Finite Group Databasebeta

← C3×#10594764 #13174361

Subgroups — C3×#10594764 #13174361

Order 1728.

1266 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64319C212yesmaximal, complemented
2C3×Q16×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×#919C212yesmaximal, complemented
4C3×#63947C214yesmaximal, complemented
5C3×#63940C212yesmaximal, complemented
6C3×#63807C212yesmaximal, complemented
7C3×#63797C211yesmaximal, characteristic, complemented
8C3×D32×C8C211yesmaximal, characteristic, complemented
9C3×D3×#135—32nomaximal
10#10594764C311yesmaximal, characteristic, direct product factor
11C3×Q8×C3⋊D3C2212yes—
12C32×#239C2214yescomplemented
13C3×#3802—22no—
14C3×#3802C2212yes—
15C3×C4×D32—22no—
16C3×C4×D32C2211yescharacteristic
17C3×#3795—24no—
18C3×#3795C2214yescomplemented
19C33×Q16C2211yescharacteristic, complemented
20C3×#3756C2212yescomplemented
21C3×#3746C2211yescharacteristic, complemented
22C3×C8×C3⋊D3C2211yescharacteristic
23C32×#215C2214yescomplemented
24C32×D24C2212yescomplemented
25C32×D3×C8C2212yescomplemented
26C3×#3716C2214yescomplemented
27C3×#3713C2212yescomplemented
28C3×#3710C2211yescharacteristic
29C3×D3×C3⋊C8C2212yescomplemented
30C3×D3×#41—34no—
31#64319C2×C312yescomplemented
32C32×#135—32no—
33Q16×C3⋊D3C2×C311yescharacteristic, complemented
34C3×#950—34no—
35C3×#919—22no—
36C3×#919D312yescomplemented
37C3×#919C2×C312yescomplemented
38C3×D3×Q16—32no—
39C3×D3×Q16—61no—
40C3×#916—34no—
41C3×D3×SD16—34no—
42C3×#912—32no—
43C3×D3×D8—32no—
44C3×#904—32no—
45C2×C3×D3×C8—32no—
46#63947C2×C314yescomplemented
47#63940C2×C312yescomplemented
48#63807C2×C312yescomplemented
49#63797C2×C311yescharacteristic, complemented
50D32×C8C2×C311yescharacteristic, complemented
51C2×C3×D32—22no—
52C2×C3×D32D411yescharacteristic
53C33×Q8C2312yes—
54C3×C4×C3⋊D3—22no—
55C3×C4×C3⋊D3C2311yescharacteristic
56C3×#634—22no—
57C3×#634C2312yes—
58C32×D12—24no—
59C32×D12C2314yes—
60C32×C4×D3—24no—
61C32×C4×D3C2312yes—
62C3×#625—24no—
63C3×#625—22no—
64C3×#624D411yescharacteristic
65C3×D3×Dic3—24no—
66C3×D3×Dic3D412yes—
67C33×C8C2311yescharacteristic
68C3×C3⋊(C3⋊C8)C2311yescharacteristic
69C32×C3⋊C8C2312yes—
70C2×C3×#135—91no—
71D3×#135—32no—
72C32×#41—34no—
73Q8×C3⋊D3C22×C312yes—
74C3×#239—24no—
75C3×#239C2×D314yescomplemented
76C3×#239C22×C314yescomplemented
77C3×D3×Q8—34no—
78C3×D3×Q8—62no—
79C3×#237—34no—
80C3×#237—64no—
81C3×D3×D4—34no—
82C3×D3×D4—64no—
83C3×#235—34no—
84C3×#235—64no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—64no—
87#3802—22no—
88#3802C22×C312yes—
89C4×D32—22no—
90C4×D32C22×C311yescharacteristic
91#3795—24no—
92#3795C22×C314yescomplemented
93C32×Q16—21no—
94C32×Q16—22no—
95C32×Q16—41no—
96C32×Q16C2×D312yescomplemented
97C32×Q16C22×C311yescharacteristic, complemented
98C32×SD16—34no—
99C32×D8—32no—
100C2×C32×C8—32no—
101#3756C22×C312yescomplemented
102#3746C22×C311yescharacteristic, complemented
103C8×C3⋊D3C22×C311yescharacteristic
104C3×#216—34no—
105C3×#216—62no—
106C3×#215—24no—
107C3×#215C2×D314yescomplemented
108C3×#215C22×C314yescomplemented
109C3×#214—34no—
110C3×#213—34no—
111C2×C3×C3⋊C8—32no—
112C3×Dic12—32no—
113C3×Dic12—61no—
114C3×D24—22no—
115C3×D24C2×D312yescomplemented
116C3×D24C22×C312yescomplemented
117C3×#204—34no—
118C3×D3×C8—22no—
119C3×D3×C8—32no—
120C3×D3×C8—32no—
121C3×D3×C8—32no—
122C3×D3×C8—61no—
123C3×D3×C8C2×D312yescomplemented
124C3×D3×C8C22×C312yescomplemented
125#3716C22×C314yescomplemented
126#3713C22×C312yescomplemented
127#3710C22×C311yescharacteristic
128D3×C3⋊C8C22×C312yescomplemented
129C2×C3×C3⋊D3C2×D411yescharacteristic
130C2×C32×D3—24no—
131C2×C32×D3C2×D412yes—
132C3×D32—22no—
133C3×D32—44no—
134C33×C4—22no—
135C33×C4C2411yescharacteristic
136C3×C3⋊Dic3—22no—
137C3×C3⋊Dic3C2×D411yescharacteristic
138C32×Dic3C2×D412yes—
139C2×C3×#41—92no—
140D3×#41—34no—
141C3×#135—32no—
142C3×#135—32no—
143C3×#135—62no—
144C3×#135—92no—
145C3×#135—94no—
146C2×C3×Q16—91no—
147C2×C3×SD16—92no—
148C2×C3×D8—91no—
149C22×C3×C8—91no—
150#950—34no—
151#919C3×D312yescomplemented
152D3×Q16—32no—
153D3×Q16—61no—
154#916—34no—
155D3×SD16—34no—
156#912—32no—
157D3×D8—32no—
158#904—32no—
159C2×D3×C8—32no—
160C22×C3×D3—32no—
161C22×C3×D3—64no—
162C2×D32—22no—
163C2×D32C3×D411yescharacteristic
164C32×Q8—22no—
165C32×Q8—24no—
166C32×Q8—42no—
167C32×Q8C22×D314yes—
168C32×Q8C23×C312yes—
169C32×D4—34no—
170C32×D4—64no—
171C2×C32×C4—32no—
172C2×C32×C4—64no—
173C4×C3⋊D3—22no—
174C4×C3⋊D3C23×C311yescharacteristic
175#634—22no—
176#634C23×C312yes—
177C3×#67—64no—
178C3×#67—64no—
179C3×#67—64no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—64no—
182C3×D12—24no—
183C3×D12—24no—
184C3×D12—24no—
185C3×D12—44no—
186C3×D12C22×D314yes—
187C3×D12C23×C314yes—
188C3×C4×D3—24no—
189C3×C4×D3—24no—
190C3×C4×D3—22no—
191C3×C4×D3—32no—
192C3×C4×D3—32no—
193C3×C4×D3—32no—
194C3×C4×D3—44no—
195C3×C4×D3—61no—
196C3×C4×D3—64no—
197C3×C4×D3—64no—
198C3×C4×D3—122no—
199C3×C4×D3C22×D312yes—
200C3×C4×D3C23×C312yes—
201C3×Dic6—34no—
202C3×Dic6—62no—
203C3×Dic6—64no—
204C3×Dic6—122no—
205#625—24no—
206#625—22no—
207#624C3×D411yescharacteristic
208D3×Dic3—24no—
209D3×Dic3C3×D412yes—
210C32×C8—21no—
211C32×C8—22no—
212C32×C8—32no—
213C32×C8—41no—
214C32×C8C22×D312yes—
215C32×C8C23×C311yescharacteristic
216C3⋊(C3⋊C8)C23×C311yescharacteristic
217C3×C3⋊C8—22no—
218C3×C3⋊C8—32no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8C22×D312yes—
221C3×C3⋊C8C23×C312yes—
222C2×#135—91noSylow
223C2×C33C22×D411yescharacteristic
224C3×C3⋊D3#13512yescomplemented
225C32×D3—22no—
226C32×D3—44no—
227C3×#41—34no—
228C3×#41—34no—
229C3×#41—64no—
230C3×#41—94no—
231C3×#41—184no—
232C2×C3×Q8—92no—
233C2×C3×D4—92no—
234C2×C3×D4—182no—
235C22×C3×C4—91no—
236C22×C3×C4—182no—
237#239C2×C3×D314yescomplemented
238D3×Q8—34no—
239D3×Q8—62no—
240#237—34no—
241#237—64no—
242D3×D4—34no—
243D3×D4—64no—
244#235—34no—
245#235—64no—
246C2×C4×D3—32no—
247C2×C4×D3—64no—
248C3×Q16—21no—
249C3×Q16—22no—
250C3×Q16—41no—
251C3×Q16—92no—
252C3×Q16—91no—
253C3×Q16D3211yescharacteristic, complemented
254C3×Q16C2×C3×D312yescomplemented
255C3×SD16—34no—
256C3×SD16—34no—
257C3×SD16—64no—
258C3×SD16—94no—
259C3×D8—32no—
260C3×D8—32no—
261C3×D8—62no—
262C3×D8—92no—
263C2×C3×C8—32no—
264C2×C3×C8—32no—
265C2×C3×C8—62no—
266C2×C3×C8—91no—
267C2×C3×C8—92no—
268C2×C3×C8—91no—
269#216—34no—
270#216—62no—
271#215C2×C3×D314yescomplemented
272#214—34no—
273#213—34no—
274C2×C3⋊C8—32no—
275Dic12—32no—
276Dic12—61no—
277D24C2×C3×D312yescomplemented
278#204—34no—
279D3×C8—32no—
280D3×C8—32no—
281D3×C8—32no—
282D3×C8—61no—
283D3×C8C2×C3×D312yescomplemented
284C22×C32—32no—
285C22×C32—64no—
286C2×C3⋊D3C2×C3×D411yescharacteristic
287C2×C3×D3—24no—
288C2×C3×D3—24no—
289C2×C3×D3—22no—
290C2×C3×D3—32no—
291C2×C3×D3—44no—
292C2×C3×D3—61no—
293C2×C3×D3—64no—
294C2×C3×D3—128no—
295C2×C3×D3D3×D412yes—
296C2×C3×D3C2×C3×D412yes—
297D32—22no—
298D32—44no—
299C32×C4—24no—
300C32×C4—22no—
301C32×C4—21no—
302C32×C4—22no—
303C32×C4—32no—
304C32×C4—41no—
305C32×C4—42no—
306C32×C4—44no—
307C32×C4—82no—
308C32×C4C23×D312yes—
309C32×C4C24×C311yescharacteristic
310C3⋊Dic3—22no—
311C3⋊Dic3C2×C3×D411yescharacteristic
312C3×Dic3—22no—
313C3×Dic3—32no—
314C3×Dic3—61no—
315C3×Dic3—64no—
316C3×Dic3—122no—
317C3×Dic3D3×D412yes—
318C3×Dic3C2×C3×D412yes—
319C2×#41—92no—
320#135—32no—
321#135—92no—
322#135—94no—
323C2×Q16—91no—
324C2×SD16—92no—
325C2×D8—91no—
326C22×C8—91no—
327C33C2×#13511yescharacteristic, complemented, Sylow
328C23×C3—91no—
329C23×C3—182no—
330C22×D3—32no—
331C22×D3—64no—
332C3×Q8—24no—
333C3×Q8—22no—
334C3×Q8—42no—
335C3×Q8—92no—
336C3×Q8—182no—
337C3×Q8C2×D3212yes—
338C3×Q8C22×C3×D314yes—
339C3×D4—34no—
340C3×D4—34no—
341C3×D4—64no—
342C3×D4—64no—
343C3×D4—64no—
344C3×D4—124no—
345C3×D4—182no—
346C3×D4—184no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—62no—
350C2×C3×C4—64no—
351C2×C3×C4—64no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—91no—
355C2×C3×C4—124no—
356C2×C3×C4—182no—
357C2×C3×C4—184no—
358#67—64no—
359#67—64no—
360#67—64no—
361C2×Dic3—32no—
362C2×Dic3—64no—
363D12—24no—
364D12C22×C3×D314yes—
365C4×D3—24no—
366C4×D3—32no—
367C4×D3—32no—
368C4×D3—32no—
369C4×D3—61no—
370C4×D3—64no—
371C4×D3—64no—
372C4×D3—122no—
373C4×D3C22×C3×D312yes—
374Dic6—34no—
375Dic6—62no—
376Dic6—64no—
377Dic6—122no—
378C3×C8—22no—
379C3×C8—21no—
380C3×C8—32no—
381C3×C8—32no—
382C3×C8—41no—
383C3×C8—62no—
384C3×C8—91no—
385C3×C8C2×D3211yescharacteristic
386C3×C8C22×C3×D312yes—
387C3⋊C8—32no—
388C3⋊C8—61no—
389C3⋊C8C22×C3×D312yes—
390C2×C32—21no—
391C2×C32—22no—
392C2×C32—41no—
393C2×C32—62no—
394C2×C32—124no—
395C2×C32C2×D3×D412yes—
396C2×C32C22×C3×D411yescharacteristic
397C3⋊D3C3×#13512yescomplemented
398C3×D3—22no—
399C3×D3—22no—
400C3×D3—34no—
401C3×D3—42no—
402C3×D3—44no—
403C3×D3—44no—
404C3×D3—62no—
405C3×D3—84no—
406#41—34no—
407#41—94no—
408#41—184no—
409C2×Q8—92no—
410C2×D4—92no—
411C2×D4—182no—
412C22×C4—91no—
413C22×C4—182no—
414Q16—92no—
415Q16—91no—
416Q16C3×D3211yescharacteristic, complemented
417SD16—34no—
418SD16—94no—
419D8—32no—
420D8—92no—
421C2×C8—32no—
422C2×C8—91no—
423C2×C8—92no—
424C2×C8—91no—
425C22×C3—32no—
426C22×C3—32no—
427C22×C3—62no—
428C22×C3—64no—
429C22×C3—64no—
430C22×C3—91no—
431C22×C3—124no—
432C22×C3—182no—
433C22×C3—364no—
434C2×D3—24no—
435C2×D3—32no—
436C2×D3—61no—
437C2×D3—64no—
438C2×D3—128no—
439C2×D3C3×D3×D412yes—
440C3×C4—22no—
441C3×C4—24no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—32no—
445C3×C4—32no—
446C3×C4—42no—
447C3×C4—44no—
448C3×C4—41no—
449C3×C4—62no—
450C3×C4—82no—
451C3×C4—91no—
452C3×C4—182no—
453C3×C4C22×D3211yescharacteristic
454C3×C4C23×C3×D312yes—
455Dic3—32no—
456Dic3—61no—
457Dic3—64no—
458Dic3—122no—
459Dic3C3×D3×D412yes—
460C32—22no—
461C32—21no—
462C32—41no—
463C32D3×#13512yescomplemented
464C32C2×C3×#13511yescharacteristic, complemented
465C23—91no—
466C23—182no—
467Q8—92no—
468Q8—182no—
469Q8C2×C3×D3212yes—
470D4—34no—
471D4—64no—
472D4—182no—
473D4—184no—
474C2×C4—32no—
475C2×C4—64no—
476C2×C4—91no—
477C2×C4—92no—
478C2×C4—91no—
479C2×C4—182no—
480C2×C4—184no—
481C8—32no—
482C8—91no—
483C8C2×C3×D3211yescharacteristic
484C2×C3—21no—
485C2×C3—22no—
486C2×C3—41no—
487C2×C3—62no—
488C2×C3—62no—
489C2×C3—92no—
490C2×C3—122no—
491C2×C3—124no—
492C2×C3—124no—
493C2×C3—244no—
494C2×C3D32×D411yescharacteristic
495C2×C3C2×C3×D3×D412yes—
496D3—22no—
497D3—34no—
498D3—44no—
499D3—62no—
500C22—32no—
501C22—64no—
502C22—91no—
503C22—182no—
504C22—364no—
505C4—22no—
506C4—32no—
507C4—91no—
508C4—182no—
509C4C22×C3×D3211yescharacteristic
510C3—21no—
511C3—22no—
512C3—41no—
513C3#1059476411yescharacteristic, direct product factor
514C3C3×D3×#13512yescomplemented
515C2—62no—
516C2—92no—
517C2—124no—
518C2C3×D32×D411yescharacteristic
subgroupG/H
centerC2×C3D32×D4
commutator, cosocleC32×C4C24×C3
FrattiniC4C22×C3×D32
FittingC33×Q16C22
socleC2×C33C22×D4
hypercenterC3×Q16D32
hypocenterC32C2×C3×#135
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×#64319C2semidirect
2. C3×Q16×C3⋊D3C2semidirect
3. C32×#919C2semidirect
4. C3×#63947C2semidirect
5. C3×#63940C2semidirect
6. C3×#63807C2semidirect
7. C3×#63797C2semidirect
8. C3×D32×C8C2semidirect
10. #10594764C3direct (N × H)
12. C32×#239C22semidirect
18. C3×#3795C22semidirect
19. C33×Q16C22semidirect
20. C3×#3756C22semidirect
21. C3×#3746C22semidirect
23. C32×#215C22semidirect
24. C32×D24C22semidirect
25. C32×D3×C8C22semidirect
26. C3×#3716C22semidirect
27. C3×#3713C22semidirect
29. C3×D3×C3⋊C8C22semidirect
31. #64319C2×C3semidirect
33. Q16×C3⋊D3C2×C3semidirect
36. C3×#919D3semidirect
37. C3×#919C2×C3semidirect
46. #63947C2×C3semidirect
47. #63940C2×C3semidirect
48. #63807C2×C3semidirect
49. #63797C2×C3semidirect
50. D32×C8C2×C3semidirect
75. C3×#239C2×D3semidirect
76. C3×#239C22×C3semidirect
92. #3795C22×C3semidirect
96. C32×Q16C2×D3semidirect
97. C32×Q16C22×C3semidirect
101. #3756C22×C3semidirect
102. #3746C22×C3semidirect
107. C3×#215C2×D3semidirect
108. C3×#215C22×C3semidirect
115. C3×D24C2×D3semidirect
116. C3×D24C22×C3semidirect
123. C3×D3×C8C2×D3semidirect
124. C3×D3×C8C22×C3semidirect
125. #3716C22×C3semidirect
126. #3713C22×C3semidirect
128. D3×C3⋊C8C22×C3semidirect
151. #919C3×D3semidirect
224. C3×C3⋊D3#135semidirect
237. #239C2×C3×D3semidirect
253. C3×Q16D32semidirect
254. C3×Q16C2×C3×D3semidirect
271. #215C2×C3×D3semidirect
277. D24C2×C3×D3semidirect
283. D3×C8C2×C3×D3semidirect
327. C33C2×#135semidirect
397. C3⋊D3C3×#135semidirect
416. Q16C3×D32semidirect
463. C32D3×#135semidirect
464. C32C2×C3×#135semidirect
513. C3#10594764direct (N × H)
514. C3C3×D3×#135semidirect

Maximal subgroups

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

subgroupindex
C3×#643192
C3×Q16×C3⋊D32
C32×#9192
C3×#639472
C3×#639402
C3×#638072
C3×#637972
C3×D32×C82
C3×D3×#1353
#105947643