Finite Group Databasebeta

← C3×#10590808 #13157782

Subgroups — C3×#10590808 #13157782

Order 1728.

594 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3746C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3C3×#63844C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic
5C32×#816C211yesmaximal, characteristic
6C32×Dic3⋊C8C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8C3×#4849—31nomaximal
9C3×#4795—31nomaximal
10#10590808C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C33×C8C2211yescharacteristic
15C3×#3746—22no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C3×#3724—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3719—22no—
24C2×#3746C2×C311yescharacteristic
25C3×#944—31no—
26C2×C3×#216—31no—
27C2×C3×Dic12—31no—
28C2×C3×Dic12—31no—
29C2×C3×Dic12—31no—
30C2×C3×Dic12—61no—
31C3×#887—31no—
32C3×#869—31no—
33C3×C4×Dic6—31no—
34#63866C2×C311yescharacteristic
35#63844C2×C311yescharacteristic
36C2×#3719C2×C311yescharacteristic
37C32×C4⋊C8—31no—
38C32×#103—31no—
39C3×#816—21no—
40C3×#816—31no—
41C3×#816D311yescharacteristic
42C3×#816C2×C311yescharacteristic
43C3×Dic3⋊C8—21no—
44C3×Dic3⋊C8D311yescharacteristic
45C3×Dic3⋊C8C2×C311yescharacteristic
46C3×#810—31no—
47#63567C2×C311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×#634—21no—
52C3×#634—21no—
53C3×#634—21no—
54C3×#634D412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×Dic6—21no—
60C32×Dic6—21no—
61C33×C8—21no—
62C32×C3⋊C8—21no—
63C3×#462—91no—
64#4849—31no—
65#4795—31no—
66C2×C32×Q8—31no—
67C2×#634C22×C311yescharacteristic
68C2×#634C22×C311yescharacteristic
69C2×C3×Dic6—21no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—61no—
75C2×C3×Dic6—61no—
76C2×C3×Dic6C2×D311yescharacteristic
77C2×C3×Dic6C22×C311yescharacteristic
78C2×C32×C8—21no—
79C2×C32×C8—21no—
80C2×C32×C8—21no—
81C2×C32×C8—31no—
82C2×C32×C8—41no—
83C2×C32×C8C2×D311yescharacteristic
84C2×C32×C8C2×D311yescharacteristic
85C2×C32×C8C22×C311yescharacteristic
86C32×C4⋊C4—31no—
87C32×C42—31no—
88#3746—22no—
89C3×#216—62no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C22×C311yescharacteristic
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—61no—
99C3×C4×Dic3—21no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3C2×D311yescharacteristic
103C3×C4×Dic3C22×C311yescharacteristic
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107C3×Dic12—62no—
108C3×Dic12—62no—
109C3×Dic12—64no—
110C3×Dic12—62no—
111#3724—21no—
112Dic3⋊Dic3—21no—
113Dic3⋊Dic3—21no—
114Dic32—21no—
115#3719—22no—
116C22×C33C2×D411yescharacteristic
117C33×C4C2×D411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—41no—
121C32×Dic3—21no—
122C32×Dic3—41no—
123C32×Dic3—41no—
124C32×Dic3Q1612yescomplemented
125C2×C3×Q16—91no—
126C2×C3×Q16—91no—
127C3×C4⋊Q8—91no—
128C3×C4×Q8—91no—
129#944—31no—
130C2×#216—31no—
131C2×Dic12—31no—
132C2×Dic12—31no—
133C2×Dic12—31no—
134C2×Dic12—61no—
135#887—31no—
136#869—31no—
137C4×Dic6—31no—
138C3×C4⋊C8—31no—
139C3×C4⋊C8—31no—
140C3×C4⋊C8—61no—
141C3×#103—31no—
142C3×#103—31no—
143C3×#103—61no—
144C3×#103—91no—
145#816—31no—
146#816C3×D311yescharacteristic
147Dic3⋊C8C3×D311yescharacteristic
148#810—31no—
149C32×Q8—61no—
150C32×Q8—61no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—31no—
155C2×C32×C4—31no—
156C2×C32×C4—41no—
157C2×C32×C4—61no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C23×C311yescharacteristic
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164#634—21no—
165#634—21no—
166#634—21no—
167#634C3×D412yes—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—121no—
181C2×C3×Dic3—121no—
182C2×C3×Dic3D1211yescharacteristic
183C2×C3×Dic3D1211yescharacteristic
184C2×C3×Dic3C3×D411yescharacteristic
185C2×C3×Dic3C3×D411yescharacteristic
186C3×Dic6—22no—
187C3×Dic6—21no—
188C3×Dic6—21no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—32no—
192C3×Dic6—32no—
193C3×Dic6—41no—
194C3×Dic6—62no—
195C3×Dic6—61no—
196C3×Dic6—61no—
197C3×Dic6—62no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—61no—
202C3×Dic6—121no—
203C3×Dic6—121no—
204C32×C8—21no—
205C32×C8—22no—
206C32×C8—21no—
207C32×C8—21no—
208C32×C8—41no—
209C32×C8—41no—
210C32×C8—42no—
211C32×C8—61no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—41no—
215#462—91noSylow
216C2×C33#12111yescharacteristic
217C2×C33C2×Q1611yescharacteristic
218C2×C33#13711yescharacteristic
219C2×C3×Q8—31no—
220C2×C3×Q8—31no—
221C2×C3×Q8—61no—
222C2×C3×Q8—91no—
223C2×C3×Q8—91no—
224C2×Dic6—31no—
225C2×Dic6—31no—
226C2×Dic6—31no—
227C2×Dic6—31no—
228C2×Dic6—61no—
229C2×Dic6—61no—
230C2×Dic6C2×C3×D311yescharacteristic
231C3×Q16—182no—
232C3×Q16—182no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—31no—
237C2×C3×C8—31no—
238C2×C3×C8—41no—
239C2×C3×C8—61no—
240C2×C3×C8D3211yescharacteristic
241C2×C3×C8C2×C3×D311yescharacteristic
242C2×C3×C8C2×C3×D311yescharacteristic
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—181no—
247C3×C4⋊C4—181no—
248C3×C4⋊C4—181no—
249C3×C42—31no—
250C3×C42—31no—
251C3×C42—61no—
252C3×C42—181no—
253#216—62no—
254C4⋊Dic3—61no—
255C4⋊Dic3—61no—
256C4⋊Dic3C2×C3×D311yescharacteristic
257Dic3⋊C4—61no—
258Dic3⋊C4—61no—
259Dic3⋊C4—61no—
260Dic3⋊C4—61no—
261C4×Dic3—61no—
262C4×Dic3—61no—
263C4×Dic3C2×C3×D311yescharacteristic
264C2×C3⋊C8C2×C3×D311yescharacteristic
265Dic12—62no—
266Dic12—62no—
267Dic12—64no—
268Dic12—62no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32C2×D1211yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32C2×C3×D411yescharacteristic
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—32no—
283C32×C4—41no—
284C32×C4—41no—
285C32×C4—61no—
286C32×C4—121no—
287C32×C4—121no—
288C32×C4#23511yescharacteristic
289C32×C4D3×D411yescharacteristic
290C32×C4D3×D411yescharacteristic
291C32×C4#23911yescharacteristic
292C32×C4C2×C3×D411yescharacteristic
293C32×C4C3×#4111yescharacteristic
294C3⋊Dic3—22no—
295C3⋊Dic3—41no—
296C3×Dic3—21no—
297C3×Dic3—21no—
298C3×Dic3—22no—
299C3×Dic3—41no—
300C3×Dic3—42no—
301C3×Dic3—41no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—81no—
308C3×Dic3—122no—
309C3×Dic3—122no—
310C3×Dic3—121no—
311C3×Dic3—121no—
312C3×Dic3Dic1212yescomplemented
313C3×Dic3C3×Q1612yescomplemented
314C2×Q16—91no—
315C2×Q16—91no—
316C4⋊Q8—91no—
317C4×Q8—91no—
318C4⋊C8—31no—
319#103—31no—
320#103—91no—
321C33#46211yescharacteristic, complemented, Sylow
322C3×Q8—62no—
323C3×Q8—61no—
324C3×Q8—61no—
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—92no—
328C3×Q8—121no—
329C3×Q8—181no—
330C3×Q8—181no—
331C3×Q8—181no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—41no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—121no—
347C2×C3×C4—121no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C22×C3×D311yescharacteristic
352C2×C3×C4C22×C3×D311yescharacteristic
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—121no—
360C2×Dic3—121no—
361C2×Dic3C3×D1211yescharacteristic
362C2×Dic3C3×D1211yescharacteristic
363Dic6—21no—
364Dic6—21no—
365Dic6—32no—
366Dic6—32no—
367Dic6—62no—
368Dic6—61no—
369Dic6—61no—
370Dic6—62no—
371Dic6—61no—
372Dic6—61no—
373Dic6—61no—
374Dic6—61no—
375Dic6—121no—
376Dic6—121no—
377C3×C8—22no—
378C3×C8—21no—
379C3×C8—21no—
380C3×C8—21no—
381C3×C8—41no—
382C3×C8—41no—
383C3×C8—42no—
384C3×C8—61no—
385C3×C8—61no—
386C3×C8—121no—
387C3⋊C8—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32#88411yescharacteristic
401C2×C32#89511yescharacteristic
402C2×C32C2×Dic1211yescharacteristic
403C2×C32#90911yescharacteristic
404C2×C32#91511yescharacteristic
405C2×C32D3×Q1611yescharacteristic
406C2×C32C3×#12111yescharacteristic
407C2×C32C2×C3×Q1611yescharacteristic
408C2×C32C3×#13711yescharacteristic
409C2×Q8—31no—
410C2×Q8—91no—
411C2×Q8—91no—
412Q16—182no—
413Q16—182no—
414C2×C8—31no—
415C2×C8C3×D3211yescharacteristic
416C4⋊C4—31no—
417C4⋊C4—181no—
418C4⋊C4—181no—
419C4⋊C4—181no—
420C42—31no—
421C42—181no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—41no—
426C22×C3D3×D1211yescharacteristic
427C22×C3C2×C3×D1211yescharacteristic
428C22×C3C3×D3×D411yescharacteristic
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—32no—
436C3×C4—32no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—62no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—121no—
443C3×C4—122no—
444C3×C4—121no—
445C3×C4—121no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—182no—
449C3×C4—241no—
450C3×C4—361no—
451C3×C4#379911yescharacteristic
452C3×C4#381111yescharacteristic
453C3×C4C3×#23511yescharacteristic
454C3×C4C3×D3×D411yescharacteristic
455C3×C4C3×D3×D411yescharacteristic
456C3×C4C3×#23911yescharacteristic
457Dic3—21no—
458Dic3—41no—
459Dic3—41no—
460Dic3—62no—
461Dic3—62no—
462Dic3—122no—
463Dic3—122no—
464Dic3—121no—
465Dic3—121no—
466Dic3C3×Dic1212yescomplemented
467C32—21no—
468C32—21no—
469C32—21no—
470C32—41no—
471C32#479511yescharacteristic, complemented
472C32#484911yescharacteristic, complemented
473C32C3×#46211yescharacteristic, complemented
474Q8—61no—
475Q8—61no—
476Q8—92no—
477Q8—181no—
478Q8—181no—
479Q8—181no—
480C2×C4—31no—
481C2×C4—31no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—181no—
485C2×C4—181no—
486C2×C4C2×C3×D3211yescharacteristic
487C8—21no—
488C8—61no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—41no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3D3×Dic1211yescharacteristic
502C2×C3#6380011yescharacteristic
503C2×C3#6391011yescharacteristic
504C2×C3C3×#88411yescharacteristic
505C2×C3C3×#89511yescharacteristic
506C2×C3C2×C3×Dic1211yescharacteristic
507C2×C3C3×#90911yescharacteristic
508C2×C3C3×#91511yescharacteristic
509C2×C3C3×D3×Q1611yescharacteristic
510C22C3×D3×D1211yescharacteristic
511C4—32no—
512C4—61no—
513C4—121no—
514C4—121no—
515C4—182no—
516C4—361no—
517C4C3×#379911yescharacteristic
518C4C3×#381111yescharacteristic
519C3—21no—
520C3—21no—
521C3—21no—
522C3—41no—
523C3#1059080811yescharacteristic, direct product factor
524C3C3×#479511yescharacteristic, complemented
525C3C3×#484911yescharacteristic, complemented
526C2C3×D3×Dic1211yescharacteristic
527C2C3×#6380011yescharacteristic
528C2C3×#6391011yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#462
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
10. #10590808C3direct (N × H)
124. C32×Dic3Q16semidirect
312. C3×Dic3Dic12semidirect
313. C3×Dic3C3×Q16semidirect
321. C33#462semidirect
466. Dic3C3×Dic12semidirect
471. C32#4795semidirect
472. C32#4849semidirect
473. C32C3×#462semidirect
523. C3#10590808direct (N × H)
524. C3C3×#4795semidirect
525. C3C3×#4849semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37462
C3×#638662
C3×#638442
C2×C3×#37192
C32×#8162
C32×Dic3⋊C82
C3×#635672
C3×#48493
C3×#47953
#105908083