Finite Group Databasebeta

← C3×#10590488 #13157457

Subgroups — C3×#10590488 #13157457

Order 1728.

556 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#888C211yesmaximal, characteristic
2C3×#63844C211yesmaximal, characteristic
3C3×#63637C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic
5C3×#63576C211yesmaximal, characteristic
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8C3×#5091—31nomaximal
9C3×#4875—31nomaximal
10#10590488C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6—21no—
13C33×C4⋊C4C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic32—21no—
22C32×C4⋊Q8—31no—
23C3×#888—21no—
24C3×#888D311yescharacteristic
25C3×#888C2×C311yescharacteristic
26C3×#887—31no—
27C3×C4×Dic6—31no—
28#63844C2×C311yescharacteristic
29C32×#107—31no—
30#63637C2×C311yescharacteristic
31C3×#816—31no—
32C3×Dic3⋊C8—31no—
33C3×#810—31no—
34C3×#810—31no—
35C3×#810—31no—
36C3×#810—61no—
37C3×#807—21no—
38C3×#807—31no—
39C3×#807—31no—
40C3×#807D311yescharacteristic
41C3×#807C2×C311yescharacteristic
42C3×C4⋊(C3⋊C8)—31no—
43#63576C2×C311yescharacteristic
44Dic3⋊(C3⋊C8)C2×C311yescharacteristic
45#63567C2×C311yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634Q812yescomplemented
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58C3×#477—91no—
59#5091—31no—
60#4875—31no—
61C2×C32×Q8—61no—
62C2×#634C22×C311yescharacteristic
63C2×C3×Dic6—22no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×C32×C8—31no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4—31no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4—41no—
76C32×C4⋊C4—61no—
77C32×C4⋊C4C2×D311yescharacteristic
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C4⋊C4C22×C311yescharacteristic
80C32×C42—31no—
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3C2×D311yescharacteristic
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C22×C311yescharacteristic
87C3×C4⋊Dic3C22×C311yescharacteristic
88C3×Dic3⋊C4—22no—
89C3×Dic3⋊C4—21no—
90C3×Dic3⋊C4—21no—
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—21no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C22×C311yescharacteristic
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C22×C311yescharacteristic
104#3724—21no—
105Dic32—21no—
106C22×C33C2×D411yescharacteristic
107C33×C4—41no—
108C33×C4C2×Q811yescharacteristic
109C33×C4#4111yescharacteristic
110C3×C3⋊Dic3—22no—
111C32×Dic3—21no—
112C32×Dic3—41no—
113C32×Dic3—41no—
114C32×Dic3Q1612yes—
115C3×C4⋊Q8—31no—
116C3×C4⋊Q8—31no—
117C3×C4⋊Q8—61no—
118C3×C4×Q8—91no—
119#888C3×D311yescharacteristic
120#887—31no—
121C4×Dic6—31no—
122C3×#107—31no—
123C3×#107—31no—
124C3×#107—61no—
125C3×#107—91no—
126C3×C4⋊C8—91no—
127C3×#103—91no—
128C3×#103—91no—
129#816—31no—
130Dic3⋊C8—31no—
131#810—31no—
132#810—31no—
133#810—31no—
134#810—61no—
135#807—31no—
136#807—31no—
137#807C3×D311yescharacteristic
138C4⋊(C3⋊C8)—31no—
139C32×Q8—122no—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—31no—
148C2×C32×C4—31no—
149C2×C32×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4—42no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C23×C311yescharacteristic
158C2×C3⋊Dic3—21no—
159#634—21no—
160#634C3×Q812yescomplemented
161C2×C3×Dic3—22no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—41no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—121no—
172C2×C3×Dic3#6711yescharacteristic
173C2×C3×Dic3#6711yescharacteristic
174C2×C3×Dic3C3×D411yescharacteristic
175C2×C3×Dic3C3×D411yescharacteristic
176C3×Dic6—32no—
177C3×Dic6—32no—
178C3×Dic6—44no—
179C3×Dic6—42no—
180C3×Dic6—42no—
181C3×Dic6—62no—
182C3×Dic6—61no—
183C3×Dic6—61no—
184C3×Dic6—121no—
185C32×C8—61no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—22no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—21no—
190C3×C3⋊C8—61no—
191C3×C3⋊C8—61no—
192C3×C3⋊C8—121no—
193#477—91noSylow
194C2×C33C22⋊Q811yescharacteristic
195C2×C33C2×Q1611yescharacteristic
196C2×C33AGL(1,ℤ/8)11yescharacteristic
197C2×C3×Q8—61no—
198C2×C3×Q8—61no—
199C2×C3×Q8—62no—
200C2×C3×Q8—91no—
201C2×Dic6—21no—
202C2×Dic6—31no—
203C2×Dic6—31no—
204C2×Dic6—61no—
205C2×C3×C8—31no—
206C2×C3×C8—31no—
207C2×C3×C8—61no—
208C2×C3×C8—91no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—31no—
216C3×C4⋊C4—41no—
217C3×C4⋊C4—61no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—61no—
220C3×C4⋊C4—61no—
221C3×C4⋊C4—62no—
222C3×C4⋊C4—181no—
223C3×C4⋊C4D3211yescharacteristic
224C3×C4⋊C4C2×C3×D311yescharacteristic
225C3×C4⋊C4C2×C3×D311yescharacteristic
226C3×C42—31no—
227C3×C42—31no—
228C3×C42—61no—
229C3×C42—181no—
230C4⋊Dic3C2×C3×D311yescharacteristic
231C4⋊Dic3C2×C3×D311yescharacteristic
232Dic3⋊C4—21no—
233Dic3⋊C4—61no—
234Dic3⋊C4—61no—
235C4×Dic3—61no—
236C4×Dic3—61no—
237C4×Dic3C2×C3×D311yescharacteristic
238C2×C3⋊C8—31no—
239C2×C3⋊C8—31no—
240C2×C3⋊C8—61no—
241C2×C3⋊C8C2×C3×D311yescharacteristic
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—41no—
246C22×C32D3×D411yescharacteristic
247C22×C32C2×#6711yescharacteristic
248C22×C32C2×C3×D411yescharacteristic
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—32no—
256C32×C4—41no—
257C32×C4—41no—
258C32×C4—41no—
259C32×C4—42no—
260C32×C4—41no—
261C32×C4—41no—
262C32×C4—61no—
263C32×C4—81no—
264C32×C4—81no—
265C32×C4—82no—
266C32×C4—121no—
267C32×C4—121no—
268C32×C4#23511yescharacteristic
269C32×C4D3×Q811yescharacteristic
270C32×C4D3×Q811yescharacteristic
271C32×C4#23911yescharacteristic
272C32×C4C2×C3×Q811yescharacteristic
273C32×C4C3×#4111yescharacteristic
274C3⋊Dic3—22no—
275C3×Dic3—22no—
276C3×Dic3—21no—
277C3×Dic3—21no—
278C3×Dic3—22no—
279C3×Dic3—42no—
280C3×Dic3—41no—
281C3×Dic3—41no—
282C3×Dic3—41no—
283C3×Dic3—41no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—81no—
287C3×Dic3—122no—
288C3×Dic3#21612yes—
289C3×Dic3C3×Q1612yes—
290C4⋊Q8—31no—
291C4×Q8—91no—
292#107—31no—
293#107—91no—
294C4⋊C8—91no—
295#103—91no—
296#103—91no—
297C33#47711yescharacteristic, complemented, Sylow
298C3×Q8—92no—
299C3×Q8—124no—
300C3×Q8—122no—
301C3×Q8—122no—
302C3×Q8—181no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—62no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—121no—
326C2×C3×C4—181no—
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C22×C3×D311yescharacteristic
329C2×C3×C4C22×C3×D311yescharacteristic
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—121no—
335C2×Dic3C3×#6711yescharacteristic
336C2×Dic3C3×#6711yescharacteristic
337Dic6—32no—
338Dic6—32no—
339Dic6—42no—
340Dic6—62no—
341Dic6—61no—
342Dic6—61no—
343Dic6—121no—
344C3×C8—62no—
345C3×C8—61no—
346C3×C8—61no—
347C3×C8—181no—
348C3⋊C8—21no—
349C3⋊C8—61no—
350C3⋊C8—61no—
351C3⋊C8—121no—
352C2×C32—21no—
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—41no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32#89611yescharacteristic
365C2×C32#91111yescharacteristic
366C2×C32D3×Q1611yescharacteristic
367C2×C32#93211yescharacteristic
368C2×C32C2×#21611yescharacteristic
369C2×C32#94611yescharacteristic
370C2×C32C3×C22⋊Q811yescharacteristic
371C2×C32C2×C3×Q1611yescharacteristic
372C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
373C2×Q8—61no—
374C2×Q8—91no—
375C2×C8—31no—
376C2×C8—91no—
377C4⋊C4—31no—
378C4⋊C4—31no—
379C4⋊C4—61no—
380C4⋊C4—181no—
381C4⋊C4C3×D3211yescharacteristic
382C42—31no—
383C42—181no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—41no—
388C22×C3D3×#6711yescharacteristic
389C22×C3C3×D3×D411yescharacteristic
390C22×C3C2×C3×#6711yescharacteristic
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—32no—
398C3×C4—32no—
399C3×C4—42no—
400C3×C4—41no—
401C3×C4—41no—
402C3×C4—41no—
403C3×C4—41no—
404C3×C4—41no—
405C3×C4—62no—
406C3×C4—62no—
407C3×C4—61no—
408C3×C4—61no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4—82no—
412C3×C4—122no—
413C3×C4—121no—
414C3×C4—121no—
415C3×C4—121no—
416C3×C4—121no—
417C3×C4—182no—
418C3×C4—241no—
419C3×C4#379711yescharacteristic
420C3×C4#379911yescharacteristic
421C3×C4C3×#23511yescharacteristic
422C3×C4C3×D3×Q811yescharacteristic
423C3×C4C3×D3×Q811yescharacteristic
424C3×C4C3×#23911yescharacteristic
425Dic3—21no—
426Dic3—41no—
427Dic3—41no—
428Dic3—62no—
429Dic3—62no—
430Dic3—122no—
431Dic3C3×#21612yes—
432C32—21no—
433C32—21no—
434C32—21no—
435C32—41no—
436C32#487511yescharacteristic, complemented
437C32#509111yescharacteristic, complemented
438C32C3×#47711yescharacteristic, complemented
439Q8—92no—
440Q8—122no—
441Q8—181no—
442C2×C4—21no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—181no—
448C2×C4C2×C3×D3211yescharacteristic
449C8—61no—
450C8—181no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3#6388811yescharacteristic
464C2×C3#6393211yescharacteristic
465C2×C3D3×#21611yescharacteristic
466C2×C3C3×#89611yescharacteristic
467C2×C3C3×#91111yescharacteristic
468C2×C3C3×D3×Q1611yescharacteristic
469C2×C3C3×#93211yescharacteristic
470C2×C3C2×C3×#21611yescharacteristic
471C2×C3C3×#94611yescharacteristic
472C22C3×D3×#6711yescharacteristic
473C4—32no—
474C4—41no—
475C4—61no—
476C4—121no—
477C4—121no—
478C4—182no—
479C4C3×#379711yescharacteristic
480C4C3×#379911yescharacteristic
481C3—21no—
482C3—21no—
483C3—21no—
484C3—41no—
485C3#1059048811yescharacteristic, direct product factor
486C3C3×#487511yescharacteristic, complemented
487C3C3×#509111yescharacteristic, complemented
488C2C3×#6388811yescharacteristic
489C2C3×#6393211yescharacteristic
490C2C3×D3×#21611yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#477
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
10. #10590488C3direct (N × H)
50. C3×#634Q8semidirect
160. #634C3×Q8semidirect
297. C33#477semidirect
436. C32#4875semidirect
437. C32#5091semidirect
438. C32C3×#477semidirect
485. C3#10590488direct (N × H)
486. C3C3×#4875semidirect
487. C3C3×#5091semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
C3×#638442
C3×#636372
C32×#8072
C3×#635762
C3×Dic3⋊(C3⋊C8)2
C3×#635672
C3×#50913
C3×#48753
#105904883