Finite Group Databasebeta

← C3×#10590411 #13157539

Subgroups — C3×#10590411 #13157539

Order 1728.

558 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#888C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3C3×#63634C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic
5C3×#63576C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic
8C3×#5078—31nomaximal
9C3×#4874—31nomaximal
10#10590411C311yesmaximal, characteristic, direct product factor
11C2×C32×Dic6—21no—
12C2×C3×#627—21no—
13C33×C4⋊C4C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×C4⋊Q8—31no—
23C3×#888—21no—
24C3×#888—31no—
25C3×#888D311yescharacteristic
26C3×#888C2×C311yescharacteristic
27C3×#869—31no—
28#63918C2×C311yescharacteristic
29C32×#107—31no—
30#63634C2×C311yescharacteristic
31C3×#818—31no—
32C3×C8×Dic3—31no—
33C3×#807—21no—
34C3×#807—31no—
35C3×#807—31no—
36C3×#807—31no—
37C3×#807—31no—
38C3×#807—31no—
39C3×#807—61no—
40C3×#807D311yescharacteristic
41C3×#807C2×C311yescharacteristic
42C3×C4×C3⋊C8—31no—
43#63576C2×C311yescharacteristic
44#63575C2×C311yescharacteristic
45Dic3×C3⋊C8C2×C311yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×Dic6—42no—
54C3×#627—42no—
55C3×C3⋊(C3⋊C8)Q812yescomplemented
56C32×C3⋊C8Q812yescomplemented
57C3×#500—91no—
58#5078—31no—
59#4874—31no—
60C2×C32×Q8—61no—
61C2×C3×Dic6—22no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—61no—
65C2×C3×Dic6—61no—
66C2×#627—21no—
67C2×C32×C8—31no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4—31no—
72C32×C4⋊C4—31no—
73C32×C4⋊C4—41no—
74C32×C4⋊C4—61no—
75C32×C4⋊C4C2×D311yescharacteristic
76C32×C4⋊C4C2×D311yescharacteristic
77C32×C4⋊C4C22×C311yescharacteristic
78C32×C42—31no—
79C4⋊(C3⋊Dic3)C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C22×C311yescharacteristic
90C3×C4⋊Dic3C22×C311yescharacteristic
91C3×Dic3⋊C4—22no—
92C3×Dic3⋊C4—21no—
93C3×Dic3⋊C4—21no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—21no—
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
104Dic3⋊Dic3—21no—
105C22×C33C2×D411yescharacteristic
106C33×C4—41no—
107C33×C4C2×Q811yescharacteristic
108C33×C4C2×Q811yescharacteristic
109C3×C3⋊Dic3—41no—
110C32×Dic3—41no—
111C32×Dic3—41no—
112C32×Dic3D812yes—
113C32×Dic3Q1612yes—
114C3×C4⋊Q8—31no—
115C3×C4⋊Q8—31no—
116C3×C4⋊Q8—61no—
117C3×C4⋊Q8—91no—
118#888—31no—
119#888C3×D311yescharacteristic
120#869—31no—
121C3×#107—31no—
122C3×#107—31no—
123C3×#107—61no—
124C3×#107—91no—
125C3×#107—91no—
126C3×#107—91no—
127C3×C4×C8—91no—
128#818—31no—
129C8×Dic3—31no—
130#807—31no—
131#807—31no—
132#807—31no—
133#807—31no—
134#807—31no—
135#807—61no—
136#807C3×D311yescharacteristic
137C4×C3⋊C8—31no—
138C32×Q8—122no—
139C2×C32×C4—21no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—31no—
147C2×C32×C4—31no—
148C2×C32×C4—41no—
149C2×C32×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4—42no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C23×C311yescharacteristic
157C2×C3⋊Dic3—21no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—41no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3#6711yescharacteristic
170C2×C3×Dic3#6711yescharacteristic
171C2×C3×Dic3C3×D411yescharacteristic
172C2×C3×Dic3C3×D411yescharacteristic
173C3×Dic6—44no—
174C3×Dic6—42no—
175C3×Dic6—42no—
176C3×Dic6—122no—
177C3×Dic6—122no—
178#627—42no—
179C32×C8—32no—
180C3⋊(C3⋊C8)C3×Q812yescomplemented
181C3×C3⋊C8—22no—
182C3×C3⋊C8—32no—
183C3×C3⋊C8—32no—
184C3×C3⋊C8—62no—
185C3×C3⋊C8Dic612yescomplemented
186C3×C3⋊C8C3×Q812yescomplemented
187#500—91noSylow
188C2×C33C4⋊Q811yescharacteristic
189C2×C33C2×D811yescharacteristic
190C2×C33C2×Q1611yescharacteristic
191C2×C3×Q8—61no—
192C2×C3×Q8—61no—
193C2×C3×Q8—62no—
194C2×C3×Q8—181no—
195C2×Dic6—21no—
196C2×Dic6—61no—
197C2×Dic6—61no—
198C2×C3×C8—31no—
199C2×C3×C8—31no—
200C2×C3×C8—61no—
201C2×C3×C8—91no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—31no—
206C3×C4⋊C4—31no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—31no—
209C3×C4⋊C4—41no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4—62no—
215C3×C4⋊C4—91no—
216C3×C4⋊C4—181no—
217C3×C4⋊C4D3211yescharacteristic
218C3×C4⋊C4C2×C3×D311yescharacteristic
219C3×C4⋊C4C2×C3×D311yescharacteristic
220C3×C42—31no—
221C3×C42—31no—
222C3×C42—61no—
223C4⋊Dic3—31no—
224C4⋊Dic3—31no—
225C4⋊Dic3—61no—
226C4⋊Dic3—61no—
227C4⋊Dic3C2×C3×D311yescharacteristic
228C4⋊Dic3C2×C3×D311yescharacteristic
229Dic3⋊C4—21no—
230Dic3⋊C4—61no—
231C4×Dic3C2×C3×D311yescharacteristic
232C2×C3⋊C8—31no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—61no—
235C2×C3⋊C8C2×C3×D311yescharacteristic
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—41no—
240C22×C32D3×D411yescharacteristic
241C22×C32C2×#6711yescharacteristic
242C22×C32C2×C3×D411yescharacteristic
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—32no—
250C32×C4—32no—
251C32×C4—41no—
252C32×C4—41no—
253C32×C4—41no—
254C32×C4—42no—
255C32×C4—41no—
256C32×C4—41no—
257C32×C4—81no—
258C32×C4—81no—
259C32×C4—82no—
260C32×C4—121no—
261C32×C4—121no—
262C32×C4C2×Dic611yescharacteristic
263C32×C4D3×Q811yescharacteristic
264C32×C4D3×Q811yescharacteristic
265C32×C4D3×Q811yescharacteristic
266C32×C4C2×C3×Q811yescharacteristic
267C32×C4C2×C3×Q811yescharacteristic
268C3⋊Dic3—41no—
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3—42no—
272C3×Dic3—41no—
273C3×Dic3—41no—
274C3×Dic3—41no—
275C3×Dic3—41no—
276C3×Dic3—81no—
277C3×Dic3—121no—
278C3×Dic3—121no—
279C3×Dic3—241no—
280C3×Dic3#21312yes—
281C3×Dic3#21612yes—
282C3×Dic3C3×D812yes—
283C3×Dic3C3×Q1612yes—
284C4⋊Q8—31no—
285C4⋊Q8—91no—
286#107—31no—
287#107—91no—
288#107—91no—
289#107—91no—
290C4×C8—91no—
291C33#50011yescharacteristic, complemented, Sylow
292C3×Q8—122no—
293C3×Q8—122no—
294C3×Q8—124no—
295C3×Q8—362no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—62no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—121no—
319C2×C3×C4—181no—
320C2×C3×C4C2×D3211yescharacteristic
321C2×C3×C4C22×C3×D311yescharacteristic
322C2×C3×C4C22×C3×D311yescharacteristic
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—61no—
326C2×Dic3—61no—
327C2×Dic3—121no—
328C2×Dic3C3×#6711yescharacteristic
329C2×Dic3C3×#6711yescharacteristic
330Dic6—42no—
331Dic6—122no—
332Dic6—122no—
333C3×C8—32no—
334C3×C8—32no—
335C3×C8—62no—
336C3×C8—92no—
337C3⋊C8—32no—
338C3⋊C8—32no—
339C3⋊C8—62no—
340C3⋊C8C3×Dic612yescomplemented
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32#88811yescharacteristic
354C2×C32D3×D811yescharacteristic
355C2×C32D3×Q1611yescharacteristic
356C2×C32C2×#21311yescharacteristic
357C2×C32C2×#21611yescharacteristic
358C2×C32#94411yescharacteristic
359C2×C32C3×C4⋊Q811yescharacteristic
360C2×C32C2×C3×D811yescharacteristic
361C2×C32C2×C3×Q1611yescharacteristic
362C2×Q8—61no—
363C2×Q8—181no—
364C2×C8—31no—
365C2×C8—91no—
366C4⋊C4—31no—
367C4⋊C4—31no—
368C4⋊C4—61no—
369C4⋊C4—91no—
370C4⋊C4—181no—
371C4⋊C4C3×D3211yescharacteristic
372C42—31no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—41no—
377C22×C3D3×#6711yescharacteristic
378C22×C3C3×D3×D411yescharacteristic
379C22×C3C2×C3×#6711yescharacteristic
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—32no—
387C3×C4—32no—
388C3×C4—32no—
389C3×C4—32no—
390C3×C4—42no—
391C3×C4—41no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—41no—
395C3×C4—41no—
396C3×C4—62no—
397C3×C4—62no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4—82no—
401C3×C4—122no—
402C3×C4—121no—
403C3×C4—121no—
404C3×C4—121no—
405C3×C4—121no—
406C3×C4—241no—
407C3×C4—361no—
408C3×C4D3×Dic611yescharacteristic
409C3×C4#379711yescharacteristic
410C3×C4C2×C3×Dic611yescharacteristic
411C3×C4C3×D3×Q811yescharacteristic
412C3×C4C3×D3×Q811yescharacteristic
413C3×C4C3×D3×Q811yescharacteristic
414Dic3—41no—
415Dic3—41no—
416Dic3—121no—
417Dic3—121no—
418Dic3—241no—
419Dic3C3×#21312yes—
420Dic3C3×#21612yes—
421C32—21no—
422C32—21no—
423C32—21no—
424C32—41no—
425C32#487411yescharacteristic, complemented
426C32#507811yescharacteristic, complemented
427C32C3×#50011yescharacteristic, complemented
428Q8—122no—
429Q8—362no—
430C2×C4—21no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—181no—
436C2×C4C2×C3×D3211yescharacteristic
437C8—32no—
438C8—92no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3#6383911yescharacteristic
452C2×C3D3×#21311yescharacteristic
453C2×C3D3×#21611yescharacteristic
454C2×C3C3×#88811yescharacteristic
455C2×C3C3×D3×D811yescharacteristic
456C2×C3C3×D3×Q1611yescharacteristic
457C2×C3C2×C3×#21311yescharacteristic
458C2×C3C2×C3×#21611yescharacteristic
459C2×C3C3×#94411yescharacteristic
460C22C3×D3×#6711yescharacteristic
461C4—32no—
462C4—32no—
463C4—41no—
464C4—121no—
465C4—121no—
466C4—361no—
467C4C3×D3×Dic611yescharacteristic
468C4C3×#379711yescharacteristic
469C3—21no—
470C3—21no—
471C3—21no—
472C3—41no—
473C3#1059041111yescharacteristic, direct product factor
474C3C3×#487411yescharacteristic, complemented
475C3C3×#507811yescharacteristic, complemented
476C2C3×#6383911yescharacteristic
477C2C3×D3×#21311yescharacteristic
478C2C3×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×#500
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. #10590411C3direct (N × H)
55. C3×C3⋊(C3⋊C8)Q8semidirect
56. C32×C3⋊C8Q8semidirect
180. C3⋊(C3⋊C8)C3×Q8semidirect
185. C3×C3⋊C8Dic6semidirect
186. C3×C3⋊C8C3×Q8semidirect
291. C33#500semidirect
340. C3⋊C8C3×Dic6semidirect
425. C32#4874semidirect
426. C32#5078semidirect
427. C32C3×#500semidirect
473. C3#10590411direct (N × H)
474. C3C3×#4874semidirect
475. C3C3×#5078semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
C3×#639182
C3×#636342
C32×#8072
C3×#635762
C3×#635752
C3×Dic3×C3⋊C82
C3×#50783
C3×#48743
#105904113