Finite Group Databasebeta

← C3×#10590367 #13157384

Subgroups — C3×#10590367 #13157384

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
1C3×#64083C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#3717C212yesmaximal, complemented
4C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63564C212yesmaximal, complemented
6C3×#5027—31nomaximal
7C3×#4848—31nomaximal
8#10590367C311yesmaximal, characteristic, direct product factor
9C2×C3×C4⋊(C3⋊D3)—21no—
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C2×C32×Dic6C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C32×C4⋊Dic3C2211yescharacteristic
15C32×Dic3⋊C4—21no—
16C32×C4×Dic3—21no—
17C2×C32×C3⋊C8C2212yes—
18C3×#3717—24no—
19C32×C4×Q8—31no—
20#64083C2×C311yescharacteristic
21C2×C3×#215—32no—
22C2×C3×#204—32no—
23C3×#874—31no—
24C3×#874—31no—
25C3×#874—61no—
26C3×C4×Dic6—21no—
27C3×C4×Dic6D311yescharacteristic, complemented
28C3×C4×Dic6C2×C311yescharacteristic, complemented
29C2×#3717C2×C312yescomplemented
30C32×C4⋊C8—31no—
31C3×#821—32no—
32C3×#809—32no—
33C3×C4⋊(C3⋊C8)—21no—
34C3×C4⋊(C3⋊C8)D311yescharacteristic, complemented
35C3×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
36#63564C2×C312yescomplemented
37C22×C3×C3⋊D3—22no—
38C2×C33×C4D411yescharacteristic
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C3×C4⋊(C3⋊D3)—22no—
42C3×C4⋊(C3⋊D3)—22no—
43C3×C4⋊(C3⋊D3)—42no—
44C3×C4⋊(C3⋊D3)—42no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—21no—
47C32×Dic6—21no—
48C32×Dic6D412yescomplemented
49C32×C3⋊C8—22no—
50C3×#460—91no—
51#5027—31no—
52#4848—31no—
53C2×C32×Q8—31no—
54C2×C4⋊(C3⋊D3)—21no—
55C2×C4⋊(C3⋊D3)—21no—
56C2×C4⋊(C3⋊D3)C22×C312yes—
57C2×C3×D12—32no—
58C2×C3×D12—32no—
59C2×C3×D12—62no—
60C2×C3×D12—62no—
61C2×C3×D12—62no—
62C2×C3×D12—61no—
63C2×C3×D12—61no—
64C2×C3×D12—61no—
65C2×C3×D12—61no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6C2×D311yescharacteristic
68C2×C3×Dic6C22×C311yescharacteristic
69C2×C32×C8—32no—
70C32×C4⋊C4—31no—
71C32×C4⋊C4—61no—
72C32×C42—21no—
73C32×C42—21no—
74C32×C42—21no—
75C32×C42—41no—
76C32×C42—61no—
77C32×C42C2×D311yescharacteristic
78C32×C42C2×D311yescharacteristic
79C32×C42C22×C311yescharacteristic
80C3×#215—64no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3C2×D311yescharacteristic
83C3×C4⋊Dic3C22×C311yescharacteristic
84C3×Dic3⋊C4—22no—
85C3×Dic3⋊C4—21no—
86C3×Dic3⋊C4—21no—
87C3×C4×Dic3—22no—
88C3×C4×Dic3—21no—
89C3×C4×Dic3—21no—
90C2×C3×C3⋊C8—22no—
91C2×C3×C3⋊C8C2×D312yes—
92C2×C3×C3⋊C8C22×C312yes—
93C3×#204—64no—
94#3717—24no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—42no—
97C2×C3×C3⋊D3—42no—
98C2×C3×C3⋊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—91no—
107C3×C4×Q8—31no—
108C3×C4×Q8—31no—
109C3×C4×Q8—61no—
110C2×#215—32no—
111C2×#204—32no—
112#874—31no—
113#874—31no—
114#874—61no—
115C4×Dic6C3×D311yescharacteristic, complemented
116C3×C4⋊C8—31no—
117C3×C4⋊C8—31no—
118C3×C4⋊C8—61no—
119C3×#102—92no—
120#821—32no—
121#809—32no—
122C4⋊(C3⋊C8)C3×D311yescharacteristic, complemented
123C22×C3⋊D3—22no—
124C22×C3×D3—64no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C32×Q8—32no—
128C32×Q8—61no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—41no—
139C2×C32×C4—41no—
140C2×C32×C4—41no—
141C2×C32×C4—61no—
142C2×C32×C4—61no—
143C2×C32×C4D1211yescharacteristic
144C2×C32×C4D1211yescharacteristic
145C2×C32×C4#6711yescharacteristic
146C2×C32×C4#6711yescharacteristic
147C2×C32×C4C3×D411yescharacteristic
148C2×C32×C4C3×D411yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152C4⋊(C3⋊D3)—22no—
153C4⋊(C3⋊D3)—22no—
154C4⋊(C3⋊D3)—42no—
155C4⋊(C3⋊D3)—42no—
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—62no—
163C3×D12—62no—
164C3×D12—62no—
165C3×D12—62no—
166C3×D12—122no—
167C3×D12—122no—
168C3×D12—124no—
169C3×D12—122no—
170C3×D12—122no—
171C3×D12—124no—
172C3×D12—122no—
173C3×D12—122no—
174C3×Dic6—22no—
175C3×Dic6—21no—
176C3×Dic6—21no—
177C3×Dic6—22no—
178C3×Dic6D1212yescomplemented
179C3×Dic6C3×D412yescomplemented
180C32×C8—62no—
181C3×C3⋊C8—22no—
182C3×C3⋊C8—22no—
183C3×C3⋊C8—42no—
184#460—91noSylow
185C2×C33#12111yescharacteristic
186C2×C33C2×SD1611yescharacteristic
187C2×C33AGL(1,ℤ/8)11yescharacteristic
188C3×C3⋊D3—82no—
189C2×C3×Q8—31no—
190C2×C3×Q8—31no—
191C2×C3×Q8—61no—
192C2×C3×D4—92no—
193C2×C3×D4—181no—
194C2×C3×D4—181no—
195C2×D12—32no—
196C2×D12—32no—
197C2×D12—62no—
198C2×D12—62no—
199C2×D12—62no—
200C2×D12—61no—
201C2×D12—61no—
202C2×D12—61no—
203C2×D12—61no—
204C2×Dic6C2×C3×D311yescharacteristic
205C3×SD16—184no—
206C2×C3×C8—32no—
207C2×C3×C8—32no—
208C2×C3×C8—62no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—62no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4—61no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42—41no—
219C3×C42—62no—
220C3×C42—61no—
221C3×C42—61no—
222C3×C42D3211yescharacteristic
223C3×C42C2×C3×D311yescharacteristic
224C3×C42C2×C3×D311yescharacteristic
225#215—64no—
226C4⋊Dic3C2×C3×D311yescharacteristic
227Dic3⋊C4—21no—
228C4×Dic3—21no—
229C2×C3⋊C8C2×C3×D312yes—
230#204—64no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—41no—
235C22×C32C2×D1211yescharacteristic
236C22×C32C2×#6711yescharacteristic
237C22×C32C2×C3×D411yescharacteristic
238C2×C3⋊D3—42no—
239C2×C3⋊D3—42no—
240C2×C3⋊D3—42no—
241C2×C3×D3—124no—
242C2×C3×D3—124no—
243C2×C3×D3—124no—
244C2×C3×D3—122no—
245C2×C3×D3—122no—
246C2×C3×D3—122no—
247C2×C3×D3—122no—
248C2×C3×D3—122no—
249C2×C3×D3—122no—
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#20412yescomplemented
272C32×C4#21512yescomplemented
273C32×C4C3×SD1612yescomplemented
274C32×C4C2×D1211yescharacteristic
275C32×C4#23511yescharacteristic
276C32×C4D3×D411yescharacteristic
277C32×C4#23911yescharacteristic
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—91no—
288C4×Q8—31no—
289C4⋊C8—31no—
290#102—92no—
291C33#46011yescharacteristic, complemented, Sylow
292C23×C3—182no—
293C22×D3—64no—
294C22×D3—62no—
295C22×D3—62no—
296C3×Q8—32no—
297C3×Q8—32no—
298C3×Q8—62no—
299C3×Q8—62no—
300C3×Q8—61no—
301C3×Q8—61no—
302C3×D4—182no—
303C3×D4—182no—
304C3×D4—362no—
305C3×D4—362no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—62no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—62no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4#62611yescharacteristic
325C2×C3×C4#62611yescharacteristic
326C2×C3×C4C3×D1211yescharacteristic
327C2×C3×C4C3×D1211yescharacteristic
328C2×C3×C4C3×#6711yescharacteristic
329C2×C3×C4C3×#6711yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C22×C3×D311yescharacteristic
332C2×C3×C4C22×C3×D311yescharacteristic
333C2×Dic3—21no—
334C2×Dic3—21no—
335D12—62no—
336D12—62no—
337D12—62no—
338D12—62no—
339D12—122no—
340D12—122no—
341D12—124no—
342D12—122no—
343D12—122no—
344D12—124no—
345D12—122no—
346D12—122no—
347Dic6—21no—
348Dic6C3×D1212yescomplemented
349C3×C8—62no—
350C3×C8—62no—
351C3×C8—122no—
352C3⋊C8—22no—
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#89511yescharacteristic
366C2×C32C2×#20411yescharacteristic
367C2×C32#90811yescharacteristic
368C2×C32#93011yescharacteristic
369C2×C32C2×#21511yescharacteristic
370C2×C32#94911yescharacteristic
371C2×C32C3×#12111yescharacteristic
372C2×C32C2×C3×SD1611yescharacteristic
373C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
374C3⋊D3—82no—
375C3×D3—244no—
376C3×D3—242no—
377C3×D3—242no—
378C2×Q8—31no—
379C2×D4—92no—
380C2×D4—181no—
381C2×D4—181no—
382SD16—184no—
383C2×C8—32no—
384C4⋊C4—31no—
385C4⋊C4—61no—
386C42—61no—
387C42C3×D3211yescharacteristic
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—41no—
392C22×C3—362no—
393C22×C3—362no—
394C22×C3—362no—
395C22×C3C2×#62611yescharacteristic
396C22×C3C2×C3×D1211yescharacteristic
397C22×C3C2×C3×#6711yescharacteristic
398C2×D3—124no—
399C2×D3—124no—
400C2×D3—124no—
401C2×D3—122no—
402C2×D3—122no—
403C2×D3—122no—
404C2×D3—122no—
405C2×D3—122no—
406C2×D3—122no—
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#371712yescomplemented
433C3×C4C3×#20412yescomplemented
434C3×C4C3×#21512yescomplemented
435C3×C4#379911yescharacteristic
436C3×C4D3×D1211yescharacteristic
437C3×C4C2×C3×D1211yescharacteristic
438C3×C4C3×#23511yescharacteristic
439C3×C4C3×D3×D411yescharacteristic
440C3×C4C3×#23911yescharacteristic
441Dic3—22no—
442Dic3—41no—
443C32—21no—
444C32—21no—
445C32—21no—
446C32—41no—
447C32#484811yescharacteristic, complemented
448C32#502711yescharacteristic, complemented
449C32C3×#46011yescharacteristic, complemented
450C23—182no—
451Q8—32no—
452Q8—61no—
453D4—182no—
454D4—182no—
455D4—362no—
456D4—362no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4C3×#62611yescharacteristic
460C2×C4C3×#62611yescharacteristic
461C2×C4C2×C3×D3211yescharacteristic
462C8—62no—
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—722no—
476C2×C3#6382511yescharacteristic
477C2×C3C2×#371711yescharacteristic
478C2×C3#6390911yescharacteristic
479C2×C3C3×#89511yescharacteristic
480C2×C3C2×C3×#20411yescharacteristic
481C2×C3C3×#90811yescharacteristic
482C2×C3C3×#93011yescharacteristic
483C2×C3C2×C3×#21511yescharacteristic
484C2×C3C3×#94911yescharacteristic
485D3—244no—
486D3—242no—
487D3—242no—
488C22—362no—
489C22—362no—
490C22—362no—
491C22C2×C3×#62611yescharacteristic
492C4—21no—
493C4—62no—
494C4—121no—
495C4C3×#371712yescomplemented
496C4C3×#379911yescharacteristic
497C4C3×D3×D1211yescharacteristic
498C3—21no—
499C3—21no—
500C3—21no—
501C3—41no—
502C3#1059036711yescharacteristic, direct product factor
503C3C3×#484811yescharacteristic, complemented
504C3C3×#502711yescharacteristic, complemented
505C2—722no—
506C2C3×#6382511yescharacteristic
507C2C2×C3×#371711yescharacteristic
508C2C3×#6390911yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC3×#64083C2
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×#3717C2semidirect
4. C32×C4⋊(C3⋊C8)C2semidirect
5. C3×#63564C2semidirect
8. #10590367C3direct (N × H)
27. C3×C4×Dic6D3semidirect
28. C3×C4×Dic6C2×C3semidirect
29. C2×#3717C2×C3semidirect
34. C3×C4⋊(C3⋊C8)D3semidirect
35. C3×C4⋊(C3⋊C8)C2×C3semidirect
36. #63564C2×C3semidirect
48. C32×Dic6D4semidirect
100. C33×C4SD16semidirect
115. C4×Dic6C3×D3semidirect
122. C4⋊(C3⋊C8)C3×D3semidirect
178. C3×Dic6D12semidirect
179. C3×Dic6C3×D4semidirect
271. C32×C4#204semidirect
272. C32×C4#215semidirect
273. C32×C4C3×SD16semidirect
291. C33#460semidirect
348. Dic6C3×D12semidirect
432. C3×C4#3717semidirect
433. C3×C4C3×#204semidirect
434. C3×C4C3×#215semidirect
447. C32#4848semidirect
448. C32#5027semidirect
449. C32C3×#460semidirect
495. C4C3×#3717semidirect
502. C3#10590367direct (N × H)
503. C3C3×#4848semidirect
504. C3C3×#5027semidirect

Maximal subgroups

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

subgroupindex
C3×#640832
C32×C4×Dic62
C2×C3×#37172
C32×C4⋊(C3⋊C8)2
C3×#635642
C3×#50273
C3×#48483
#105903673