—
325C3×D4—34no—
326C3×D4—34no—
327C3×D4—64no—
328C3×D4—64no—
329C3×D4—68no—
330C3×D4—68no—
331C3×D4—64no—
332C3×D4—64no—
333C3×D4—64no—
334C3×D4—64no—
335C3×D4—62no—
336C3×D4—62no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—68no—
352C2×C3×C4—68no—
353C2×C3×C4—64no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—64no—
357C2×C3×C4—64no—
358C2×C3×C4—64no—
359C2×C3×C4—64no—
360C2×C3×C4#62511yescharacteristic
361C2×C3×C4#62512yes—
362C2×C3×C4C3×#6712yes—
363C2×C3×C4C3×#6712yes—
364C2×C3×C4C3×#6712yes—
365C2×C3×C4C2×D3212yes—
366C2×C3×C4C2×D3211yescharacteristic
367C2×C3×C4C22×C3×D312yes—
368C2×C3×C4C22×C3×D312yes—
369C2×C3×C4C22×C3×D312yes—
370#67—24no—
371#67—24no—
372C2×Dic3—22no—
373D12—22no—
374D12C22×C3×D314yes—
375C4×D3—24no—
376C4×D3—24no—
377Dic6—22no—
378Dic6C22×C3×D314yes—
379C3×C8—94no—
380C3⋊C8—38no—
381C3⋊C8—64no—
382C2×C32—21no—
383C2×C32—22no—
384C2×C32—24no—
385C2×C32—22no—
386C2×C32—22no—
387C2×C32—22no—
388C2×C32—22no—
389C2×C32—21no—
390C2×C32—22no—
391C2×C32—41no—
392C2×C32—42no—
393C2×C32—42no—
394C2×C32—42no—
395C2×C32—44no—
396C2×C32—42no—
397C2×C32—124no—
398C2×C32C22×#6712yes—
399C2×C32C22×C3×D411yescharacteristic
400C3×D3—48no—
401C3×D3—44no—
402C3×D3—44no—
403#41—34no—
404#41—34no—
405#41—62no—
406#41—62no—
407C2×Q8—32no—
408C2×D4—32no—
409C2×D4—62no—
410C22×C4—62no—
411C22×C4C3×D3211yescharacteristic
412Q16—94no—
413SD16—98no—
414D8—94no—
415M16—94no—
416C2×C8—92no—
417C22×C3—22no—
418C22×C3—21no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—41no—
423C22×C3—42no—
424C22×C3—44no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—42no—
428C22×C3—41no—
429C22×C3—68no—
430C22×C3—64no—
431C22×C3—64no—
432C22×C3—128no—
433C22×C3—124no—
434C22×C3—124no—
435C22×C3C2×#62512yes—
436C22×C3C2×#62511yescharacteristic
437C22×C3C2×C3×#6712yes—
438C22×C3C2×C3×#6712yes—
439C22×C3C2×C3×#6712yes—
440C2×D3—24no—
441C2×D3—44no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—22no—
446C3×C4—22no—
447C3×C4—21no—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—42no—
452C3×C4—68no—
453C3×C4—64no—
454C3×C4—64no—
455C3×C4C2×#62512yes—
456C3×C4C2×#62511yescharacteristic
457C3×C4C2×C3×#6712yes—
458C3×C4C2×C3×#6712yes—
459C3×C4C2×C3×#6712yes—
460C3×C4C22×D3211yescharacteristic
461C3×C4C23×C3×D312yes—
462Dic3—24no—
463C32—22no—
464C32—21no—
465C32—41no—
466C32#582212yescomplemented
467C32C3×#57511yescharacteristic, complemented
468C23—62no—
469C23C3×#62511yescharacteristic
470Q8—34no—
471Q8—62no—
472D4—34no—
473D4—64no—
474D4—64no—
475D4—62no—
476C2×C4—62no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4C3×#62512yes—
480C2×C4C3×#62511yescharacteristic
481C2×C4C2×C3×D3211yescharacteristic
482C2×C4C2×C3×D3212yes—
483C8—94no—
484C2×C3—24no—
485C2×C3—22no—
486C2×C3—22no—
487C2×C3—22no—
488C2×C3—22no—
489C2×C3—22no—
490C2×C3—21no—
491C2×C3—22no—
492C2×C3—21no—
493C2×C3—42no—
494C2×C3—42no—
495C2×C3—42no—
496C2×C3—44no—
497C2×C3—42no—
498C2×C3—41no—
499C2×C3—128no—
500C2×C3—124no—
501C2×C3—124no—
502C2×C3C22×#62511yescharacteristic
503C2×C3C22×C3×#6712yes—
504D3—44no—
505C22—41no—
506C22—64no—
507C22—124no—
508C22C2×C3×#62512yes—
509C22C2×C3×#62511yescharacteristic
510C4—64no—
511C4C2×C3×#62511yescharacteristic
512C4C2×C3×#62512yes—
513C4C22×C3×D3211yescharacteristic
514C3—22no—
515C3—21no—
516C3—41no—
517C3#1059478611yescharacteristic, direct product factor
518C3C3×#582212yescomplemented
519C2—22no—
520C2—21no—
521C2—124no—
522C2C22×C3×#62511yescharacteristic
subgroupG/H
centerC3×C4C2×#625
commutator, cosocleC32×C4C24×C3
FrattiniC4C22×C3×D32
FittingC22×C33×C4C22
socleC2×C33C22×D4
hypercenterC22×C3×C4D32
hypocenterC32C3×#575
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#235C2semidirect
2. C2×C3×#3748C2semidirect
3. C3×#63827C2semidirect
4. C3×#63823C2semidirect
5. C3×#63822C2semidirect
7. #10594786C3direct (N × H)
13. C32×#235C22semidirect
14. C2×C32×D12C22semidirect
16. C2×C32×Dic6C22semidirect
17. C3×#3748C22semidirect
18. C2×C3×C3⋊(C3⋊C8)C22semidirect
19. C3×#3718C22semidirect
20. C3×#3715C22semidirect
21. C3×#3713C22semidirect
24. C2×C3×#235D3semidirect
25. C2×C3×#235C2×C3semidirect
26. C2×#3748C2×C3semidirect
34. #63827C2×C3semidirect
35. #63823C2×C3semidirect
36. #63822C2×C3semidirect
80. C3×#235C2×D3semidirect
82. C3×#235C22×C3semidirect
84. C2×C3×D12C2×D3semidirect
85. C2×C3×D12C22×C3semidirect
90. C2×C3×Dic6C2×D3semidirect
91. C2×C3×Dic6C22×C3semidirect
92. #3748C22×C3semidirect
93. C2×C3⋊(C3⋊C8)C22×C3semidirect
103. #3718C22×C3semidirect
104. #3715C22×C3semidirect
105. #3713C22×C3semidirect
118. C2×#235C3×D3semidirect
240. #235C2×C3×D3semidirect
241. C2×D12C2×C3×D3semidirect
243. C2×Dic6C2×C3×D3semidirect
309. C33#575semidirect
466. C32#5822semidirect
467. C32C3×#575semidirect
517. C3#10594786direct (N × H)
518. C3C3×#5822semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×C3×#37482
C3×#638272
C3×#638232
C3×#638222
C3×#58223
#105947863