Finite Group Databasebeta

← C3×#10590364 #13157380

Subgroups — C3×#10590364 #13157380

Order 1728.

758 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#874C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#3715C212yesmaximal, complemented
4C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
5C3×#63562C212yesmaximal, complemented
6C3×#5075—31nomaximal
7C3×#5027—31nomaximal
8#10590364C311yesmaximal, characteristic, direct product factor
9C2×C32×D12—21no—
10C2×C32×D12—21no—
11C2×C32×D12C2212yes—
12C2×C32×Dic6C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15C32×C4⋊Dic3C2211yescharacteristic
16C32×Dic3⋊C4—21no—
17C32×C4×Dic3—21no—
18C3×#3715—24no—
19C32×#127—31no—
20C32×C4×Q8—31no—
21C2×C3×#215—32no—
22C2×C3×#214—32no—
23C3×#874—21no—
24C3×#874D311yescharacteristic
25C3×#874C2×C311yescharacteristic
26C3×C4×Dic6—21no—
27C3×C4×Dic6D311yescharacteristic, complemented
28C3×C4×Dic6C2×C311yescharacteristic, complemented
29C2×#3715C2×C312yescomplemented
30C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
31C3×#832—32no—
32C3×#809—32no—
33C3×C4⋊(C3⋊C8)—31no—
34C3×C4⋊(C3⋊C8)—31no—
35C3×C4⋊(C3⋊C8)—61no—
36#63562C2×C312yescomplemented
37C22×C32×D3—22no—
38C2×C33×C4D411yescharacteristic
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C32×Dic3—21no—
42C2×C32×Dic3—21no—
43C32×D12—22no—
44C32×D12—22no—
45C32×D12—42no—
46C32×D12—42no—
47C32×Dic6—21no—
48C32×Dic6D412yescomplemented
49C3×C3⋊(C3⋊C8)—22no—
50C3×#460—91no—
51#5075—31no—
52#5027—31no—
53C2×C32×Q8—31no—
54C2×C32×D4—32no—
55C2×C32×D4—61no—
56C2×C32×D4—61no—
57C2×C3×D12—22no—
58C2×C3×D12—22no—
59C2×C3×D12—22no—
60C2×C3×D12—21no—
61C2×C3×D12—21no—
62C2×C3×D12—21no—
63C2×C3×D12—21no—
64C2×C3×D12C2×D312yes—
65C2×C3×D12C22×C312yes—
66C2×C3×Dic6—21no—
67C2×C3×Dic6C2×D311yescharacteristic
68C2×C3×Dic6C22×C311yescharacteristic
69C32×C4⋊C4—31no—
70C32×C4⋊C4—61no—
71C32×C42—21no—
72C32×C42—21no—
73C32×C42—21no—
74C32×C42—41no—
75C32×C42—61no—
76C32×C42C2×D311yescharacteristic
77C32×C42C2×D311yescharacteristic
78C32×C42C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C312yes—
80C3×#215—64no—
81C3×#214—64no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3C2×D311yescharacteristic
84C3×C4⋊Dic3C22×C311yescharacteristic
85C3×Dic3⋊C4—22no—
86C3×Dic3⋊C4—21no—
87C3×Dic3⋊C4—21no—
88C3×C4×Dic3—22no—
89C3×C4×Dic3—21no—
90C3×C4×Dic3—21no—
91C2×C3×C3⋊C8—32no—
92C2×C3×C3⋊C8—32no—
93C2×C3×C3⋊C8—62no—
94#3715—24no—
95C22×C33C2×D411yescharacteristic
96C2×C32×D3—42no—
97C2×C32×D3—42no—
98C2×C32×D3—42no—
99C33×C4—21no—
100C33×C4SD1612yescomplemented
101C33×C4C2×D411yescharacteristic
102C33×C4#4111yescharacteristic
103C32×Dic3—22no—
104C32×Dic3—41no—
105C2×C3×SD16—92no—
106C3×#127—31no—
107C3×#127—31no—
108C3×#127—61no—
109C3×C4×Q8—31no—
110C3×C4×Q8—31no—
111C3×C4×Q8—61no—
112C2×#215—32no—
113C2×#214—32no—
114#874C3×D311yescharacteristic
115C4×Dic6C3×D311yescharacteristic, complemented
116C3×C4⋊C8—91no—
117C3×#102—92no—
118#832—32no—
119#809—32no—
120C4⋊(C3⋊C8)—31no—
121C4⋊(C3⋊C8)—31no—
122C4⋊(C3⋊C8)—61no—
123C23×C32—62no—
124C22×C3×D3—24no—
125C22×C3×D3—22no—
126C22×C3×D3—22no—
127C32×Q8—32no—
128C32×Q8—61no—
129C32×D4—62no—
130C32×D4—62no—
131C32×D4—122no—
132C32×D4—122no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4—41no—
144C2×C32×C4—41no—
145C2×C32×C4—61no—
146C2×C32×C4—61no—
147C2×C32×C4#6711yescharacteristic
148C2×C32×C4#6711yescharacteristic
149C2×C32×C4#6711yescharacteristic
150C2×C32×C4#6711yescharacteristic
151C2×C32×C4C3×D411yescharacteristic
152C2×C32×C4C3×D411yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C23×C311yescharacteristic
156C2×C3×Dic3—22no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C3×D12—22no—
163C3×D12—22no—
164C3×D12—22no—
165C3×D12—22no—
166C3×D12—42no—
167C3×D12—42no—
168C3×D12—44no—
169C3×D12—42no—
170C3×D12—42no—
171C3×D12—44no—
172C3×D12—42no—
173C3×D12—42no—
174C3×Dic6—22no—
175C3×Dic6—21no—
176C3×Dic6—21no—
177C3×Dic6—22no—
178C3×Dic6D1212yescomplemented
179C3×Dic6C3×D412yescomplemented
180C3⋊(C3⋊C8)—22no—
181C3×C3⋊C8—62no—
182C3×C3⋊C8—62no—
183C3×C3⋊C8—122no—
184#460—91noSylow
185C2×C33#12111yescharacteristic
186C2×C33C2×SD1611yescharacteristic
187C2×C33AGL(1,ℤ/8)11yescharacteristic
188C32×D3—82no—
189C2×C3×Q8—31no—
190C2×C3×Q8—31no—
191C2×C3×Q8—61no—
192C2×C3×D4—32no—
193C2×C3×D4—32no—
194C2×C3×D4—62no—
195C2×C3×D4—62no—
196C2×C3×D4—62no—
197C2×C3×D4—61no—
198C2×C3×D4—61no—
199C2×C3×D4—61no—
200C2×C3×D4—61no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12C2×C3×D312yes—
204C2×Dic6C2×C3×D311yescharacteristic
205C3×SD16—184no—
206C2×C3×C8—92no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—31no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—62no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42—41no—
217C3×C42—62no—
218C3×C42—61no—
219C3×C42—61no—
220C3×C42D3211yescharacteristic
221C3×C42C2×C3×D311yescharacteristic
222C3×C42C2×C3×D311yescharacteristic
223#215—64no—
224#214—64no—
225C4⋊Dic3C2×C3×D311yescharacteristic
226Dic3⋊C4—21no—
227C4×Dic3—21no—
228C2×C3⋊C8—32no—
229C2×C3⋊C8—32no—
230C2×C3⋊C8—62no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—41no—
235C22×C32—122no—
236C22×C32—122no—
237C22×C32—122no—
238C22×C32C2×#6711yescharacteristic
239C22×C32C2×#6711yescharacteristic
240C22×C32C2×C3×D411yescharacteristic
241C2×C3×D3—44no—
242C2×C3×D3—44no—
243C2×C3×D3—44no—
244C2×C3×D3—42no—
245C2×C3×D3—42no—
246C2×C3×D3—42no—
247C2×C3×D3—42no—
248C2×C3×D3—42no—
249C2×C3×D3—42no—
250C32×C4—21no—
251C32×C4—22no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—42no—
264C32×C4—41no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4—42no—
269C32×C4—62no—
270C32×C4—121no—
271C32×C4#21412yescomplemented
272C32×C4#21512yescomplemented
273C32×C4C3×SD1612yescomplemented
274C32×C4C2×D1211yescharacteristic
275C32×C4#23511yescharacteristic
276C32×C4D3×D411yescharacteristic
277C32×C4#23711yescharacteristic
278C32×C4C2×C3×D411yescharacteristic
279C32×C4C3×#4111yescharacteristic
280C3×Dic3—24no—
281C3×Dic3—22no—
282C3×Dic3—22no—
283C3×Dic3—42no—
284C3×Dic3—41no—
285C3×Dic3—41no—
286C2×SD16—92no—
287#127—31no—
288C4×Q8—31no—
289C4⋊C8—91no—
290#102—92no—
291C33#46011yescharacteristic, complemented, Sylow
292C23×C3—64no—
293C23×C3—62no—
294C23×C3—62no—
295C22×D3—22no—
296C3×Q8—32no—
297C3×Q8—32no—
298C3×Q8—62no—
299C3×Q8—62no—
300C3×Q8—61no—
301C3×Q8—61no—
302C3×D4—62no—
303C3×D4—62no—
304C3×D4—62no—
305C3×D4—62no—
306C3×D4—122no—
307C3×D4—122no—
308C3×D4—124no—
309C3×D4—122no—
310C3×D4—122no—
311C3×D4—124no—
312C3×D4—122no—
313C3×D4—122no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—62no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—62no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4#62511yescharacteristic
333C2×C3×C4#62511yescharacteristic
334C2×C3×C4C3×#6711yescharacteristic
335C2×C3×C4C3×#6711yescharacteristic
336C2×C3×C4C3×#6711yescharacteristic
337C2×C3×C4C3×#6711yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C22×C3×D311yescharacteristic
340C2×C3×C4C22×C3×D311yescharacteristic
341C2×Dic3—21no—
342C2×Dic3—21no—
343D12—22no—
344D12—22no—
345D12—42no—
346D12—42no—
347Dic6—21no—
348Dic6C3×D1212yescomplemented
349C3×C8—182no—
350C3⋊C8—62no—
351C3⋊C8—62no—
352C3⋊C8—122no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—242no—
366C2×C32#93011yescharacteristic
367C2×C32#93211yescharacteristic
368C2×C32C2×#21411yescharacteristic
369C2×C32#93811yescharacteristic
370C2×C32C2×#21511yescharacteristic
371C2×C32#94911yescharacteristic
372C2×C32C3×#12111yescharacteristic
373C2×C32C2×C3×SD1611yescharacteristic
374C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
375C3×D3—84no—
376C3×D3—82no—
377C3×D3—82no—
378C2×Q8—31no—
379C2×D4—32no—
380C2×D4—61no—
381C2×D4—61no—
382SD16—184no—
383C2×C8—92no—
384C4⋊C4—31no—
385C4⋊C4—61no—
386C42—61no—
387C42C3×D3211yescharacteristic
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—41no—
392C22×C3—124no—
393C22×C3—124no—
394C22×C3—124no—
395C22×C3—122no—
396C22×C3—122no—
397C22×C3—122no—
398C22×C3—122no—
399C22×C3—122no—
400C22×C3—122no—
401C22×C3C2×#62511yescharacteristic
402C22×C3C2×C3×#6711yescharacteristic
403C22×C3C2×C3×#6711yescharacteristic
404C2×D3—42no—
405C2×D3—42no—
406C2×D3—42no—
407C3×C4—22no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—42no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4—42no—
426C3×C4—64no—
427C3×C4—62no—
428C3×C4—62no—
429C3×C4—122no—
430C3×C4—121no—
431C3×C4—121no—
432C3×C4#371512yescomplemented
433C3×C4C3×#21412yescomplemented
434C3×C4C3×#21512yescomplemented
435C3×C4#379511yescharacteristic
436C3×C4D3×D1211yescharacteristic
437C3×C4C2×C3×D1211yescharacteristic
438C3×C4C3×#23511yescharacteristic
439C3×C4C3×D3×D411yescharacteristic
440C3×C4C3×#23711yescharacteristic
441Dic3—22no—
442Dic3—41no—
443C32—21no—
444C32—21no—
445C32—21no—
446C32—41no—
447C32#502711yescharacteristic, complemented
448C32#507511yescharacteristic, complemented
449C32C3×#46011yescharacteristic, complemented
450C23—62no—
451Q8—32no—
452Q8—61no—
453D4—62no—
454D4—62no—
455D4—122no—
456D4—122no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4C3×#62511yescharacteristic
460C2×C4C3×#62511yescharacteristic
461C2×C4C2×C3×D3211yescharacteristic
462C8—182no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—244no—
476C2×C3—242no—
477C2×C3—242no—
478C2×C3#6382311yescharacteristic
479C2×C3C2×#371511yescharacteristic
480C2×C3#6390811yescharacteristic
481C2×C3C3×#93011yescharacteristic
482C2×C3C3×#93211yescharacteristic
483C2×C3C2×C3×#21411yescharacteristic
484C2×C3C3×#93811yescharacteristic
485C2×C3C2×C3×#21511yescharacteristic
486C2×C3C3×#94911yescharacteristic
487D3—82no—
488C22—122no—
489C22—122no—
490C22—122no—
491C22C2×C3×#62511yescharacteristic
492C4—21no—
493C4—62no—
494C4—121no—
495C4C3×#371512yescomplemented
496C4C3×#379511yescharacteristic
497C4C3×D3×D1211yescharacteristic
498C3—21no—
499C3—21no—
500C3—21no—
501C3—41no—
502C3#1059036411yescharacteristic, direct product factor
503C3C3×#502711yescharacteristic, complemented
504C3C3×#507511yescharacteristic, complemented
505C2—242no—
506C2C3×#6382311yescharacteristic
507C2C2×C3×#371511yescharacteristic
508C2C3×#6390811yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC32×#874C2
hypercenterC3×C42D32
hypocenterC32C3×#460
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×C4×Dic6C2semidirect
3. C2×C3×#3715C2semidirect
4. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
5. C3×#63562C2semidirect
8. #10590364C3direct (N × H)
27. C3×C4×Dic6D3semidirect
28. C3×C4×Dic6C2×C3semidirect
29. C2×#3715C2×C3semidirect
30. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
36. #63562C2×C3semidirect
48. C32×Dic6D4semidirect
100. C33×C4SD16semidirect
115. C4×Dic6C3×D3semidirect
178. C3×Dic6D12semidirect
179. C3×Dic6C3×D4semidirect
271. C32×C4#214semidirect
272. C32×C4#215semidirect
273. C32×C4C3×SD16semidirect
291. C33#460semidirect
348. Dic6C3×D12semidirect
432. C3×C4#3715semidirect
433. C3×C4C3×#214semidirect
434. C3×C4C3×#215semidirect
447. C32#5027semidirect
448. C32#5075semidirect
449. C32C3×#460semidirect
495. C4C3×#3715semidirect
502. C3#10590364direct (N × H)
503. C3C3×#5027semidirect
504. C3C3×#5075semidirect

Maximal subgroups

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

subgroupindex
C32×#8742
C32×C4×Dic62
C2×C3×#37152
C3×C4⋊(C3⋊(C3⋊C8))2
C3×#635622
C3×#50753
C3×#50273
#105903643