Finite Group Databasebeta

← C3×#10591624 #13158627

Subgroups — C3×#10591624 #13158627

Order 1728.

598 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#947C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic, complemented
3C3×#63662C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5C3×#63563C211yesmaximal, characteristic
6C3×#63562C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic, complemented
8C3×#5055—31nomaximal
9C3×#4794—31nomaximal
10#10591624C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#627—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×(C2×D3)⋊C4—21no—
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×#124—31no—
23C3×#947—21no—
24C3×#947D311yescharacteristic
25C3×#947C2×C311yescharacteristic
26C3×#888—31no—
27C3×#869—31no—
28#63918C2×C311yescharacteristic, complemented
29C32×#103—31no—
30#63662C2×C311yescharacteristic, complemented
31C3×#835—21no—
32C3×#835—31no—
33C3×#835—31no—
34C3×#835—61no—
35C3×#835D311yescharacteristic, complemented
36C3×#835C2×C311yescharacteristic, complemented
37C3×#832—31no—
38C3×#832—31no—
39C3×#821—31no—
40C3×#815—31no—
41C3×#809—31no—
42C3×#803—31no—
43#63563C2×C311yescharacteristic
44#63562C2×C311yescharacteristic
45#63554C2×C311yescharacteristic, complemented
46C22×C32×D3—21no—
47C33×Q8—21no—
48C33×Q8—21no—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57C3×#627—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60C3×#489—91no—
61#5055—31no—
62#4794—31no—
63C2×C32×Q8—21no—
64C2×C32×Q8—21no—
65C2×C32×Q8—21no—
66C2×C32×Q8—41no—
67C2×C32×Q8C2×D311yescharacteristic
68C2×C32×Q8C2×D311yescharacteristic
69C2×C32×Q8C22×C311yescharacteristic
70C2×C32×D4—31no—
71C2×C3×D12—21no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×D12C22×C311yescharacteristic
74C2×C3×Dic6—61no—
75C2×C3×Dic6—61no—
76C2×#627—21no—
77C2×C32×C8—31no—
78C32×C4⋊C4—31no—
79C32×C22⋊C4—61no—
80C32×C22⋊C4—61no—
81C32×C42—31no—
82C4⋊(C3⋊Dic3)C22×C311yescharacteristic
83C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
84C3×(C2×D3)⋊C4—22no—
85C3×(C2×D3)⋊C4—22no—
86C3×(C2×D3)⋊C4—21no—
87C3×(C2×D3)⋊C4—21no—
88C3×(C2×D3)⋊C4—21no—
89C3×(C2×D3)⋊C4—21no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×Dic3⋊C4—61no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3C2×D311yescharacteristic
100C3×C4×Dic3C22×C311yescharacteristic
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107Dic3⋊Dic3—21no—
108C22×C33C2×D411yescharacteristic
109C2×C32×D3—41no—
110C2×C32×D3—41no—
111C2×C32×D3—41no—
112C33×C4—41no—
113C33×C4#4111yescharacteristic
114C33×C4#4111yescharacteristic
115C3×C3⋊Dic3—41no—
116C32×Dic3—21no—
117C32×Dic3—21no—
118C32×Dic3—41no—
119C3×C4⋊Q8—91no—
120C3×#124—31no—
121C3×#124—31no—
122C3×#124—61no—
123#947C3×D311yescharacteristic
124#888—31no—
125#869—31no—
126C3×#103—31no—
127C3×#103—31no—
128C3×#103—61no—
129C3×#103—91no—
130C3×#102—91no—
131C3×#102—91no—
132C3×#97—91no—
133#835—31no—
134#835—31no—
135#835—61no—
136#835C3×D311yescharacteristic, complemented
137#832—31no—
138#832—31no—
139#821—31no—
140#815—31no—
141#809—31no—
142#803—31no—
143C23×C32—61no—
144C22×C3×D3—22no—
145C22×C3×D3—21no—
146C22×C3×D3—21no—
147C32×Q8—21no—
148C32×Q8—21no—
149C32×Q8—22no—
150C32×Q8—22no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—21no—
154C32×Q8—21no—
155C32×Q8—41no—
156C32×Q8—41no—
157C32×Q8—41no—
158C32×Q8—41no—
159C32×Q8—42no—
160C32×Q8—42no—
161C32×D4—61no—
162C32×D4—61no—
163C2×C32×C4—21no—
164C2×C32×C4—22no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—31no—
171C2×C32×C4—31no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—42no—
176C2×C32×C4—61no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C23×C311yescharacteristic
180C2×C3⋊Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—41no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—121no—
189C2×C3×Dic3#6711yescharacteristic
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3C3×D411yescharacteristic
192C2×C3×Dic3C3×D411yescharacteristic
193C3×D12—21no—
194C3×D12—21no—
195C3×D12—21no—
196C3×D12—21no—
197C3×D12—41no—
198C3×D12—41no—
199C3×Dic6—122no—
200C3×Dic6—122no—
201#627—42no—
202C32×C8—61no—
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210#489—91noSylow
211C2×C33#12411yescharacteristic
212C2×C33AGL(1,ℤ/8)11yescharacteristic
213C2×C33#13711yescharacteristic
214C32×D3—81no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—41no—
219C2×C3×Q8—181no—
220C2×C3×Q8D3211yescharacteristic
221C2×C3×Q8C2×C3×D311yescharacteristic
222C2×C3×Q8C2×C3×D311yescharacteristic
223C2×C3×D4—31no—
224C2×C3×D4—31no—
225C2×C3×D4—61no—
226C2×D12C2×C3×D311yescharacteristic
227C2×Dic6—61no—
228C2×Dic6—61no—
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C2×C3×C8—61no—
232C2×C3×C8—91no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—181no—
238C3×C22⋊C4—61no—
239C3×C22⋊C4—61no—
240C3×C22⋊C4—61no—
241C3×C22⋊C4—61no—
242C3×C22⋊C4—62no—
243C3×C22⋊C4—62no—
244C3×C42—31no—
245C3×C42—31no—
246C3×C42—61no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—61no—
252C4⋊Dic3—61no—
253C4⋊Dic3C2×C3×D311yescharacteristic
254Dic3⋊C4—61no—
255C4×Dic3C2×C3×D311yescharacteristic
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8C2×C3×D311yescharacteristic
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—41no—
264C22×C32—121no—
265C22×C32—121no—
266C22×C32—121no—
267C22×C32D3×D411yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C22×C32C2×C3×D411yescharacteristic
270C2×C3×D3—42no—
271C2×C3×D3—42no—
272C2×C3×D3—42no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—42no—
288C32×C4—42no—
289C32×C4—42no—
290C32×C4—44no—
291C32×C4—41no—
292C32×C4—41no—
293C32×C4—41no—
294C32×C4—61no—
295C32×C4—61no—
296C32×C4—121no—
297C32×C4#23511yescharacteristic
298C32×C4#23711yescharacteristic
299C32×C4#23711yescharacteristic
300C32×C4#23711yescharacteristic
301C32×C4C3×#4111yescharacteristic
302C32×C4C3×#4111yescharacteristic
303C3⋊Dic3—41no—
304C3×Dic3—21no—
305C3×Dic3—21no—
306C3×Dic3—21no—
307C3×Dic3—21no—
308C3×Dic3—41no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—81no—
313C3×Dic3—121no—
314C3×Dic3—121no—
315C3×Dic3—241no—
316C4⋊Q8—91no—
317#124—31no—
318#103—31no—
319#103—91no—
320#102—91no—
321#102—91no—
322#97—91no—
323C33#48911yescharacteristic, complemented, Sylow
324C23×C3—62no—
325C23×C3—61no—
326C23×C3—61no—
327C22×D3—21no—
328C3×Q8—22no—
329C3×Q8—22no—
330C3×Q8—21no—
331C3×Q8—21no—
332C3×Q8—21no—
333C3×Q8—21no—
334C3×Q8—21no—
335C3×Q8—21no—
336C3×Q8—41no—
337C3×Q8—41no—
338C3×Q8—41no—
339C3×Q8—41no—
340C3×Q8—42no—
341C3×Q8—42no—
342C3×Q8—362no—
343C3×D4—61no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—61no—
347C3×D4—121no—
348C3×D4—121no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—121no—
369C2×C3×C4—181no—
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C22×C3×D311yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×Dic3—21no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—121no—
377C2×Dic3C3×#6711yescharacteristic
378C2×Dic3C3×#6711yescharacteristic
379D12—21no—
380D12—21no—
381Dic6—122no—
382Dic6—122no—
383C3×C8—61no—
384C3×C8—61no—
385C3×C8—121no—
386C3×C8—181no—
387C3⋊C8—21no—
388C3⋊C8—61no—
389C3⋊C8—61no—
390C3⋊C8—121no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—41no—
403C2×C32—241no—
404C2×C32#88511yescharacteristic
405C2×C32#91411yescharacteristic
406C2×C32#91811yescharacteristic
407C2×C32#93611yescharacteristic
408C2×C32#94911yescharacteristic
409C2×C32#95111yescharacteristic
410C2×C32C3×#12411yescharacteristic
411C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
412C2×C32C3×#13711yescharacteristic
413C3×D3—82no—
414C3×D3—81no—
415C3×D3—81no—
416C2×Q8—181no—
417C2×Q8C3×D3211yescharacteristic
418C2×D4—31no—
419C2×C8—31no—
420C2×C8—91no—
421C4⋊C4—31no—
422C4⋊C4—91no—
423C4⋊C4—181no—
424C22⋊C4—61no—
425C22⋊C4—61no—
426C42—31no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—41no—
431C22×C3—122no—
432C22×C3—122no—
433C22×C3—122no—
434C22×C3—121no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3—121no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3D3×#6711yescharacteristic
441C22×C3C3×D3×D411yescharacteristic
442C22×C3C2×C3×#6711yescharacteristic
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—41no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—44no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—41no—
461C3×C4—61no—
462C3×C4—61no—
463C3×C4—61no—
464C3×C4—61no—
465C3×C4—121no—
466C3×C4—121no—
467C3×C4—121no—
468C3×C4—121no—
469C3×C4—241no—
470C3×C4—361no—
471C3×C4#380411yescharacteristic
472C3×C4#380511yescharacteristic
473C3×C4C3×#23511yescharacteristic
474C3×C4C3×#23711yescharacteristic
475C3×C4C3×#23711yescharacteristic
476C3×C4C3×#23711yescharacteristic
477Dic3—21no—
478Dic3—21no—
479Dic3—41no—
480Dic3—121no—
481Dic3—121no—
482Dic3—241no—
483C32—21no—
484C32—21no—
485C32—21no—
486C32—41no—
487C32#479411yescharacteristic, complemented
488C32#505511yescharacteristic, complemented
489C32C3×#48911yescharacteristic, complemented
490C23—61no—
491Q8—21no—
492Q8—21no—
493Q8—362no—
494D4—61no—
495D4—61no—
496C2×C4—21no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—61no—
500C2×C4—181no—
501C2×C4C2×C3×D3211yescharacteristic
502C8—61no—
503C8—181no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—41no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—242no—
517C2×C3—241no—
518C2×C3—241no—
519C2×C3#6393911yescharacteristic
520C2×C3#6394311yescharacteristic
521C2×C3#6395911yescharacteristic
522C2×C3C3×#88511yescharacteristic
523C2×C3C3×#91411yescharacteristic
524C2×C3C3×#91811yescharacteristic
525C2×C3C3×#93611yescharacteristic
526C2×C3C3×#94911yescharacteristic
527C2×C3C3×#95111yescharacteristic
528D3—81no—
529C22—121no—
530C22—121no—
531C22—121no—
532C22C3×D3×#6711yescharacteristic
533C4—41no—
534C4—61no—
535C4—61no—
536C4—121no—
537C4—361no—
538C4C3×#380411yescharacteristic
539C4C3×#380511yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059162411yescharacteristic, direct product factor
545C3C3×#479411yescharacteristic, complemented
546C3C3×#505511yescharacteristic, complemented
547C2—241no—
548C2C3×#6393911yescharacteristic
549C2C3×#6394311yescharacteristic
550C2C3×#6395911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×C3×Q8D32
hypocenterC32C3×#489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#63918C2semidirect
3. C3×#63662C2semidirect
4. C32×#835C2semidirect
7. C3×#63554C2semidirect
10. #10591624C3direct (N × H)
28. #63918C2×C3semidirect
30. #63662C2×C3semidirect
35. C3×#835D3semidirect
36. C3×#835C2×C3semidirect
45. #63554C2×C3semidirect
136. #835C3×D3semidirect
323. C33#489semidirect
487. C32#4794semidirect
488. C32#5055semidirect
489. C32C3×#489semidirect
544. C3#10591624direct (N × H)
545. C3C3×#4794semidirect
546. C3C3×#5055semidirect

Maximal subgroups

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

subgroupindex
C32×#9472
C3×#639182
C3×#636622
C32×#8352
C3×#635632
C3×#635622
C3×#635542
C3×#50553
C3×#47943
#105916243